Hansen-Jagannathan Distance: Geometry and Exact Distribution

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 lies in how the zero-beta rate is estimated. For the statistical properties, we provide the exact distribution of the sample HJdistance and also a simple numerical procedure for computing its distribution function. Simulation evidence shows that the asymptotic distribution for the sample HJ-distance is grossly inappropriate for typical number of test assets and time series observations, making the small sample analysis empirically relevant. Asset pricing models are at best approximations. Therefore, although it is of interest to test whether a particular asset pricing model is literally true or not, a more interesting task for empirical researchers is to find out how wrong a model is and to compare the performance of different asset pricing models. For the latter task, we need to establish a scalar measure of model misspecification. While there are many reasonable measures that can be used, the one recently introduced by Hansen and Jagannathan (1997) has gained tremendous popularity in the empirical asset pricing literature. Their proposed measure, called the HJ-distance, has been used both as a model diagnostic and as a tool for model selection by many researchers. Examples include Jagannathan and Wang (1996), Jagannathan, Kubota, and Takehara (1998), Campbell and Cochrane (2000), Lettau and Ludvigson (2001), Hodrick and Zhang (2001), and Dittmar (2002), among others. In this paper, we attempt to provide an improved understanding of the HJ-distance by focusing on the case of linear beta pricing models. In order to gain some intuition on what the HJ-distance is attempting to measure, we provide a geometric interpretation of the HJ-distance in terms of the minimum-variance frontiers of the test assets and the factor mimicking positions. We then provide a comparison of the HJ-distance with the cross-sectional regression test (CSRT) statistic of Shanken (1985) and the GMM over-identification test statistic of Hansen (1982). This comparison allows us to better understand how the HJ-distance is different from the traditional test statistics. While the HJ-distance has emerged as one of the most dominant measure of model misspecification, there does not appear to be a very good understanding of the statistical behavior of the sample HJ-distance. In most cases, statistical inference on the HJ-distance is based on asymptotic distribution. Little is known about the finite sample distribution of the sample HJ-distance. However, asymptotic distribution can be grossly misleading. For example, using simulation evidence, Ahn and Gadarowski (1999) find that the asymptotic distribution of the sample HJ-distance rejects the correct model too often. Therefore, it is important for us to obtain the finite sample distribution of the HJ-distance. Under the normality assumption, we present the exact distribution of the sample HJ-distance under both the null that the model is correctly specified and under the alternatives that the model is misspecified. Our analysis on the exact distribution not only helps us to understand what are the parameters that determine the distribution, but also provides a simple numerical method to compute the distribution in practice. For model comparison, one is more interested in testing if