# Type 2 error in t-test on time series

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.

• What about looking into autocorrelation-resistant standard errors? See this paper Nov 23, 2020 at 16:23
• @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. Nov 23, 2020 at 17:08

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))
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 $$\alpha$$.