Modeling Ordered Choices William H. Greene1 David A. Hensher2 January, 2009 1Department of Economics, Stern School of Business, New York University, New York, NY 10012, [email protected] 2Institute of Transport and Logistics Studies, Faculty of Economics and Business, University of Sydney, NSW 2006 Australia [email protected] Modeling Ordered Choices Brief Contents List of Tables List of Figures Preface Chapter 1 Introduction Chapter 2 Modeling Binary Choices Chapter 3 An Ordered Choice Model for Social Science Applications Chapter 4 Antecedents and Contemporary Counterparts Chapter 5 Estimation, Inference and Analysis Using the Ordered Choice Model Chapter 6 Specification Issues in Ordered Choice Models Chapter 7 Accommodating Individual Heterogeneity Chapter 8 Parameter Variation and a Generalized Ordered Choice Model Chapter 9 Ordered Choice Modeling with Panel and Time Series Data Chapter 10 Bivariate and Multivariate Ordered Choice Models Chapter 11 Two Part and Sample Selection Models Chapter 12 Semiparametric and Nonparametric Estimators and Analyses References Index 2 Modeling Ordered Choices Contents List of Tables List of Figures Preface Chapter 1 Introduction: Random Utility Models Chapter 2 Modeling Binary Choices 2.1 Random Utility Formulation of a Model for Binary Choice 2.2 Probability Models for Binary Choices 2.2.1 Nonparametric and Semiparametric Specifications 2.2.2 The Linear Probability Model 2.2.3 The Probit and Logit Models 2.3 Estimation and Inference 2.3.1 Maximum Likelihood Estimation 2.3.2 Maximizing the Log Likelihood Function 2.3.3 The EM Algorithm 2.3.4 Bayesian Estimation by Gibbs Sampling and MCMC 2.3.5 Estimation with Grouped Data and Iteratively Reweighted Least Squares 2.3.6 The Minimum Chi Squared Estimator 2.4 Covariance Matrix Estimation Robust Covariance Matrix Estimation 2.5 Application of the Binary Choice Model to Health Satisfaction 2.6 Partial Effects in a Binary Choice Model 2.6.1 Partial Effect for a Dummy Variable 2.6.2 Odds Ratios 2.6.3 Elasticities 2.6.4 Inference for Partial Effects 2.6.5 Standard Errors for Estimated Odds Ratios 2.6.6 Average Partial Effects 2.6.7 Standard Errors for Marginal Effects Using the Krinsky and Robb Method 2.6.8 Fitted Probabilities 2.7 Hypothesis Testing 2.7.1 Wald Tests 2.7.2 Likelihood Ratio Tests 2.7.3 Lagrange Mltiplier Tests 2.7.4 Application of Hypothesis Tests 2.8 Goodness of Fit Measures 2.8.1 Perfect Prediction 2.8.2 Dummy Variables with Empty Cells 2.8.3 Explaining Variation in the Implied Regression 2.8.4 Fit Measures Based on Predicted Probabilities 2.8.5 Assessing the Model’s Ability to Predict 2.8.6 A Specification Test Based on Fit 2.8.7 ROC Plots for Binary Choice Models 2.9 Heteroscedasticity 2.10 Panel Data 2.10.1 Pooled Estimation, Clustering and Robust Covariance Matrix Estimation 2.10.2 Fixed Effects 2.10.3 Random Effects 3 Modeling Ordered Choices The Pooled Estimator The Maximum Likelihood Estimator GMM Estimation Heckman ad Singer’s Semiparametric Approach 2.10.4 Mundlak’s Correction for the Probit and Logit Models 2.10.5 Testing for Heterogeneity 2.10.6 Testing for Fixed or Random Effects 2.11 Parameter Heterogeneity 2.12 Endogeneity of a Right Hand Side variable 2.13 Bivariate Probit Models 2.13.1 Tetrachoric Correlation 2.13.2 Testing for Zero Correlation 2.13.3 Marginal Effects in a Bivariate Probit Model 2.13.4 Recursive