Transcript Chapter 9
Chapter 9 Regression with Time Series Data: Stationary Variables
Principles of Econometrics, 4t h Edition Walter R. Paczkowski Rutgers University Chapter 9: Regression with Time Series Data: Stationary Variables Page 1
Chapter Contents 9.1 Introduction 9.2 Finite Distributed Lags 9.3 Serial Correlation 9.4 Other Tests for Serially Correlated Errors 9.5 Estimation with Serially Correlated Errors 9.6 Autoregressive Distributed Lag Models 9.7 Forecasting 9.8 Multiplier Analysis Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 2
9.1 Introduction Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 3
9.1
Introduction When modeling relationships between variables, the nature of the data that have been collected has an important bearing on the appropriate choice of an econometric model – Two features of time-series data to consider: 1. Time-series observations on a given economic unit, observed over a number of time periods, are likely to be correlated 2. Time-series data have a natural ordering according to time Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 4
9.1
Introduction There is also the possible existence of dynamic relationships between variables – A dynamic relationship is one in which the change in a variable now has an impact on that same variable, or other variables, in one or more future time periods – These effects do not occur instantaneously but are spread, or
distributed
, over future time periods Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 5
9.1
Introduction FIGURE 9.1 The distributed lag effect Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 6
9.1
Introduction 9.1.1
Dynamic Nature of Relationships Eq. 9.1
Ways to model the dynamic relationship: 1. Specify that a dependent variable
y
is a function of current and past values of an explanatory variable
x y t
( ,
t t
1 ,
x t
2 ,...) • Because of the existence of these lagged effects, Eq. 9.1 is called a distributed lag model Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 7
9.1
Introduction 9.1.1
Dynamic Nature of Relationships Eq. 9.2
Eq. 9.3
Ways to model the dynamic relationship (Continued) : 2. Capturing the dynamic characteristics of time series by specifying a model with a lagged dependent variable as one of the explanatory variables • Or have:
y t
t
1
x t
Principles of Econometrics, 4t h Edition
y t
t
1 , ,
t t
1 ,
x t
2 ) – Such models are called
autoregressive distributed lag
(
ARDL
) models, with ‘‘autoregressive’’ meaning a regression of
y
t on its own lag or lags Chapter 9: Regression with Time Series Data: Stationary Variables Page 8
9.1
Introduction 9.1.1
Dynamic Nature of Relationships Eq. 9.4
Ways to model the dynamic relationship (Continued) : 3. Model the continuing impact of change over several periods via the error term
y t
f x t
e t e t
t
1 ) • • In this case
e
t is correlated with We say the errors are
e
t - 1
serially correlated
or
autocorrelated
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 9
9.1
Introduction 9.1.2
Least Squares Assumptions The primary assumption is Assumption MR4: cov
i j
cov
i j
0 for
i
j
• For time series, this is written as: cov
t s
cov
t s
0 for
t
s
– The dynamic models in Eqs. 9.2, 9.3 and 9.4 imply correlation between
y
t
e
t - 1 and
y
t - 1 or
e
t and or both, so they clearly violate assumption MR4 Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 10
9.1
Introduction 9.1.2a
Stationarity A stationary variable is one that is not explosive, nor trending, and nor wandering aimlessly without returning to its mean Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 11
9.1
Introduction 9.1.2a
Stationarity FIGURE 9.2 (a) Time series of a stationary variable Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 12
9.1
Introduction 9.1.2a
Stationarity FIGURE 9.2 (b) time series of a nonstationary variable that is ‘‘slow-turning’’ or ‘‘wandering’’ Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 13
9.1
Introduction 9.1.2a
Stationarity FIGURE 9.2 (c) time series of a nonstationary variable that ‘‘trends” Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 14
9.1
Introduction 9.1.3
Alternative Paths Through the Chapter FIGURE 9.3 (a) Alternative paths through the chapter starting with finite distributed lags Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 15
9.1
Introduction 9.1.3
Alternative Paths Through the Chapter FIGURE 9.3 (b) Alternative paths through the chapter starting with serial correlation Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 16
9.2 Finite Distributed Lags Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 17
9.2
Finite Distributed Lags Eq. 9.5
Consider a linear model in which, after
q
time periods, changes in
x
no longer have an impact on
y
y
t
0
x
t
1
x
t
1
2
x
t
2
q
x
e
t
– Note the notation change: β s the coefficient of
x
t-s denote the intercept is used to denote and α is introduced to Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 18
9.2
Finite Distributed Lags Eq. 9.6
Eq. 9.7
Model 9.5 has two uses: – Forecasting
y
T
1
0
x
T
1
1
x
T
2
x
T
1
x
q T q
1
e
T
1 – Policy analysis • What is the effect of a change in
x
on
y
?
( )
t
x
x t
)
s
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 19
9.2
Finite Distributed Lags Assume
x
t is increased by one unit and then maintained at its new level in subsequent periods – The immediate impact will be β 0 – the total effect in period period •
t
+ 2 it will be β 0
t
+ 1 will be β 0 + β 1 These quantities are called + β 2
interim
+ β 1 , and so on , in
multipliers
– The
total multiplier
is the final effect on
y
of the sustained increase after
q q
have elapsed
s
0 β
s
or more periods Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 20
9.2
Finite Distributed Lags The effect of a one-unit change in
x
t is
distributed
over the current and next
q
periods, from which we get the term ‘‘distributed lag model’’ – It is called a
order q
•
finite distributed lag model of
It is assumed that after a finite number of periods
q
, changes in
x
impact on
y
no longer have an – The coefficient β
weight
or an s is called a
distributed-lag s-period delay multiplier
– The coefficient β
multiplier
0 (
s
= 0) is called the
impact
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 21
9.2
Finite Distributed Lags 9.2.1
Assumptions
ASSUMPTIONS OF THE DISTRIBUTED LAG MODEL
TSMR1.
y t
β 0
x t
β 1
x t
1 β 2
x t
2 β
q x
TSMR2.
y
and
x
are stationary random variables, and
e
t current, past and future values of
x
.
e t
,
t
1, , is independent of
T
TSMR3.
