# Effect autocorrelation on bias in (sample) standard deviation

I am trying to quantify the effect of autocorrelation on my estimates of the standard deviation.

Let's say I have a variable $$x = (x_1, x_2, ..., x_n)$$ of which I want to estimate its standard deviation via $$\hat{\sigma} = \sqrt{\frac{1}{n-1}\sum_{i}^{n} (x_i - \bar{x})^2}$$.

Now, if my variables follow an AR(1) process: $$x_t = 0.2x_{t-1} + e_t$$, with $$e_t$$ and $$e_{t-1}$$ uncorrelated, my standard deviation estimate will be (upwards) biased.

I have carried out some simulation studies to quantify this bias, but struggle to find literature and/or clear-cut formulas on this matter.

Can someone provide of some?

## 1 Answer

The introductory example of Jan Beran: Statistics for Long-Memory Processes (CRC, 1994) is essentially this! (Section 1.1.) Although it is a rather standard derivation, so I assume you can find it in many other places, perhaps also at less specific ones too.