Bivariate Probit Models 2.13.5 A Sample Selection Model 2.14 The Multivariate Probit and Panel Probit Models 2.15 Endogenous Sampling and Case Control Studies Chapter 3 An Ordered Choice Model for Social Science Applications 3.1 A Latent Regression Model for a Continuous Measure 3.2 Ordered Choice as an Outcome of Utility Maximization 3.3 The Observed Discrete Outcome 3.4 Probabilities 3.5 Log Likelihood Function 3.6 Analysis of Data on Ordered Choices Chapter 4 Antecedents and Contemporary Counterparts 4.1 The Origin of Probit Analysis: Bliss (1934), Finney (1947) 4.2 Social Science Data and Regression Analysis for Binary Outcomes 4.3 Analysis of Binary Choice 4.4 Ordered Outcomes: Aitchison and Silvey (1957), Snell (1964) 4.5 Minimum Chi Squared Estimation of an Ordered Response Model: Gurland et al. (1960) 4.6 Individual Data and Polychotomous Outcomes: Walker and Duncan (1967) 4.7 McElvey and Zavoina (1975) 4.8 Developments Since McElvey and Zavoina 4.9 Other Related Models 4.9.1 Known Thresholds 4.9.2 Nonparallel Regressions Chapter 5 Estimation, Inference and Analysis Using the Ordered Choice Model 5.1 Application of the Ordered Choice Model to Self Assessed Health Status 5.2 Distributional Assumptions 5.3 The Estimated Ordered Probit (Logit) Model 5.4 The Estimated Threshold Parameters 5.5 Interpretation of the Model – Partial Effects and Scaled Coefficients 5.5.1 Nonlinearities in the Variables 5.5.2 Average Partial Effects 5.5.3 Interpreting the Threshold Parameters 5.5.4 The Underlying Regression 5.6 Inference 5.6.1 Inference about Coefficients 5.6.2 Testing for Structural Change or Homogeneity of Strata 5.6.3 Robust Covariance Matrix Estimation 4 Modeling Ordered Choices 5.6.4 Inference About Partial Effects 5.7 Prediction – Computing Probabilities 5.8 Measuring Fit 5.9 Estimation Issues 5.9.1 Grouped Data 5.9.2 Perfect Prediction 5.9.3 Different Normalizations 5.9.4 Censoring of the Dependent Variable 5.9.5 Maximum Likelihood Estimation of the Ordered Choice Model 5.9.6 Bayesian (MCMC) Estimation of Ordered Choice Models 5.9.7 Software For Estimation of Ordered Choice Models Chapter 6 Specification Issues in Ordered Choice Models 6.1 Functional Form Issues and the Generalized Ordered Choice Model (1) 6.1.1 Parallel Regressions 6.1.2 Testing the Parallel Regressions Assumption – The Brant (1990) Test 6.1.3 Generalized Ordered Logit Model (1) 6.2 Model Implications for Partial Effects 6.2.1 The Single Crossing Feature of the Ordered Choice Model 6.2,2 Choice Invariant Ratios of Partial Effects 6.3 Methodological Issues 6.4 Specification Tests for Ordered Choice Models 6.4.1 Model Specifications – Missing Variables and Heteroscedasticity 6.4.2 Testing Against the Logistic and Normal Distribution 6.4.3 Unspecified Alternatives Chapter 7 Accommodating Individual Heterogeneity 7.1 Threshold Models – The Generalized Ordered Probit Model (2) 7.2 Nonlinear Specifications – A Hierarchical Ordered Probit Model 7.3 Thresholds and Heterogeneity – Anchoring Vignettes 7.3.1 Using Anchoring Vignettes in the Ordered Probit Model Self Assessment Component Vignette Component 7.3.2 Log Likelihood and Model Identification Through the Anchoring Vignettes 7.3.3 Testing the Assumptions of the Model 7.3.4 Application 7.3.5 Multiple Self-Assessment Equations 7.4 Heterogeneous Scaling (Heteroscedasticity) of Random Utility 7.4 Individually Heterogeneous