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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 College London)

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.

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File URL: http://cemmap.ifs.org.uk/wps/cwp0311.pdf
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Bibliographic Info

Paper provided by Centre for Microdata Methods and Practice, Institute for Fiscal Studies in its series CeMMAP working papers with number CWP03/11.

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Date of creation: Feb 2011
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Handle: RePEc:ifs:cemmap:03/11

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  1. Martin Burda & Matthew 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.
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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. Matthew Gentzkow & Jesse M. Shapiro & Michael Sinkinson, 2012. "Competition and Ideological Diversity: Historical Evidence from US Newspapers," NBER Working Papers 18234, National Bureau of Economic Research, Inc.
  3. Peter Davis & Pasquale Schiraldi, 2013. "The flexible coefficient multinomial logit (FC-MNL) model of demand for differentiated products," LSE Research Online Documents on Economics 54252, London School of Economics and Political Science, LSE Library.
  4. 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.

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