Appendix 22.1: Covariance Matrix Estimation

Consider the simplest case with two return series for and for where . Truncation to S would imply that means be equal to their maximum likelihood estimates:

(22.1)

Here, and are the sample MLE for the truncated sample of size S. Similarly, the MLE covariance matrix based upon S is equal to:

(22.2)

where are scalars. The parameters and are typically replaced with their MLE counterparts based on the full sample of size T.

Consider now the likelihood function for these series, which can be written as a combination of the information unique to (that is, the observations in not common to ) and the information common to both series, for example, the joint likelihood given by:

(22.3)

The second part of the likelihood is the distribution of joint information while the first part describes the contribution to the likelihood function from the information unique to r_1,t. The truncated estimator ignores the first half of the likelihood function. Ordinarily, one would maximize this likelihood with respect to E and V, but unfortunately these two sets of parameters do not appear in both halves of the likelihood. Stambaugh uses a result from Anderson (1957) that rewrites the joint likelihood as the product of a marginal and a conditional density . Assuming returns to be multivariate normal, we get the familiar result that the conditional mean of (conditional on ) is equal to:

(22.4)

where , is the regression coefficient of on . If and are positively correlated and a return is observed to be above its mean , then is adjusted downward from its unconditional value. (This is what statisticians mean by “regression toward the mean.”) Furthermore, the conditional covariance given by the multivariate normal is:

(22.5)

In the more general multivariate case, is a vector of returns on assets all with observations and is an vector of returns on the shorter time series. In that case, and are mean return vectors of size and , respectively, so that E is an vector, is: is is and . Furthermore, is now a matrix of regression coefficients.

The objective is to derive estimates for these covariance matrices. To estimate the maximum likelihood estimators for the moments of the conditional density, regress on using S observations, saving the covariance matrix of the residuals as . Likewise, estimate mean returns for and using T and S observations, respectively. Then, applying the results in (22.3) and (22.4), adjust the truncated mean for given in (22.1) by conditioning on the information in :

(22.6)

Therefore, if the two returns series are positively correlated and the mean of the longer series exceeds its truncated mean, the mean for the shorter series is adjusted upward. That is, the truncated mean is most likely biased downward. Likewise, the truncated covariance matrices are adjusted according to (see Stambaugh):

thus

(7)

It is easy to show that is identical to in equation (22.5). In equation (22.7), it is also true that the covariance between and is a linear rescaling of the covariation in with the magnitude depending on the strength of their covariation. Moreover, revisions to depend on the how much the covariation in the longer series changes over the time interval .

Removing the Effects of Smoothing

Consider, for example, the appraised value P which is a moving average of current and past comps :

(22.8)

Rewrite the weights to be geometrically declining such that for some scalar value of . Let a = 1−w₀. Note as well that . Substituting into (8) yields:

Now, replace P with its natural logarithm, lag one period, and subtract the resulting expression from (22.8), yielding a moving average of returns:

(22.9)

where . Obviously, returns are a weighted average of past market rates of return and this is the source of the smoothing. Solving (22.9) for gives us:

(22.10)

We seek the unsmoothed component , which is:

(22.11)

Thus, we use the observed smoothed returns to recover the market return. An estimator for can be obtained from an OLS regression of observed on its lagged value. This is a special case of an ARMA model for which the moving average component, under certain stationarity conditions, is invertible. An invertible infinite order moving average is equivalent to a first-order autoregressive (AR) model whose parameter is estimated using ordinary least squares. The series can then be unsmoothed and covariances estimated thereafter.

Notes

1. Much of this chapter appeared in Peterson and Grier 2006. Asset allocation in the presence of covariance misspecification. Financial Analysts Journal 62(4): 76–85.