“Hansen-Jagannathan Distance: Geometry and Exact Distribution” by Raymond KAN
University of Toronto
This paper provides an in-depth analysis of the Hansen-Jagannathan (HJ) distance, which is a measure that is widely used for diagnosis of asset pricing models, and also as a tool for model selection. In the mean and standard deviation space of portfolio returns, we provide a geometric interpretation of the HJ-distance. In relation to the traditional regression approach of testing asset pricing models, we show that the sample HJ-distance is a scaled version of Shanken’s (1985) crosssectional regression test (CSRT) statistic, with the major difference being the way the zero-beta rate is estimated. For the statistical properties, we provide an exact distribution of the sample HJ-distance under both the null and the alternative hypotheses. In addition, we also suggest a simple numerical procedure for computing its distribution function. Simulation evidence shows that for a typical length of time series, the asymptotic distribution for the sample HJ-distance is grossly inappropriate when the number of factors or the number of test assets is large, making the small sample analysis in this paper empirically relevant.