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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?
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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.

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