A new measure of model misspecification with the no-arbitrage constraint: extending the second Hansen–Jagannathan distance

Abstract

This paper proposes a new measure for evaluating asset pricing models with the no-arbitrage constraint which naturally extends the classical (second) distance of Hansen and Jagannathan (J Polit Econ 99(2):225–262, 1991, J Finance 52(2):57–590, 1997). The new measure is designed to capture model misspecifications in terms of arbitrary moments/co-moments in the stochastic discount factors in contrast to the classical Hansen–Jagannathan distance which only uses information contained in the first two moments/co-moments. The new measure $$D_{p}^{+}$$ is defined for any $$1<p<+\infty$$ , which reduces to the Hansen–Jagannathan distance when $$p=2;$$ but for other values of p, the new measure captures model misspecifications in terms of higher or lower moments/co-moments of the model stochastic discount factors and asset returns. An explicit representation of the new measure is obtained by solving the conjugate problem using convexity arguments. The relationship between the new distance and pricing errors is also established. An empirical analysis of the Fama–French 6-portfolios is conducted to illustrate the applications of the new measure.