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

Abstract

Standard-Nutzungsbedingungen:Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden.Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen.Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. 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. Terms of use: Documents in

Asymptotic theory for differentiated products demand models with many markets

“…The results in the aforementioned papers do not directly apply to the setup presented here because there is no closed form analytic expression for the objective function which is constructed by solving a system of equations. The results are also in line with simulation results in a recent study by Judd and Skrainka (2011) who find among others finite sample bias and excessively tight standard errors when using Monte Carlo integration. Other recent contributions to literature on estimation of discrete choice demand models include Gandhi andKim (2011) andArmstrong (2012).…”

“…For these distributions there is no closed form expression for the density function. An interesting alternative is to use non-stochastic approximations, such as quadrature rules recently advocated by Judd and Skrainka (2011). These approximations are shown to perform well in simulations when integrating over a normal distribution.…”

Asymptotic theory for differentiated products demand models with many markets

“…The setup for the Monte Carlo simulation is adapted from Dubé, Fox, and Su (2009) with very few changes to accommodate the asymptotics in the number of markets. This setup is also used by Judd and Skrainka (2011). The number of products is set to 4 and I vary the number of markets, T , and draws, R. I use a constant term and three product characteristics next to the price.…”

Finite purchasing power and computations of Bertrand–Nash equilibrium prices

“…In this case a uniform Law of Large Numbers holds for prices in any compact subset of P J by Proposition 7 in [35]. There are quasi-random, non-random, and weighted sampling schemes that lead to more efficient simulation procedures than sample-path sampling [37] and recent work concerning monomial approximates [20]. Note that the convergence (with probability one) of the simulated choice probabilities to the true choice probabilities alone does not ensure the convergence of equilibrium prices for simulators to equilibrium prices for the true model.…”

“…Let B ∈ F denote some limit that describes how small we want computed exponentials to get. For example, 20 where B 20 ≈ 46.051701859960303. We thus want to define somep j,s such that u j (θ s , p j ) > −B for all p j <p j,s .…”