Improving the Numerical Performance of Static and Dynamic Aggregate Discrete Choice Random Coefficients Demand Estimation

The widely used estimator of Berry, Levinsohn, and Pakes (1995) produces estimates of consumer preferences from a discrete‐choice demand model with random coefficients, market‐level demand shocks, and endogenous prices. We derive numerical theory results characterizing the properties of the nested fixed point algorithm used to evaluate the objective function of BLP's estimator. We discuss problems with typical implementations, including cases that can lead to incorrect parameter estimates. As a solution, we recast estimation as a mathematical program with equilibrium constraints, which can be faster and which avoids the numerical issues associated with nested inner loops. The advantages are even more pronounced for forward‐looking demand models where the Bellman equation must also be solved repeatedly. Several Monte Carlo and real‐data experiments support our numerical concerns about the nested fixed point approach and the advantages of constrained optimization. For static BLP, the constrained optimization approach can be as much as ten to forty times faster for large‐dimensional problems with many markets.     

Markov Perfect Industry Dynamics With Many Firms

We propose an approximation method for analyzing Ericson and Pakes (1995)‐style dynamic models of imperfect competition. We define a new equilibrium concept that we call oblivious equilibrium, in which each firm is assumed to make decisions based only on its own state and knowledge of the long‐run average industry state, but where firms ignore current information about competitors' states. The great advantage of oblivious equilibria is that they are much easier to compute than are Markov perfect equilibria. Moreover, we show that, as the market becomes large, if the equilibrium distribution of firm states obeys a certain “light‐tail” condition, then oblivious equilibria closely approximate Markov perfect equilibria. This theorem justifies using oblivious equilibria to analyze Markov perfect industry dynamics in Ericson and Pakes (1995)‐style models with many firms.     

Sequential Estimation of Structural Models With a Fixed Point Constraint

This paper considers the estimation problem of structural models for which empirical restrictions are characterized by a fixed point constraint, such as structural dynamic discrete choice models or models of dynamic games. We analyze a local condition under which the nested pseudo likelihood (NPL) algorithm converges to a consistent estimator, and derive its convergence rate. We find that the NPL algorithm may not necessarily converge to a consistent estimator when the fixed point mapping does not have a local contraction property. To address the issue of divergence, we propose alternative sequential estimation procedures that can converge to a consistent estimator even when the NPL algorithm does not.     

Finite Mixture Distributions,Sequential Likelihood and the EM Algorithm

A popular way to account for unobserved heterogeneity is to assume that the data are drawn from a finite mixture distribution. A barrier to using finite mixture models is that parameters that could previously be estimated in stages must now be estimated jointly: using mixture distributions destroys any additive separability of the log‐likelihood function. We show, however, that an extension of the EM algorithm reintroduces additive separability, thus allowing one to estimate parameters sequentially during each maximization step. In establishing this result, we develop a broad class of estimators for mixture models. Returning to the likelihood problem, we show that, relative to full information maximum likelihood, our sequential estimator can generate large computational savings with little loss of efficiency.     

Asymptotic Variance of Semiparametric Estimators With Generated Regressors

We study the asymptotic distribution of three‐step estimators of a finite‐dimensional parameter vector where the second step consists of one or more nonparametric regressions on a regressor that is estimated in the first step. The first‐step estimator is either parametric or nonparametric. Using Newey's (1994) path‐derivative method, we derive the contribution of the first‐step estimator to the influence function. In this derivation, it is important to account for the dual role that the first‐step estimator plays in the second‐step nonparametric regression, that is, that of conditioning variable and that of argument.     

Swapping the Nested Fixed Point Algorithm: A Class of Estimators for Discrete Markov Decision Models

This paper proposes a new nested algorithm (NPL) for the estimation of a class of discrete Markov decision models and studies its statistical and computational properties. Our method is based on a representation of the solution of the dynamic programming problem in the space of conditional choice probabilities. When the NPL algorithm is initialized with consistent nonparametric estimates of conditional choice probabilities, successive iterations return a sequence of estimators of the structural parameters which we call K–stage policy iteration estimators. We show that the sequence includes as extreme cases a Hotz–Miller estimator (for K=1) and Rust's nested fixed point estimator (in the limit when K→∞). Furthermore, the asymptotic distribution of all the estimators in the sequence is the same and equal to that of the maximum likelihood estimator. We illustrate the performance of our method with several examples based on Rust's bus replacement model. Monte Carlo experiments reveal a trade–off between finite sample precision and computational cost in the sequence of policy iteration estimators.     

