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318.4Contents16.8.3 Dynamic Unobserved E ects Tobit ModelsProblems542544Sample Selection, Attrition, and Stratified SamplingIntroductionWhen Can Sample Selection Be Ignored?17.2.1 Linear Models: OLS and 2SLS17.2.2 Nonlinear ModelsSelection on the Basis of the Response Variable: TruncatedRegressionA Probit Selection Equation17.4.1 Exogenous Explanatory Variables17.4.2 Endogenous Explanatory Variables17.4.3 Binary Response Model with Sample SelectionA Tobit Selection Equation17.5.1 Exogenous Explanatory Variables17.5.2 Endogenous Explanatory VariablesEstimating Structural Tobit Equations with Sample SelectionSample Selection and Attrition in Linear Panel Data Models17.7.1 Fixed E ects Estimation with Unbalanced Panels17.7.2 Testing and Correcting for Sample Selection Bias17.7.3 AttritionStratified Sampling17.8.1 Standard Stratified Sampling and Variable ProbabilitySampling17.8.2 Weighted Estimators to Account for Stratification17.8.3 Stratification Based on Exogenous VariablesProblems551551552552556Estimating Average Treatment E ectsIntroductionA Counterfactual Setting and the Self-Selection ProblemMethods Assuming Ignorability of Treatment18.3.1 Regression Methods18.3.2 Methods Based on the Propensity ScoreInstrumental Variables Methods18.4.1 Estimating the ATE Using 596598603603603607608614621621

Contents18.51919.119.219.319.419.519.6xv18.4.2 Estimating the Local Average Treatment E ect by IVFurther Issues18.5.1 Special Considerations for Binary and Corner SolutionResponses18.5.2 Panel Data18.5.3 Nonbinary Treatments18.5.4 Multiple TreatmentsProblems633636Count Data and Related ModelsWhy Count Data Models?Poisson Regression Models with Cross Section Data19.2.1 Assumptions Used for Poisson Regression19.2.2 Consistency of the Poisson QMLE19.2.3 Asymptotic Normality of the Poisson QMLE19.2.4 Hypothesis Testing19.2.5 Specification TestingOther Count Data Regression Models19.3.1 Negative Binomial Regression Models19.3.2 Binomial Regression ModelsOther QMLEs in the Linear Exponential Family19.4.1 Exponential Regression Models19.4.2 Fractional Logit RegressionEndogeneity and Sample Selection with an Exponential RegressionFunction19.5.1 Endogeneity19.5.2 Sample SelectionPanel Data Methods19.6.1 Pooled QMLE19.6.2 Specifying Models of Conditional Expectations withUnobserved E ects19.6.3 Random E ects Methods19.6.4 Fixed E ects Poisson Estimation19.6.5 Relaxing the Strict Exogeneity 76678

xvi2020.120.220.320.420.5ContentsDuration AnalysisIntroductionHazard Functions20.2.1 Hazard Functions without Covariates20.2.2 Hazard Functions Conditional on Time-InvariantCovariates20.2.3 Hazard Functions Conditional on Time-VaryingCovariatesAnalysis of Single-Spell Data with Time-Invariant Covariates20.3.1 Flow Sampling20.3.2 Maximum Likelihood Estimation with Censored FlowData20.3.3 Stock Sampling20.3.4 Unobserved HeterogeneityAnalysis of Grouped Duration Data20.4.1 Time-Invariant Covariates20.4.2 Time-Varying Covariates20.4.3 Unobserved HeterogeneityFurther Issues20.5.1 Cox’s Partial Likelihood Method for the ProportionalHazard Model20.5.2 Multiple-Spell Data20.5.3 Competing Risks 694695700703706707711713714714714715715721737

AcknowledgmentsMy interest in panel data econometrics began in earnest when I was an assistantprofessor at MIT, after I attended a seminar by a graduate student, Leslie Papke,who would later become my wife. Her empirical research using nonlinear panel datamethods piqued my interest and eventually led to my research on estimating nonlinear panel data models without distributional assumptions. I dedicate this text toLeslie.My former colleagues at MIT, particularly Jerry Hausman, Daniel McFadden,Whitney Newey, Danny Quah, and Thomas Stoker, played significant roles in encouraging my interest in cross section and panel data econometrics. I also havelearned much about the modern approach to panel data econometrics from GaryChamberlain of Harvard University.I cannot discount the excellent training I received from Robert Engle, CliveGranger, and especially Halbert White at the University of California at San Diego. Ihope they are not too disappointed that this book excludes time series econometrics.I did not teach a course in cross section and panel data methods until I startedteaching at Michigan State. Fortunately, my colleague Peter Schmidt encouraged meto teach the course at which this book is aimed. Peter also suggested that a text onpanel data methods that uses ‘‘vertical bars’’ would be a worthwhile contribution.Several classes of students at Michigan State were subjected to this book in manuscript form at various stages of development. I would like to thank these students fortheir perseverance, helpful comments, and numerous corrections. I want to specificallymention Scott Baier, Linda Bailey, Ali Berker, Yi-Yi Chen, William Horrace, RobinPoston, Kyosti Pietola, Hailong Qian, Wendy Stock, and Andrew Toole. Naturally,they are not responsible for any remaining errors.I was fortunate to have several capable, conscientious reviewers for the manuscript.Jason Abrevaya (University of Chicago), Joshua Angrist (MIT), David Drukker(Stata Corporation), Brian McCall (University of Minnesota), James Ziliak (University of Oregon), and three anonymous reviewers provided excellent suggestions,many of which improved the book’s organization and coverage.The people at MIT Press have been remarkably patient, and I have very muchenjoyed working with them. I owe a special debt to Terry Vaughn (now at PrincetonUniversity Press) for initiating this project and then giving me the time to produce amanuscript with which I felt comfortable. I am grateful to Jane McDonald andElizabeth Murry for reenergizing the project and for allowing me significant leewayin crafting the final manuscript. Finally, Peggy Gordon and her crew at P. M. GordonAssociates, Inc., did an expert job in editing the manuscript and in producing thefinal text.

