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Asymptotic theory for differentiated products demand models with many markets

  • Joachim Freyberger

    (Institute for Fiscal Studies)

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    This paper develops asymptotic theory for estimated parameters in differentiated product demand systems with a fixed number of products, as the number of markets T increases, taking into account that the market shares are approximated by Monte Carlo integration. It is shown that the estimated parameters are vT consistent and asymptotically normal as long as the number of simulations R grows fast enough relative to T. Monte Carlo integration induces both additional variance as well additional bias terms in the asymptotic expansion of the estimator. If R does not increase as fast as T, the leading bias term dominates the leading variance term and the asymptotic distribution might not be centered at 0. This paper suggests methods to eliminate the leading bias term from the asymptotic expansion. Furthermore, an adjustment to the asymptotic variance is proposed that takes the leading variance term into account. Monte Carlo results show that these adjustments, which are easy to compute, should be used in applications to avoid severe undercoverage caused by the simulation error.

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    Paper provided by Centre for Microdata Methods and Practice, Institute for Fiscal Studies in its series CeMMAP working papers with number CWP19/12.

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    Date of creation: Aug 2012
    Date of revision:
    Handle: RePEc:ifs:cemmap:19/12
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    1. Villas-Boas, Sofia B., 2006. "Vertical relationships between manufacturers and retailers: inference with limited data," Department of Agricultural & Resource Economics, UC Berkeley, Working Paper Series qt0z26d2v9, Department of Agricultural & Resource Economics, UC Berkeley.
    2. Berry, Steven & Levinsohn, James & Pakes, Ariel, 1995. "Automobile Prices in Market Equilibrium," Econometrica, Econometric Society, vol. 63(4), pages 841-90, July.
    3. repec:clu:wpaper:0910-15 is not listed on IDEAS
    4. Nevo, Aviv, 2001. "Measuring Market Power in the Ready-to-Eat Cereal Industry," Econometrica, Econometric Society, vol. 69(2), pages 307-42, March.
    5. 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.
    6. Donald W. K. Andrews & Xu Cheng, 2012. "Estimation and Inference With Weak, Semiā€Strong, and Strong Identification," Econometrica, Econometric Society, vol. 80(5), pages 2153-2211, 09.
    7. Pakes, Ariel & Pollard, David, 1989. "Simulation and the Asymptotics of Optimization Estimators," Econometrica, Econometric Society, vol. 57(5), pages 1027-57, September.
    8. McFadden, Daniel, 1989. "A Method of Simulated Moments for Estimation of Discrete Response Models without Numerical Integration," Econometrica, Econometric Society, vol. 57(5), pages 995-1026, September.
    9. Lee, Lung-Fei, 1995. "Asymptotic Bias in Simulated Maximum Likelihood Estimation of Discrete Choice Models," Econometric Theory, Cambridge University Press, vol. 11(03), pages 437-483, June.
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