Average and Quantile Effects in Nonseparable Panel Models

Nonseparable panel models are important in a variety of economic settings, including discrete choice. This paper gives identification and estimation results for nonseparable models under time‐homogeneity conditions that are like “time is randomly assigned” or “time is an instrument.” Partial‐identification results for average and quantile effects are given for discrete regressors, under static or dynamic conditions, in fully nonparametric and in semiparametric models, with time effects. It is shown that the usual, linear, fixed‐effects estimator is not a consistent estimator of the identified average effect, and a consistent estimator is given. A simple estimator of identified quantile treatment effects is given, providing a solution to the important problem of estimating quantile treatment effects from panel data. Bounds for overall effects in static and dynamic models are given. The dynamic bounds provide a partial‐identification solution to the important problem of estimating the effect of state dependence in the presence of unobserved heterogeneity. The impact of T, the number of time periods, is shown by deriving shrinkage rates for the identified set as T grows. We also consider semiparametric, discrete‐choice models and find that semiparametric panel bounds can be much tighter than nonparametric bounds. Computationally convenient methods for semiparametric models are presented. We propose a novel inference method that applies in panel data and other settings and show that it produces uniformly valid confidence regions in large samples. We give empirical illustrations.     

Sequential Estimation of Dynamic Discrete Games

This paper studies the estimation of dynamic discrete games of incomplete information. Two main econometric issues appear in the estimation of these models: the indeterminacy problem associated with the existence of multiple equilibria and the computational burden in the solution of the game. We propose a class of pseudo maximum likelihood (PML) estimators that deals with these problems, and we study the asymptotic and finite sample properties of several estimators in this class. We first focus on two‐step PML estimators, which, although they are attractive for their computational simplicity, have some important limitations: they are seriously biased in small samples; they require consistent nonparametric estimators of players' choice probabilities in the first step, which are not always available; and they are asymptotically inefficient. Second, we show that a recursive extension of the two‐step PML, which we call nested pseudo likelihood (NPL), addresses those drawbacks at a relatively small additional computational cost. The NPL estimator is particularly useful in applications where consistent nonparametric estimates of choice probabilities either are not available or are very imprecise, e.g., models with permanent unobserved heterogeneity. Finally, we illustrate these methods in Monte Carlo experiments and in an empirical application to a model of firm entry and exit in oligopoly markets using Chilean data from several retail industries.     

Estimation of Nonparametric Models With Simultaneity

We introduce methods for estimating nonparametric, nonadditive models with simultaneity. The methods are developed by directly connecting the elements of the structural system to be estimated with features of the density of the observable variables, such as ratios of derivatives or averages of products of derivatives of this density. The estimators are therefore easily computed functionals of a nonparametric estimator of the density of the observable variables. We consider in detail a model where to each structural equation there corresponds an exclusive regressor and a model with one equation of interest and one instrument that is included in a second equation. For both models, we provide new characterizations of observational equivalence on a set, in terms of the density of the observable variables and derivatives of the structural functions. Based on those characterizations, we develop two estimation methods. In the first method, the estimators of the structural derivatives are calculated by a simple matrix inversion and matrix multiplication, analogous to a standard least squares estimator, but with the elements of the matrices being averages of products of derivatives of nonparametric density estimators. In the second method, the estimators of the structural derivatives are calculated in two steps. In a first step, values of the instrument are found at which the density of the observable variables satisfies some properties. In the second step, the estimators are calculated directly from the values of derivatives of the density of the observable variables evaluated at the found values of the instrument. We show that both pointwise estimators are consistent and asymptotically normal.     

