# Estimation of mean of AR(1) process

It is well known that for the basic AR(1) model $X_k=\phi X_{k-1}+\varepsilon_k$ where $\vert\phi\vert<1$ and $\varepsilon\propto\mathrm{WN}(0,\sigma^2)$ the sample mean $\bar{X}_n$ (after being scaled properly) is asymptotically normal with mean 0 and variance $\sigma^2/(1-\phi)^2$. This fact can be used to construct confidence intervals for the true mean. What I find curious though is that $1-\phi>1$ if $\phi<0$. This means that if the observations are negatively correlated ($\phi<0$) then the standard error that determines the width of the confidence interval is actually smaller than the standard error in the iid case (when $\phi=0$). In other words, negative correlation turns out to be a good thing, for it results in tighter confidence bounds. Is there a simple intuitive way to explain this?

That precision of the sample mean is decreasing in $\phi$ might be motivated by the thoroughness with which the sample space is explored for different values of $\phi$. For $\phi$ close to $+1$, we are likely to see more or less the same values of $X_k$ for several periods in a row, whereas for "antipersistent" processes with $\phi<0$, we are effectively guaranteed to many different values of $X_k$ in a given trajectory. This should allow us to pin down more precisely the location parameter of the process.