High Performance Quadrature Rules: How Numerical Integration Affects a Popular Model of Product Differentiation

Benjamin S. Skrainka, Kenneth L. Judd
2011 Social Science Research Network  
Numerically approximating multi-dimensional integrals has become an increasingly important part of an economist's toolbox because heterogeneity, uncertainty, and incomplete information -often key factors in modern models -require integrating accurately over some probability density function. This paper demonstrates that polynomial-based rules out-perform number-theoretic quadratures rules, namely pseudo-Monte Carlo, for Berry, Levinsohn, and Pakes (1995)'s model of product differentiation. In
more » ... fferentiation. In addition, we show how Monte Carlo methods introduce considerable numerical error and instability into the computations in this model. These problems include inaccurate point estimates, instability of the inner loop 'contraction' mapping for inverting market shares, and poor convergence of several state of the art solvers when computing point estimates. We show how both monomial rules and sparse grids methods provide more accurate, cheaper numerical multi-dimensional integration than the traditional pseudo-Monte Carlo techniques. Finally, we demonstrate how easily researchers can implement high quality, high dimensional quadrature rules.
doi:10.2139/ssrn.1870703 fatcat:7h6qhu7shnf2hbmpksu7qviwji