E
(
e
t ) = 0 TSMR4. var(
e
t ) = σ 2 TSMR5. cov(
e
t ,
e
s ) = 0
t
TSMR6.
e t
~
N
(0, σ 2 ) ≠
s
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 22
9.2
Finite Distributed Lags 9.2.2
An Example: Okun’s Law Eq. 9.8
Eq. 9.9
Consider Okun’s Law – In this model the change in the unemployment rate from one period to the next depends on the rate of growth of output in the economy:
U t
U t
1 – We can rewrite this as:
G t
G N
DU t
β 0
G t
e t
where
DU
α = γ
G
N = Δ
U
=
U
t -
U
t-1 , β 0 = -γ, and Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 23
9.2
Finite Distributed Lags 9.2.2
An Example: Okun’s Law Eq. 9.10
We can expand this to include lags:
DU t
β 0
G t
β 1
G t
1 β 2
G t
2 β
q G
e t
Eq. 9.11
We can calculate the growth in output,
G
, as:
G t
GDP t
GDP t
1 100
GDP t
1 Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 24
9.2
Finite Distributed Lags 9.2.2
An Example: Okun’s Law FIGURE 9.4 (a) Time series for the change in the U.S. unemployment rate: 1985Q3 to 2009Q3 Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 25
9.2
Finite Distributed Lags 9.2.2
An Example: Okun’s Law FIGURE 9.4 (b) Time series for U.S. GDP growth: 1985Q2 to 2009Q3 Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 26
9.2
Finite Distributed Lags 9.2.2
An Example: Okun’s Law Table 9.1 Spreadsheet of Observations for Distributed Lag Model Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 27
9.2
Finite Distributed Lags 9.2.2
An Example: Okun’s Law Table 9.2 Estimates for Okun’s Law Finite Distributed Lag Model Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 28
9.3 Serial Correlation Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 29
9.3
Serial Correlation
t
When is assumption
TSMR5
, cov(
e
t ,
e
s ) = 0 for ≠
s
likely to be violated, and how do we assess its validity? – When a variable exhibits correlation over time, we say it is
autocorrelated
or
serially correlated
• These terms are used interchangeably Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 30
9.3
Serial Correlation 9.3.1
Serial Correlation in Output Growth FIGURE 9.5 Scatter diagram for G t and G t-1 Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 31
9.3
Serial Correlation 9.3.1a
Computing Autocorrelation Recall that the population correlation between two variables
x
and
y
ρ
xy
is given by: cov var
x
y
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 32
9.3
Serial Correlation 9.3.1a
Computing Autocorrelation Eq. 9.12
For the Okun’s Law problem, we have: ρ 1 cov
var
G t t t
1
G t
1
cov
var
t
t
1
The notation ρ 1 is used to denote the population correlation between observations that are one period apart in time – This is known also as the
population autocorrelation of order one
. – The second equality in Eq. 9.12 holds because var(
G
t ) = var(
G
t-1 ) , a property of time series that are stationary Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 33
9.3
Serial Correlation 9.3.1a
Computing Autocorrelation The first-order sample autocorrelation for
G
is obtained from Eq. 9.12 using the estimates: cov
t t
1
T
1 1
t T
2
G t
var
t
T
1 1
t T
1
G t
G
2
t
1
G
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 34
9.3
Serial Correlation 9.3.1a
Computing Autocorrelation Eq. 9.13
Making the substitutions, we get:
r
1
t T
2
G t
t T
1
G t
t
1
G
2
G
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 35
9.3
Serial Correlation 9.3.1a
Computing Autocorrelation Eq. 9.14
More generally, the
k-th order sample autocorrelation
for a series
y
that gives the correlation between observations that are
k
periods apart is:
r k
T
t k
1
y t
t T
1
y t y
y y
2
y
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 36
9.3
Serial Correlation 9.3.1a
Computing Autocorrelation Eq. 9.15
Because (
T
-
k
) observations are used to compute the numerator and
T
observations are used to compute the denominator, an alternative that leads to larger estimates in finite samples is:
r k
T
1
k T
t k
1
y t
1
T t T
1
y t
y
y
y
2
y
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 37
9.3
Serial Correlation 9.3.1a
Computing Autocorrelation Eq. 9.16
Applying this to our problem, we get for the first four autocorrelations:
r
1 0.494
r
2 0.411
r
3 0.154
r
4 0.200
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 38
9.3
Serial Correlation 9.3.1a
Computing Autocorrelation Eq. 9.17
How do we test whether an autocorrelation is significantly different from zero?
– The null hypothesis is
H
0 : ρ
k
– A suitable test statistic is: = 0
Z
r k
0 1
T T r k N
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 39
9.3
Serial Correlation 9.3.1a
Computing Autocorrelation For our problem, we have:
Z
1 4.89,
Z
2 4.10
Z
3 –
Z
4 We reject the hypotheses
H
0 : ρ 1
H
0 : ρ 2 = 0 = 0 and – – – We have insufficient evidence to reject
H
0 : ρ 3 ρ 4 We conclude that
G
, the quarterly growth rate in U.S. GDP, exhibits significant serial correlation at lags one and two Principles of Econometrics, 4t h Edition = 0 is on the borderline of being significant.
