Koyck has proposed an ingenious method of estimating distributed-lag models. Suppose we start with the infinite lag distributed-lag model (17.3.1).
Assuming that the fa's are all of the same sign, Koyck assumes that they decline geometrically as follows.10
where X, such that 0 < X < 1, is known as the rate of decline, or decay, of the distributed lag and where 1 — X is known as the speed of adjustment.
What (17.4.1) postulates is that each successive fa coefficient is numerically less than each preceding fa (this statement follows since X < 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. Geometrically, the Koyck scheme is depicted in Figure 17.5.
10L. M. Koyck, Distributed Lags and Investment Analysis, North Holland Publishing Company, Amsterdam, 1954.
11Sometimes this is also written as
for reasons given in footnote 12.
Models: Autoregressive and Distributed-Lag Models
666 PART THREE: TOPICS IN ECONOMETRICS
As this figure shows, the value of the lag coefficient jk depends, apart from the common j0; on the value of X. The closer X is to 1, the slower the rate of decline in jk, whereas the closer it is to zero, the more rapid the decline in jk. In the former case, distant past values of X will exert sizable impact on Yt, whereas in the latter case their influence on Yt will peter out quickly. This pattern can be seen clearly from the following illustration:
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