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Unformatted text preview: Estimation of Random Coe cient Demand Models: Challenges, Di culties and Warnings Christopher R. Knittel and Konstantinos Metaxoglou & October 1, 2008 Abstract Empirical exercises in economics frequently involve estimation of highly nonlinear models. The criterion function may not be globally concave or convex and exhibit many local ex- trema. Choosing among these local extrema is non-trivial for a variety of reasons. In this paper, we analyze the sensitivity of parameter estimates, and most importantly of eco- nomic variables of interest, to both starting values and the type of non-linear optimization algorithm employed. We focus on a class of demand models for di/erentiated products that have been used extensively in industrial organization, and more recently in public and labor. We &nd that convergence may occur at a number of local extrema, at saddles and in regions of the objective function where the &rst-order conditions are not satis&ed. We &nd own- and cross-price elasticities that di/er by a factor of over 100 depending on the set of candidate parameter estimates. In an attempt to evaluate the welfare e/ects of a change in an industrys structure, we undertake a hypothetical merger exercise. Our calculations indicate consumer welfare e/ects can vary between positive values to negative seventy billion dollars depending on the set of parameter estimates used. & Knittel: Department of Economics, University of California, Davis, CA and NBER. Email: crknit- firstname.lastname@example.org. Metaxoglou: Bates White LLC. Email: email@example.com. We have bene&tted greatly from conversations with Steve Berry, Severin Borenstein, Michael Greenstone, Phil Haile, Aviv Nevo, Hal White, Frank Wolak, Catherine Wolfram, and seminar participants at the University of Cal- gary, University of California at Berkeley, the University of California Energy Institute, and the 2008 NBER Winter IO meeting. Metaxoglou aknowledges &nancial support from Bates White, LLC. We are also grateful to Bates White, LLC for making their computing resources available. All remaining errors are ours. 1 1 Introduction 1.1 What this paper is about Empirical research in economics often requires estimating highly nonlinear models, where the objective function may not be globally concave or convex. Obtaining parameter estimates in these cases requires a nonlinear search algorithm along with sets of starting values and stopping rules. For a common class of demand models used in industrial organization, and more recently in labor and public economics, we &nd that termination of many popular nonlinear search algo- rithms may occur at local extrema, saddles points, as well as in regions of the objective function where the &rst-order conditions for optimality fail. Furthermore, parameter estimates and mea- sures of market performance, such as price elasticities, exhibit notable variation depending on the combination of the algorithm and starting values in the optimization exercise at hand.the combination of the algorithm and starting values in the optimization exercise at hand....
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