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 (SDFs) 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 only exist 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 allows us to show that nontrivial dierences between HJ-distances with and without no-arbitrage constraints can only arise 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.

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