Abstract

Asset return histories can greatly vary in length. Such data are said to follow a monotone missingness pattern, which leads to a convenient factorization of the likelihood for the purposes of inference. Under an MVN assumption, MLEs and samples from a Bayesian posterior can be obtained by repeated OLS regression, one for each asset. When there are more assets than historical returns (a “big p small n problem”), however, OLS becomes unstable. We explore remedies that apply shrinkage, like ridge regression or the lasso, which have a natural Bayesian implementation, and can offer improvements in accuracy and interpretation. We focus on the Bayesian approach and thus improve upon the work of Stambaugh (1997) to provide full posterior inference rather than just moments, in addition to extending its reach to the “big p small n” setting. We show how, in this framework, it is straightforward to relax the MVN and monotonicity assumptions, incorporate known factors, test hypotheses about correlation, obtain credible intervals all parameters in the model and, in particular, obtain uncertainty bands for the solutions to the quadratic programs using samples from the posterior to, e.g., balance portfolios. Where possible, we provide detailed comparisons to the alternative ML approach. We conclude with an investment exercise and descriptive analysis based on real financial returns data. An R package implementing all of the methods described herein is available on CRAN .