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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 ...