Chapter 9: Regression with Time Series Data: Stationary Variables Page 40
9.3
Serial Correlation 9.3.1b
The Correlagram The
correlogram
, also called the
sample autocorrelation function
, is the sequence of autocorrelations
r
1 ,
r
2 ,
r
3 , … – It shows the correlation between observations that are one period apart, two periods apart, three periods apart, and so on Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 41
9.3
Serial Correlation 9.3.1b
The Correlagram FIGURE 9.6 Correlogram for
G
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 42
9.3
Serial Correlation 9.3.2
Serially Correlated Errors The correlogram can also be used to check whether the multiple regression assumption cov(
e t
,
e s
) = 0 for
t
≠
s
is violated Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 43
9.3
Serial Correlation 9.3.2a
A Phillips Curve Eq. 9.18
Eq. 9.19
Consider a model for a Phillips Curve:
INF t
INF t E
γ
U t
U t
1 – If we initially assume that inflationary expectations are constant over time (β 1 set β 2 = -γ, and add an error term: = I
NF E
t )
INF t
1 β 2
DU t
e t
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 44
9.3
Serial Correlation 9.3.2a
A Phillips Curve FIGURE 9.7 (a) Time series for Australian price inflation Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 45
9.3
Serial Correlation 9.3.2a
A Phillips Curve FIGURE 9.7 (b) Time series for the quarterly change in the Australian unemployment rate Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 46
9.3
Serial Correlation 9.3.2a
A Phillips Curve Eq. 9.20
To determine if the errors are serially correlated, we compute the least squares residuals:
e
ˆ
t
INF t
1
b DU
2
t
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 47
9.3
Serial Correlation 9.3.2a
A Phillips Curve FIGURE 9.8 Correlogram for residuals from least-squares estimated Phillips curve Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 48
9.3
Serial Correlation 9.3.2a
A Phillips Curve Eq. 9.21
Eq. 9.22
The
k
-th order autocorrelation for the residuals can be written as: –
r k
T
t k
1
t T
1
e
ˆ
t
2 The least squares equation is:
INF
DU
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 49
9.3
Serial Correlation 9.3.2a
A Phillips Curve The values at the first five lags are:
r
1 0.549
r
2 0.456
r
3 0.433
r
4 0.420
r
5 0.339
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 50
9.4 Other Tests for Serially Correlated Errors Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 51
9.4
Other Tests for Serially Correlated Errors 9.4.1
A Lagrange Multiplier Test An advantage of this test is that it readily generalizes to a joint test of correlations at more than one lag Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 52
9.4
Other Tests for Serially Correlated Errors 9.4.1
A Lagrange Multiplier Test Eq. 9.23
Eq. 9.24
If
e
t and
e
t-1 are correlated, then one way to model the relationship between them is to write:
e t
ρ
e t
1
v t
– We can substitute this into a simple regression equation:
y t
1
β
2
x t
ρ
e t
1
v t
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 53
9.4
Other Tests for Serially Correlated Errors 9.4.1
A Lagrange Multiplier Test
e
ˆ
– Two ways to handle this are: 1. Delete the first observation and use a total of
T
observations
e
ˆ
0
0
T
observations Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 54
9.4
Other Tests for Serially Correlated Errors 9.4.1
A Lagrange Multiplier Test For the Phillips Curve:
t t
6.219
F
6.202
F
0.000
0.000
– The results are almost identical – The null hypothesis
H
0 : ρ = 0 is rejected at all conventional significance levels – We conclude that the errors are serially correlated Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 55
9.4
Other Tests for Serially Correlated Errors 9.4.1
A Lagrange Multiplier Test Eq. 9.25
To derive the relevant auxiliary regression for the autocorrelation LM test, we write the test equation as:
y t
1 β 2
x t
ρ
e
ˆ
t
1
v t
– get:
y t
1
b x
2
t
e
ˆ
t b
1
b x
2
t
ˆ
t
β 1 β 2
x t
ρ
e
ˆ
t
1
v t
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 56
9.4
Other Tests for Serially Correlated Errors 9.4.1
A Lagrange Multiplier Test Eq. 9.26
Rearranging, we get:
e
ˆ
t
β
1
b
1
1
γ
2
x
t
β
2
2
ρ
e
ˆ
t
1
v
t
ρ
e
ˆ
t
1
v
t
– If
H
0 : ρ = 0 is true, then LM =
T
approximate χ 2 (1) distribution •
T
and
R
2 x
R
2 has an are the sample size and goodness of-fit statistic, respectively, from least squares estimation of Eq. 9.26
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 57
9.4
Other Tests for Serially Correlated Errors 9.4.1
A Lagrange Multiplier Test
e
ˆ
0
LM
LM
T
R
2
R
2 27.61
27.59
– These values are much larger than 3.84, which is the 5% critical value from a χ 2 (1) -distribution • We reject the null hypothesis of no autocorrelation – Alternatively, we can reject
H
0 by examining the
p
-value for LM = 27.61, which is 0.000
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 58
9.4
Other Tests for Serially Correlated Errors 9.4.1a
Testing Correlation at Longer Lags For a four-period lag, we obtain:
LM
LM
T
R
2
R
2 36.7
33.4
– Because the 5% critical value from a χ 2 (4) distribution is 9.49, these LM values lead us to conclude that the errors are serially correlated Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 59
9.4
Other Tests for Serially Correlated Errors 9.4.2
The Durbin Watson Test This is used less frequently today because its critical values are not available in all software packages, and one has to examine upper and lower critical bounds instead – Also, unlike the LM and correlogram tests, its distribution no longer holds when the equation contains a lagged dependent variable Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 60
9.5 Estimation with Serially Correlated Errors Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 61
9.5
Estimation with Serially Correlated Errors Three estimation procedures are considered: 1. Least squares estimation 2. An estimation procedure that is relevant when the errors are assumed to follow what is known as a first-order autoregressive model
e t
ρ
e t
1
v t
3. A general estimation strategy for estimating models with serially correlated errors Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 62
9.5
Estimation with Serially Correlated Errors We will encounter models with a lagged dependent variable, such as:
y t
1
y t
1 δ 0
x t
δ 1
x t
1
v t
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 63
9.5
Estimation with Serially Correlated Errors
ASSUMPTION FOR MODELS WITH A LAGGED DEPENDENT VARIABLE
TSMR2A In the multiple regression model Where some of the
x
tk
y t
1 may be lagged values of
y
,
v
t β 2
x t
2 β
K x K
is uncorrelated with all
v t x
tk and their past values.
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 64
9.5
Estimation with Serially Correlated Errors 9.5.1
Least Squares Estimation Suppose we proceed with least squares estimation without recognizing the existence of serially correlated errors. What are the consequences?
