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Statistical Analysis of Discrete Probability Models Alternative Estimators and Sample Designs for Discrete Choice Analysis Charles F. Manski and Daniel McFadden Introduction The Likelihood of an Observation under Alternative Stratified Sampling Processes Estimation of the Choice Model Parameters Estimation with p and Q Both Known Estimation with p Known and Q Unknown Estimation with p Unknown and Q Known Estimation with p and Q Both Unknown Estimation in a General Stratified Sample Selection of a Sample Design and Estimation Method Conclusion Appendix : Consistency of the Estimators Appendix : Asymptotic Normality References Efficient Estimation of Discrete-Choice Models Stephen R. Cosslett Introduction Discrete Choice Models Stratified Sampling and Choice-Based Sampling Generalized Choice-Based Sample Sample with Known Aggregate Shares Aggregate Shares Estimated from an Auxiliary Sample Supplemented Sample General Considerations in Maximum Likelihood Estimation Notation for a General Choice-Based Sample The Likelihood Function for Choice-Based Samples .. XII xv xvii Contents Maximization of the Likelihood Asymptotic Properties of the Unconstrained Estimator Estimation of Aggregate Shares The Unconstrained Maximum Likelihood Estimator The Logit Model as a Special Case Estimation with Known Aggregate Shares Consistency of the Constraint Equations Asymptotic Properties with Known Aggregate Shares The Constrained Maximum Likelihood Estimator Estimation of the Logit Model with Known Aggregate Shares Estimation with Aggregate Shares Inferred from an Auxiliary Sample Asymptotic Variance of the Auxiliary Sample Estimator Special Cases of the Auxiliary Sample Estimator Estimation with a Supplementary Sample Comparison of Estimators and Sample Designs Appendix : Conditions on the Choice Probability Model Appendix : Derivation of Asymptotic Properties Appendix : Alternative Estimators for Generalized Choice-Based Samples with Known Aggregate Shares References Dynamic Discrete Probability Models Statistical Models for Discrete Panel Data James J. Heckman Introduction A Framework for Analyzing Dynamic Choice The General Model An Independent Trials Bernoulli Model A Random Effect Bernoulli Model and One-Factor Schemes A Fixed Effect Bernoulli Model Models with General Correlation in the Errors: The Concept of Heterogeneity Extended Models with Structural State Dependence A Renewal Model Contents vii A Model with Habit Persistence Computation in the General Model A Summary of Sections 3.2 through 3.11 Heterogeneity versus Structural State Dependence: An Application of the Preceding Models Testing for Heterogeneity versus State Dependence Analogies with Time-Series Models Examples of Models that Generate Structural State Dependence Summary and Conclusion Appendix : Factor Analytic Probit Models References 4 The Incidental Parameters Problem and the Problem of Initial Conditions in Estimating a Discrete Time-Discrete Data Stochastic Process James J. Heckman 4.1 Introduction 179 4.2 The Problem of Initial Conditions and Some Formal Solutions 181 4.3 Simpler Solutions and the Problem of Incidental Parameters 185 4.4 Some Monte Carlo Evidence 189 4.5 Conclusions 194 References 195 Structural Discrete Probability Models Derived from Theories of Choice Econometric Models of Probabilistic Choice Daniel McFadden Economic Man Discrete Choice Probabilistic Consumer Theory Probabilistic Choice Systems The Random Utility Maximization Hypothesis Stochastic Revealed Preference Contents Aggregation of Preferences The Williams-Daly-Zachary Theorem Criteria for Parametric Probabilistic Choice Systems Specification of Variables Functional Form The Luce Model Thurstone's Model V Tversky Elimination Models Generalized Extreme Value Models Preference Trees Estimation of Tree Extreme Value Models Sequential Estimation An Application Appendix: Normalization in MNL and MNP Models Appendix: Computational Formulas for a Simple Model Appendix : Computational Formulas for the Nested Multinornial Logit Model Appendix : Proof of Theorem 5.1 Appendix : The Elimination-by-Strategy Model References Random versus Fixed Coefficient Quantal Choice Models Gregory W. Fischer and Daniel Nagin Introduction Quantal Choice Theory and Variation in Tastes An Empirical Comparison of the LPIID and RCCD Models Details of the Experiment Results Analysis of Individual Respondents A Comparison of LPIID Probit and RCCD Probit Estimation Conclusions Appendix : The Unestimable Models Appendix: Mean Taste Estimates in the LPIID and RCCD Models References Contents On the Use of Simulated Frequencies to Approximate Choice Probabilities Steven R. Lerman and Charles F. Manski Introduction The Simulated Frequency Method Bayesian Approach Estimation of a Function of a Collection of Probabilities Application to the Calculation of Multinomial Probit Choice Probabilities The Simulation Routine The Clark Method Numerical Test Objectives and Design Test Results and Analysis Conclusions References Application of a Continuous Spatial Choice Logit Model Moshe Ben-Akiva and Thawat Watanatada Introduction Basic Definitions Spatial Aggregation The Discrete Logit Model Spatial Aggregation Using Continuous Functions The Continuous Logit Model A Parametric Example of Spatial Aggregation Continuous Logit with Featureless Plane Basic Operations of the MIT-TRANS Model References Simultaneous Equations Models with Discrete Endogenous Variables Simultaneous Equations Models with Discrete and Censored Variables Lung-Fei Lee Introduction Contents Two-Stage Methods and Amemiya's Principle Structural Equations with Probit Structure Structural Equations with Observable Continuous Endogenous Variables Structural Equations with Censored Dependent Variables Structural Equations with Tobit Structure Switching and Censored Models with Sample Separation Information Conclusion References Stratification on Endogenous Variables and Estimation: The Gary Income Maintenance Experiment Jerry A. Hausman and David A. Wise Introduction The Problem of Endogenous Sampling and Estimation Methods Relative Efficiencies of Weighted Least Squares versus Maximum Likelihood Estimates Empirical Results of the Selection Bias in the Gary Income Maintenance Experiment Alternative Sampling Procedures Conclusion Appendix: Extension of the Analysis to Two Time Periods and Two Equations References A Switching Simultaneous Equations Model of Physician Behavior in Ontario Dale J. Poirier Introduction Econometric Model Estimation Estimation of the Switching (Option) Equation Empirical Results Estimated Option Equation Estimated Referral Equation Contents 11.8 Concluding Remarks References 12 Constraints on the Parameters in Simultaneous Tobit and Probit Models Peter Schmidt 12.1 Introduction 12.2 Simultaneous Tobit Models 12.3 All Endogenous Variables Truncated 12.4 Some Endogenous Variables Truncated 12.5 Both Y and Y* as Explanatory Variables 12.6 Simultaneous Probit Models 12.7 All Endogenous Variables Truncated 12.8 Some Endogenous Variables Truncated 12.9 Both Y and Y* as Explanatory Variables 12.10 Conclusions References Estimating Credit Constraints by Switching Regressions Robert B. Avery Introduction The Supply of Debt The Model and Data Simultaneous Switching Regression and Linear Equations The Evidence Qualifications and Evaluations Appendix : Proof of Theorem 13.1 Appendix : Empirical Reduced Form Equations References Index