# Distributed-Lag Models

5.1.5 The General Lag

If we add an exogenous variable to the right-hand side of Eq. (5.3.2), we obtain

(*L)y, = ax, + р(Це,. (5.6.1)

Such a model is called a distributed-lag model by econometricians because solving (5.6.1) for y, yields

where p~l(L) describes a distribution of the effects of the lagged values of x, (that is, xt, xt-i, xt-2, • • .) on y,. There is a vast amount of econometric literature on distributed-lag models. The interested reader is referred to the aforementioned works by Fuller (1976); Nerlove, Grether, and Carvalho

(1979) ; and Dhrymes (1971). A model of the form (5.6.2) also arises as a solution of the rational expectations model (see Shiller, 1978; Wallis, 1980). We shall discuss briefly two of the most common types of distributed-lag models: the geometric lag and the Almon lag.

5.1.6 The Geometric Lag

The geometric lag model is defined by

(5.6.3)

This model was first used in econometric applications by Koyck (1954) (hence, it is sometimes referred to as the Koyck lag) and by Nerlove (1958). Griliches’s survey article (1967) contains a discussion of this model and other types of distributed-lag models. This model has the desirable property of having only two parameters, but it cannot deal with more general types of lag distribution where the effect of x on у attains its peak after a certain number of lags and then diminishes.

By inverting the lag operator 2“_0 MU, we can write (5.6.3) equivalently as

У, = Му,- і + ax,+ ut, (5.6.4)

where u, = vt — . If {«,} are i. i.d., the estimation of Я and a is not much

more difficult than the estimation in the purely autoregressive model dis­cussed in Section 5.4. Under general conditions the least squares estimator can be shown to be consistent and asymptotically normal (see Crowder, 1980). For a discussion of the estimation of Я and a when (i>f), rather than {«,}, are

i. i. d., see Amemiya and Fuller (1967) and Hatanaka (1974) (also see Section 6.3.7 for further discussion of this model).