Robust Rolling Estimates of Mean Stock Returns

Summary

The document discusses whether rolling sample means are useful when stock returns are autocorrelated and may have heavy tails. It describes rolling windows with equal weights, as well as exponentially decaying weights, as practical ways to emphasize more recent observations when estimating a return mean. The response frames these approaches as accounting for dependence in the data, though it does not explain a formal correction for autocorrelation or establish that a rolling mean is unbiased under general dependence.

For heavy-tailed returns, it notes that the sample mean may be less efficient than alternatives. It gives the median as an efficient estimator under a double-exponential distribution and mentions the trimmed mean as a robust practical choice. These recommendations are brief and partly reflect the respondent’s personal preference. No data, simulations, or comparisons are provided, and the best estimator depends on the return distribution and the inferential goal.

Key ideas

• Rolling windows can weight recent return observations equally or with exponential decay.
• Autocorrelation affects the statistical assumptions behind standard mean estimation.
• Under double-exponential tails, the median can be more efficient than the mean.
• A trimmed mean is presented as a robust alternative, without a general optimality claim.
• Estimator choice depends on the return distribution and the purpose of the estimate.

Tags

# What is a robust estimator of stock return mean?


# What is a robust estimator of stock return mean?












I recently started a mathematical statistics course and learned that the sample mean assumes the set of samples $ X_1, X_2, \dots, X_n $ to be i.i.d. With stock returns, this is clearly not the case, as they exhibit autocorrelation. So why are things like rolling sample means so commonly used? Is this practice incorrect?

## Answer by Brian B (score 0)

https://quant.stackexchange.com/a/81965

Rolling sample means with a boxcar (or better yet, exponential-decay) weighting are a way to help account for the autocorrelation.

Since you ask about robust, you also need to account for the fat tails, and the mean is no longer the most efficient estimator. For example, if your tails are double-exponential ($e^{-c\cdot x}$ in shape) then the median is the most efficient estimator.

In practice, though, I personally prefer a trimmed mean, which is also quite robust. I'm not even sure what tail shape it is the most efficient estimator for, but there's got to be one.

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.