1. The least squares estimator is still a linear unbiased estimator, but it is no longer best 2. The formulas for the standard errors usually computed for the least squares estimator are no longer correct • Confidence intervals and hypothesis tests that use these standard errors may be misleading Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 65
9.5
Estimation with Serially Correlated Errors 9.5.1
Least Squares Estimation It is possible to compute correct standard errors for the least squares estimator: –
HAC (heteroskedasticity and autocorrelation consistent) standard errors
, or
Newey-West standard errors
• These are analogous to the heteroskedasticity consistent standard errors Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 66
9.5
Estimation with Serially Correlated Errors 9.5.1
Least Squares Estimation Eq. 9.27
Consider the model
y
t – The variance of b 2 = β is: 1 + β 2
x
t +
e
t var 2
t w t
2 var
t
w w t s
cov
t
t w t
2 var
t
1
w w t s
t w t
2 cov var
t t s
s
where
w t
x t
x
t
x t
x
2 Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 67
9.5
Estimation with Serially Correlated Errors 9.5.1
Least Squares Estimation When the errors are not correlated, cov(
e
t ,
e
s ) = 0, and the term in square brackets is equal to one. – The resulting expression var 2
t w t
2 var
t
is the one used to find heteroskedasticity consistent (HC) standard errors – When the errors are correlated, the term in square brackets is estimated to obtain HAC standard errors Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 68
9.5
Estimation with Serially Correlated Errors 9.5.1
Least Squares Estimation Eq. 9.28
If we call the quantity in square brackets
g
and its estimated variances is:
var
HAC
2
var
HC
2
g
ˆ
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 69
9.5
Estimation with Serially Correlated Errors 9.5.1
Least Squares Estimation Eq. 9.29
Let’s reconsider the Phillips Curve model:
INF
DU
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 70
9.5
Estimation with Serially Correlated Errors 9.5.1
Least Squares Estimation The
t
and
p
-values for testing
H
0 : β 2 = 0 are:
t
0.5279 0.2294
2.301
p
t
0.5279 0.3127
1.688
p
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 71
9.5
Estimation with Serially Correlated Errors 9.5.2
Estimating an AR(1) Error Model Eq. 9.30
Eq. 9.31
Return to the Lagrange multiplier test for serially correlated errors where we used the equation:
e t
ρ
e t
1
v t
– Assume the
v
t are uncorrelated random errors with zero mean and constant variances:
t
0 var
t
v
2 cov
t s
0 for
t
s
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 72
9.5
Estimation with Serially Correlated Errors 9.5.2
Estimating an AR(1) Error Model Eq. 9.30 describes a
first-order autoregressive model
or a
first-order autoregressive process
for
e
t – The term
AR(1) model
is used as an abbreviation for first-order autoregressive model – It is called an
autoregressive
model because it can be viewed as a regression model – It is called
first-order
because the right-hand side variable is
e
t lagged one period Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 73
9.5
Estimation with Serially Correlated Errors 9.5.2a
Properties of an AR(1) Error Eq. 9.32
We assume that: Eq. 9.33
Eq. 9.34
The mean and variance of
e
t are:
t
0 var
t
e
2
2
v
2 The covariance term is:
cov
t
ρ
k
v
2 2
,
k
0
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 74
9.5
Estimation with Serially Correlated Errors 9.5.2a
Properties of an AR(1) Error Eq. 9.35
The correlation implied by the covariance is: ρ
k
corr
t
cov var
e t
t
var
e
cov ρ
k
v
var 2
v
2
t
2
2 ρ
k
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 75
9.5
Estimation with Serially Correlated Errors 9.5.2a
Properties of an AR(1) Error Eq. 9.36
Setting
k
= 1:
ρ
1
corr
t t
1
ρ
– ρ represents the correlation between two errors that are one period apart • It is the
first-order autocorrelation
for
e
, sometimes simply called the autocorrelation coefficient • It is the population autocorrelation at lag one for a time series that can be described by an AR(1) model •
r
1 is an estimate for ρ when we assume a series is AR(1) Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 76
9.5
Estimation with Serially Correlated Errors 9.5.2a
Properties of an AR(1) Error Eq. 9.37
Each
e
t depends on all past values of the errors
v
t :
e t
t
ρ
v t
1 ρ 2
v t
2 ρ 3
v t
3 – For the Phillips Curve, we find for the first five lags:
r
1 0.549
r
2 0.456
r
3 0.433
r
4 0.420
r
5 0.339
– For an AR(1) model, we have: ˆ 1 1 0.549
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 77
9.5
Estimation with Serially Correlated Errors 9.5.2a
Properties of an AR(1) Error For longer lags, we have: 2 3 4 5 3 2
0.549
2
0.549
3 0.301
0.165
5 4
0.549
4
0.549
5 0.091
0.050
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 78
9.5
Estimation with Serially Correlated Errors 9.5.2b
Nonlinear Least Squares Estimation Eq. 9.38
Eq. 9.39
Our model with an AR(1) error is:
y t
1 β 2
x t
e t
with
e t
ρ
e t
1
v t
with -1 < ρ < 1 – For the
v
t , we have:
t
0 var
t
v
2 cov
t t
1 0 for
t
s
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 79
9.5
Estimation with Serially Correlated Errors 9.5.2b
Nonlinear Least Squares Estimation Eq. 9.40
Eq. 9.41
With the appropriate substitutions, we get:
y t
1 β 2
x t
ρ
e t
1
v t
– For the previous period, the error is:
e t
1
y t
1 1 β 2
x t
1 – Multiplying by ρ: Eq. 9.42
ρ
e t
1
e y t t
1 ρβ 1 ρβ 2
x t
1 Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 80
9.5
Estimation with Serially Correlated Errors 9.5.2b
Nonlinear Least Squares Estimation Eq. 9.43
Substituting, we get:
y t
1 β 2
x t
ρ
y t
1 ρβ 2
x t
1
v t
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 81
9.5
Estimation with Serially Correlated Errors 9.5.2b
Nonlinear Least Squares Estimation The coefficient of
x
t-1 – equals -ρβ 2 Although Eq. 9.43 is a linear function of the variables
x
t ,
y
t-1 and
x
t-1 , it is not a linear function of the parameters (β 1 , β 2 , ρ) – The usual linear least squares formulas cannot be obtained by using calculus to find the values of (β 1 , β 2 , ρ) that minimize
