Length of Lag DW

0 0.38

4 0.58

8 0.35

12 0.44

"L. G. Godfrey and D. S. Poskitt, "Testing the Restrictions of the Almon Lag Technique," Journal of the American Statistical Association, Vol. 70, March 1975, pp. 105-108.

quence of tests are independent and that the significance level for the 7th test is

1 - (1 - 7,)

where 7, is the significance level chosen at the /th step in the sequence. If 7, = 0.05 for all /, the significance levels for four successive tests are 0.05, 0.0975 , 0.1426, and 0.1855, respectively. Godfrey and Poskitt use this procedure suggested by Anderson to choose the degree of the polynomial for the Almon lag.

We will now illustrate this procedure with an example.

Illustrative Example

Consider the data in Table 10.3. The data are on capital expenditures ( ) and approximations {X) for the years 1953-1967 on a quarterly basis. These data are from the National Industrial Conference Board. Since they have been used very often (in fact, too often) in the illustration of different distributed lag models, they are being reproduced here. It is a good data set for some beginning exercises in distributed lag estimation.

Before we impose any functional forms on the data, it would be interesting to see what some OLS estimates of unrestricted distributed lag equations look like. There are a total of 60 observations and it was decided to estimate 12 lagged coefficients. The implicit assumptions is that the maximum time lag between appropriations and expenditures is 3 years. To make the estimated comparable, the equations were all estimated using the last 48 observations of the dependent variable Y. The results are presented in Table 10.4.

There are several features of the lag distributions in Table 10.4 that are interesting. The steadily increases until we use seven lags. The sum of the coefficients also increases steadily. Overall, from the results presented there, it appears that a lag distribution using seven lags is appropriate. This corresponds to a maximum of 1.75-year lag between capital appropriations and expenditures. Further, about 98.6% of all appropriations are eventually spent.

One disturbing feature of the OLS estimates is that no matter how many lags we include, the DW test statistic shows significant positive correlation. For instance, the following are the DW test statistics for different lengths of the lag distribution.

Table 10.3 Capital Expenditures (Y) and Appropriations {X) for 1953-1967 (Quarterly Data)

This suggests that we should be estimating some more general dynamic models, allowing for autocorrelated errors.

Choosing the Degree of the Polynomial

Consider the data in Table 10.3. We will consider a lag length of 12 and consider the choice of the degrees of the polynomial by the sequential testing method outlined here. We start with a fourth-degree polynomial.

The results are as follows (figures in parentheses are t-ratios, not standard errors):

rnffirij>r,t Equation

10.9 RATIONAL LAGS

First we test the coefficient of Z4, at the 5% level and do not reject the hypothesis that it is zero. Next we test the coefficient of Zy, and we reject the hypothesis that its coefficient is zero. Since this is the first hypothesis rejected, we use a polynomial of the third degree. The results of the other lower-degree polynomials are not needed and are just presented for the sake of curiosity. Actually, for the fourth-degree polynomial, the high R and the low f-ratios indicates that there is high multicoUinearity among the variables.

We now estimate the coefficients of the lag distribution using the formula

P, = 0.1108 + 0.3305 X 10 - 0.8549 x lO/z + 0.4011 x 10~4 The coefficients are:

The sum of the coefficients is 0.9017, and furthermore, the last three coefficients are negative. This is the reason that, as mentioned earlier, some endpoint restrictions are often imposed in estimating the polynomial distributed lags. Comparing these results with the unrestricted OLS estimates in Table 10.4, it appears that not much is to be gained (in this particular example) by using the polynomial lag.

In any case, the example illustrates the procedure of choice of the degree of the polynomial for given lag length.

10.9 Rational Lags

In Section 10.8 we discussed finite lag distributions. We will now discuss infinite lag distributions. Actually, we considered one earlier: the geometric or Koyck lag. A straightforward generalization of this is the rational lag distribution.

*D. W. Jorgenson, "Rational Distributed Lag Functions," Econometnca, Vol. 34, January 1966, pp. 135-149.