Conditional Choice Probability Estimation of Dynamic Discrete Choice Models With Unobserved Heterogeneity

We adapt the expectation–maximization algorithm to incorporate unobserved heterogeneity into conditional choice probability (CCP) estimators of dynamic discrete choice problems. The unobserved heterogeneity can be time‐invariant or follow a Markov chain. By developing a class of problems where the difference in future value terms depends on a few conditional choice probabilities, we extend the class of dynamic optimization problems where CCP estimators provide a computationally cheap alternative to full solution methods. Monte Carlo results confirm that our algorithms perform quite well, both in terms of computational time and in the precision of the parameter estimates.     

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.     

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.     

Nonparametric Identification of Finite Mixture Models of Dynamic Discrete Choices

In dynamic discrete choice analysis, controlling for unobserved heterogeneity is an important issue, and finite mixture models provide flexible ways to account for it. This paper studies nonparametric identifiability of type probabilities and type‐specific component distributions in finite mixture models of dynamic discrete choices. We derive sufficient conditions for nonparametric identification for various finite mixture models of dynamic discrete choices used in applied work under different assumptions on the Markov property, stationarity, and type‐invariance in the transition process. Three elements emerge as the important determinants of identification: the time‐dimension of panel data, the number of values the covariates can take, and the heterogeneity of the response of different types to changes in the covariates. For example, in a simple case where the transition function is type‐invariant, a time‐dimension of T = 3 is sufficient for identification, provided that the number of values the covariates can take is no smaller than the number of types and that the changes in the covariates induce sufficiently heterogeneous variations in the choice probabilities across types. Identification is achieved even when state dependence is present if a model is stationary first‐order Markovian and the panel has a moderate time‐dimension (T 6).     

Nonparametric Estimation in Random Coefficients Binary Choice Models

This paper considers random coefficients binary choice models. The main goal is to estimate the density of the random coefficients nonparametrically. This is an ill‐posed inverse problem characterized by an integral transform. A new density estimator for the random coefficients is developed, utilizing Fourier–Laplace series on spheres. This approach offers a clear insight on the identification problem. More importantly, it leads to a closed form estimator formula that yields a simple plug‐in procedure requiring no numerical optimization. The new estimator, therefore, is easy to implement in empirical applications, while being flexible about the treatment of unobserved heterogeneity. Extensions including treatments of nonrandom coefficients and models with endogeneity are discussed.     

Classification Error in Dynamic Discrete Choice Models: Implications for Female Labor Supply Behavior

Two key issues in the literature on female labor supply are (i) whether persistence in employment status is due to unobserved heterogeneity or state dependence, and (ii) whether fertility is exogenous to labor supply. Until recently, the consensus was that unobserved heterogeneity is very important and fertility is endogenous. Hyslop (1999) challenged this. Using a dynamic panel probit model of female labor supply including heterogeneity and state dependence, he found that adding autoregressive errors led to a substantial diminution in the importance of heterogeneity. This, in turn, meant he could not reject that fertility is exogenous. Here, we extend Hyslop (1999) to allow classification error in employment status, using an estimation procedure developed by Keane and Wolpin (2001) and Keane and Sauer (2005). We find that a fairly small amount of classification error is enough to overturn Hyslop's conclusions, leading to overwhelming rejection of the hypothesis of exogenous fertility.     

Bounds on Parameters in Panel Dynamic Discrete Choice Models

Identification of dynamic nonlinear panel data models is an important and delicate problem in econometrics. In this paper we provide insights that shed light on the identification of parameters of some commonly used models. Using these insights, we are able to show through simple calculations that point identification often fails in these models. On the other hand, these calculations also suggest that the model restricts the parameter to lie in a region that is very small in many cases, and the failure of point identification may, therefore, be of little practical importance in those cases. Although the emphasis is on identification, our techniques are constructive in that they can easily form the basis for consistent estimates of the identified sets.     

Constrained Optimization Approaches to Estimation of Structural Models

Estimating structural models is often viewed as computationally difficult, an impression partly due to a focus on the nested fixed‐point (NFXP) approach. We propose a new constrained optimization approach for structural estimation. We show that our approach and the NFXP algorithm solve the same estimation problem, and yield the same estimates. Computationally, our approach can have speed advantages because we do not repeatedly solve the structural equation at each guess of structural parameters. Monte Carlo experiments on the canonical Zurcher bus‐repair model demonstrate that the constrained optimization approach can be significantly faster.     

Instrumental Variable Models for Discrete Outcomes

Single equation instrumental variable models for discrete outcomes are shown to be set identifying, not point identifying, for the structural functions that deliver the values of the discrete outcome. Bounds on identified sets are derived for a general nonparametric model and sharp set identification is demonstrated in the binary outcome case. Point identification is typically not achieved by imposing parametric restrictions. The extent of an identified set varies with the strength and support of instruments, and typically shrinks as the support of a discrete outcome grows. The paper extends the analysis of structural quantile functions with endogenous arguments to cases in which there are discrete outcomes.     

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.     

Bayesian and Frequentist Inference in Partially Identified Models

A large‐sample approximation of the posterior distribution of partially identified structural parameters is derived for models that can be indexed by an identifiable finite‐dimensional reduced‐form parameter vector. It is used to analyze the differences between Bayesian credible sets and frequentist confidence sets. We define a plug‐in estimator of the identified set and show that asymptotically Bayesian highest‐posterior‐density sets exclude parts of the estimated identified set, whereas it is well known that frequentist confidence sets extend beyond the boundaries of the estimated identified set. We recommend reporting estimates of the identified set and information about the conditional prior along with Bayesian credible sets. A numerical illustration for a two‐player entry game is provided.     

