DISTRIBUTED-LAG MODELS

1. DISTRIBUTED-LAG MODELS SUBJECT in ECONOMETRICS DARMANTO STATISTICS UNIVERSITY of BRAWIJAYA

2. PREFACE... • Distributed-lag Model is a model in regression where the modelincludes not only the current but also the lagged (past) values of the explanatory variables (the X’s). • Autoregressive Modelis a model thath themodel includes one or more lagged values of the dependent variable amongits explanatory variables.

3. THE ROLE of LAG in ECONOMICS • In economics the dependence of a variable Y (the dependent variable) onanother 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 iscalled 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)

4. ESTIMATION of DISTRIBUTED-LAG MODEL • Let: • How do we estimate the α and β’s of (1)? • ad hoc estimation • a priori restrictions on the β’s byassuming that the β’s follow some systematic pattern. ...(1)

5. 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 canbe applied (taken by Altand 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 onXt , Xt−1, and Xt−2, and so on. • This sequential procedure stops when theregression coefficients of the lagged variables start becoming statisticallyinsignificant and/or the coefficient of at least one of the variables changessigns from positive to negative or vice versa.

6. AD HOC ESTIMATION: 2 • Example: • Alt chose the second regression as the “best’’ one because in the last twoequations the sign of Xt−2 was not stable and in the last equation the sign ofXt−3 was negative, which may be difficult to interpret economically.

7. THE KOYCK APPROACH: 1 • Suppose we start with the infinite lag distributed-lagmodel. 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 thedistributed lag and where 1 − λ is known asthe speed of adjustment. ...(2)

8. 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.

9. THE KOYCK APPROACH: 3 • Koyck transformation: where vt = (ut − λut−1), a moving average of ut and ut−1. The median and mean lagsserve as a summary measure of the speed with which Y responds to X. ...(3)

10. THE KOYCK APPROACH: 4

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12. THE ALMON APPROACH: 1

13. THE ALMON APPROACH: 2 • To illustrate her technique, let us revert to the finitedistributed-lag model considered previously,

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