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The Structural Estimation of Behavioral Models: Discrete Choice Dynamic Programming Methods and Applications

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The Structural Estimation of Behavioral Models: Discrete Choice Dynamic Programming Methods and Applications CHAPTER 44 The Structural Estimation of Behavioral Models: Discrete Choice Dynamic Programming Methods and Applications Michael P.Keane* , Petra E. Todd**, Kenneth I. Wolpin** * University of Technology, Sydney and Arizona State University ** University of Pennsylvania Contents 1. Introduction 332 2. The Latent Variable Framework for Discrete Choice Problems 335 3. The Common Empirical Structure of Static and Dynamic Discrete Choice Models 336 3.1. Married woman's labor force participation 336 3.1.1. Static model 336 3.1.2. Dynamic model 342 3.2. The multinomial dynamic discrete choice problem 357 3.2.1. Alternative estimation approaches 367 4. Applications 371 4.1. Labor supply 372 4.1.1. Female labor supply 372 4.1.2. Mincer's (1962) life cycle model 373 4.1.3. Non-full solution methods of estimation 374 4.1.4. DCDP models 385 4.1.5. Male labor supply 400 4.2. Job search 407 4.2.1. The standard discrete-time job search model 410 4.3. Dynamic models of schooling and occupational choices 429 4.3.1. Foundational literature 430 4.3.2. DCDP models 432 4.3.3. The use of DCDP models in related contexts 445 4.3.4. Summary 452 5. Concluding Remarks—How Credible are DCDP Models? 452 References 455 Abstract The purpose of this chapter is twofold: (1) to provide an accessible introduction to the methods of structural estimation of discrete choice dynamic programming (DCDP) models and (2) to survey the contributions of applications of these methods to substantive and policy issues in labor economics. Handbook of Labor Economics, Volume 4a ISSN 0169-7218, DOI 10.1016/S0169-7218(11)00410-2 c 2011 Elsevier B.V. All rights reserved. 331 332 Michael P. Keane et al. The first part of the chapter describes solution and estimation methods for DCDP models using, for expository purposes, a prototypical female labor force participation model. The next part reviews the contribution of the DCDP approach to three leading areas in labor economics: labor supply, job search and human capital. The final section discusses approaches to validating DCDP models. JEL classification: J; C51; C52; C54 Keywords: Structural estimation; Discrete choice; Dynamic programming; Labor supply; Job search; Human capital 1. INTRODUCTION The purpose of this chapter is twofold: (1) to provide an accessible introduction to the methods of structural estimation of discrete choice dynamic programming (DCDP) models and (2) to survey the contributions of applications of these methods to substantive and policy issues in labor economics.1 The development of estimation methods for DCDP models over the last 25 years has opened up new frontiers for empirical research in labor economics as well as other areas such as industrial organization, economic demography, health economics, development economics and political economy.2 ReXecting the generality of the methodology, the Wrst DCDP papers, associated with independent contributions by Gotz and McCall (1984), Miller (1984), Pakes (1986), Rust (1987) and Wolpin (1984), addressed a variety of topics, foreshadowing the diverse applications to come in labor economics and other Welds. Gotz and McCall considered the sequential decision to re-enlist in the military, Miller the decision to change occupations, Pakes the decision to renew a patent, Rust the decision to replace a bus engine and Wolpin the decision to have a child. The Wrst part of this chapter provides an introduction to the solution and estimation methods for DCDP models. We begin by placing the method within the general latent variable framework of discrete choice analysis. This general framework nests static and dynamic models and nonstructural and structural estimation approaches. Our discussion of DCDP models starts by considering an agent making a binary choice. For concreteness, and for simplicity, we take as a working example the unitary model of a married couple’s decision about the woman’s labor force participation. To Wx ideas, we use the static model with partial wage observability, that is, when wage oVers are observed only for women who are employed, to draw the connection between theory, data and estimation approaches. In that context, we delineate several goals of estimation, for example, testing theory or evaluating counterfactuals, and discuss the ability of alternative estimation approaches, encompassing those that are parametric or nonparametric and 1 More technical discussions can be found in the surveys by Rust (1993, 1994), Miller (1997) and Aguirregebaria and Mira (forthcoming), as well as in a number of papers cited throughout this chapter. 