S
v • These are
nonlinear least squares estimates
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 82
9.5
Estimation with Serially Correlated Errors 9.5.2b
Nonlinear Least Squares Estimation Eq. 9.44
Eq. 9.45
Our Phillips Curve model assuming AR(1) errors is:
INF t
1 β 2
DU t
ρ
INF t
1 ρβ 2
DU t
1
v t
– Applying nonlinear least squares and presenting the estimates in terms of the original untransformed model, we have:
INF
DU e t
0.557
e t
1
v t
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 83
9.5
Estimation with Serially Correlated Errors 9.5.2c
Generalized Least Squares Estimation Nonlinear least squares estimation of Eq. 9.43 is equivalent to using an iterative generalized least squares estimator called the Cochrane-Orcutt procedure Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 84
9.5
Estimation with Serially Correlated Errors 9.5.3
Estimating a More General Model Eq. 9.46
Eq. 9.47
We have the model:
y t
1 β 2
x t
ρ
y t
1 ρβ 2
x t
1
v t
– Suppose now that we consider the model:
y t
1
y t
1 δ 0
x t
δ 1
x t
1
v t
• This new notation will be convenient when we discuss a general class of
autoregressive distributed lag (ARDL) models
– Eq. 9.47 is a member of this class Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 85
9.5
Estimation with Serially Correlated Errors 9.5.3
Estimating a More General Model Eq. 9.48
Note that Eq. 9.47 is the same as Eq. 9.47 since: 1 0 β δ 2 1 ρβ θ 2 1 ρ – Eq. 9.46 is a restricted version of Eq. 9.47 with the restriction δ 1 = -θ 1 δ 0 imposed Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 86
9.5
Estimation with Serially Correlated Errors 9.5.3
Estimating a More General Model Eq. 9.49
Applying the least squares estimator to Eq. 9.47 using the data for the Phillips curve example yields:
INF t
INF t
1
0.6882
DU t
0.3200
DU t
1 Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 87
9.5
Estimation with Serially Correlated Errors 9.5.3
Estimating a More General Model The equivalent AR(1) estimates are: 1
θ ˆ 1 ˆ 1 0 β ˆ ˆ 2 0.6944
2 0.5574
0.6944
0.3368
0.3871
– These are similar to our other estimates Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 88
9.5
Estimation with Serially Correlated Errors 9.5.3
Estimating a More General Model Eq. 9.50
Eq. 9.51
The original economic model for the Phillips Curve was:
INF t
INF t E
γ
U t
U t
1 – Re-estimation of the model after omitting
DU
t-1 yields:
INF t
INF t
1
0.4909
DU t
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 89
9.5
Estimation with Serially Correlated Errors 9.5.3
Estimating a More General Model In this model inflationary expectations are given by:
INF t E
INF t
1 – A 1% rise in the unemployment rate leads to an approximate 0.5% fall in the inflation rate Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 90
9.5
Estimation with Serially Correlated Errors 9.5.4
Summary of Section 9.5 and Looking Ahead We have described three ways of overcoming the effect of serially correlated errors: 1. Estimate the model using least squares with
HAC
standard errors 2. Use nonlinear least squares to estimate the model with a lagged x, a lagged y, and the restriction implied by an AR(1) error specification 3. Use least squares to estimate the model with a lagged
x
and a lagged
y
, but without the restriction implied by an AR(1) error specification Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 91
9.6 Autoregressive Distributed Lag Models Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 92
9.6
Autoregressive Distributed Lag Models Eq. 9.52
An autoregressive distributed lag (ARDL) model is one that contains both lagged
x
t ’s and lagged
y
t ’s
y t
0
x t
1
x t
1
x
1
y t
1
p y
v t
– Two examples:
INF t
INF t
INF t
1 0.6882
DU t
0.3200
DU t
1
INF t
1 0.4909
DU t
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 93
9.6
Autoregressive Distributed Lag Models Eq. 9.53
An ARDL model can be transformed into one with only lagged
x
’s which go back into the infinite past:
y t
0
x t
β 1
x t
1 β 2
x t
2 β 3
x t
3
e t
s
0 β
s x
e t
– This model is called an
infinite distributed lag model
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 94
9.6
Autoregressive Distributed Lag Models Four possible criteria for choosing
p
and
q
: 1. Has serial correlation in the errors been eliminated?
2. Are the signs and magnitudes of the estimates consistent with our expectations from economic theory? 3. Are the estimates significantly different from zero, particularly those at the longest lags? 4. What values for
p
and
q
minimize information criteria such as the
AIC
and
SC
?
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 95
9.6
Autoregressive Distributed Lag Models Eq. 9.54
Eq. 9.55
The
Akaike information criterion
(
AIC
) is: AIC ln
SSE T
2
K T
where
K
=
p
+
q
+ 2 The Schwarz criterion (
SC
), also known as the Bayes information criterion (
BIC
), is: ln SC ln
SSE
K T T T
– Because
K
ln(
T
)/
T
> 2
K
/
T
for
T
≥ 8, the
SC
penalizes additional lags more heavily than does the
AIC
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 96
9.6
Autoregressive Distributed Lag Models 9.6.1
The Phillips Curve Eq. 9.56
Consider the previously estimated ARDL(1,0) model:
INF t
INF t
1
0.4909
DU t
, obs 90 Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 97
9.6
Autoregressive Distributed Lag Models 9.6.1
The Phillips Curve FIGURE 9.9 Correlogram for residuals from Phillips curve ARDL(1,0) model Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 98
9.6
Autoregressive Distributed Lag Models 9.6.1
The Phillips Curve Table 9.3
p
-values for LM Test for Autocorrelation Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 99
9.6
Autoregressive Distributed Lag Models 9.6.1
The Phillips Curve Eq. 9.57
For an ARDL(4,0) version of the model:
INF t
INF t
1
0.1213
INF t
2
INF t
-4
0.7902
DU t
0.1677
INF t
3 obs 87 Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 100
9.6
Autoregressive Distributed Lag Models 9.6.1
The Phillips Curve Inflationary expectations are given by:
INF t E
INF t
1 0.1213
INF t
2 0.1677
INF t
3 0.2819
INF t
-4 Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 101
9.6
Autoregressive Distributed Lag Models 9.6.1
The Phillips Curve Table 9.4
AIC
and
SC
Values for Phillips Curve ARDL Models Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 102
9.6
Autoregressive Distributed Lag Models 9.6.2
Okun’s Law Eq. 9.58
Recall the model for Okun’s Law:
DU t
se
G t
0.1653
G t
1 0.0700G
t
2 , obs 96 Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 103
9.6
Autoregressive Distributed Lag Models 9.6.2
Okun’s Law FIGURE 9.10 Correlogram for residuals from Okun’s law ARDL(0,2) model Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 104
9.6
Autoregressive Distributed Lag Models 9.6.2
Okun’s Law Table 9.5
AIC
and
SC
Values for Okun’s Law ARDL Models Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 105
9.6
Autoregressive Distributed Lag Models 9.6.2
Okun’s Law Now consider this version: Eq. 9.59
DU t
DU t
1
0.1841
G t
0.0992G , obs
t
1
96 Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 106
9.6
Autoregressive Distributed Lag Models 9.6.3
Autoregressive Models Eq. 9.60
An autoregressive model of order
p
, denoted AR(
p
), is given by:
y t
1
y t
1 θ 2
y t
2 θ
p y
v t
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 107
9.6
Autoregressive Distributed Lag Models 9.6.3
Autoregressive Models Eq. 9.61
Consider a model for growth in real GDP:
G t
G t
1
0.2462
G t
2 Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 108
9.6
Autoregressive Distributed Lag Models 9.6.3
Autoregressive Models FIGURE 9.11 Correlogram for residuals from AR(2) model for GDP growth Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 109
9.6
Autoregressive Distributed Lag Models 9.6.3
Autoregressive Models Table 9.6
AIC
and
SC
Values for AR Model of Growth in U.S. GDP Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 110
9.7 Forecasting Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 111
9.7
Forecasting We consider forecasting using three different models: 1. AR model 2. ARDL model 3. Exponential smoothing model Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 112
9.7
Forecasting 9.7.1
Forecasting with an AR Model Eq. 9.62
Consider an AR(2) model for real GDP growth:
G t
1
G t
1 θ 2
G t
2
v t
The model to forecast
G T
+1 is:
G T
1 1
G T
θ 2
G T
1
v T
1 Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 113
9.7
Forecasting 9.7.1
Forecasting with an AR Model Eq. 9.63
The growth values for the two most recent quarters are:
G T
=
G
2009Q3 = 0.8
G T-1
=
G
2009Q2 = -0.2
The forecast for
G
2009Q4 is:
G T
1 ˆ 1
G T
0.7181
2
G T
1
0.2
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 114
9.7
Forecasting 9.7.1
Forecasting with an AR Model Eq. 9.64
Eq. 9.65
For two quarters ahead, the forecast for
G
2010Q1 is:
G T
2 ˆ 1
G T
1 θ ˆ 2
G T
0.9334
For three periods out, it is:
G T
3 ˆ 1
G T
2 0.9945
2
G T
1 Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 115
9.7
Forecasting 9.7.1
Forecasting with an AR Model Summarizing our forecasts: – Real GDP growth rates for 2009Q4, 2010Q1, and 2010Q2 are approximately 0.72%, 0.93%, and 0.99%, respectively Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 116
9.7
Forecasting 9.7.1
Forecasting with an AR Model A 95% interval forecast for
j
periods into the future is given by:
T
j
t
0.975,
df
ˆσ
j
ˆσ
error and
df
is the number of degrees of freedom in the estimation of the AR model Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 117
9.7
Forecasting 9.7.1
Forecasting with an AR Model Eq. 9.66
The first forecast error, occurring at time
T
+1, is:
u
1
G T
1
G T
1 θ 1 ˆ 1
G T
θ 2 2
G T
1
v T
1 Ignoring the error from estimating the coefficients, we get:
u
1
v T
1 Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 118
9.7
Forecasting 9.7.1
Forecasting with an AR Model Eq. 9.67
Eq. 9.68
The forecast error for two periods ahead is:
u
2 θ 1
G T
1
G T
1
v T
2 θ
u
1 1
v T
2 θ 1
v T
1
v T
2 The forecast error for three periods ahead is:
u
3 θ
u
1 2 θ
u
2 1
v T
3 θ 1 2 θ 2
v T
1 θ 1
v T
2
v T
3 Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 119
9.7
Forecasting 9.7.1
Forecasting with an AR Model Because the
v
2
v
t ’s are uncorrelated with constant σ 1 2 σ 2 2 σ 2 3 var var var 3 σ 2
v v
σ 2
v
θ 1 2 1 2 θ 2 2 θ 1 2 1 Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 120
9.7
Forecasting 9.7.1
Forecasting with an AR Model Table 9.7 Forecasts and Forecast Intervals for GDP Growth Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 121
9.7
Forecasting 9.7.2
Forecasting with an ARDL Model Eq. 9.69
Consider forecasting future unemployment using the Okun’s Law ARDL(1,1):
DU
t
1
DU
t
1
δ
0
G
t
δ
1
G
t
1
v
t
Eq. 9.70
The value of
DU
in the first post-sample quarter is:
DU T
1 1
DU T
δ 0
G T
1 δ 1
G T
v T
1 – But we need a value for
G
T+1 Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 122
9.7
Forecasting 9.7.2
Forecasting with an ARDL Model Eq. 9.71
Now consider the
change
in unemployment – Rewrite Eq. 9.70 as:
U T
1
U T
1
U T
U T
1 δ 0
G T
1 δ 1
G T
v T
1 – Rearranging:
U T
1 θ 1 1
U T
θ 1
U T
1 δ 0
G T
1 δ 1
G T
v T
1 * 1
U T
θ * 2
U T
1 δ 0
G T
1 δ 1
G T
v T
1 Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 123
9.7
Forecasting 9.7.2
Forecasting with an ARDL Model For the purpose of computing point and interval forecasts, the ARDL(1,1) model for a change in unemployment can be written as an ARDL(2,1) model for the level of unemployment – This result holds not only for ARDL models where a dependent variable is measured in terms of a change or difference, but also for pure AR models involving such variables Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 124
9.7
Forecasting 9.7.3
Exponential Smoothing Another popular model used for predicting the future value of a variable on the basis of its history is the exponential smoothing method – Like forecasting with an AR model, forecasting using exponential smoothing does not utilize information from any other variable Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 125
9.7
