# HAC estimator in GARCH models

In my time-series class we learned that the HAC estimator is only applicable to correct the standard error (SE) if the underlying series is stationary. Now, GARCH models are unconditionally stationary (under certain conditions that are given let's say), however, they are conditionally non-stationary. Does this mean that we can apply the HAC estimator to the coefficients in the GARCH equations or not?

Furthermore, there was a question in my class that I cannot find an answer to:
Consider the model $$X_{t} = \mu + Y_{t}$$, where $$Y_{t}$$ is a stationary GARCH(1,1) process with mean zero. When computing the confidence interval for $$\mu$$, can the HAC adjusted SE for the mean of $$X_{t}$$ be applied or not?

• What do you mean by conditionally non-stationary? Commented Jun 19, 2022 at 11:41
• By conditionally non-stationary I mean that the conditional variance $\sigma_{t}^2$ in the GARCH model $Y_{t}=\sigma_{t}\epsilon_{t}$ depends on time.
– DLTS
Commented Jun 19, 2022 at 11:49
• In a GARCH model, the dependence of $\sigma_t$ is on past values rather than time. It is true that $\sigma_t$ varies over time, though. By analogy, consider regression with ARMA errors. ARMA models could be called conditionally non-stationary by your definition, as the conditional mean is determined by past values. We can use HAC in that case. (I do not have a definite answer to your question, but I am trying to draw some parallels.) Commented Jun 19, 2022 at 12:02
• I see. But doesn't the H in HAC stand for heteroskedasticity? It should then work for heteroskedastic errors, too. Commented Jun 19, 2022 at 14:42
• Be careful. You usually need to make assumptions about fourth moments when you prove the consistency of HAC. And the fourth moments of GARCH are not always finite (depends on the parameters; there are several papers on finiteness conditions) Commented Jun 19, 2022 at 21:37