On the properties of the constrained Hansen-Jagannathan distance

We provide an in-depth analysis of the theoretical properties of the Hansen-Jagannathan (HJ) distance that incorporates a no-arbitrage constraint. Under a multivariate elliptical distribution assumption, we present explicit expressions for the HJ-distance with a no-arbitrage constraint, the associated Lagrange multipliers, and the stochastic discount factor (SDF) parameters in the case of linear SDFs. This allows us to analyze the benefits and costs of using the HJ-distance with a no-arbitrage constraint to evaluate and rank models. We also study the asymptotic and finite-sample properties of the sample constrained HJ-distance. Finally, we demonstrate the practical relevance of our theoretical findings in an empirical illustration of some popular asset-pricing models.