  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.
