Testing disturbances in the dynamic linear regression model
Finally we turn our attention to the problem of testing for autocorrelation in the disturbances of the dynamic regression model (3.19). Whether the DW test can
be used in these circumstances has been an area of controversy in the literature. Durbin and Watson (1950) and others have warned against its use in the dynamic model and this is a theme taken up by many textbooks until recently. The problem has been one of finding appropriate critical values.
Based on the work of Inder (1985, 1986), King and Wu (1991) observed that the small disturbance distribution (the limiting distribution as о2 ^ 0) of the DW statistic is the exact distribution of d1 for the corresponding regression with the lagged dependent variables replaced by their expected values. This provides a justification for the use of the familiar tables of bounds when the DW test is applied to a dynamic regression model. It also highlights a further difficulty. Because E(yt_1) is a function of the regression parameters, it is clear the null distribution of the DW test also depends on these parameters, so the test can have different sizes for different parameter values under the null hypothesis. It appears this is a property shared by many alternative tests.
Durbin (1970) suggested his й-test and t-test as alternatives to the DW test. The й-test can suffer from problems in small samples caused by the need to take the square root of what can sometimes be a negative estimate of a variance. The t-test appears more reliable and can be conducted in a simple manner. Let e1,..., en be the OLS residuals from (3.19). H0 : p = 0 can be tested using OLS regression to test the significance of the coefficient of et-1 in the regression of et on et-1, yt-1, yt_2,..., xt.
That this can be a difficult testing problem is best illustrated by considering
(3.18) and (3.13). If xt is lag invariant then if we switch p and a1, we will end up with the same value of the likelihood function which indicates a local identification problem. For near lag invariant xt vectors, it is therefore difficult to distinguish between p and a1. This causes problems for the small-sample properties of asymptotic tests in this case and explains why it has been difficult to find a satisfactory test for this testing problem.