Characteristic-sorted portfolios: Estimation and inference

Portfolio sorting is ubiquitous in the empirical finance literature, where it has been widely used to identify pricing anomalies in different asset classes. Despite the popularity of portfolio sorting, little attention has been paid to the statistical properties of the procedure or to the conditions under which it produces valid inference. We develop a general, formal framework for portfolio sorting by casting it as a nonparametric estimator. We give precise conditions under which the portfolio sorting estimator is consistent and asymptotically normal, and we also establish consistency of both the Fama-MacBeth variance estimator and a new plug-in estimator. Our framework bridges the gap between portfolio sorting and cross-sectional regressions by allowing for linear conditioning variables when sorting. In addition, we obtain a valid mean square error expansion of the sorting estimator, which we employ to develop optimal choices for the number of portfolios. We show that the choice of the number of portfolios is crucial for drawing accurate conclusions from the data and we provide a simple, data-driven procedure that balances higherorder bias and variance. In many practical settings the optimal number of portfolios varies substantially across applications and subsamples and is, in many cases, much larger than the standard choices of five or ten portfolios used in the literature. We give formal and intuitive justifications for this finding based on the bias-variance trade-off underlying the portfolio sorting estimator. To illustrate the relevance of our results, we revisit the size and momentum anomalies.