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.
 A: 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 $\alpha$.
