Analytical Solution for the Constrained Hansen-Jagannathan Distance under Multivariate Ellipticity

Nikolay Gospodinov, Raymond Kan, and Cesare Robotti
Working Paper 2012-18
November 2012

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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 SDF parameters in the case of linear SDFs. This approach allows us to analyze the benefits and costs of using the HJ-distance with a no-arbitrage constraint to rank asset pricing models.

JEL classification: G12

Key words: Hansen-Jagannathan distance, no-arbitrage, model ranking, multivariate elliptical distributions

The authors thank Esther Eiling, Wayne Ferson, Jonathan Fletcher, Eric Jondeau, Francisco Penaranda, B. Ravikumar, Sergei Sarkissian, Enrique Sentana, Jonathan Wright, Chu Zhang, Guofu Zhou, seminar participants at the Chinese University of Hong Kong, EDHEC Business School, Emory University, HEC Lausanne, National University of Singapore, Singapore Management University, University of Montreal, University of South Australia, and participants at the 2010 IFM2 Symposium on Mathematical Finance, the 2010 Meetings of the Society for Nonlinear Dynamics and Econometrics, the 2010 International Symposium on Econometric Theory and Applications, and the 2010 Third Annual SoFiE Conference for helpful discussions and comments. Gospodinov gratefully acknowledges financial support from Fonds de recherche sur la société et la culture (FQRSC), Institut de Finance Mathematique de Montreal (IFM2), and the Social Sciences and Humanities Research Council of Canada. 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, Department of Economics, Concordia University, 1455 de Maisonneuve Boulevard West, Montreal, Quebec H3G 1M8, Canada, and CIREQ, 514-848-2424, 514-848-4536 (fax),; Raymond Kan, Joseph L. Rotman School of Management, University of Toronto, 105 St. George Street, Toronto, Ontario M5S 3E6, Canada,; or Cesare Robotti, Research Department, Federal Reserve Bank of Atlanta, 1000 Peachtree Street N.E., Atlanta, GA 30309, and EDHEC Risk Institute,

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