11.943 /9

8.387 \10 = 1 - 0.2172 = 0.7828

(0.89)(1 - 0.8096)

This is to be compared with R\ = 0.8096 from the equation in first differences. For the production function data we get

= 1 - 0.2079 = 0.7921

This is to be compared with R = 0.8405 from the equation in first differences. Harvey gives a different definition of R\). He defines it as:

This does not adjust for the fact that the error variances in the levels equations and the first difference equation are not the same. The arguments for his suggestion are given in his paper. In the example with the Friedman-Meiselman data his measure of R\) is given by

119 430

= 1 - -(1 - 0-8096) = 0.7289

Although R]) cannot be greater than 1, it can be negative. This would be the case when 2 ( / - < RSSq, that is, when the levels model is giving a poorer explanation than the naive model, which says that Ay, is a constant.

Usually, with time-series data, one gets high R values if the regressions are estimated with the levels y, and x, but one gets low R values if the regressions are estimated in first differences (y, - y, ,) and (x, - x, ,). Since a high R is usually considered as proof of a strong relationship between the variables under investigation, there is a strong tendency to estimate the equations in levels rather than in first differences. This is sometimes called the "R syndrome."

Harvey, "On Comparing Regression Models," p. 711.

Again, even though the R is larger for the equation in levels, the equation in first differences is better than the equation in levels, because it gives a lower RSS (even after the adjustments described) and a higher DW statistic. The estimate of returns to scale is 0.987 + 0.502 = 1.489 in the first difference equation compared to 1.451 + 0.384 = 1.835 in the levels equation.

We can also compute R\,, the comparable from the equation in levels and see how it compares with R], the R from the equation in first differences. In the example with the Friedman-Meiselman data the value of is given by

However, if the DW statistic is very low, it often implies a misspecified equation, no matter what the value of the is. In such cases one should estimate the regression equation in first differences and if the R is low, this merely indicates that the variables and x are not related to each other. Granger and Newbold* present some examples with artificially generated data where y, x, and the error are each generated independently so that there is no relationship between and x, but the correlations between y, and y,i, x, and x, y, and u, and are very high. Although there is no relationship between and x the regression of on x gives a high R but a low DW statistic. When the regression is run in first differences, the R is close to zero and the DW statistic is close to 2, thus demonstrating that there is indeed no relationship between and x and that the R obtained earlier is spurious. Thus regressions in first differences might often reveal the true nature of the relationship between and x. (Further discussion of this problem is in Sections 6.10 and 14.7.)

Finally, it should be emphasized that all this discussion of the Durbin-Watson statistic, first differences, and quasi-first differences is relevant only if we believe that the correlation structure between the errors can be entirely described in terms of p, the correlation coefficient between u, and u, i. This may not always be the case. We will discuss some general formulations of the correlation structure of the errors in Section 6.9 after we analyze the simple case thoroughly. Also, even if the correlation structure can be described in terms of just one parameter, this need not be the correlation between u, and For instance, suppose that we have quarterly data; then it is possible that the errors in any quarter this year are most highly correlated with the errors in the corresponding quarter last year rather than the errors in the preceding quarter; that is, u, could be uncorrelated with but it could be highly correlated with M, 4. If this is the case, the DW statistic will fail to detect it. What we should be using is a modified statistic defined as

I ( , - , ,)2 2uj

Also, instead of using first differences or quasi first differences in the regressions, we should be using fourth differences or quasi fourth differences, that is, y, - y,~4 and X, - x,4 or y, - py,4 and x, - px,4, where p is the correlation coefficient between the estimated residuals u, and , 4.

6.4 Estimation Procedures with Autocorrelated Errors

In Section 6.3 we considered estimation in first differences. We will now consider estimation with quasi first differences, that is, regressing y, ~ py,„, on X, - px, i. As we said earlier, we will be discussing the simplest case where

C. W. J. Granger and P. Newbold, "Spurious Regressions in Econometrics," Journal of Econometrics. Vol. 2, No. 2, July 1976, pp. III-I20.

Thus we have

1 - p2

This gives the variance of u, in terms of variance of e, and the parameter p. Let us now derive the correlations. Denoting the correlation between u, and (which is called the correlation of lag s) by p, we get

Hence

E{u,u,,) = o-2p E{u,u,..,) = ( , , m, j + E(e,u, J = P • Ps-I + 0 P. = P • P*-i

the entire correlation structure of the errors u, can be summarized in a single parameter p. This would be the case if the errors u, are first-order autoregres-sive, that is,

h, = , , + e, (6.1)

where e, are serially uncorrelated, with mean zero and common variance ]. Equation (6.1) is called an autoregression because it is the usual regression model with u, regressed on ,,. It is called first-order autoregression because M, is regressed on its past with only one lag. If there are two lags, it is called second-order autoregression. If there are three lags, it is called third-order autoregression, and so on. If the errors u, satisfy equation (6.1), we say u, are AR(1) (i.e., autoregressive of first order). If the errors u, satisfy the equation

U, = p,M,. I + P2M, 2 + e,

then we say that , are AR(2), and so on.

Now we will derive var(m,) and the correlations between u, and lagged values of u,. From (6.1) note that u, depends on e, and , , depends on and u,2> and so on. Thus u, depends on e„ , 2>- • • • Since e, are serially independent, and u, , depends on e, ,, e, 2 and so on, but not e„ we have

£(m, ,e,) = 0

Since E(e,) = 0, we have E{u,) = 0 for all t. If we denote var(m,) by cr, we have

cr2 = var(«,) = £(«?)

= £( , , -I- e,f

- pV + ul since cov(m, ,, e,) = 0