Nº 21-45: Constrained Polynomial Likelihood

We develop a non-negative polynomial minimum-norm likelihood ratio (PLR) of two distributions of which only moments are known. The PLR converges to the true, unknown, likelihood ratio. We show consistency, obtain the asymptotic distribution for the PLR coefficients estimated with sample moments, and present two 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 polynomial return models consistent with no-arbitrage while simultaneously nesting the linear return model. In the S\&P 500 market, this modification entails sizable positions in option contracts necessary to implement the optimal trading strategy suggested by its dual portfolio formulation.