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

46 Pages Posted: 24 Jun 2011

See all articles by Benjamin S. Skrainka

Benjamin S. Skrainka

affiliation not provided to SSRN

Kenneth L. Judd

Stanford University - The Hoover Institution on War, Revolution and Peace; Center for Robust Decisionmaking on Climate & Energy Policy (RDCEP); National Bureau of Economic Research (NBER)

Date Written: February 1, 2011

Abstract

Efficient, accurate, multi-dimensional, numerical integration has become an important tool for approximating the integrals which arise in modern economic models built on unobserved heterogeneity, incomplete information, and uncertainty. This paper demonstrates that polynomial-based rules out-perform number-theoretic quadrature (Monte Carlo) rules both in terms of efficiency and accuracy. To show the impact a quadrature method can have on results, we examine the performance of these rules in the context of Berry, Levinsohn, and Pakes (1995)’s model of product differentiation, where Monte Carlo methods introduce considerable numerical error and instability into the computations. These problems include inaccurate point estimates, excessively tight standard errors, 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. Both monomial rules and sparse grid methods lack these problems and provide a more accurate, cheaper method for quadrature. Finally, we demonstrate how researchers can easily utilize high quality, high dimensional quadrature rules in their own work.

Keywords: Numerical Integration, Monomial Rules, Gauss-Hermite Quadrature, Sparse Grid Integration, Monte Carlo Integration, pseudo-Monte Carlo, Product Differentiation, Econometrics, Random Coefficients, Discrete Choice

JEL Classification: C1, C5, C6, L00, M3

Suggested Citation

Skrainka, Benjamin S. and Judd, Kenneth L., High Performance Quadrature Rules: How Numerical Integration Affects a Popular Model of Product Differentiation (February 1, 2011). Available at SSRN: https://ssrn.com/abstract=1870703 or http://dx.doi.org/10.2139/ssrn.1870703

Benjamin S. Skrainka (Contact Author)

affiliation not provided to SSRN

Kenneth L. Judd

Stanford University - The Hoover Institution on War, Revolution and Peace ( email )

Stanford, CA 94305-6010
United States

Center for Robust Decisionmaking on Climate & Energy Policy (RDCEP) ( email )

5735 S. Ellis Street
Chicago, IL 60637
United States

National Bureau of Economic Research (NBER) ( email )

1050 Massachusetts Avenue
Cambridge, MA 02138
United States

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