Estimating Derivatives in Nonseparable Models With Limited Dependent Variables

We present a simple way to estimate the effects of changes in a vector of observable variables X on a limited dependent variable Y when Y is a general nonseparable function of X and unobservables, and X is independent of the unobservables. We treat models in which Y is censored from above, below, or both. The basic idea is to first estimate the derivative of the conditional mean of Y given X at x with respect to x on the uncensored sample without correcting for the effect of x on the censored population. We then correct the derivative for the effects of the selection bias. We discuss nonparametric and semiparametric estimators for the derivative. We also discuss the cases of discrete regressors and of endogenous regressors in both cross section and panel data contexts.     

Solving,Estimating, and Selecting Nonlinear Dynamic Models Without the Curse of Dimensionality

We present a comprehensive framework for Bayesian estimation of structural nonlinear dynamic economic models on sparse grids to overcome the curse of dimensionality for approximations. We apply sparse grids to a global polynomial approximation of the model solution, to the quadrature of integrals arising as rational expectations, and to three new nonlinear state space filters which speed up the sequential importance resampling particle filter. The posterior of the structural parameters is estimated by a new Metropolis–Hastings algorithm with mixing parallel sequences. The parallel extension improves the global maximization property of the algorithm, simplifies the parameterization for an appropriate acceptance ratio, and allows a simple implementation of the estimation on parallel computers. Finally, we provide all algorithms in the open source software JBendge for the solution and estimation of a general class of models.     

基于离散选择模型的电信资费套餐选择行为研究

以大学生为例,采用离散选择模型对影响电信客户选择资费套餐行为的因素进行了研究。结果表明,运用嵌套Logit模型比多元Logit模型更能解释影响客户选择资费套餐行为的因素,品牌对客户是否选择中国移动和中国联通的影响大于是否选择中国电信的影响,应付通信费对客户是否选择中国联通和中国电信的影响大于是否选择中国移动的影响,并且中国联通和其他运营商的套餐之间具有高度的替代性。     

Identification and Inference in Nonlinear Difference‐in‐Differences Models

This paper develops a generalization of the widely used difference‐in‐differences method for evaluating the effects of policy changes. We propose a model that allows the control and treatment groups to have different average benefits from the treatment. The assumptions of the proposed model are invariant to the scaling of the outcome. We provide conditions under which the model is nonparametrically identified and propose an estimator that can be applied using either repeated cross section or panel data. Our approach provides an estimate of the entire counterfactual distribution of outcomes that would have been experienced by the treatment group in the absence of the treatment and likewise for the untreated group in the presence of the treatment. Thus, it enables the evaluation of policy interventions according to criteria such as a mean–variance trade‐off. We also propose methods for inference, showing that our estimator for the average treatment effect is root‐N consistent and asymptotically normal. We consider extensions to allow for covariates, discrete dependent variables, and multiple groups and time periods.     

Sharp Identification Regions in Models With Convex Moment Predictions

We provide a tractable characterization of the sharp identification region of the parameter vector θ in a broad class of incomplete econometric models. Models in this class have set‐valued predictions that yield a convex set of conditional or unconditional moments for the observable model variables. In short, we call these models with convex moment predictions. Examples include static, simultaneous‐move finite games of complete and incomplete information in the presence of multiple equilibria; best linear predictors with interval outcome and covariate data; and random utility models of multinomial choice in the presence of interval regressors data. Given a candidate value for θ, we establish that the convex set of moments yielded by the model predictions can be represented as the Aumann expectation of a properly defined random set. The sharp identification region of θ, denoted Θ_I, can then be obtained as the set of minimizers of the distance from a properly specified vector of moments of random variables to this Aumann expectation. Algorithms in convex programming can be exploited to efficiently verify whether a candidate θ is in Θ_I. We use examples analyzed in the literature to illustrate the gains in identification and computational tractability afforded by our method.     

An Adaptive Simulation Framework for the Exploration of Extreme and Unexpected Events in Dynamic Engineered Systems

The end states reached by an engineered system during an accident scenario depend not only on the sequences of the events composing the scenario, but also on their timing and magnitudes. Including these additional features within an overarching framework can render the analysis infeasible in practical cases, due to the high dimension of the system state‐space and the computational effort correspondingly needed to explore the possible system evolutions in search of the interesting (and very rare) ones of failure. To tackle this hurdle, in this article we introduce a framework for efficiently probing the space of event sequences of a dynamic system by means of a guided Monte Carlo simulation. Such framework is semi‐automatic and allows embedding the analyst's prior knowledge about the system and his/her objectives of analysis. Specifically, the framework allows adaptively and intelligently allocating the simulation efforts preferably on those sequences leading to outcomes of interest for the objectives of the analysis, e.g., typically those that are more safety‐critical (and/or rare). The emerging diversification in the filling of the state‐space by the preference‐guided exploration allows also the retrieval of critical system features, which can be useful to analysts and designers for taking appropriate means of prevention and mitigation of dangerous and/or unexpected consequences. A dynamic system for gas transmission is considered as a case study to demonstrate the application of the method.