The Costs of Environmental Regulation in a Concentrated Industry

The typical cost analysis of an environmental regulation consists of an engineering estimate of the compliance costs. In industries where fixed costs are an important determinant of market structure, this static analysis ignores the dynamic effects of the regulation on entry, investment, and market power. I evaluate the welfare costs of the 1990 Amendments to the Clean Air Act on the U.S. Portland cement industry, accounting for these effects through a dynamic model of oligopoly in the tradition of Ericson and Pakes (1995). Using the two‐step estimator of Bajari, Benkard, and Levin (2007), I recover the entire cost structure of the industry, including the distributions of sunk entry costs and capacity adjustment costs. My primary finding is that the Amendments have significantly increased the sunk cost of entry, leading to a loss of between $810M and $3.2B in product market surplus. A static analysis misses the welfare penalty on consumers, and obtains the wrong sign of the welfare effects on incumbent firms.     

Least Squares Model Averaging

This paper considers the problem of selection of weights for averaging across least squares estimates obtained from a set of models. Existing model average methods are based on exponential Akaike information criterion (AIC) and Bayesian information criterion (BIC) weights. In distinction, this paper proposes selecting the weights by minimizing a Mallows criterion, the latter an estimate of the average squared error from the model average fit. We show that our new Mallows model average (MMA) estimator is asymptotically optimal in the sense of achieving the lowest possible squared error in a class of discrete model average estimators. In a simulation experiment we show that the MMA estimator compares favorably with those based on AIC and BIC weights. The proof of the main result is an application of the work of Li (1987).     

Consistent Estimation of Models Defined by Conditional Moment Restrictions

In econometrics, models stated as conditional moment restrictions are typically estimated by means of the generalized method of moments (GMM). The GMM estimation procedure can render inconsistent estimates since the number of arbitrarily chosen instruments is finite. In fact, consistency of the GMM estimators relies on additional assumptions that imply unclear restrictions on the data generating process. This article introduces a new, simple and consistent estimation procedure for these models that is directly based on the definition of the conditional moments. The main feature of our procedure is its simplicity, since its implementation does not require the selection of any user‐chosen number, and statistical inference is straightforward since the proposed estimator is asymptotically normal. In addition, we suggest an asymptotically efficient estimator constructed by carrying out one Newton–Raphson step in the direction of the efficient GMM estimator.     

Estimation of Semiparametric Models when the Criterion Function Is Not Smooth

We provide easy to verify sufficient conditions for the consistency and asymptotic normality of a class of semiparametric optimization estimators where the criterion function does not obey standard smoothness conditions and simultaneously depends on some nonparametric estimators that can themselves depend on the parameters to be estimated. Our results extend existing theories such as those of Pakes and Pollard (1989), Andrews (1994a), and Newey (1994). We also show that bootstrap provides asymptotically correct confidence regions for the finite dimensional parameters. We apply our results to two examples: a ‘hit rate’ and a partially linear median regression with some endogenous regressors.     

Generalized Method of Integrated Moments for High‐Frequency Data

We propose a semiparametric two‐step inference procedure for a finite‐dimensional parameter based on moment conditions constructed from high‐frequency data. The population moment conditions take the form of temporally integrated functionals of state‐variable processes that include the latent stochastic volatility process of an asset. In the first step, we nonparametrically recover the volatility path from high‐frequency asset returns. The nonparametric volatility estimator is then used to form sample moment functions in the second‐step GMM estimation, which requires the correction of a high‐order nonlinearity bias from the first step. We show that the proposed estimator is consistent and asymptotically mixed Gaussian and propose a consistent estimator for the conditional asymptotic variance. We also construct a Bierens‐type consistent specification test. These infill asymptotic results are based on a novel empirical‐process‐type theory for general integrated functionals of noisy semimartingale processes.     

Identification and Estimation of a Discrete Game of Complete Information

We discuss the identification and estimation of discrete games of complete information. Following Bresnahan and Reiss (1990, 1991), a discrete game is a generalization of a standard discrete choice model where utility depends on the actions of other players. Using recent algorithms to compute all of the Nash equilibria to a game, we propose simulation‐based estimators for static, discrete games. We demonstrate that the model is identified under weak functional form assumptions using exclusion restrictions and an identification at infinity approach. Monte Carlo evidence demonstrates that the estimator can perform well in moderately sized samples. As an application, we study entry decisions by construction contractors to bid on highway projects in California. We find that an equilibrium is more likely to be observed if it maximizes joint profits, has a higher Nash product, uses mixed strategies, and is not Pareto dominated by another equilibrium.     

