Covariance Estimation

The covariance estimator for returns series of differing lengths was first introduced by Stambaugh (1997), and the methodology was extended in Pastor and Stambaugh (2002). We summarize and discuss Stambaugh's derivations in Appendix 22.1 at the end of this chapter and the reader is referred there for details. The intuition, however, is based on an application of the multivariate normal distribution for which the conditional moments of the distribution of returns to shorter history assets depend on moments for the longer-lived assets.

Consider, for example, a bivariate case consisting of two assets, J and K, but with J having a longer history. The truncated maximum likelihood estimator (MLE) uses the assets’ separate histories to estimate unconditional means and variances but uses the history truncated at K to estimate the covariance. As such, asset K's moment estimates are not only inefficient, as is the covariance estimator, but there is no guarantee that the covariance matrix will be a positive definite. Stambaugh shows that these estimates can be improved by appealing to the properties of the bivariate normal, that is, the conditional distribution of asset K (conditional on information contained in asset J returns) has mean and variance that are linear functions of the information contained in the longer return history. If the return histories are independent, there is no informational gain generated by using the MLE for the conditional distribution of returns. In this case, the conditional distribution yields the truncated estimator.

Although the conditional distribution provides exact solutions for estimators of first and second moments, these estimators implicitly assume that the linear relationships among asset returns are constant over time. For example, the conditional mean for asset K in the preceding example is a linear combination of its unconditional mean and the product of its beta with asset J and asset J's mean return in the period preceding asset K's return history—see equation (22.6). All equation references refer to Appendix 22.1. Thus, the unconditional mean is augmented with the information contained in the longer series return history not observed for asset K with the magnitude of this extra information determined by its time-invariant beta. Time invariance holds for conditional second moments as well—see equation (22.7). In effect, MLE averages the impact of structural changes in the linear relationships among returns series.

There are really two issues here. One is the implied time-invariance, which, if overly restrictive, suggests some degree of estimation error that contributes to poor out-of-sample performance. Sample moment estimators, in general, often produce extreme portfolio positions that are inconsistent with equilibrium market capitalization weights (Black and Litterman 1992). The other issue relates to sample size itself; for very short histories (like TIPS), conditional moment estimates fall victim to a degrees-of-freedom problem in the unconditional distribution that could translate into less precise estimates of the conditional moments. In these cases, the conditional moments may not be representative of the characteristics of the shorter series. This might be especially relevant in the event that shocks are peculiar to a single series, say, or institutional changes alter the structural relationships among series. Contagion, for example, diffuses across markets, and depending on the rate of diffusion, will alter the structural relationships between various series, but its impact will continue to be averaged with older data. The point is that perceived gains in efficiency depend on being able to extract stable and meaningful linear relationships between series. We note that time-varying estimation schemes are available using Bayesian dynamic linear methods in which both means and covariances are recursively updated as new information on returns becomes available. See, for example, Kling and Novemestky (1999).

Estimation risk, which complicates this process, arises when sample estimates of parameters of return distributions are implicitly assumed to be the true parameters. Consequent portfolios may be, quite plausibly, inadmissible once estimation risk is explicitly incorporated into the analysis (Klein and Bawa 1976). More recently, Jorion (1986) and Frost and Savarino (1986) introduce Bayesian estimates of multivariate unconditional returns based on informative priors. Stambaugh shows that the Bayesian predictions (with a diffuse prior), relative to MLE, do not alter estimates of mean returns but scale up MLE covariance estimates due to estimation error. Because the difference between the MLE and Bayesian covariance estimates is shown to be small for portfolios consisting of relatively few assets, I report and discuss MLE only.

In a multivariate world, asset K's moments are functions of its betas with all other assets of equal or longer returns duration and the conditional maximum likelihood estimators, though still assumed to be time-invariant, use information contained in all the longer return histories. The exact multivariate estimation procedure is described further on in the results section. I do not address data problems pertaining to missing observations or gaps in returns series but note that these issues are adequately addressed using data-augmentation methods such as the EM algorithm (an iterative MLE method that effectively treats missing data as parameters to be estimated) and Gibbs sampling (to bootstrap the Bayesian PDF).

Returns smoothing only complicates attempts to resolve the truncation problem. Both are information problems; truncation throws information away, while smoothing filters it. In general, affected series must be unsmoothed beforehand. Working (1960) first commented on the impact that aggregation has on smoothing, noting that a random walk, when averaged, induces serial correlation but with an upper bound . One would expect returns averaging to induce some degree of serial correlation in an otherwise efficient market. Nevertheless, observed levels of correlation, especially for real estate and private equity returns, appear too high to be explained by simple aggregation. The effects of smoothing are especially well documented in the real estate literature (Geltner [1991, 1993a, 1993b], Quan and Quigley [1991], and Ross and Zisler [1991]) but correlation may also be the consequence of nonsynchronous trading (see Campbell, Lo, and MacKinlay [1997] and the references therein), or nonperiodic marking to market. The smoothing of real estate returns is tied largely to the appraisal process, in which estimated property values are linear combinations of past subjective appraisals. Similarly for private equity, returns are based on subjective valuations of nonpublicly traded firms. In both cases, market returns are not observed, that is, valuations are not tied to a unique market-determined price for a single publicly traded security.

In Appendix 22.1, I summarize and discuss a method, proposed by Fisher and Geltner, to unsmooth real estate returns in which observed returns are assumed to be an infinite order moving average of past market returns (based on property appraisals). If the moving average process is stationary and invertible, then the unsmoothed returns series can be recovered from the observed lagged, but smoothed, series of returns (see Appendix 22.1 for details). We will use this method to unsmooth both real estate and private equity returns in our seven-asset study. The unsmoothing parameters for real estate and private equity in equation (22.10) are estimated to be 0.728 (t-stat = 10.5) and 0.436 (t-stat = 4.39), respectively.

Other approaches to unsmooth return series can be found in Shilling (1993) and Wang (2001). These are multivariate approaches; Shilling implies the degree to which the variance in observed returns has been smoothed by exploiting the properties of a biased ordinary least squares (OLS) estimator (true returns are not directly observed, thus creating an errors-in-variables problem that biases OLS estimates) and its consistent instrumental variables counterpart. Wang, on the other hand, exploits certain co-integrating relationships to get at the degree to which returns variability has been smoothed.