#### Transcript autoregressive and distributed-lag models

```DISTRIBUTED-LAG MODELS
SUBJECT in ECONOMETRICS
DARMANTO
STATISTICS
UNIVERSITY of BRAWIJAYA
PREFACE...
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
Distributed-lag Model is a model in regression
where the model includes not only the current but
also the lagged (past) values of the explanatory
variables (the X’s).
Autoregressive Model is a model thath the model
includes one or more lagged values of the
dependent variable among its explanatory
variables.
THE ROLE of LAG in ECONOMICS
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
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In economics the dependence of a variable Y (the dependent
variable) on another variable(s) X (the explanatory variable) is
rarely instantaneous.
Very often, Y responds to X with a lapse of time. Such a lapse of
time is called a lag.
Example:
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
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The consumption function
Lag between R&D expenditure and productivity
The relationship between trade balance and depreciation of currency
(J-curve)
ESTIMATION of DISTRIBUTED-LAG MODEL

Let:
...(1)

How do we estimate the α and β’s of (1)?
1.
2.
a priori restrictions on the β’s by assuming that the β’s

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Since the explanatory variable Xt is assumed to be
nonstochastic, Xt−1, Xt−2, and so on, are non-stochastic, too.
OLS can be applied (taken by Alt and Tinbergen).
They suggest that to estimate (1) one may proceed
sequentially; that is, ﬁrst regress Yt on Xt , then regress Yt on
Xt and Xt−1, then regress Yt on Xt , Xt−1, and Xt−2, and so on.
This sequential procedure stops when the regression
coefﬁcients of the lagged variables start becoming
statistically insigniﬁcant and/or the coefﬁcient of at least
one of the variables changes signs from positive to negative
or vice versa.

Example:
 Alt
chose the second regression as the “best’’ one because in
the last two equations the sign of Xt−2 was not stable and in
the last equation the sign of Xt−3 was negative, which may
be difﬁcult to interpret economically.
THE KOYCK APPROACH: 1

Suppose we start with the inﬁnite lag distributedlag model. Assuming that the β’s are all of the same
sign, Koyck assumes that they decline geometrically
as follows.
...(2)

where λ, such that 0 <λ< 1, is known as the rate of
decline, or decay, of the distributed lag and where
1 − λ is known as the speed of adjustment.
THE KOYCK APPROACH: 2

Postulates (2):
Each successive β coefﬁcient is numerically less than each
preceding β (this statement follows since λ< 1), implying
that as one goes back into the distant past, the effect of
that lag on Yt becomes progressively smaller, a quite
plausible assumption. After all, current and recent past
incomes are expected to affect current consumption
expenditure more heavily than income in the distant past.
THE KOYCK APPROACH: 3

Koyck transformation:
...(3)
where vt = (ut − λut−1), a moving average of ut and ut−1.
The median and mean lags serve as a summary measure of
the speed with which Y responds to X.
THE KOYCK APPROACH: 4
THE KOYCK APPROACH: 5
THE ALMON APPROACH: 1
THE ALMON APPROACH: 2

To illustrate her technique, let us revert to the ﬁnite
distributed-lag model considered previously,
THE ALMON APPROACH: 3
THE ALMON APPROACH: 4
THE ALMON APPROACH: 5
THE ALMON APPROACH: 6
THE ALMON APPROACH: 7
THE ALMON APPROACH: 8
```