PrefaceThis book is intended primarily for use in a second-semester course in graduateeconometrics, after a first course at the level of Goldberger (1991) or Greene (1997).Parts of the book can be used for special-topics courses, and it should serve as ageneral reference.My focus on cross section and panel data methods—in particular, what is oftendubbed microeconometrics—is novel, and it recognizes that, after coverage of thebasic linear model in a first-semester course, an increasingly popular approach is totreat advanced cross section and panel data methods in one semester and time seriesmethods in a separate semester. This division reflects the current state of econometricpractice.Modern empirical research that can be fitted into the classical linear model paradigm is becoming increasingly rare. For instance, it is now widely recognized that astudent doing research in applied time series analysis cannot get very far by ignoringrecent advances in estimation and testing in models with trending and strongly dependent processes. This theory takes a very di erent direction from the classical linear model than does cross section or panel data analysis. Hamilton’s (1994) timeseries text demonstrates this di erence unequivocally.Books intended to cover an econometric sequence of a year or more, beginningwith the classical linear model, tend to treat advanced topics in cross section andpanel data analysis as direct applications or minor extensions of the classical linearmodel (if they are treated at all). Such treatment needlessly limits the scope of applications and can result in poor econometric practice. The focus in such books on thealgebra and geometry of econometrics is appropriate for a first-semester course, butit results in oversimplification or sloppiness in stating assumptions. Approaches toestimation that are acceptable under the fixed regressor paradigm so prominent in theclassical linear model can lead one badly astray under practically important departures from the fixed regressor assumption.Books on ‘‘advanced’’ econometrics tend to be high-level treatments that focus ongeneral approaches to estimation, thereby attempting to cover all data configurations—including cross section, panel data, and time series—in one framework, without givingspecial attention to any. A hallmark of such books is that detailed regularity conditions are treated on par with the practically more important assumptions that haveeconomic content. This is a burden for students learning about cross section andpanel data methods, especially those who are empirically oriented: definitions andlimit theorems about dependent processes need to be included among the regularityconditions in order to cover time series applications.In this book I have attempted to find a middle ground between more traditionalapproaches and the more recent, very unified approaches. I present each model and

xviiiPrefacemethod with a careful discussion of assumptions of the underlying population model.These assumptions, couched in terms of correlations, conditional expectations, conditional variances and covariances, or conditional distributions, usually can be givenbehavioral content. Except for the three more technical chapters in Part III, regularityconditions—for example, the existence of moments needed to ensure that the centrallimit theorem holds—are not discussed explicitly, as these have little bearing on applied work. This approach makes the assumptions relatively easy to understand, whileat the same time emphasizing that assumptions concerning the underlying populationand the method of sampling need to be carefully considered in applying any econometric method.A unifying theme in this book is the analogy approach to estimation, as expositedby Goldberger (1991) and Manski (1988). [For nonlinear estimation methods withcross section data, Manski (1988) covers several of the topics included here in a morecompact format.] Loosely, the analogy principle states that an estimator is chosen tosolve the sample counterpart of a problem solved by the population parameter. Theanalogy approach is complemented nicely by asymptotic analysis, and that is the focushere.By focusing on asymptotic properties I do not mean to imply that small-sampleproperties of estimators and test statistics are unimportant. However, one typicallyfirst applies the analogy principle to devise a sensible estimator and then derives itsasymptotic properties. This approach serves as a relatively simple guide to doinginference, and it works well in large samples (and often in samples that are not solarge). Small-sample adjustments may improve performance, but such considerationsalmost always come after a large-sample analysis and are often done on a case-bycase basis.The book contains proofs or outlines the proofs of many assertions, focusing on therole played by the assumptions with economic content while downplaying or ignoringregularity conditions. The book is primarily written to give applied researchers a veryfirm understanding of why certain methods work and to give students the backgroundfor developing new methods. But many of the arguments used throughout the bookare representative of those made in modern econometric research (sometimes withoutthe technical details). Students interested in doing research in cross section or paneldata methodology will find much here that is not available in other graduate texts.I have also included several empirical examples with included data sets. Most ofthe data sets come from published work or are intended to mimic data sets used inmodern empirical analysis. To save space I illustrate only the most commonly usedmethods on the most common data structures. Not surprisingly, these overlap con-

