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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?).

Dayne
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1 Answers1

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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.

Christoph Hanck
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  • 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](https://stats.stackexchange.com/q/399681/239481) 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... – Dayne Mar 01 '21 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? – Dayne Mar 01 '21 at 12:53
  • Also, can I relax the normality assumption? Since this is a limiting distribution, CLT would apply nevertheless? – Dayne Mar 01 '21 at 12:56
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    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. – Christoph Hanck Mar 01 '21 at 13:31
  • I posted an answer to your other question there. – Christoph Hanck Mar 01 '21 at 13:48
  • Sorry, I am still confused about the normality assumption. In the first line of my question, I have assumed that Yt follows normal distribution. If we relax this assumption, how can I still claim z-stat will follow normality? I am unable to see how (and which) CLT can be used here. – Dayne Mar 02 '21 at 05:43
  • See e.g. here https://personal.utdallas.edu/~d.sul/Econo2/andrews91.pdf – Christoph Hanck Mar 02 '21 at 05:48
  • Sorry for bothering again but this seems to be about *estimation* of LRV. I am saying even if we knew true LRV, can make make assertion about normality of the z-statistic? Again, I am sorry I couldn't see where this is addressed in the Andrews paper. – Dayne Mar 02 '21 at 05:54
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    The book by White (Asymptotic Theory for Econometricians) for example discusses a wide range of CLTs covering many dependent and heterogeneous processes. – Christoph Hanck Mar 02 '21 at 08:26
  • Thank you so much for the reference. I had been looking for a book on exact title! – Dayne Mar 02 '21 at 09:46