date when a long enough history is available for all the assets of interest (so-called truncated-sample estimation). A more appealing alternative was proposed in a paper by Stambaugh (1997). Since his approach requires several technical steps, we start by describing the method in words and then proceed to a formal description:

1.    Estimate the truncated-sample moments for both sets of assets.

2.    Estimate a regression of each of the assets with a shorter history on all the assets with a longer history (use the truncated sample for this step). The regression coefficients identify the statistical relationship between the two sets of data.

3.    For the assets with a longer history:

a. Estimate the moments for the entire sample.

b. Measure the difference between the moments computed over the entire sam ple and the moments computed using the truncated sample. If the difference is positive, this means that the moments computed over the shorter sample underestimate the more precise estimates obtained using the entire sample (and vice versa).

4. Using the results from the regressions and the measures from step 3b, adjust the moment estimates for the series with a shorter history.

The method proposed by Stambaugh was not originally developed to accommodate some of the features that we have incorporated into our estimator (serial correlation correction and a weighting scheme that assigns more weight to more recent observations). However, since the case of no serial correlation and constant weight is a special case of our estimator, we proceed to a formal presentation of Stambaugh's method using our notation, which is more general.

Start by defining two sets of assets, and group them into two matrices RA{d) and Rg(d). The first matrix contains T observations on NA assets, whereas the second matrix contains S observations on NB assets. If S < T, then the second matrix contains the assets with a shorter history. Assuming that the assets have already been premultiplied by a vector of weights, we proceed according to the steps described earlier.

First, estimate the truncated-sample moments for both groups of assets using the estimator in equation (16.6). Let SAAS(m) and ^BS(m) be the covariance matrices for the two sets of data, based on the truncated sample.

Second, run a regression for each of the assets in Rgid) on the entire set of assets in RA(d). For each regression, use the truncated sample (i.e., the last S observations). Since the parameters of a regression can be estimated using variances and covari-ances, this is easily accomplished using the covariance matrix estimator proposed in (16.6) and selecting the appropriate components:

Bs=SAAiS(m)-1SA3iS(m) (16.9)

where SAB s (m) is the covariance matrix between the returns in RA(d) and the returns in Rgld), estimated using the last S observations.