Constrained Polynomial Likelihood

Abstract

We develop a non-negative polynomial minimum-norm likelihood ratio (PLR) of two distributions of which only moments are known under shape restrictions. The PLR converges to the true, unknown, likelihood ratio under mild conditions. We establish asymptotic theory for the PLR coefficients and present two empirical applications. The first develops a PLR for the unknown transition density of a jump-diffusion process. The second modifies the Hansen-Jagannathan pricing kernel framework to accommodate non-negative polynomial return models consistent with no-arbitrage while simultaneously nesting the linear return model. In both cases, we show the value of implementing the non-negative restriction.

Keywords: Likelihood ratio, positive polynomial, Reproducing Kernel Hilbert Space (RKHS)

JEL Classification: C13, C51, C61