When estimating an AR, ADL and VAR model, should I use robust standard errors or HAC errors? In an exercise, I used the robust standard error, and then check for autocorrelation in the residuls (there is none). Is this approach ok?
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For dealing with autocorrelation one typically uses either a model that appropriately incorporates the autoregressive structure (e.g., AR, ADL, VAR, ARIMA, etc.) or treats it as a nuisance parameter (i.e., ignores it in the estimation) and corrects the standard errors afterwards.
So in your case (AR/ADL/VAR), there should be no remaining autocorrelation in the residuals and then you also don't need HAC standard errors. If there were remaining autocorrelation, the model itself would need to be improved and not just the standard errors adjusted.
Heteroscedasticity consistent/robust standard errors might make sense in addition to an autoregressive model. However, rather than adjusting for unstructured heteroscedastiticy it is often more natural to check for autoregressive heteroscedasticity (i.e., GARCH-type) effects. If necessary one could then use an AR-GARCH model etc.
answered Jan 9, 2017 at 21:54
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$\begingroup$ Both the question and your answer mention HAC and robust standard errors in simultaneous. Can the term be used interchangeably or does your answer apply to both? Could you please disambiguate technical jargon? Thanks! $\endgroup$– mugenCommented Jan 9, 2017 at 23:07
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$\begingroup$ Achim, what about deliberately choosing a model that has some remaining autocorrelation (because accounting for it fully raises AIC or BIC) and then using autocorrelation-robust standard errors for inference? It does makes sense to me, but I am not entirely sure. $\endgroup$ Commented Jan 10, 2017 at 2:26
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$\begingroup$ @RichardHardy The question is whether you want a model for predictions or for inference about certain parameters. In the former case, model selection based on some performance measure (AIC, BIC, cross-validation, etc.) will be preferable but then you won't care about significances. In the latter case you will want to capture autocorrelation in the "right" way even if some other model performs slightly better in predictions. $\endgroup$ Commented Jan 10, 2017 at 23:48
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$\begingroup$ @mugen The term robust standard errors is sometimes used as an umbrella term for HC, HAC, and other sandwich standard errors. However, more often than not robust standard errors means the HC0 standard errors, originally developed by Eicker and Huber, and later popularized by White. Not least due to the "robust" option in Stata (that computes these basic HC0 standard errors) this latter interpretation is quite widely used. $\endgroup$ Commented Jan 10, 2017 at 23:53
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$\begingroup$ Achim, I am not 100% sure about the case of modelling for inference. If we want high power of our tests, we want to avoid high variance of estimators, so leaving some small patterns in the data unaccounted four sounds like a reasonable solution. I am thinking of Galit Smhueli "To Explain or To Predict" and Rob J. Hyndman's blog post on the same topic where BIC is recommended for model selection in explanatory modelling. $\endgroup$ Commented Jan 11, 2017 at 6:36