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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?

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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.

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