Estimating Covariance When Return Histories Have Different Lengths
Summary
The document asks how to estimate relationships among return series that begin at different times, a common issue in portfolio risk analysis. It contrasts filling missing observations with index returns scaled by estimated beta against a more integrated covariance approach.
The accepted answer describes a full information maximum likelihood style procedure: estimate covariance from the longest history, regress the next-longest series on the longer-history data, and combine the results into an updated covariance estimate. Principal component analysis can be added when the number of assets exceeds the available observations, and factor models can be incorporated as well. The answer also notes that Fama–MacBeth methods can handle uneven datasets. These are brief pointers rather than a derivation or empirical comparison; the document does not specify assumptions, implementation details, or how the alternatives perform in practice.
Key ideas
- Unequal return histories complicate covariance estimation for portfolio risk calculations.
- A basic risk-system approach fills missing observations with index returns multiplied by estimated beta.
- A full information maximum likelihood approach can combine series by adding shorter histories through regression on longer-history data.
- Principal component analysis may help when the asset count is large relative to the observation count.
- Factor models and Fama–MacBeth methods are mentioned as possible extensions or alternatives.
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Full text
# Estimate correlation of time series whose histories differ in length
# Estimate correlation of time series whose histories differ in length
Very often in quantitative analysis (e.g. calculating portfolio volatility) we have to analyze various time series - mostly returns - whose lenghts differ.
Risk systems usually apply a one-factor model in order to create a generic history for such time series.
- choose a market index with returns $r_{index}$
- estimate the beta $\beta$ of the time series to this index
- insert $\beta * r_{index,t}$ for all missing dates $t$.
My attention was drawn to the paper Analyzing investments whose histories differ in length by Robert F. Stambaugh. It seems to use a more sophisticated approach.
My question:
- Does anyone here have access to a description of the approach taken by Stambaugh which is not behind a pay-wall?
- What other approaches were published that address this issue?
## Answer by John (score 2, accepted)
https://quant.stackexchange.com/a/14565
The technique is sometimes referred to as full information maximum likelihood. It is more general than the technique you describe, but it is similar. Basically you start with the data with the longest horizon and get the covariance matrix, then for the data with the next longest horizon you regress them against the data with the longest horizon, finally you combine them together for a new combined covariance matrix.
Meucci has some code that does it in this package. http://www.mathworks.com/matlabcentral/fileexchange/9061-risk-and-asset-allocation
Sometimes the technique is expanded so that you use PCA at each step. This is important when the number of stocks increases larger than the number of observations. There's also no reason that this can't be combined with a factor model.
More generally, techniques like Fama-Macbeth can be used with uneven data sets.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.