Discrete Choice Methods with Simulation, Second Edition

This book describes the new generation of discrete choice methods, focusing on the many advances that are made possible by simulation. Researchers use these statistical methods to examine the choices that consumers, households, firms, and other agents make. Each of the major models is covered: logit, generalized extreme value, or GEV (including nested and cross-nested logits), probit, and mixed logit, plus a variety of specifications that build on these basics. Simulation-assisted estimation procedures are investigated and compared, including maximum stimulated likelihood, method of simulated moments, and method of simulated scores. Procedures for drawing from densities are described, including variance reduction techniques such as anithetics and Halton draws. Recent advances in Bayesian procedures are explored, including the use of the Metropolis-Hastings algorithm and its variant Gibbs sampling. The second edition adds chapters on endogeneity and expectation-maximization (EM) algorithms. No other book incorporates all these fields, which have arisen in the past 25 years. The procedures are applicable in many fields, including energy, transportation, environmental studies, health, labor, and marketing Cover 1 Half-title 3 Title 5 Copyright 6 Dedication 7 Contents 9 1 Introduction 13 1.1 Motivation 13 1.2 Choice Probabilities and Integration 15 1.2.1. Complete Closed-Form Expression 16 1.2.2. Complete Simulation 17 1.2.3. Partial Simulation, Partial Closed Form 18 1.3 Outline of Book 19 1.4 A Couple of Notes 20 Part I Behavioral Models 21 2 Properties of Discrete Choice Models 23 2.1 Overview 23 2.2 The Choice Set 23 2.3 Derivation of Choice Probabilities 26 2.4 Specific Models 29 2.5 Identification of Choice Models 31 2.5.1. Only Differences in Utility Matter 31 Alternative-Specific Constants 32 Sociodemographic Variables 33 Number of Independent Error Terms 34 2.5.2. The Overall Scale of Utility Is Irrelevant 35 Normalization with iid Errors 36 Normalization with Heteroskedastic Errors 37 Normalization with Correlated Errors 39 2.6 Aggregation 41 2.6.1. Sample Enumeration 43 2.6.2. Segmentation 43 2.7 Forecasting 44 2.8 Recalibration of Constants 45 3 Logit 46 3.1 Choice Probabilities 46 3.2 The Scale Parameter 52 3.3 Power and Limitations of Logit 54 3.3.1. Taste Variation 54 3.3.2. Substitution Patterns 57 The Property of Independence 57 Proportional Substitution 59 Advantages of IIA 60 Tests of IIA 61 3.3.3. Panel Data 62 3.4 Nonlinear Representative Utility 64 Example 1: The Goods–Leisure Tradeoff 65 Example 2: Geographic Aggregation 66 3.5 Consumer Surplus 67 3.6 Derivatives and Elasticities 69 3.7 Estimation 72 3.7.1. Exogenous Sample 72 Estimation on a Subset of Alternatives 76 3.7.2. Choice-Based Samples 78 3.8 Goodness of Fit and Hypothesis Testing 79 3.8.1. Goodness of Fit 80 3.8.2. Hypothesis Testing 82 3.9 Case Study: Forecasting for a New Transit System 83 3.10 Derivation of Logit Probabilities 86 4 GEV 88 4.1 Introduction 88 4.2 Nested Logit 89 4.2.1. Substitution Patterns 89 4.2.2. Choice Probabilities 91 4.2.3. Decomposition into Two Logits 93 4.2.4. Estimation 96 4.2.5. Equivalence of Nested Logit Formulas 98 4.3 Three-Level Nested Logit 98 4.4 Overlapping Nests 101 4.4.1. Paired Combinatorial Logit 102 4.4.2. Generalized Nested Logit 103 4.5 Heteroskedastic Logit 104 4.6 The GEV Family 105 5 Probit 109 5.1 Choice Probabilities 109 5.2 Identification 112 5.3 Taste Variation 118 5.4 Substitution Patterns and Failure of IIA 120 Full Covariance: Unrestricted Substitution Patterns 120 Structured Covariance: Restricted 121 Substitution Patterns 121 5.5 Panel Data 122 5.6 Simulation of the Choice Probabilities 126 5.6.1. Accept–Reject Simulator 127 5.6.2. Smoothed AR Simulators 132 5.6.3. GHK Simulator 134 Three Alternatives 135 General Model 138 GHK Simulator with Maximum Likelihood Estimation 141 GHK as Importance Sampling 143 6 Mixed Logit 146 6.1 Choice Probabilities 146 6.2 Random Coefficients 149 6.3 Error Components 151 6.4 Substitution Patterns 153 6.5 Approximation to Any Random Utility Model 153 6.6 Simulation 156 6.7 Panel Data 157 6.8 Case Study 159 7 Variations on a Theme 163 7.1 Introduction 163 7.2 Stated-Preference and Revealed-Preference Data 164 7.3 Ranked Data 168 7.3.1. Standard and Mixed Logit 168 7.3.2. Probit 170 7.4 Ordered Responses 171 7.4.1. Multiple Ordered Responses 175 7.5 Contingent Valuation 176 7.6 Mixed Models 178 7.6.1. Mixed Nested Logit 179 7.6.2. Mixed Probit 180 7.7 Dynamic Optimization 181 7.7.1. Two Periods, No Uncertainty about Future Effects 183 7.7.2. Multiple Periods 187 7.7.3. Uncertainty about Future Effects 190 Part II Estimation 195 8 Numerical Maximization 197 8.1 Motivation 197 8.2 Notation 197 8.3 Algorithms 199 