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

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?).

• In the limit, divide by standard deviation, not variance. Mar 1, 2021 at 12:36
• @RichardHardy: thanks for pointing out the error. Corrected now. Mar 1, 2021 at 12:41

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.

• Thanks for including some imp details in the answer. On your comment about exploiting dependence structure if it's known, I have had a related question here which unfortunately remains unanswered. Although the finite sample performance be poor, I am inclined to use this for practical reasons. Model selection while fitting a time series is often based on information criteria which often selects model with insignificant coefficients (which maybe good for forecasting though). When the interest is to check statistical significance of mean... Mar 1, 2021 at 12:51
• ...that is often not helpful. Then we end up selecting a model based on PACF ad even after that the standard errors of the coefficients are based on asymptomatic distribution. Wouldn't there also we have problem of finite sample performance? Mar 1, 2021 at 12:53
• Also, can I relax the normality assumption? Since this is a limiting distribution, CLT would apply nevertheless? Mar 1, 2021 at 12:56
• Yes, the conditions on the errors are indeed far weaker than normality. The papers in the linked answers by Andrews and Newey and West provide (technical) detail. As to the first part, I am not sure I completely understand - yes, indeed, poor finite-sample inference can pop up in other contexts, too. Mar 1, 2021 at 13:31
• The book by White (Asymptotic Theory for Econometricians) for example discusses a wide range of CLTs covering many dependent and heterogeneous processes. Mar 2, 2021 at 8:26