Forecasting 9.7.3
Exponential Smoothing One possible forecasting method is to use the average of past information, such as:
y
ˆ
T
1
y T
y T
1
y T
2 3 – This forecasting rule is an example of a simple (equally-weighted) moving average model with
k
= 3 Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 126
9.7
Forecasting 9.7.3
Exponential Smoothing Now consider a form in which the weights decline exponentially as the observations get older: Eq. 9.72
y T
1 αy T
1
y T
1
– We assume that 0 < α < 1 – Also, it can be shown that:
s
0
s
1 Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables
2
y T
2 Page 127
9.7
Forecasting 9.7.3
Exponential Smoothing Eq. 9.73
For forecasting, recognize that:
ˆ
y T
y T
1
2
y T
2
3
y T
3 Eq. 9.74
– We can simplify to:
y
ˆ
T
1
α
y
T
y
ˆ
T
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 128
9.7
Forecasting 9.7.3
Exponential Smoothing Eq. 9.75
Eq. 9.76
The value of α can reflect one’s judgment about the relative weight of current information – It can be estimated from historical information by obtaining
within-sample forecasts
:
y
ˆ
t
α
y
t
1
y
ˆ
t
1
t
2,3, ,
T
• Choosing α that minimizes the sum of squares of the
one-step forecast errors
:
v t t y
ˆ
t y t
α
y t
1
ˆ
y t
1 Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 129
9.7
Forecasting 9.7.3
Exponential Smoothing FIGURE 9.12 (a) Exponentially smoothed forecasts for GDP growth with α = 0.38
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 130
9.7
Forecasting 9.7.3
Exponential Smoothing FIGURE 9.12 (b) Exponentially smoothed forecasts for GDP growth with α = 0.8
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 131
9.7
Forecasting 9.7.3
Exponential Smoothing The forecasts for 2009Q4 from each value of α are:
G
ˆ
T
1 α
G T
= 0.0536
G
ˆ
T
1 α
G T
= 0.5613
G
ˆ
T G
ˆ
T
0.403921
0.393578
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 132
9.8 Multiplier Analysis Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 133
9.8
Multiplier Analysis Multiplier analysis refers to the effect, and the timing of the effect, of a change in one variable on the outcome of another variable Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 134
9.8
Multiplier Analysis Eq. 9.77
Eq. 9.78
Let’s find multipliers for an ARDL model of the form:
y t
1
y t
1
p y
0
x t
1
x t
1
q x
v t
– We can transform this into an infinite distributed lag model:
y t
0 t 1
x t
1 β 2
x t
2 β 3
x t
3
e t
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 135
9.8
Multiplier Analysis The multipliers are defined as: β
s
y t
x
s
period delay multiplier
j s
0 β
j
s
period interim multiplier
j
0 β
j
total multiplier Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 136
9.8
Multiplier Analysis The
lag operator
is defined as:
Ly t
y t
1 – Lagging twice, we have:
t
Ly t
1 – Or: 2
L y t
y t
2
y t
2 – More generally, we have:
s L y t
y
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 137
9.8
Multiplier Analysis Now rewrite our model as: Eq. 9.79
y t
1
Ly t
2 2
L y t
p p L y t
0
x t
1
Lx t
2 2
L x t
q q L x t
v t
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 138
9.8
Multiplier Analysis Rearranging terms: Eq. 9.80
1 1
L
2
L
2
p L p
y t
0 1
L
2
L
2
q q
t
v t
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 139
9.8
Multiplier Analysis Eq. 9.81
Eq. 9.82
Let’s apply this to our Okun’s Law model – The model:
DU t
1
DU t
1 δ 0
G t
δ 1
G t
1
v t
can be rewritten as: 1
t
δ 0 δ 1
t
v t
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 140
9.8
Multiplier Analysis Define the inverse of (1 – θ 1
L
) as (1 – θ 1
L
) -1 that: such
1
L
1
L
1 Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 141
9.8
Multiplier Analysis Eq. 9.83
Eq. 9.84
Multiply both sides of Eq. 9.82 by (1 – θ 1
L
) -1 :
DU t
1
L
1 δ
1
L
δ 0 δ 1
t
1
L
1
v t
– Equating this with the infinite distributed lag representation:
DU t
β 0
G t
0 β 1 β 1
G t L
1 β 2
L
2 β 2
G t
β 3 2
L
3 β 3
G t
3
G t
e t e t
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 142
9.8
Multiplier Analysis Eq. 9.85
Eq. 9.86
Eq. 9.87
For Eqs. 9.83 and 9.84 to be identical, it must be true that:
1
L
1 δ β 0 β 1
L
β 2
L
2 β 3
L
3
e t
1
L
1
v t
1
L
δ 0 δ 1
L
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 143
9.8
Multiplier Analysis Multiply both sides of Eq. 9.85 by (1 – θ 1
L
) to obtain (1 – θ 1
L
)α = δ. – Note that the lag of a constant that does not change so
L
α = α – Now we have:
1
δ and α δ 1 Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 144
9.8
Multiplier Analysis Eq. 9.88
Multiply both sides of Eq. 9.86 by (1 – θ 1
L
): δ 0 δ 1
L
1
L
β 0 β 1
L
β 2
L
2 β 3
L
3 β 0 β 1
L
β 2
L
2 β 3
L
3 β 0 0 1
L
β θ 1 1
L
2 0 1 β θ 2 1
L
3 β 2
L
2
β 3
L
3 Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 145
9.8
Multiplier Analysis Eq. 9.89
Rewrite Eq. 9.86 as: δ 0 δ 1
L
0
L
2 0
L
3 β 0 1 0 1 2 1 1
L
2 3 2 1
L
3 – Equating coefficients of like powers in
L
yields: δ 1 0 0 1 0 β 2 β θ 1 1 0 β 3 β θ 2 1 and so on Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 146
9.8
Multiplier Analysis Eq. 9.90
We can now find the β’s using the recursive equations: β β = δ 0 0 β
j
1 1 0 1 β θ for
j
1 1
j
2 Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 147
9.8
Multiplier Analysis Eq. 9.91
You can start from the equivalent of Eq. 9.88 which, in its general form, is: δ 0 δ 1
L
δ 2
L
2 δ
q L q
1 β 0
L
θ 2
L
2 β 1
L
β 2
L
2 θ
p L p
β 3
L
3 – Given the values
p
and
q
for your ARDL model, you need to multiply out the above expression, and then equate coefficients of like powers in the lag operator Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 148
9.8
Multiplier Analysis For the Okun’s Law model:
DU t
DU t
1 0.1841
G t
0.0992
G t
1 – The impact and delay multipliers for the first β ˆ 1 β ˆ 2 β ˆ 3 β ˆ 4 four quarters are: ˆ 0 ˆ 0 0.1841
1 ˆ ˆ 0 1 β θ 1 1 β θ 2 1 β θ 3 1 Principles of Econometrics, 4t h Edition 0.0573
0.0201
0.0070
Chapter 9: Regression with Time Series Data: Stationary Variables 0.1636
Page 149
9.8
Multiplier Analysis FIGURE 9.13 Delay multipliers from Okun’s law ARDL(1,1) model Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 150
