When an AR model is correctly specified, the residual terms will not exhibit serial correlation.
But why we can’t use Durbin-Watson to test if its residual terms exist the serial correlation, then conclude AR model is correctly specified or not? Thanks.
I think the Durbin-Watson test is inappropriate for AR models; they are only applicable to non-AR linear regression models. I believe you have to use Durbin’s h test for AR models (not covered in CFAI material)
Thanks, I understand we can’t use it but just curious:
If AR model is correctly specified, the residual terms will not exhibit serial correlation.
Why we can’t check the residual terms directly?
If AR model is correctly specified, there won’t be serial correlation.
It is generally the other way …you check for autocorrelation and if you dont mind any, then you check for ARCH. If none if present, then the model is correctly specified.
The DW test is most powerful for detecting first-order autocorrelation. If you have accounted for 1st order correlation, it is not likely that you will detect higher order autocorrelations with the DW test. Additionally, an aspect of serial correlation is the time-ordering of the data. The order of the observations (and obviously corresponding residuals) is important, which is why the DW test should only be applied to time series data.
The DW test can be used on randomized data (random order/not time series) and still get a “significant” result, but this does not indicate autocorrelation (since the data are randomized the test is meaningless).
Basically, if you account for the correlation properly, the problem is solved. However, incorrectly applying the DW test to the new (correctly specified AR model) residuals can still show a “significant” test statistic, but that doesnt mean that 1st order correlation still exists.