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I have an AR(1) time series with $1>\phi>0$. If I naively use t-test to check $H_0:\mu=0$ and it does not reject the null, then can I accept the result?

I think yes because for a time-series with positive autocorrelation, $\hat{\mu}$ would have a higher variance $({\sigma}^2(1+\phi)/(1-\phi))$ and therefore, the odds of making a type II error would be even smaller. Is this right thinking? I am looking for a more rigorous proof/reasoning?

The reason I want to confirm this intuition is that in t-test we are making the conclusion based on t-distribution which is the exact distribution of the test statistic under null. However, the above argument is based on the asymptotic distribution. So we need asymptotic distribution of t-statistic in case of an AR(1) time-series.

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    $\begingroup$ What about looking into autocorrelation-resistant standard errors? See this paper $\endgroup$ Nov 23, 2020 at 16:23
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    $\begingroup$ @kjetilbhalvorsen: Thanks! Yes this paper is certainly helpful. I do remember finding some more papers when I was working on this. But eventually, to avoid explaining lot of details to my superiors at work, I stuck to regular t-test. Though I showed that my time series had positive autocorrelation and t-test didn't reject the null, so I concluded that $\mu=0$ in the corrected test would also not be rejected. $\endgroup$
    – Dayne
    Nov 23, 2020 at 17:08

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I agree that the naive (not accounting for autocorrelation) t-test would tend to reject far too often under the null. For example,

mean(replicate(500, t.test(arima.sim(list(ar=0.8), n = 1000))$p.value<.05))

returns rejection frequencies around 50% in my runs, so that half of all mean-zero series are false declared to have a nonzero mean.

However, generating time series with a small non-zero mean, as in

mean(replicate(500, t.test(.01+arima.sim(list(ar=0.8), n = 1000))$p.value<.05))

does not generate very different rejection frequencies, so that the type-II error - at least for alternatives close to the null - is not much different, and for any given non-rejection observed for some real time series fow which you do not know the true DGP you are not going to know if you just made a correct decision under the null or a type-II error.

Of course, not knowing if, when you did not reject, you made a correct decision or a type-II error, is not fundamentally different when your test is sized correctly, but at least the rejection frequencies under the null are then controlled at some (typically small) level $\alpha$.

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  • $\begingroup$ Thanks for taking out the time to answer this. If I understand correctly, in your first example the rejection frequency is type 1 error but in second (1-rejection frequency) would be type 2 error and so shouldn't we replicate t-test with white noise errors for comparing frequency of type 2 error in naive t-test? BTW, I didn't think of trying simulations to test my assertion so it's a great idea! $\endgroup$
    – Dayne
    Mar 1, 2021 at 14:01
  • $\begingroup$ Type-II error means not rejecting a wrong null, which in this case means not rejecting the wrong null that the series has mean zero. The (auto-)correlation structure of the series basically is a "nuisance" that affects the properties of our test, but not what determines whether a (non-)rejection is a type I error, type II error or a correct decision. $\endgroup$ Mar 1, 2021 at 14:04
  • $\begingroup$ Ah! I got the logic of your second simulation now. So basically, type 2 error is also high in naive t-test. I think what was wrong in my intuition was that the sample variance estimated in t-test also gets influenced by the dependence structure and apparently it widens the confidence interval to include zero, so not rejecting the incorrect null as often as it should. $\endgroup$
    – Dayne
    Mar 1, 2021 at 14:30

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