8.3.1. Newton–Raphson 199 Quadratics 200 Step Size 201 Concavity 203 8.3.2. BHHH 204 8.3.3. BHHH-2 207 8.3.4. Steepest Ascent 208 8.3.5. DFP and BFGS 209 8.4 Convergence Criterion 210 8.5 Local versus Global Maximum 211 8.6 Variance of the Estimates 212 8.7 Information Identity 214 9 Drawing from Densities 217 9.1 Introduction 217 9.2 Random Draws 217 9.2.1. Standard Normal and Uniform 217 9.2.2. Transformations of Standard Normal 218 9.2.3. Inverse Cumulative for Univariate Densities 218 9.2.4. Truncated Univariate Densities 219 9.2.5. Choleski Transformation for Multivariate Normals 220 9.2.6. Accept–Reject for Truncated Multivariate Densities 221 9.2.7. Importance Sampling 222 9.2.8. Gibbs Sampling 224 9.2.9. Metropolis–Hastings Algorithm 225 9.3 Variance Reduction 226 9.3.1. Antithetics 228 9.3.2. Systematic Sampling 230 9.3.3. Halton Sequences 233 9.3.4. Randomized Halton Draws 243 9.3.5. Scrambled Halton Draws 245 9.3.6. Other Procedures 248 10 Simulation-Assisted Estimation 249 10.1 Motivation 249 10.2 Definition of Estimators 250 10.2.1. Maximum Simulated Likelihood 250 10.2.2. Method of Simulated Moments 252 10.2.3. Method of Simulated Scores 255 10.3 The Central Limit Theorem 257 10.4 Properties of Traditional Estimators 259 10.5 Properties of Simulation-Based Estimators 262 10.5.1. Maximum Simulated Likelihood 267 10.5.2. Method of Simulated Moments 268 10.5.3. Method of Simulated Scores 269 10.6 Numerical Solution 269 11 Individual-Level Parameters 271 11.1 Introduction 271 11.2 Derivation of Conditional Distribution 274 11.3 Implications of Estimation of 276 11.4 Monte Carlo Illustration 279 11.5 Average Conditional Distribution 281 11.6 Case Study: Choice of Energy Supplier 282 11.6.1. Population Distribution 282 11.6.2. Conditional Distributions 286 11.6.3. Conditional Probability for the Last Choice 290 11.7 Discussion 292 12 Bayesian Procedures 294 12.1 Introduction 294 12.2 Overview of Bayesian Concepts 296 12.2.1. Bayesian Properties of ? 298 12.2.2. Classical Properties of ?: The Bernstein–von Mises Theorem 300 12.3 Simulation of the Posterior Mean 303 12.4 Drawing from the Posterior 305 12.5 Posteriors for the Mean and Variance of a Normal Distribution 306 12.5.1. Result A: Unknown Mean, Known Variance 307 12.5.2. Result B: Unknown Variance, Known Mean 309 12.5.3. Unknown Mean and Variance 311 12.6 Hierarchical Bayes for Mixed Logit 311 12.6.1. Succinct Restatement 316 12.7 Case Study: Choice of Energy Supplier 317 12.7.1. Independent Normal Coefficients 317 12.7.2. Multivariate Normal Coefficients 319 12.7.3. Fixed Coefficients for Some Variables 320 12.7.4. Lognormals 322 12.7.5. Triangulars 323 12.7.6. Summary of Results 324 12.8 Bayesian Procedures for Probit Models 325 13 Endogeneity 327 13.1 Overview 327 13.2 The BLP Approach 330 13.2.1. Specification 331 13.2.2. The Contraction 334 13.2.3. Estimation by Maximum Simulated Likelihood and Instrumental Variables 336 13.2.4. Estimation by GMM 338 13.3 Supply Side 340 13.3.1. Marginal Cost 341 13.3.2. MC Pricing 342 MSL and IV with MC Pricing 342 GMM with MC Pricing 343 13.3.3. Fixed Markup over Marginal Cost 343 13.3.4. Monopoly Pricing and Nash Equilibrium for Single-Product Firms 344 13.3.5. Monopoly Pricing and Nash Equilibrium for Multiproduct Firms 345 13.4 Control Functions 346 13.4.1. Relation to Pricing Behavior 350 13.5 Maximum Likelihood Approach 352 13.6 Case Study: Consumers' Choice among New Vehicles 354 14 EM Algorithms 359 14.1 Introduction 359 14.2 General Procedure 360 14.2.1. Why the EM Algorithm Works 363 14.2.2. Convergence 366 14.2.3. Standard Errors 367 14.3 Examples of EM Algorithms 367 14.3.1. Discrete Mixing Distribution with Fixed Points 367 14.3.2. Discrete Mixing Distribution with Points as Parameters 371 14.3.3. Normal Mixing Distribution with Full Covariance 373 14.4 Case Study: Demand for Hydrogen Cars 377 Bibliography 383 Index 397 "This book describes the new generation of discrete choice methods, focusing on the many advances that are made possible by simulation. Researchers use these statistical methods to examine the choices that consumers, households, firms, and other agents make. Each of the major models is covered: logit, generalized extreme value, or GEV (including nested and cross-nested logits), probit, and mixed logit, plus a variety of specifications that build on these basics. Recent advances in Bayesian procedures are explored, including the use of the Metropolis-Hastings algorithm and its variant Gibbs sampling. This second edition adds chapters on endogeneity and expectation-maximization (EM) algorithms. No other book incorporates all these fields, which have arisen in the past 25 years. The procedures are applicable in many fields, including energy, transportation, environmental studies, health, labor, and marketing."--Pub. desc