2 Their use has spread to areas outside of traditional economics, such as marketing, in which it is arguably now the predominant approach to empirical research. The Structural Estimation of Behavioral Models: DCDP Methods and Applications 333 structural or nonstructural, to achieve those goals. We show how identiWcation issues relate to what one can learn from estimation. The discussion of the static model sets the stage for dynamics, which we introduce again, for expository purposes, within the labor force participation example by incorporating a wage return to work experience (learning by doing).3 A comparison of the empirical structure of the static and dynamic models reveals that the dynamic model is, in an important sense, a static model in disguise. In particular, the essential element in the estimation of both the static and dynamic model is the calculation of a latent variable representing the diVerence in payoVs associated with the two alternatives (in the binary case) that may be chosen. In the static model, the latent variable is the diVerence in alternative-speciWc utilities. In the case of the dynamic model, the latent variable is the diVerence in alternative-speciWc value functions (expected discounted values of payoVs). The only essential diVerence between the static and dynamic cases is that alternative-speciWc utilities are more easily calculated than alternative-speciWc value functions, which require solving a dynamic programming problem. In both cases, computational considerations play a role in the choice of functional forms and distributional assumptions. There are a number of modeling choices in all discrete choice analyses, although some are more important in the dynamic context because of computational issues. Modeling choices include the number of alternatives, the size of the state space, the error structure and distributional assumptions and the functional forms for the structural relationships. In addition, in the dynamic case, one must make an assumption about how expectations are formed.4 To illustrate the DCDP methodology, the labor force participation model assumes additive, normally distributed, iid over time errors for preferences and wage oVers. We Wrst discuss the role of exclusion restrictions in identiWcation, and work through the solution and estimation procedure. We then show how a computational simpliWcation can be achieved by assuming errors to be independent type 1 extreme value (Rust, 1987) and describe the model assumptions that are consistent with adopting that simpliWcation. Although temporal independence of the unobservables is often assumed, the DCDP methodology does not require it. We show how the solution and estimation of DCDP models is modiWed to allow for permanent unobserved heterogeneity and for serially correlated errors. In the illustrative model, the state space was chosen to be of a small Wnite dimension. We then describe the practical problem that arises in implementing the DCDP methodology as the state space expands, the well-known curse of dimensionality (Bellman, 1957), and describe suggested practical solutions found in the literature including discretization, approximation and randomization. 3 Most applications of DCDP models assume that agents, usually individuals or households, solve a Wnite horizon problem in discrete time. For the most part, we concentrate on that case and defer discussion of inWnite horizon models to the discussion of the special case of job search models. We do not discuss continuous time models except in passing. 4 The conventional approach assumes that agents have rational expectations. An alternative approach directly elicits subjective expectations (see, e.g., Dominitz and Manski, 1996, 1997; Van der Klaauw, 2000; Manski, 2004). 334 Michael P. Keane et al. To illustrate the DCDP framework in a multinomial choice setting, we extend the labor force participation model to allow for a fertility decision at each period and for several levels of work intensity. In that context, we also consider the implications of introducing nonadditive errors (that arise naturally within the structure of models that fully specify payoVs and constraints) and general functional forms. It is a truism that any dynamic optimization model that can be (numerically) solved can be estimated. Throughout the presentation, the estimation approach is assumed to be maximum likelihood or, as is often the case when there are many alternatives, simulated maximum likelihood. However, with simulated data from the solution to the dynamic programming problem, other methods, such as minimum distance estimation, are also available. We do not discuss those methods because, except for solving the dynamic programming model, their application is standard. Among the more recent developments in the DCDP literature is a Bayesian approach to the solution and estimation of DCDP models. Although the method has the potential to reduce the computational burden associated with DCDP
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