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?