Marginal Utilities Appendix: Equivalence of the Vignette and HOPIT Models Chapter 8 Parameter Variation and a Generalized Ordered Choice Model 8.1 Random Parameters Models 8.1.1 Implied Heteroscedasticity 8.1.2 Maximum Simulated Likelihood Estimation 8.1.3 Conditional Mean Estimation in the Random Parameters Model 8.2 Latent Class and Finite Mixture Modeling 8.2.1 The Latent Class Ordered Choice Model 8.2.2 Estimation by Maximum Likelihood 8.2.3 The EM Algorithm 8.2.4 Estimating the Class Assignments 8.2.5 A Latent Class Model Extension 8.2.6 Application 8.2.7 Endogenous Class Assignment and A Generalized Ordered Choice Model 8.3 Generalized Ordered Choice Model with Random Thresholds (3) 5 Modeling Ordered Choices Chapter 9 Ordered Choice Modeling with Panel and Time Series Data 9.1 Ordered Choice Models with Fixed Effects 9.2 Ordered Choice Models with Random Effects 9.3 Testing for Random or Fixed Effects 9.4 Extending Parameter Heterogeneity Models to Ordered Choices 9.5 Dynamic Models Chapter 10 Bivariate and Multivariate Ordered Choice Models 10.1 Bivariate Ordered Probit Models 10.2 Polychoric Correlation 10.3 Semi-Ordered Bivariate Probit Model 10.4 Applications of the Bivariate Ordered Probit Model 10.5 A Panel Data Version of the Bivariate Ordered Probit Model 10.6 Trivariate and Multivariate Ordered Probit Models Chapter 11 Two Part and Sample Selection Models 11.1 Inflation Models 11.2 Sample Selection Models 11.2.1 A Sample Selected Ordered Probit Model 11.2.2 Models of Sample Selection with an Ordered Probit Selection Rule 11.2.3 A Sample Selected Bivariate Ordered Probit Model 11.3 An Ordered Probit Model with Endogenous Treatment Effects Chapter 12 Semiparametric and Nonparametric Estimators and Analyses 12.1 Heteroscedasticity 12.2 A Distribution Free Estimator with Unknown Heteroscedasticty 12.3 A Semi-nonparametric Approach 12.4 A Partially Linear Model 12.5 Semiparametric Analysis 12.6 A Nonparametric Duration Model 12.6.1 Unobserved Heterogeneity 12.6.2 Application References Index 6 Modeling Ordered Choices List of Tables 2.1 Data Used in Binary Choice Application 2.2 Estimated Probit and Logit Models 2.3 Alternative Estimated Standard Errors for the Probit Model 2.4 Partial Effects for Probit and Logit Models at Means of x 2.5 Marginal Effects and Average Partial Effects 2.6 Hypothesis Tests 2.7 Homogeneity Test 2.8 Fit Measures for Probit Model 2.9 Prediction Success for Probit Model 2.10 Success Measures for Predictions by Estimated Probit Model 2.11 Heteroscedastic Probit Model 2.12 Cluster Corrected Covariance Matrix (7293 Groups) 2.13 Fixed Effects Probit Model 2.14 Estimated Fixed Effects Logit Models 2.15 Estimated Random Effects Probit Models 2.16a Semiparametric Random Effects Probit Model 2.16b Estimated Parameters for 4 Class Latent Class Model 2.17 Random Effects Model with Mundlak Correction 2.18 Estimated Random Parameter Models 2.19 Estimated Partial Effects 2.20 Cross Tabulation of Healthy and Working 2.21 Estimated Bivariate Probit Model 2.22 Estimated Sample Selection Model 5.1 Estimated Ordered Choice Models: Probit and Logit 5.2 Estimated Partial Effects for Ordered Choice Models 5.3 Estimated Expanded Ordered Probit Model 5.4 Transformed Latent Regression Coefficients 5.5 Estimated Partial Effects with Asymptitic Standard Errors 5.6 Mean Predicted Probabilities by Kids 5.7 Predicted vs.