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

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  • 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

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    1. Martin Burda & Matthew C. Harding & Jerry Hausman, 2008. "A Bayesian mixed logit-probit model for multinomial choice," CeMMAP working papers CWP23/08, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    2. 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.
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    Cited by:

    1. Reynaert, Mathias & Verboven, Frank, 2014. "Improving the performance of random coefficients demand models: The role of optimal instruments," Journal of Econometrics, Elsevier, vol. 179(1), pages 83-98.
    2. Freyberger, Joachim, 2015. "Asymptotic theory for differentiated products demand models with many markets," Journal of Econometrics, Elsevier, vol. 185(1), pages 162-181.
    3. Matthew Gentzkow & Jesse M. Shapiro & Michael Sinkinson, 2014. "Competition and Ideological Diversity: Historical Evidence from US Newspapers," American Economic Review, American Economic Association, vol. 104(10), pages 3073-3114, October.
    4. Timothy Derdenger, 2014. "Technological tying and the intensity of price competition: An empirical analysis of the video game industry," Quantitative Marketing and Economics (QME), Springer, vol. 12(2), pages 127-165, June.
    5. André De Palma & Julien Monardo, 2017. "The General Nesting Logit (GNL) Model using Aggregate Data," Working Papers hal-01552455, HAL.
    6. Peter Davis & Pasquale Schiraldi, 2014. "The flexible coefficient multinomial logit (FC-MNL) model of demand for differentiated products," RAND Journal of Economics, RAND Corporation, vol. 45(1), pages 32-63, March.
    7. Santiago Pereda Fernández, 2016. "Copula-based random effects models for clustered data," Temi di discussione (Economic working papers) 1092, Bank of Italy, Economic Research and International Relations Area.
    8. Joachim Freyberger, 2012. "Asymptotic theory for differentiated products demand models with many markets," CeMMAP working papers CWP19/12, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    9. Philipp Eisenhauer & James J. Heckman & Stefano Mosso, 2015. "Estimation Of Dynamic Discrete Choice Models By Maximum Likelihood And The Simulated Method Of Moments," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 56, pages 331-357, May.
    10. W. Morrow, 2015. "Finite purchasing power and computations of Bertrand–Nash equilibrium prices," Computational Optimization and Applications, Springer, vol. 62(2), pages 477-515, November.
    11. Eggleston, Jonathan, 2016. "An efficient decomposition of the expectation of the maximum for the multivariate normal and related distributions," Journal of Econometrics, Elsevier, vol. 195(1), pages 120-133.
    12. Malina, Christiane, 2016. "The environmental impact of vehicle circulation tax reform in Germany," CAWM Discussion Papers 86, University of Münster, Center of Applied Economic Research Münster (CAWM).

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