Durbin-Watson Critical Values for Large Sample Sizes

Question

My sample includes 3,627 observations but I can only find tables displaying critical values for the Durbin-Watson test for sample sizes 2,000 and below.

Where can I find tables for sample sizes exceeeding 2,000 observations?

Answer 1

For such a large sample size, the normal approximation of the Durbin-Watson distribution is working well enough. You just need to compute the specific expectiation and variance of the Durbin Watson statistic $d$ given your regressor matrix $X$ under the null distribution. For example, this is implemented in the dwtest() function in the R package lmtest. Also, that function interfaces a numerical algorithm to compute the exact distribution (depending on $X$) rather than using the tables of upper and lower bounds of critical values.

The dwtest() function implements the following formulas, originally taken from the econometrics book of Griffiths, Hill, and Judge, I think. With the usual transformation matrix $A$ (with 2 on the diagonal, except the first and last element, and -1 on the off-diagonals), the test statistic $d$ can be written as: $d = \frac{e^\top A e}{e^\top e}$.

Then, we define to auxiliary quantities $P$ and $Q$:

$P = 2 \cdot (n - 1) - tr(X^\top A X (X^\top X)^{-1})$

$Q = 2 \cdot (3 \cdot n - 4) - 2 \cdot tr(X^\top A^\top A X (X^\top X)^{-1}) + tr(X^\top A X (X^\top X)^{-1} X^\top A X (X^\top X)^{-1})$

And with these we can easily compute $E(d) = \frac{1}{n-k} \cdot P$ and $Var(d) = \frac{2}{(n - k) (n - k + 2)} \cdot (Q - P \cdot E(d))$. The variables $n$ and $k$ are the number of observations and regressors in $X$, respectively, and $tr(\cdot)$ is the trace of a matrix.