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

Question

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?

Thanks for your help.

Answer 1

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.