Overlapping Rolling Returns and Effective Sample Size
Summary
The document considers autocorrelation in overlapping ten-year rolling returns and asks how many independent observations a series of such returns represents. The answer clarifies that overlapping windows are not independent when they share even one underlying observation; this follows from how the samples are constructed, rather than from a significance test. Consequently, counting every rolling window as an independent ten-year return is not justified.
The answer reframes the question in terms of effective sample size, which depends on the quantity being estimated. For uncertainty around a sample mean, it describes comparing a naive variance estimate with a variance estimate adjusted for autocorrelation, such as one obtained using Newey–West, then scaling their ratio by the number of rolling observations. This is an illustrative approach, not a universal effective-sample-size formula. The appropriate adjustment depends on the estimand and assumptions, and robust variance estimation may already provide the uncertainty measure of interest.
Key ideas
- Overlapping return windows that share observations are not independent.
- Effective sample size depends on the target quantity being estimated.
- For a sample mean, the answer describes comparing naive and autocorrelation-adjusted variance estimates.
- An effective sample size is not a universal property of a dataset independent of its use.
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Full text
# At which Lag the AutoCorrelation of Rolling 10 Year Returns is not significant?
# At which Lag the AutoCorrelation of Rolling 10 Year Returns is not significant?
I believe there are two ways to measure Confidence Intervals of Autocorrelation one assumption is assuming the Autocorrelation is following Gaussian Distribution and assuming Lags other than Lag 0 are Equal to 0 and Bartlett’s formula which assumes Moving Average Process.
I want to measure the extent of the Autocorrelation of 10 Year Rolling Returns of Sensex. I have 404 10 Year Rolling Returns which I calculated from April 1979-Nov 2022 and I want to know what is the number of Independent Samples I am having of 10 Year Returns. The simple answer is 4 but that is assuming a very strong Autocorrelation and considering the Start Date Sensitivity of Financial Returns Data I wanted to check how the strong the Autocorrelation is.
I used the ACF function in R to see at what Lag the Autocorrelation is not significant from 0 but I am seeing different results depending on what formula I am using for calculating the ACF Confidence Intervals. If I use the first assumption, then at Lag 46 there is no Autocorrelation (meaning 10 Year Return which was calculated 46 Months Ago does not influence 10 Year Return which was calculated Today) but Bartlett's Formula is showing there is no Autocorrelation at Lag 26.
I believe the later is correct because the former assumes there is no Autocorrelation for all Lags other than Lag 0 which is incorrect in this case due to the 10 Year Rolling Returns being constructed using overlapping Monthly Periods (i.e. First 10 Year Return is calculated From Jan 1979-Jan 1989,Second From Feb 1979-Feb 1989, Third From March 1979-March 1989,etc) so the Autocorrelation of First 10 Lags will definitely not be Equal to 0 but I am not sure on this hence the question.
Thanks and Regards,
Anon9001.
## Answer by Richard Hardy (score 1)
https://quant.stackexchange.com/a/74994
> I want to know what is the number of Independent Samples I am having of 10 Year Returns.
Overlapping samples are not independent. If you have two overlapping samples that share even a single observation, they are not independent. You do not need to test this statistically, as this is an algebraic fact.
On the other hand, you could ask what your effective sample size is, as in how many independent 10-year observations your sample corresponds to. This has to be qualified by stating the target of your estimation, as I think the answer may depend on that. E.g. if you wanted to estimate the uncertainty around your estimate of the sample mean (a 10-year mean), you could obtain the effective sample size in the following way:
- Estimate the variance by adjusting for autocorrelation using Newey-West or some other method to obtain $\tilde\sigma^2$,
- Estimate the variance in the naive way to obtain $\hat\sigma^2$,
- Take the ratio $\frac{ \hat\sigma^2 }{ \tilde\sigma^2 }$ and multiply it by the number of 10-year periods your data covers, i.e. 404 in your case.
(The example is probably not a great one, as you would not care about the effective number of observations once you have the robust estimate for variance $\tilde\sigma^2$. The latter gives you the uncertainty you are after.)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.