We provide an in-depth analysis of the theoretical and statistical properties of the Hansen-Jagannathan (HJ) distance that incorporates a no-arbitrage constraint. We show that for stochastic discount factors (SDF) that are spanned by the returns on the test assets, testing the equality of HJ distances with no-arbitrage constraints is the same as testing the equality of HJ distances without no-arbitrage constraints. A discrepancy can exist only when at least one SDF is a function of factors that are poorly mimicked by the returns on the test assets. Under a joint normality assumption on the SDF and the returns, we derive explicit solutions for the HJ distance with a no-arbitrage constraint, the associated Lagrange multipliers, and the SDF parameters in the case of linear SDFs. This solution allows us to show that nontrivial differences between HJ distances with and without no-arbitrage constraints can arise only when the volatility of the unspanned component of an SDF is large and the Sharpe ratio of the tangency portfolio of the test assets is very high. Finally, we present the appropriate limiting theory for estimation, testing, and comparison of SDFs using the HJ distance with a no-arbitrage constraint.
JEL classification: G12, C12, C13
Key words: Hansen-Jagannathan distance, no-arbitrage constraint, stochastic discount factor, specification tests, model selection tests
The authors thank Esther Eiling, Wayne Ferson, Sergei Sarkissian, Jonathan Wright, Chu Zhang, Guofu Zhou, and seminar participants at Emory University and the University of Montreal for helpful discussions and comments. Kan gratefully acknowledges financial support from the National Bank Financial of Canada, the Social Sciences and Humanities Research Council of Canada, and the Center for Financial Innovation and Stability at the Federal Reserve Bank of Atlanta. The views expressed here are the authors' and not necessarily those of the Federal Reserve Bank of Atlanta or the Federal Reserve System. Any remaining errors are the authors' responsibility.
Please address questions regarding content to Nikolay Gospodinov, Associate Professor of Economics, Department of Economics, Concordia University, 1455 de Maisonneuve Boulevard, West Montreal, Quebec, Canada H3G 1M8, 514-848-2424, ext. 3935, firstname.lastname@example.org; Raymond M. Kan, Joseph L. Rotman School of Management, University of Toronto, 105 St. George Street, Toronto, Ontario, Canada M5S 3E6, 416-978-4291, email@example.com; or Cesare Robotti (corresponding author), Research Department, Federal Reserve Bank of Atlanta, 1000 Peachtree Street, N.E., Atlanta, GA 30309-4470; 404-498-8543, .