# How to have an unbiased estimation of the standard deviation when using rolling returns?

I want to estimate the weekly standard deviation of a lognormal process in a usual setup.

$$\frac{dS}{S} = (\dots) dt + \sigma dW$$ where $\sigma$ is a constant and $W$ a brownian motion.

The usual estimator of the standard deviation is $$\hat{s} = \sqrt{\frac{\sum_{i=1}^n (r_i - \overline{r})^2}{n-1}}$$ where $r_i = \ln{\frac{S_i*5\ days}{S_{(i-1)*5\ days}}}$ and $\overline{r}$ the average of those returns.

I have a daily timeseries and I am not trying to capture some kind of "day specific effect", so I'd like to use all the rolling increments $r_i = \ln{ \frac{S_i}{S_{i-5\ days}} }$ to have more samples. My issue is that those returns are correlated.

Is there an unbiased estimator for the correlated returns?

Your estimator $$\hat{s_i}$$ for stock $$i$$ is an unbiased estimator of its latent standard deviation $$\sigma_i$$ (which is constant for your model). When applying your "window rolling" for calculating $$\hat{s_i}$$, you get a time-series $$ts_{\sigma_i}$$ for each stock $$i$$.

With an intercept-only OLS-regression for each time-series $$ts_{\sigma_i}$$,

$$\hat{s_{it}} = a + \epsilon_{it}$$

you receive the respective mean standard deviation $$a$$. While auto-correlation does not bias your (point-)estimate of $$a$$, the standard errors tend to be underestimated (and the t-scores overestimated) when the auto-correlations of the errors at low lags are positive.

### How to account for auto-correlation?

• Apply Newey/West (1987) HAC standard errors, which corrects for both heteroskedasticity and auto-correlation. The appropriate time lag may be the number of overlapping time periods.

• Cochrane–Orcutt estimation, which adjusts the linear model for serial correlation in the error term. Be aware that you have to assume a particular form for the structure of the auto-correlation (typically a first-order AR-process). This method is well described in Introductory Econometrics for Finance by Chris Brooks, pp. 199.

• Hansen Hodrick (1980) standard errors with $$k-1$$ overlapping periods. A good starting point is this excellent answer.