if we are estimating the first difference equation, we can use the Berenblut-Webb test as well, without any extra computational burden.

B. There are many tables other than those reprinted at the end of this book for the DW test that have been prepared to take care of special situations. Some of these are:

1. R. W. Farebrother in Econometrica, Vol. 48, September 1980, pp. 1553-1563, gives tables for regression models with no intercept term.

2. N. E. Savin and K. J. White, in Econometrica, Vol. 45, No. 8, November 1977, pp. 1989-19%, present tables for the DW test for samples with 6 to 200 observations and for as many as 20 regressors.

3. K. F Wallis in Econometrica, Vol. 40, 1972, pp. 617-636, gives tables for regression models with quarterly data. Here one would like to test for fourth-order autocorrelation rather than first-order autocorrelation. In this case the DW statistic is

d,i=-

Wallis provides 5% critical values and dy for two situations: where the regressors include an intercept (but not a full set of seasonal dummy variables) and another where the regressors include four quarterly seasonal dummy variables. In each case the critical values are for testing : p = 0 against ,: p > 0. For the hypothesis ,: p < 0 Wallis suggests that the appropriate critical values are (4 - djj) and (4 - J). M. L. King and D. E. A. Giles in Journal of Econometrics, Vol. 8, 1978, pp. 255-260, give further significance points for WaUiss test.

4. M. L. King in Econometrica, Vol. 49, November 1981, pp. 1571-1581, gives the 5% points for dj and dy for quarterly time-series data with trend and/or seasonal dummy variables. These tables are for testing first-order autocorrelation.

5. M. L. King in Journal of Econometrics, Vol. 21, 1983, pp. 357-366, gives tables for the DW test for monthly data. In the case of monthly data we would want to test for twelfth-order autocorrelation.

C. All the elaborate tables mentioned in have been prepared for 5% level of significance (and 1% level of significance) and a question arises as to what the appropriate level of significance is for the DW test. Given that the test for serial correlation is a prelude to further estimation and not an end in itself, the theory of pretest estimation suggests that a significance level of, say, 0.35 or 0.4 is more appropriate than the conventional 0.05 significance level.

-"See, for instance, T. B. Fomby and D. K. Guilkey, "On Choosing the Optimal Level of Significance for the Durbin-Watson Test and a Bayesian Alternative," Journal of Econometrics, Vol. 8, 1978, pp. 203-214.

(I - ap)D

"The proofs are somewhat long and are omitted. For a first-order autoregressive x, they can be found in G. S. Maddala and A. S. Rao, "Tests for Serial Correlation in Regression Models with Lagged Dependent Variables and Serially Correlated Errors," Econometrica, Vol. 47, No. 4, July 1973, App. A, pp. 761-774.

D. A significant DW statistic can arise fiom a lot of different sources. The DW statistic can detect moving average errors, AR(2) errors, or just the effect of omitted variables that are themselves autocorrelated. This raises the question of what the appropriate strategy should be when the DW statistic is significant. It does not necessarily imply that the errors are AR(1). One can proceed in different directions. The different strategies are discussed in Section 6.9.

Finally, the DW test is not applicable if the explanatory variables contain lagged dependent variables. The appropriate tests are discussed in the next section.

6.7 Tests for Serial Correlation in Models with Lagged Dependent Variables

In previous sections we considered explanatory variables that were uncorrelated with the error term. This will not be the case if we have lagged dependent variables among the explanatory variables and we have serially correlated errors. There are several situations under which we would be considering lagged dependent variables as explanatory variables. These could arise through expectations, adjustment lags, and so on. The various situations and models are explained in Chapter 10. For the present we will not be concerned with how the models arise. We will merely study the problem of testing for autocorrelation in these models. Let us consider a simple model

y, = ay, , + Px, + u, (6.12)

u, = ,„, + e, (6.13)

e, are independent with mean 0 and variance and a < 1, p < 1. Because u, depends on «, , and y,„, depends on the two variables y, , and u, will be correlated. The least squares estimator a will be inconsistent. It can be shown that

plim a = a -I- A

plim p = p - A

where

6.7 TESTS IN MODELS WITH LAGGED DEPENDENT VARIABLES 249

D = var(y ,) var(jc) - coviyi, jc) > 0

aj = var(jc) al = var(M)

Thus if p is positive, the estimate of a is biased upward and the estimate of p is biased downward. Hence the DW statistic, which is = 2(1 - p) is biased toward 2 and we would not find any significant serial correlation even if the errors are serially correlated.

Durbins /7-Test

Since the DW test is not applicable in these models, Durbin suggests an alternative test, called the /i-test.- This test uses

= p V1 - &}

as a standard normal variable. Here the p is the estimated first-order serial correlation from the OLS residuals V(a) is the estimated variance of the OLS estimate of a, and n is the sample size. If nVia) > 1, the test is not applicable. In this case Durbin suggests the following test:

Durbins Alternative Test

From the OLS estimation of (6.12) compute the residuals ,. Then regress , on M, ,, y, ,, and JC,. The test for p = 0 is carried out by testing the significance of the coefficient of , , in the latter regression.

Illustrative Example

An equation of demand for food estimated from 50 observations gave the following results.

log q, = const. - 0.31 log P, + 0.45 log y, -I- 0.65 log q, ,

(0.05) (0.20) (0.14)

= 0.90 DW = 1.8 (Figures in parentheses are standard errors.) q, - food consumption per capita

p, = food price (retail price deflated by the Consumer Price Index)

y, = per capita disposable income deflated by the Consumer Price Index

Durbin, "Testing for Serial Correlation in Least Squares Regression When Some of the Regressors Are Lagged Dependent Variables," Econometrica, 1970, pp. 410-421. Durbins paper was prompted by a note by Nerlove and Wallis that argued that the DW statistic is not applicable when lagged dependent variables are present. See M. Nerlove and K. F. Wallis, "Use of DW Statistic in Inappropriate Situations," Econometrica, Vol. 34, 1966, pp. 235-238.