Can long run variance of a time series be used to test mean of the series?

Question

Let $Y_t$ be a stationary time series with $Y_t \sim N(\mu, \gamma_0) \,\, \forall t$.

Further define, $$\bar Y_T \equiv (1/T)\sum\limits_{t=1}^TY_t$$

Further, let $\sigma^2_l$ be the long-run variance this time series. From this answer, I think we can say:

$$\lim_{T \to \infty} \frac{\bar Y_T-\mu}{\sigma_l/\sqrt{T}} \sim N(0,1)$$

This is because $\bar Y_T$ is a sum of $T$ normal random variables with (unconditional) mean $\mu$. Now say we have a consistent estimate of $\sigma^2_l, \hat \sigma^2_l$. Can we use the following statistic to test for hypothesis such as $H_0: \mu = \mu_0$?

I feel that if the true dependence structure of the time series was known (say for example, we know that $Y_t$ follows $AR(p)$), then the standard error of the residuals of the $AR(p)$ model, estimated using MLE, is nothing but an estimate of long-run variance. If true, then the above allows us to test for mean without having to fit a model (although the bandwidth selection is perhaps a similar exercise?).

Answer 1

Basically, yes and yes - you can replace the long-run variance with a consistent estimator thereof and, by Slutzky's theorem, the test statistic will still be standard normal under the null.

And indeed, kernel-based long-run variance estimators are sometimes also referred to as nonparametric estimators that do not (there still are some assumptions, of course) require you to postulate a parametric model for the dependence.

That said, indeed, if you knew that your series follows a specific structure, you could exploit that. Why if in OLS the autocorrelation between residuals is positive, it will lead to inflated t-stats? discusses that the long-run variance of an AR(1) is $$ \sigma^2/(1-\phi)^2, $$ which you could estimate parametrically.

And indeed, as discussed for example in Newey-West t-stats and critical values, there is a price to pay for the above nonparametric inference, namely relatively poor finite-sample performance.