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High performance quadrature rules: how numerical integration affects a popular model of product differentiation

Author

Listed:
  • Kenneth L. Judd

    (Institute for Fiscal Studies and Hoover Institution, Stanford University)

  • Ben Skrainka

    (Institute for Fiscal Studies and University of Chicago)

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 polynomialbased 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.

Suggested Citation

  • Kenneth L. Judd & Ben Skrainka, 2011. "High performance quadrature rules: how numerical integration affects a popular model of product differentiation," CeMMAP working papers CWP03/11, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    • Handle: RePEc:ifs:cemmap:03/11
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    File URL: http://cemmap.ifs.org.uk/wps/cwp0311.pdf
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    References listed on IDEAS

    as
    1. Steve Berry & Oliver B. Linton & Ariel Pakes, 2004. "Limit Theorems for Estimating the Parameters of Differentiated Product Demand Systems," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 71(3), pages 613-654.
    2. Steven Berry & James Levinsohn & Ariel Pakes, 2004. "Differentiated Products Demand Systems from a Combination of Micro and Macro Data: The New Car Market," Journal of Political Economy, University of Chicago Press, vol. 112(1), pages 68-105, February.
    3. Burda, Martin & Harding, Matthew & Hausman, Jerry, 2008. "A Bayesian mixed logit-probit model for multinomial choice," Journal of Econometrics, Elsevier, vol. 147(2), pages 232-246, December.
    4. Steven Berry & Ariel Pakes, 2007. "The Pure Characteristics Demand Model," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 48(4), pages 1193-1225, November.
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