# Infinite Distributed Lag

So far we have been dealing with a finite number of lags imposed on Xt. Some lags may be infinite. For example, the investment in building highways and roads several decades ago may still have an effect on today’s growth in GNP. In this case, we write equation (6.1) as

Yt _ a + £І 3iXt-i + ut t _1, 2,…,T. (6.7)

There are an infinite number of 3i s to estimate with only T observations. This can only be feasible if more structure is imposed on the 3fs. First, we normalize these 3fs by their sum, i. e., let wi _ 3i/3 where 3 _ i=0 3i. If all the 3i’ s have the same sign, then the 3i’ s take the sign of 3 and 0 < wi < 1 for all i, with ^°=0 wi _ 1. This means that the wi’ s can be interpreted as probabilities. In fact, Koyck (1954) imposed the geometric lag on the wi’ s, i. e., wi _ (1 — Л)Лі for i _ 0,1,…, те1. Substituting

3i _ 3wi _ 3(1 — л)лі

in (6.7) we get

Yt _ a + 3(1 — л) І=о лXt-i + ut (6.8)

Equation (6.8) is known as the infinite distributed lag form of the Koyck lag. The short-run effect of a unit change in Xt on Yt is given by 3(1 — Л); whereas the long-run effect of a unit change in Xt on Yt is X}І=0 3i _ 3^2І=0 wi _ 3. Implicit in the Koyck lag structure is that the effect of a unit change in Xt on Yt declines the further back we go in time. For example, if Л _ 1/2, then 30 _ 3/2, 3i _ 3/4, 32 _ 3/8, etc. Defining LXt _ Xt-1, as the lag operator, we have L1 Xt _ Xt-i, and (6.8) reduces to Yt _ a + 3(1 — л) T=0(LiXt + ut _ a + 3(1 — л)Xt/(1 — л^ + ut

where we have used the fact that ^°=0 сг — 1/(1 — c). Multiplying the last equation by (1 — XL) one gets

Yt — XYt-i — a(1 — X)+ в (1 — X)Xt + ut — Xut-i

or

Yt — XYt-i + a(1 — X)+ в (1 — X)Xt + ut — Xut-i (6.10)

This is the autoregressive form of the infinite distributed lag. It is autoregressive because it includes the lagged value of Yt as an explanatory variable. Note that we have reduced the problem of estimating an infinite number of в/s into estimating X and в from (6.10). However, OLS would lead to biased and inconsistent estimates, because (6.10) contains a lagged dependent variable as well as serially correlated errors. In fact the error in (6.10) is a Moving Average process of order one, i. e., MA(1), see Chapter 14. We digress at this stage to give two econometric models which would lead to equations resembling (6.10).

Suppose that output Yt is a function of expected sales Xf and that the latter is unobservable, i. e.,

Yt — a + вх; + ut

where expected sales are updated according to the following method

Xf — X-i — 6(Xt — X-i) (6.11)

that is, expected sales at time t is a weighted combination of expected sales at time t — 1 and actual sales at time t. In fact,

Xf — 6Xt + (1 — S)Xf-l (6.12)

Equation (6.11) is also an error learning model, where one learns from past experience and adjust expectations after observing current sales. Using the lag operator L, (6.12) can be rewritten as Xf — 8Xt/[1 — (1 — 6)L]. Substituting this last expression in the above relationship, we get

Yt — a + в8Xt/[1 — (1 — S)L]+ut (6.13)

Multiplying both sides of (6.13) by [1 — (1 — S)L], we get

Yt — (1 — 6)Yt-i — a[(1 — (1 — 6)] + в8Xt + ut — (1 — 6)ut-i (6.14)

(6.14) looks exactly like (6.10) with X — (1 — 6).

Under this model there is a cost of being out of equilibrium and a cost of adjusting to that equilibrium, i. e., Cost — o(Yt — Yt*)2 + b(Yt — Yt-i)2

where Yt* is the target or equilibrium level for Y, whereas Yt is the current level of Y. The first term of (6.15) gives a quadratic loss function proportional to the distance of Yt from the equilibrium level Yt*. The second quadratic term represents the cost of adjustment. Minimizing this quadratic cost function with respect to Y, we get Yt = 7Yt*+(1—Y)Yt_i, where 7 = a/(a+b). Note that if the cost of adjustment was zero, then b = 0, 7 = 1, and the target is reached immediately. However, there are costs of adjustment, especially in building the desired capital stock. Hence,

Yt = YY* + (1 — 7) Yt_i + ut (6.16)

where we made this relationship stochastic. If the true relationship is Yt* = a + /Xt, then from (6.16)

Yt = 7a + YPXt + (1— y)Y_i + ut (6.17)

and this looks like (6.10) with A = (1 — 7), except for the error term, which is not necessarily MA(1) with the Moving Average parameter A.