Inference with Overlapping Returns and Serially Correlated Estimates
Summary
The document considers estimating a constant return volatility from overlapping multi-day returns. Because adjacent rolling windows share observations, the resulting estimates are serially correlated. Its central point is that this dependence does not by itself bias the average of unbiased volatility estimates, but it does affect uncertainty estimates: conventional standard errors can be too small and t-statistics too large when nearby errors are positively correlated.
It suggests several ways to handle inference under serial correlation: Newey–West heteroskedasticity and autocorrelation consistent standard errors, Cochrane–Orcutt estimation under an assumed error structure, and Hansen–Hodrick standard errors for overlapping periods. The discussion is brief and does not derive the methods or prescribe a universally correct lag length. It also assumes the volatility model is constant, so it does not resolve how to estimate changing volatility or whether overlapping returns are appropriate for a particular research question.
Key ideas
- Overlapping return windows create serial dependence among volatility estimates.
- Serial correlation can leave a mean estimate unbiased while making ordinary standard errors unreliable.
- Newey–West errors adjust inference for heteroskedasticity and autocorrelation.
- Cochrane–Orcutt relies on a specified serial-correlation model.
- Hansen–Hodrick errors are another option for inference with overlapping observations.
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Full text
# How to have an unbiased estimation of the standard deviation when using rolling returns?
# 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?
Thanks for your help.
## Answer by skoestlmeier (score 2)
https://quant.stackexchange.com/a/41647
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.