Module 9.2 LOS 9.e: Testing AR models for improper specifications using t-tests

We cannot use DW tests to test for autocorrelation (serial correlation in an AR model). To identify autocorrelation we use the following procedure:

1. Estimate the AR model being evaluated using linear regression:
   Start with a first-order AR model (i.e., AR(1)) using x_t = b₀ + b₁x_t–1 + ε_t.
2. Calculate the autocorrelations of the model’s residuals (i.e., the level of correlation between the forecast errors from one period to the next).
3. Test whether the autocorrelations are significantly different from zero.

If the model is correctly specified, none of the autocorrelations will be statistically significant, or above the critical value at the level of significance we are testing. To test for significance, a t-test is used to test the hypothesis that the correlations of the residuals are zero. The t-statistic is the estimated autocorrelation divided by the standard error. The standard error is:

where T is the number of observations, so the test statistic for each autocorrelation is:

with (T − 2) degrees of freedom.

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