Panel Data Discrete Choice Models with Lagged Dependent Variables

In this paper, we consider identification and estimation in panel data discrete choice models when the explanatory variable set includes strictly exogenous variables, lags of the endogenous dependent variable as well as unobservable individual‐specific effects. For the binary logit model with the dependent variable lagged only once, Chamberlain (1993) gave conditions under which the model is not identified. We present a stronger set of conditions under which the parameters of the model are identified. The identification result suggests estimators of the model, and we show that these are consistent and asymptotically normal, although their rate of convergence is slower than the inverse of the square root of the sample size. We also consider identification in the semiparametric case where the logit assumption is relaxed. We propose an estimator in the spirit of the conditional maximum score estimator (Manski (1987)) and we show that it is consistent. In addition, we discuss an extension of the identification result to multinomial discrete choice models, and to the case where the dependent variable is lagged twice. Finally, we present some Monte Carlo evidence on the small sample performance of the proposed estimators for the binary response model.     

基于不同预期报价决策的电力市场动态演化及其混沌控制

基于市场参与者不同预期的报价决策方式,提出了考虑输电网约束的电力市场动态模型,即内嵌市场清算优化问题的差分动态模型.该模型刻画出发电方和需求方同时报价的不用决策行为,并准确反映出独立系统调度员ISO的统一市场清算过程,考虑了输电网固有物理特性所赋予电力市场的复杂约束.借助非线性互补函数,对应不同的输电网运行状态:阻塞和不阻塞,分析比较了电力市场处于Nash均衡、周期和混沌的经济表现.针对经济表现差的市场混沌态,提出电力市场状态和参数时滞反馈控制方法,给出电力市场由混沌到Nash均衡的调控措施和手段,从而为有效提高电力市场的经济效益提供了理论依据.     

Matching on the Estimated Propensity Score

Propensity score matching estimators (Rosenbaum and Rubin (1983)) are widely used in evaluation research to estimate average treatment effects. In this article, we derive the large sample distribution of propensity score matching estimators. Our derivations take into account that the propensity score is itself estimated in a first step, prior to matching. We prove that first step estimation of the propensity score affects the large sample distribution of propensity score matching estimators, and derive adjustments to the large sample variances of propensity score matching estimators of the average treatment effect (ATE) and the average treatment effect on the treated (ATET). The adjustment for the ATE estimator is negative (or zero in some special cases), implying that matching on the estimated propensity score is more efficient than matching on the true propensity score in large samples. However, for the ATET estimator, the sign of the adjustment term depends on the data generating process, and ignoring the estimation error in the propensity score may lead to confidence intervals that are either too large or too small.     

Bayesian Estimation of Dynamic Discrete Choice Models

We propose a new methodology for structural estimation of infinite horizon dynamic discrete choice models. We combine the dynamic programming (DP) solution algorithm with the Bayesian Markov chain Monte Carlo algorithm into a single algorithm that solves the DP problem and estimates the parameters simultaneously. As a result, the computational burden of estimating a dynamic model becomes comparable to that of a static model. Another feature of our algorithm is that even though the number of grid points on the state variable is small per solution‐estimation iteration, the number of effective grid points increases with the number of estimation iterations. This is how we help ease the “curse of dimensionality.” We simulate and estimate several versions of a simple model of entry and exit to illustrate our methodology. We also prove that under standard conditions, the parameters converge in probability to the true posterior distribution, regardless of the starting values.     

STATE-OF-THE-ART SURVEY: OPEN QUEUEING NETWORKS: OPTIMIZATION AND PERFORMANCE EVALUATION MODELS FOR DISCRETE MANUFACTURING SYSTEMS

In this survey we review methods to analyze open queueing network models for discrete manufacturing systems. We focus on design and planning models for job shops. The survey is divided in two parts: in the first we review exact and approximate decomposition methods for performance evaluation models for single and multiple product class networks. The second part reviews optimization models of three categories of problems: the first minimizes capital investment subject to attaining a performance measure (WIP or lead time), the second seeks to optimize the performance measure subject to resource constraints, and the third explores recent research developments in complexity reduction through shop redesign and products partitioning.