Estimating Monthly Return Variance from Mixed-Frequency Data
Summary
The document asks how to estimate the variance of monthly returns when a return series includes both monthly observations and one quarterly aggregate. The proposed response describes a moment-based approach under temporal independence: infer the quarterly period's average monthly return by dividing its log return across the three months, then use the mean and a sum of squared returns to form a variance estimate. It relies on the relationship that a three-month aggregate has three times the monthly variance, or equivalently that volatility scales with the square root of time.
A Bayesian maximum-likelihood route under a normality assumption is also mentioned, but characterized as iterative and less straightforward. Another response assumes stationary variance and reports that excluding the quarter yields the same estimate as its calculation with the aggregate. The discussion does not fully derive or reconcile these procedures, and its independence and stationarity assumptions may fail when returns are serially dependent or volatility changes over time. Thus it offers a conditional method, not a generally unbiased solution for every return process.
Key ideas
- A quarterly aggregate can be related to monthly variance under temporal independence.
- The proposed moment method splits the quarterly log return across its component months.
- Under independence, quarterly variance is three times monthly variance.
- A Bayesian normal-model approach is mentioned as an iterative alternative.
- The approach depends on assumptions such as independence and stationary variance.
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Full text
# Estimating the variance of returns with aggregated data # Estimating the variance of returns with aggregated data Say I have an asset return time series: Jan2020: -5% Feb2020: +5% Mar2020: -5% Apr2020: +5% May2020: -5% Jun2020: +5% Q3 2020: +20% Oct2020: +5 Nov2020: -5 Dec2020: +5 Note that 3 months of data is an aggregate i.e., we don't have July, Aug, Sept. Is there a simple unbiased estimator for the variance of monthly returns of such a series? ## Answer by demully (score 1) https://quant.stackexchange.com/a/60641 This can be done through Bayesian inference, on a maximum likelihood estimation basis. Assuming a (log)normal distribution, you could calculate the likelihood that any prior (however stupid) estimate of mu and sigma (and thus variance). Since the normal distribution is its own conjugate prior, one can always come up with an iterative better guess for the parameters. But it's iterative; and the update formulae ain't pretty. https://www.statlect.com/fundamentals-of-statistics/normal-distribution-Bayesian-estimation A simpler way - that is unbiased - would be fall back on the observation that the variance calculation can be decomposed into a sum of the squares versus mean formulation. Both of which are easy to meausre. Your mean is easy to compute, give the aggregation problem. Just third/log-third the quarterly one. Then calculate the sum of the squares of your returns - irrespective of timeframe. This just assumes temporall return independence, which is implicit measuring monthly returns in the first place. If this, then quarterly returns should be root-3 more volatile than monthly returns; and monthly variances 3x. The sum-of-squares follows. https://www.sciencebuddies.org/science-fair-projects/science-fair/variance-and-standard-deviation#:~:text=The%20variance%20(%CF%832)%2C,in%20the%20distribution%20(N).&text=You%20take%20the%20sum%20of,in%20the%20distribution%20(N). Given this average and sum of squares, the variance is as below: ## Answer by Sergei Rodionov (score 0) https://quant.stackexchange.com/a/60649 If variance is stationary, specifically sample variance in Q3 was the same as in other periods: - Determine sample mean including missing observations as product^1/12: 1.9% - Calculate unbiased variance for known observations: -11.7%, or stdev 5.3% This is the same as ignoring Q3 altogether. I'm curious for my own education, how is this straight-forward estimate suboptimal to the maximum likelihood calculation?
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