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$x_t$ on variable y_t$y_t$ in a time series context. A simple model is y_t = a + b * x_t + error term$y_t = a + b x_t + \varepsilon_t$. 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 estimateestimates, while arguing that the parameter estimate isestimates are still consistent in the presence of residual serial correlationscorrelation.
However, in this case, one could often add lags of y_t$y_t$ to the model to remove the autocorrelations in the residuals, e.g. y_t = a + b * x_t + c * y_{t-1} + error term$y_t = a + b x_t + c y_{t-1} + \varepsilon_t$. Since the lag y_{t-1}$y_{t-1}$ could be correlated with the regressor x_t$x_t$, either by economic reasoning (x_t$x_t$ is often serially correlated) or by sample correlation between the lag y_{t-1}$y_{t-1}$ and the regressor x_t$x_t$, it would be simply wrong to ignore the lag y_{t-1}$y_{t-1}$ 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.
