Chi-Squared Tests for Evaluation and Comparison of Asset Pricing Models
Gospodinov, Nikolay, Kan, Raymond, Robotti, Cesare, Federal Reserve Bank of Atlanta, Working Paper Series
Working Paper 2011-8 March 2011
Abstract: This paper presents a general statistical framework for estimation, testing, and comparison of asset pricing models using the unconstrained distance measure of Hansen and Jagannathan (1997). The limiting results cover both linear and nonlinear models that could be correctly specified or misspecified. We propose new pivotal specification and model comparison tests that are asymptotically chi-squared distributed. In addition, we develop modified versions of the existing model selection tests with improved finite-sample properties. Finally, we fill an important gap in the literature by providing formal tests of multiple model comparison.
JEL classification: C12, C13, G12
Key words: asset pricing models, Hansen-Jagannathan distance, model selection, model misspecification
It is common for financial economists to view all asset pricing models only as approximations of reality. Although these models are likely to be misspecified, it is still useful to empirically evaluate the degree of misspecification and their relative pricing performance using actual data. In their seminal paper, Hansen and Jagannathan (1997, HJ hereafter) propose measures of model misspecification that are now routinely used for parameter estimation, specification testing and comparison of competing asset pricing models. The unconstrained (constrained) H J-distance measures the distance between the stochastic discount factor (SDF) of a proposed model and the set of (nonnegative) admissible stochastic discount factors. But despite the recent advances in developing the appropriate econometric theory for comparing asset pricing models based on the H J-distance, a general statistical procedure for model selection in this context is still missing (Chen and Ludvigson, 2009, p. 1080). As a result, researchers are still ranking alternative models by comparing their corresponding sample HJ-distances (see, for example, Parker and Julliard, 2005 and Chen and Ludvigson, 2009, among others) without any use of a formal statistical criterion that takes into account the sampling and model misspecification uncertainty. In this paper, we provide a fully-fledged statistical framework for estimation, evaluation and comparison of linear and nonlinear (potentially misspecified) asset pricing models based on the unconstrained HJ-distances. Given some unappealing theoretical properties of the constrained HJ-distance (Gospodinov, Kan and Robotti, 2010a), we do not consider explicitly the sample constrained HJ-distance but the generality of our analytical framework allows us to easily extend the main results for the unconstrained HJ-distance that we derive in this paper to its constrained analog (see Gospodinov, Kan and Robotti, 2010a, for details).
The econometric methodology for using the unconstrained HJ-distance as a specification test for linear and nonlinear models is developed by Hansen, Heaton and Luttmer (1995), Jagannathan and Wang (1996) and Parker and Julliard (2005). Kan and Robotti (2009) provide a statistical procedure for comparing linear asset pricing models based on the unconstrained HJ-distance. Furthermore, Kan and Robotti (2009) propose standard errors for the SDF parameter estimates and the sample HJ-distance that are valid for misspecified models. The objective of this paper is to provide a unifying framework for improved statistical inference, specification testing and (pairwise and multiple) model comparison based on the sample HJ-distances of competing linear and nonlinear asset pricing models.
Our main contributions can be summarized as follows. First, we propose new Lagrange multiplier tests for individual and joint testing of correct specification of one or more asset pricing models. These new specification tests are asymptotically chi-squared distributed and enjoy improved finite-sample properties compared to the specification test based on the HJ-distance. …