Transcript MODELS WITH A LAGGED DEPENDENT VARIABLE ADL(p, q) Y t 1 2 X t 3Y t.
Slide 1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
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Subject to respect for copyright and, where appropriate, attribution, they may be
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Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
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or the University of London International Programmes distance learning course
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2013.01.20
Slide 2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
Slide 3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
Slide 4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
Slide 5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
Slide 6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
Slide 7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
Slide 8
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
Slide 9
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
Slide 10
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
Slide 11
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
Slide 12
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
Slide 13
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
Slide 14
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
Slide 15
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
Slide 16
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
Slide 17
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
Slide 18
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
Slide 19
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
Slide 20
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
Slide 21
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
Slide 22
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
Slide 23
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
Slide 24
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
Slide 25
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
Slide 26
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
Slide 27
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
Slide 28
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
Slide 29
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
Slide 2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
Slide 3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
Slide 4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
Slide 5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
Slide 6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
Slide 7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
Slide 8
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
Slide 9
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
Slide 10
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
Slide 11
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
Slide 12
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
Slide 13
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
Slide 14
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
Slide 15
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
Slide 16
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
Slide 17
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
Slide 18
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
Slide 19
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
Slide 20
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
Slide 21
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
Slide 22
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
Slide 23
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
Slide 24
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
Slide 25
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
Slide 26
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
Slide 27
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
Slide 28
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own who feel that they might benefit
from participation in a formal course should consider the London School of
Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
2013.01.20
Slide 29
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
A widely-used solution to the problem of including dynamics in a model while mitigating the
problem of multicollinearity is to employ an autoregressive distributed lag model, often
written ADL(p, q).
1
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent
variable are included on the right side as explanatory variables.
2
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
p is the maximum number of lags of the dependent variable used in this way. q is the
maximum lag of the X variable, or variables if there are several.
3
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
The ADL model is particularly appealing when the dependent variable exhibits a high degree
of dependence because then, as a matter of common sense, its value in one observation is
likely to be influenced by its value in the previous one.
4
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
It is econometrically attractive because it can accommodate a broad range of dynamic
patterns with relatively few lag terms and parameters. (It is parsimonious, to use the
technical term.)
5
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(p, q)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = maximum number of lags of the dependent variable
q = maximum number of lags of the X variable(s)
This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.
6
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
We will start with the simplest model of all, the ADL(1,0) model where the only lagged
variable is the lagged dependent variable.
7
MODELS WITH A LAGGED DEPENDENT VARIABLE
ADL(1, 0)
Y t 1 2 X t 3Y t 1 u t
ADL = autoregressive distributed lag
'autoregressive' because Yt depends on previous values of Y.
p = 1 in this specification
q=0
Given the continuity of many time series processes, Yt–1, the value of a time series at time
t – 1, is often the most important determinant of its value Yt at time t. and it makes sense to
include it explicitly in the model as an explanatory variable.
8
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will begin by investigating the dynamics implicit in the model graphically. We will
suppose, for convenience, that 2 is positive and that X increases with time, and we will
neglect the effect of the disturbance term.
9
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We shall suppose throughout this section that │3│ < 1. This is a stability condition for the
process. We will discuss the consequences of violations of this condition in Chapter 13.
We will in fact assume 0 < 3 < 1 because Yt and Yt–1 are typically positively correlated.
10
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
At time t, Yt is given by the equation at the top. It is represented by the point corresponding
to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t,
so the term 3Yt–1 is fixed.
11
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
The equation thus may be viewed as giving the short-run relationship between Yt and Xt for
period t. (1 + 3Yt–1) is effectively the intercept and 2, the slope coefficient, gives the shortrun effect of X on Y.
12
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept
is now (1 + 3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that
for Yt and the short-run relationship has shifted upwards. The slope is unchanged, 2.
13
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t u t 1
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Thus two factors are responsible for the growth of Y over time: the direct effect of the
increase in X, and the gradual upward shift of the short-run relationship. The figure shows
the outcomes for time t as far as time t + 4.
14
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
You can see that the actual relationship between Y and X, traced out by the markers
representing the observations, is steeper than the short-run relationship for each time
period.
15
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
We will determine the long-run relationship between Y and X by performing a comparative
statics analysis described in the previous slideshow.
16
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
1 + 3Yt+3
1 + 3Yt+2
1 + 3Yt+1
short-run relationship
at time t
1 + 3Yt
1 + 3Yt–1
Xt
Xt+1 Xt+2
Xt+3
Xt+4
X
Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence,
ignoring the transient effect of the disturbance term, the equilibrium relationship is as
shown above.
17
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The
factor 2 / (1 – 3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We
will describe this as the long-run effect.
18
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y 1 2 X 3Y
Y
2
1 3
2
1 3
1
1 3
2
1 3
X
long-run effect of X on Y
2
if
0 3 1
In the present context, with 0 < 3 < 1, it will be greater than 2 because 1 – 3 will also lie
between 0 and 1.
19
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Another way of exploring the dynamics is to look at the implicit relationship between Yt and
current and lagged values of X. If the relationship is true for time period t, it is also true for
time period t – 1.
20
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
We substitute for Yt–1 in the first equation.
21
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Continuing to lag and substitute, one obtains the equation shown.
22
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag
distribution that consists of geometrically declining weights: 2, 23, 232, ...
23
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We have thus found a way of allowing lagged values of X to influence Y without introducing
them into the model explicitly and giving rise to multicollinearity.
24
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
Koyck lag distribution: geometrically declining weights
One should, however, note that this particular pattern of weights, known as a Koyck lag
distribution, embodies the assumption that more recent values of X have more influence
than older ones, and that the rate of decline in the weights is constant.
25
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
We will see in due course that we can relax both of these constraints.
26
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
2 short-run effect of X on Y
From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined,
the only influence of X on Y is via Xt. For this reason, again, we describe 2 as the short-run
effect.
27
MODELS WITH A LAGGED DEPENDENT VARIABLE
Y t 1 2 X t 3Y t 1 u t
Y t 1 1 2 X t 1 3Y t 2 u t 1
Y t 1 2 X t 3 1 2 X t 1 3Y t 2 u t 1 u t
1 1 3 2 X t 2 3 X t 1 3 Y t 2 u t 3 u t 1
2
Y t 1 1 3 3 ... 2 X t 2 3 X t 1 2 3 X t 2 ...
2
2
u t 3 u t 1 3 u t 1 ...
2
This representation of the model also yields the same long-run effect, as before. The proof
is left as an exercise.
28
Copyright Christopher Dougherty 2013.
These slideshows may be downloaded by anyone, anywhere for personal use.
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used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 11.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
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Individuals studying econometrics on their own who feel that they might benefit
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or the University of London International Programmes distance learning course
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2013.01.20