Weak (Proxy) Factors Robust Hansen-Jagannathan Distance For Linear Asset Pricing Models

Abstract: The Hansen-Jagannathan (HJ) distance statistic is one of the most dominant measures of model misspecification. However, the conventional HJ specification test procedure has poor finite sample performance, and we show that it can be size distorted even in large samples when (proxy) factors exhibit small correlations with asset returns. In other words, applied researchers are likely to falsely reject a model even when it is correctly specified. We provide two alternatives for the HJ statistic and two corresponding novel procedures for model specification tests, which are robust against the presence of weak (proxy) factors, and we also offer a novel robust risk premia estimator. Simulation exercises support our theory. Our empirical application documents the non-reliability of the traditional HJ test since it may produce counter-intuitive results when comparing nested models by rejecting a four-factor model but not the reduced three-factor model. At the same time, our proposed methods are practically more appealing and show support for a four-factor model for Fama French portfolios.