# HAC standard error or missing ARMA terms

In the context of regressions, it seems a convention that the HAC estimator should be applied when the residual is serially correlated. But isn't the presence of residual autocorrelations an indication that the model misses lagged dependent variables (or MA terms for that matter)?

For example, suppose one is investigating the impact of variable $x_t$ on variable $y_t$ in a time series context. A simple model is $y_t = a + b x_t + \varepsilon_t$. It is often the case that the resulting error term $\varepsilon_t$ is serially correlated. One response to this problem could be just applying the HAC estimator to correct the standard errors of the parameter estimates, while arguing that the parameter estimates are still consistent in the presence of residual serial correlation.

However, in this case, one could often add lags of $y_t$ to the model to remove the autocorrelations in the residuals, e.g. $y_t = a + b x_t + c y_{t-1} + \varepsilon_t$. Since the lag $y_{t-1}$ could be correlated with the regressor $x_t$, either by economic reasoning ($x_t$ is often serially correlated) or by sample correlation between the lag $y_{t-1}$ and the regressor $x_t$, it would be simply wrong to ignore the lag $y_{t-1}$ while just correcting the standard errors by HAC.

This makes me wonder why one would ever use HAC estimator in regressions? At most, use the White estimator to correct for heteroskedasticidy if it is present. Otherwise, whenever there is residual autocorrelation, investigate adding AR or MA terms but never simply use HAC estimator.

A great question! I was pondering upon this myself some time ago.

If you do not care about inference on the original model's parameters (which could sometimes be called "structural parameters"), then you can afford tweaking the model to make it represent the data better. In such a setting altering the model specification to achieve well-behaved residuals makes more sense than resorting to HAC standard errors (which then seems like a lazy solution), and your criticism is spot on.

If, on the other hand, you care about inference on the original model's parameters, you will perhaps not be willing to give up the current model specification that produces ill-behaved errors. Then you may either (1) leave the specification completely unchanged and use HAC standard errors or (2) try including an extra equation for the error process, which would yield regression with ARMA errors (see more in Rob J. Hyndman's blog post). I am in favour of the second solution as long as it produces well-behaved residuals; but it may not, and then using HAC standard errors seems reasonable.

So after all the answer depends on what you are interested in and what you are going to use the model for.

Edit 1: later I found Diebold's blog posts on a similar topic, "The HAC Emperor Has No Clothes" and "The HAC Emperor Has No Clothes: Part II", where he argues forcefully in favour of modelling errors as ARMA processes rather than resorting to HAC (because of higher estimation accuracy, more powerful inference, more accurate predictions, etc.). I side with him completely.

Edit 2: If the modelling goal is statistical inference, tweaking the model specification to match the patterns in the observed data undermines the standard practice of hypothesis testing by making the $p$-values conditional on the changes in model specification after the data has been observed. As I have not received any answers to this related question, there does not seem to be an easy way out. So HAC (which avoids tweaking the model, unlike adding some ARMA terms) could be the preferred solution for inference.

• Thanks for your help, Richard =) But I don't think that including an extra equation for the error process can solve the problem. – Frank Jan 21 '16 at 18:10
• @Frank, How come? As long as the model is well specified, what is the problem with that? The two equations would be estimated simultaneously so that the estimation is efficient etc. Regression with ARMA errors is a valid model by itself, the only question is whether it makes sense in a given application. – Richard Hardy Jan 21 '16 at 18:17
• Thanks for your help, Richard =) But I don't think an extra equation for the error can solve the problem. The key thing is that the lag could be correlated with x_t. If the lag is left out in the residual, the estimate of the coef on x_t is no longer consistent. It is an omitted-variable problem. The suggestion of adding an AR equation for the residual doesn't help. For eample, y_t = a + b * x_t + u_t, u_t = c * u_{t-1} + v_t. Combining this two gives y_t = a(1-c) + c * y_{t-1} + b * x_t - b * c * x_{t-1} + v_t. It still shows that the lag should be in the model to estimate the coef b. – Frank Jan 21 '16 at 18:23
• I wasn't saying adding the AR equation for the residual is wrong. It is correct. But estimation could be tougher than just combining the two equations, which can be estimated by the usual OLS. – Frank Jan 21 '16 at 18:27
• @Frank, next time you type an equation, put dollar signs in front and at the end. Then you will get proper-looking output. – Richard Hardy Jan 21 '16 at 18:27