ßo 0.75ßo ß0 0.25ß0
■ 0.06ß0 ■ 0.0
Note these features of the Koyck scheme: (1) By assuming nonnegative values for X, Koyck rules out the 0's from changing sign; (2) by assuming X < 1, he gives lesser weight to the distant 0's than the current ones; and (3) he ensures that the sum of the 0's, which gives the long-run multiplier, is finite, namely,
As a result of (17.4.1), the infinite lag model (17.3.1) may be written as
As it stands, the model is still not amenable to easy estimation since a large (literally infinite) number of parameters remain to be estimated and the parameter X enters in a highly nonlinear form: Strictly speaking, the method of linear (in the parameters) regression analysis cannot be applied to such a model. But now Koyck suggests an ingenious way out. He lags (17.4.3) by one period to obtain
Yt = a + ßo Xt + ßoX Xt-i + ßoX2 X— + ■■■+ u (17.4.3)
Yt_i = a + ßo Xt_i + ßoX Xt-2 + ßoX2 X— + ■ ■ ■ + u— (17.4.4)
He then multiplies (17.4.4) by X to obtain
XYt_i = Xa + Xß o Xt-i + ßoX2 Xt-2 + ßoX3 Xt-3 + ■ ■ ■ + Xut-i (17.4.5)
12This is because since the expression in the parentheses on the right side is an infinite geometric series whose sum is 1/(1 — X) provided 0 < X < 1. In passing, note that if 0k is as defined in footnote 11, Y^ 0k = 0o(1 — X)/(1 — X) = 0o thus ensuring that the weights (1 — X)Xk sum to one.
since the expression in the parentheses on the right side is an infinite geometric series whose sum is 1/(1 — X) provided 0 < X < 1. In passing, note that if 0k is as defined in footnote 11, Y^ 0k = 0o(1 — X)/(1 — X) = 0o thus ensuring that the weights (1 — X)Xk sum to one.
Models: Autoregressive and Distributed-Lag Models
CHAPTER SEVENTEEN: DYNAMIC ECONOMETRIC MODELS 667
where vt = (ut — Xut-1), a moving average of ut and ut-1.
The procedure just described is known as the Koyck transformation. Comparing (17.4.7) with (17.3.1), we see the tremendous simplification accomplished by Koyck. Whereas before we had to estimate a and an infinite number of ft's, we now have to estimate only three unknowns: a, ft0, and X. Now there is no reason to expect multicollinearity. In a sense multicollinearity is resolved by replacing Xt—1, Xt—2,..., by a single variable, namely, Yt—1. But note the following features of the Koyck transformation:
1. We started with a distributed-lag model but ended up with an autoregressive model because Yt— 1 appears as one of the explanatory variables. This transformation shows how one can "convert" a distributed-lag model into an autoregressive model.
2. The appearance of Yt—1 is likely to create some statistical problems. Yt—1, like Yt, is stochastic, which means that we have a stochastic explanatory variable in the model. Recall that the classical least-squares theory is predicated on the assumption that the explanatory variables either are non-stochastic or, if stochastic, are distributed independently of the stochastic disturbance term. Hence, we must find out if Yt—1 satisfies this assumption. (We shall return to this point in Section 17.8.)
3. In the original model (17.3.1) the disturbance term was ut, whereas in the transformed model it is vt = (ut — Xut—1). The statistical properties of vt depend on what is assumed about the statistical properties of ut, for, as shown later, if the original ut's are serially uncorrelated, the vt's are serially correlated. Therefore, we may have to face up to the serial correlation problem in addition to the stochastic explanatory variable Yt—1. We shall do that in Section 17.8.
4. The presence of lagged Y violates one of the assumptions underlying the Durbin-Watson d test. Therefore, we will have to develop an alternative to test for serial correlation in the presence of lagged Y. One alternative is the Durbin h test, which is discussed in Section 17.10.
As we saw in (17.1.4), the partial sums of the standardized ft tell us the proportion of the long-run, or total, impact felt by a certain time period. In
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Econometrics, Fourth Models: Autoregressive Companies, 2004 Edition and Distributed-Lag
668 PART THREE: TOPICS IN ECONOMETRICS
practice, though, the mean or median lag is often used to characterize the nature of the lag structure of a distributed lag model.
The median lag is the time required for the first half, or 50 percent, of the total change in Y following a unit sustained change in X. For the Koyck model, the median lag is as follows (see exercise 17.6):
Thus, if X = 0.2 the median lag is 0.4306, but if X = 0.8 the median lag is 3.1067. Verbally, in the former case 50 percent of the total change in Y is accomplished in less than half a period, whereas in the latter case it takes more than 3 periods to accomplish the 50 percent change. But this contrast should not be surprising, for as we know, the higher the value of X the lower the speed of adjustment, and the lower the value of X the greater the speed of adjustment.
The Mean Lag
Provided all Pk are positive, the mean, or average, lag is defined as
which is simply the weighted average of all the lags involved, with the respective p coefficients serving as weights. In short, it is a lag-weighted average of time. For the Koyck model the mean lag is (see exercise 17.7)
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