DISTRIBUTED-LAG MODELSSUBJECT in ECONOMETRICSDARMANTOSTATISTICSUNIVERSITY of BRAWIJAYA

PREFACE... 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 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: The consumption function Link between money and price Lag between R&D expenditure and productivity The relationship between trade balance and depreciation of

currency (J-curve)

ESTIMATION of DISTRIBUTED-LAG MODEL

Let:

How do we estimate the α and β’s of (1)? 1. ad hoc estimation2. a priori restrictions on the β’s by

assuming that the β’s follow some systematic pattern.

...(1)

AD HOC ESTIMATION: 1 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.

AD HOC ESTIMATION: 2 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

distributed-lag model. Assuming that the β’s are all of the same sign, Koyck assumes that they decline geometrically as follows.

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.

...(2)

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:

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.

...(3)

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