Prefacexixsiderably with methods that are packaged in econometric software programs. Otherexamples are of models where, given access to the appropriate data set, one couldundertake an empirical analysis.The numerous end-of-chapter problems are an important component of the book.Some problems contain important points that are not fully described in the text;others cover new ideas that can be analyzed using the tools presented in the currentand previous chapters. Several of the problems require using the data sets that areincluded with the book.As with any book, the topics here are selective and reflect what I believe to be themethods needed most often by applied researchers. I also give coverage to topics thathave recently become important but are not adequately treated in other texts. Part Iof the book reviews some tools that are elusive in mainstream econometrics books—in particular, the notion of conditional expectations, linear projections, and variousconvergence results. Part II begins by applying these tools to the analysis of singleequation linear models using cross section data. In principle, much of this materialshould be review for students having taken a first-semester course. But starting withsingle-equation linear models provides a bridge from the classical analysis of linearmodels to a more modern treatment, and it is the simplest vehicle to illustrate theapplication of the tools in Part I. In addition, several methods that are used oftenin applications—but rarely covered adequately in texts—can be covered in a singleframework.I approach estimation of linear systems of equations with endogenous variablesfrom a di erent perspective than traditional treatments. Rather than begin with simultaneous equations models, we study estimation of a general linear system by instrumental variables. This approach allows us to later apply these results to modelswith the same statistical structure as simultaneous equations models, includingpanel data models. Importantly, we can study the generalized method of momentsestimator from the beginning and easily relate it to the more traditional three-stageleast squares estimator.The analysis of general estimation methods for nonlinear models in Part III beginswith a general treatment of asymptotic theory of estimators obtained from nonlinear optimization problems. Maximum likelihood, partial maximum likelihood,and generalized method of moments estimation are shown to be generally applicableestimation approaches. The method of nonlinear least squares is also covered as amethod for estimating models of conditional means.Part IV covers several nonlinear models used by modern applied researchers.Chapters 15 and 16 treat limited dependent variable models, with attention given to

xxPrefacehandling certain endogeneity problems in such models. Panel data methods for binaryresponse and censored variables, including some new estimation approaches, are alsocovered in these chapters.Chapter 17 contains a treatment of sample selection problems for both cross section and panel data, including some recent advances. The focus is on the case wherethe population model is linear, but some results are given for nonlinear models aswell. Attrition in panel data models is also covered, as are methods for dealing withstratified samples. Recent approaches to estimating average treatment e ects aretreated in Chapter 18.Poisson and related regression models, both for cross section and panel data, aretreated in Chapter 19. These rely heavily on the method of quasi-maximum likelihood estimation. A brief but modern treatment of duration models is provided inChapter 20.I have given short shrift to some important, albeit more advanced, topics. Thesetting here is, at least in modern parlance, essentially parametric. I have not includeddetailed treatment of recent advances in semiparametric or nonparametric analysis.In many cases these topics are not conceptually di‰cult. In fact, many semiparametricmethods focus primarily on estimating a finite dimensional parameter in the presenceof an infinite dimensional nuisance parameter—a feature shared by traditional parametric methods, such as nonlinear least squares and partial maximum likelihood.It is estimating infinite dimensional parameters that is conceptually and technicallychallenging.At the appropriate point, in lieu of treating semiparametric and nonparametricmethods, I mention when such extensions are possible, and I provide references. Abenefit of a modern approach to parametric models is that it provides a seamlesstransition to semiparametric and nonparametric methods. General surveys of semiparametric and nonparametric methods are available in Volume 4 of the Handbookof Econometrics—see Powell (1994) and Härdle and Linton (1994)—as well as inVolume 11 of the Handbook of Statistics—see Horowitz (1993) and Ullah and Vinod(1993).I only briefly treat simulation-based methods of estimation and inference. Computer simulations can be used to estimate complicated nonlinear models when traditional optimization methods are ine ective. The bootstrap method of inference andconfidence interval construction can improve on asymptotic analysis. Volume 4 ofthe Handbook of Econometrics and Volume 11 of the Handbook of Statistics containnice surveys of these topics (Hajivassilou and Ruud, 1994; Hall, 1994; Hajivassilou,1993; and Keane, 1993).

PrefacexxiOn an organizational note, I refer to sections throughout the book first by chapternumber followed by section number and, sometimes, subsection number. Therefore,Section 6.3 refers to Section 3 in Chapter 6, and Section 13.8.3 refers to Subsection 3of Section 8 in Chapter 13. By always including the chapter number, I hope tominimiz

10.2.3 Some Examples of Unobserved E ects Panel Data Models 254 10.3 Estimating Unobserved E ects Models by Pooled OLS 256 10.4 Random E ects Methods 257 10.4.1 Estimation and Inference under the Basic Random E ects Assumptions 257 10.4.2 Robust Variance Matrix Estimator 262 10.4.3 A General FGLS Analysis 263