9.8
Multiplier Analysis We can estimate the total multiplier given by:
j
0 β
j
and the normal growth rate that is needed to maintain a constant rate of unemployment:
G N
α
j
0 β
j
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 151
9.8
Multiplier Analysis
j
0 We can show that: β
j
δ ˆ 0 ˆ 1 ˆ ˆ 0 1 ˆ 1 0.184084
– 0.163606
An estimate for α is given by: ˆα ˆδ ˆ 1 0.37801
0.649884
0.5817
– Therefore, normal growth rate is: 0.4358
ˆG N 0.5817
0.4358
1.3% per quarter Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 152
Key Words Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 153
Keywords AIC criterion AR(1) error AR(p) model ARDL(p,q) model autocorrelation Autoregressive distributed lags autoregressive error autoregressive model BIC criterion correlogram delay multiplier distributed lag weight Principles of Econometrics, 4t h Edition dynamic models exponential smoothing finite distributed lag forecast error forecast intervals forecasting HAC standard errors impact multiplier infinite distributed lag interim multiplier lag length lag operator lagged dependent variable Chapter 9: Regression with Time Series Data: Stationary Variables LM test multiplier analysis nonlinear least squares out-of-sample forecasts sample autocorrelations serial correlation standard error of forecast error SC criterion total multiplier T x R 2 test form of LM within-sample forecasts Page 154
Appendices Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 155
9A The Durbin Watson Test Eq. 9A.1
For the Durbin-Watson test, the hypotheses are:
H
0 : 0
H
1 : 0 The test statistic is:
d
t T
2
e
ˆ
t
e
ˆ
t
1
2
t T
1
e
ˆ
t
2 Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 156
9A The Durbin Watson Test Eq. 9A.2
We can expand the test statistic as:
d
t T
2
e
ˆ
t
2
t T
2
e
ˆ
t
2 1
t T
1
e
ˆ
t
2 2
t T
2
e e t t
1
t T
2
e
ˆ
t
2
t T
1
e
ˆ
t
2
t T
2
e
ˆ
t
2 1
t T
1
e
ˆ
t
2 2
t T
2
t T
1
e
ˆ
t
2 1 Principles of Econometrics, 4t h Edition
r
1 Chapter 9: Regression with Time Series Data: Stationary Variables Page 157
9A The Durbin Watson Test We can now write: Eq. 9A.3
d
r
1 – If the estimated value of ρ is
r
1 Durbin-Watson statistic
d
≈ 2 • errors are not autocorrelated = 0, then the This is taken as an indication that the model – If the estimate of ρ happened to be
r
1
d
≈ 0 • = 1 then A low value for the Durbin-Watson statistic implies that the model errors are correlated, and ρ > 0 Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 158
9A The Durbin Watson Test FIGURE 9A.1 Testing for positive autocorrelation Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 159
9A The Durbin Watson Test 9A.1
The Durbin Watson Bounds Test FIGURE 9A.2 Upper and lower critical value bounds for the Durbin Watson test Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 160
9A The Durbin Watson Test 9A.1
The Durbin Watson Bounds Test Decision rules, known collectively as the Durbin Watson bounds test: – If
d
<
d
Lc : reject
H
0 : ρ = 0 and accept H 1 : ρ > 0 – If
d
>
d
Uc do not reject
H
0 : ρ = 0 – If
d
Lc <
d
<
d
Uc , the test is inconclusive Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 161
9B Properties of the AR(1) Error Eq. 9B.1
Eq. 9B.2
Note that:
e t
ρ
e t
1
e t
2
v t
v t
1
v t
ρ
2
e t
2
ρ
v t
1
v t
Further substitution shows that:
e
t
e
t
3
v
t
2
ρ
v
t
1
v
t
ρ
3
e
t
3
ρ
2
v
t
2
ρ
v
t
1
v
t
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 162
9B Properties of the AR(1) Error Eq. 9B.3
Eq. 9B.4
Repeating the substitution
k
times and rearranging:
e t
ρ
k e
t
ρ
v t
1
ρ
2
v t
2
ρ
k
1
v t k
1 If we let
k
→ ∞, then we have:
e t
t
ρ
v t
1
ρ
2
v t
2
ρ
3
v t
3 Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 163
9B Properties of the AR(1) Error We can now find the properties of
e
t :
ρE
ρ
2
ρ
3
0 ρ 0 ρ var
t
0 var
t
v v
2
v
ρ
2
1 ρ
2
v
2 2
ρ
4
0 ρ
3
ρ
4 2
v
t
1
ρ
6
ρ
6 2
v
0
t
2
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 164
t
3
9B Properties of the AR(1) Error The covariance for one period apart is:
cov
t t
1
E
v
t
v
t
1
ρ
v
t
ρ
v
t
1 2
ρ
2
v
t
2
ρ
2
v
t
3
ρ
3
v
t
3
ρ
3
v
t
4
ρ
E v
t
2 1 2
v
2
ρ
3
ρ
E v
t
2 4
2
ρ
5
t
2 3
ρ
v
2 2 Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 165
9B Properties of the AR(1) Error Similarly, the covariance for
k
periods apart is:
cov
t
ρ
k
2
v
2
k
0
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 166
9C Generalized Least Squares Estimation Eq. 9C.1
Eq. 9C.2
We are considering the simple regression model with
AR
(1) errors:
y t
1 2
x t
e t e t
e t
1
v t
To specify the transformed model we begin with:
y t
1 2
x t
y t
1 1 2
x t
1
v t
– Rearranging terms:
y t
y t
1 1
1 2
x t
x t
1
v t
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 167
9C Generalized Least Squares Estimation Eq. 9C.3
Defining the following transformed variables:
y t
t y t
1
x t
2
t x t
1
x
t
1 Substituting the transformed variables, we get:
y t
x
t
1 1
x t
2 2
v t
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 168
9C Generalized Least Squares Estimation There are two problems: 1. Because lagged values of
y
t formed, only (
T
and
x
t had to be - 1) new observations were created by the transformation 2. The value of the autoregressive parameter ρ is not known Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 169
9C Generalized Least Squares Estimation Eq. 9C.4
For the second problem, we can write Eq. 9C.1 as:
y t
1 2
x t
(
y t
1 1 2
x t
1 )
v t
For the first problem, note that:
y
1 1
x
1 2
e
1 and that 1 2
y
1 1 1 1 1 2 1 2
e
1 Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 170
9C Generalized Least Squares Estimation Eq. 9C.5
Eq. 9C.6
Or:
y
1
x
11 1
x
12 2
e
1 where
y
1 1 2
y
1
x
11 1 2
x
12 1 2
x
1
e
1 1 2
e
1 Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 171
9C Generalized Least Squares Estimation To confirm that the variance of
e
* 1 is the same as that of the errors (
v
2 ,
v
3 ,…,
v
T ), note that: var( ) 1 (1 2
e
1 (1 2 ) 1 2
v
2 2
v
Principles of Econometrics, 4t h Edition Chapter 9: Regression with Time Series Data: Stationary Variables Page 172