Estimating Covariance Matrices with Unequal Return Histories
Summary
The document examines Stambaugh’s method for combining asset returns when one asset has a longer history than another. It considers a full-history asset X and a later-starting asset Y, focusing on how the shorter sample’s regression of Y on X contributes to inferred full-sample means and covariance estimates. The author describes the off-diagonal covariance adjustment and a variance expression that combines the regression residual variance with a term based on X’s variance and the regression coefficient.
The question seeks an intuitive explanation for that variance decomposition and asks whether the residual term already incorporates the short-sample variance of Y without a separate full-history adjustment. No answer or resolution is included, so the document presents a technical inference question rather than a completed explanation. It gives the regression setup and formulas as context but offers no empirical example or guidance on applying the estimator.
Key ideas
- Stambaugh inference can use a longer return history to inform estimates involving an asset with a shorter history.
- A regression of the truncated-history asset on the full-history asset informs inferred covariances.
- The inferred variance is expressed through a residual variance and a regression-based variance component.
- The document poses questions about the interpretation of this decomposition but does not answer them.
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# Stambaugh inference for Investment Analysis when History Lengths Differ
# Stambaugh inference for Investment Analysis when History Lengths Differ
This pertains to Stambaugh in the JFE (vol. 45, 1997 pp 285-331), and I have a question about Proposition 1 results (page 292). (link)
To set the background, let's take the smallest relevant application of two assets, one with full return history, say X, and another with truncated (start date is later) history, say Y. In this set up, a number of variables denoted by capital letters are scalars below.
It is clear to perceive that the combined sample estimates for first moment of the truncated series Y depends on the full sample estimate for X and so does the the off-diagonal in the ($2 \times 2$) second moment estimate, e.g., the off diagonal estimate is $V_{21}=B_\textrm{tr}V_{11,\textrm{fl}}$ where $B$ is a regression coefficient (in a truncated sample coefficient of Y returns on X returns) and $V$ is the full sample variance of X. This is therefore, an adjustment of the covariance between the two series (truncated) by the ratio between the full sample and truncated sample variances for X returns. So full sample information is incorporated in the off-diagonal in this manner.
Turning to the full-sample inferred variance of Y returns, this is $V_{22}=\Sigma+BV_{11}B$, where the first term is the mean square error of the residual in the same regression. The second term has no information about the (truncated) sample estimate of the variance of Y returns, but has information about the inferred full sample covariance since it is essentially $BV_{21}$. So the truncated sample variance estimate for Y returns enters from the first term. We have, $$\Sigma=\textrm{Var}(r_Y-A-Br_X)=\textrm{Var}(r_Y)+B^2\textrm{Var}(r_X)-2B\textrm{Cov}(r_Y,r_X)\textrm{.}$$ These are all truncated sample estimates (i.e., Var$(\cdot)$) and $A$ is the intercept in the regression.
- Is there an intuitive explanation for why we are adding the truncated sample variance for the full history asset, X, and then subtracting the covariance?
- Is there no adjustment to the truncated sample estimate $\textrm{Var}(r_y)$ in the first term that includes full sample information, as I have written it out?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.