Seminar

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

Benjamin S. Skrainkai (University College London)

April 1, 2010, 12:45–14:00

Toulouse

Room MF 323

Brown Bag Seminar

Abstract

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 a some probability density function. This paper demonstrates that monomial rules out-perform other quadratures rules, including pseudo-Monte Carlo, quasi-Monte Carlo, and sparse grid integration for Berry, Levinsohn, and Pakes (1995)’s model of product differentiation. In addition, we show how Monte Carlo methods introduce numerical error and instability into the computations in this model. These problems include inaccurate market share integrals, excess sensitivity of these share values to small perturbations in parameters, instability of Berry’s mapping for inverting market shares, and, ultimately, poor convergence of several state of the art solvers when computing point estimates. We show how both monomial rules and quasi-Monte Carlo methods provide more accurate, cheaper numerical multi-dimensional integration than the traditional pseudo-Monte Carlo techniques. We conclude with a discussion of how to further increase performance with custom monomial rules which exploit the properties of the multinomial logit function.