Econometric Analysis of Count Data (eBook)

Preface 5
Contents 7
List of Figures 13
List of Tables 15
1 Introduction 16
1.1 Poisson Regression Model 16
1.2 Examples 17
1.3 Organization of the Book 19
2 Probability Models for Count Data 22
2.1 Introduction 22
2.2 Poisson Distribution 22
2.2.1 De.nitions and Properties 22
2.2.2 Genesis of the Poisson Distribution 25
2.2.3 Poisson Process 26
2.2.4 Generalizations of the Poisson Process 29
2.2.5 Poisson Distribution as a Binomial Limit 30
2.2.6 Exponential Interarrival Times 31
2.2.7 Non-Poissonness 32
2.3 Further Distributions for Count Data 35
2.3.1 Negative Binomial Distribution 35
2.3.2 Binomial Distribution 40
2.3.3 Logarithmic Distribution 42
2.3.4 Summary 43
2.4 Modified Count Data Distributions 45
2.4.1 Truncation 45
2.4.2 Censoring and Grouping 46
2.4.3 Altered Distributions 47
2.5 Generalizations 48
2.5.1 Mixture Distributions 48
2.5.2 Compound Distributions 51
2.5.3 Birth Process Generalizations 54
2.5.4 Katz Family of Distributions 55
2.5.5 Additive Log-Di.erenced Probability Models 56
2.5.6 Linear Exponential Families 57
2.5.7 Summary 59
2.6 Distributions for Over- and Underdispersion 60
2.6.1 Generalized Event Count Model 60
2.6.2 Generalized Poisson Distribution 61
2.6.3 Poisson Polynomial Distribution 62
2.6.4 Double Poisson Distribution 64
2.6.5 Summary 64
2.7 Duration Analysis and Count Data 65
2.7.1 Distributions for Interarrival Times 67
2.7.2 Renewal Processes 69
2.7.3 Gamma Count Distribution 71
2.7.4 Duration Mixture Models 74
3 Poisson Regression 78
3.1 Specification 78
3.1.1 Introduction 78
3.1.2 Assumptions of the Poisson Regression Model 78
3.1.3 Ordinary Least Squares and Other Alternatives 80
3.1.4 Interpretation of Parameters 85
3.1.5 Period at Risk 89
3.2 Maximum Likelihood Estimation 92
3.2.1 Introduction 92
3.2.2 Likelihood Function and Maximization 92
3.2.3 Newton-Raphson Algorithm 93
3.2.4 Properties of the Maximum Likelihood Estimator 95
3.2.5 Estimation of the Variance Matrix 97
3.2.6 Approximate Distribution of the Poisson Regression Coefficients 98
3.2.7 Bias Reduction Techniques 99
3.3 Pseudo-Maximum Likelihood 102
3.3.1 Linear Exponential Families 104
3.3.2 Biased Poisson Maximum Likelihood Inference 105
3.3.3 Robust Poisson Regression 106
3.3.4 Non-Parametric Variance Estimation 110
3.3.5 Poisson Regression and Log-Linear Models 112
3.3.6 Generalized Method of Moments 113
Application to the Poisson Model 116
3.4 Sources of Misspecification 117
3.4.1 Mean Function 117
3.4.2 Unobserved Heterogeneity 118
3.4.3 Measurement Error 120
3.4.4 Dependent Process 122
3.4.5 Selectivity 122
3.4.6 Simultaneity and Endogeneity 123
3.4.7 Underreporting 124
3.4.8 Excess Zeros 124
3.4.9 Variance Function 125
3.5 Testing for Misspecification 127
3.5.1 Classical Speci.cation Tests 127
3.5.2 Regression Based Tests 133
3.5.3 Goodness-of-Fit Tests 133
3.5.4 Tests for Non-Nested Models 135
3.6 Outlook 140
4 Unobserved Heterogeneity 142
4.1 Introduction 142
4.1.1 Conditional Mean Function 142
4.1.2 Partial E.ects with Unobserved Heterogeneity 143
4.1.3 Unobserved Heterogeneity in the Poisson Model 144
4.1.4 Parametric and Semi-Parametric Models 145
4.2 Parametric Mixture Models 145
4.2.1 Gamma Mixture 146
4.2.2 Inverse Gaussian Mixture 146
4.2.3 Log-Normal Mixture 147
4.3 Negative Binomial Models 149
4.3.1 Negbin II Model 150
4.3.2 Negbin I Model 151
4.3.3 Negbink Model 151
4.3.4 NegbinX Model 152
4.4 Semiparametric Mixture Models 153
4.4.1 Series Expansions 153
4.4.2 Finite Mixture Models 154
5 Sample Selection and Endogeneity 158
5.1 Censoring and Truncation 158
5.1.1 Truncated Count Data Models 159
5.1.2 Endogenous Sampling 159
5.1.3 Censored Count Data Models 161
5.1.4 Grouped Poisson Regression Model 162
5.2 Incidental Censoring and Truncation 163
5.2.1 Outcome and Selection Model 163
5.2.2 Models of Non-Random Selection 164
5.2.3 Bivariate Normal Error Distribution 165
5.2.4 Outcome Distribution 167
5.2.5 Incidental Censoring 168
5.2.6 Incidental Truncation 169
5.3 Endogeneity in Count Data Models 171
5.3.1 Introduction and Examples 171
5.3.2 Parameter Ancillarity 172
5.3.3 Endogeneity and Mean Function 174
5.3.4 A Two-Equation Framework 176
5.3.5 Instrumental Variable Estimation 177
5.3.6 Estimation in Stages 180
5.4 Switching Regression 182
5.4.1 Full Information Maximum Likelihood Estimation 183
5.4.2 Moment-Based Estimation 185
5.4.3 Non-Normality 186
5.5 Mixed Discrete-Continuous Models 186
6 Zeros in Count Data Models 188
6.1 Introduction 188
6.2 Zeros in the Poisson Model 189
6.2.1 Excess Zeros and Overdispersion 189
6.2.2 Two-Crossings Theorem 190
6.2.3 Effects at the Extensive Margin 191
6.2.4 Multi-Index Models 192
6.2.5 A General Decomposition Result 192
6.3 Hurdle Count Data Models 193
6.3.1 Hurdle Poisson Model 196
6.3.2 Marginal E.ects 197
6.3.3 Hurdle Negative Binomial Model 198
6.3.4 Non-nested Hurdle Models 198
6.3.5 Unobserved Heterogeneity in Hurdle Models 200
6.3.6 Finite Mixture Versus Hurdle Models 201
6.3.7 Correlated Hurdle Models 202
6.4 Zero-Inflated Count Data Models 203
6.4.1 Introduction 203
6.4.2 Zero-Inflated Poisson Model 204
6.4.3 Zero-Inflated Negative Binomial Model 206
6.4.4 Marginal Effets 206
6.5 Compound Count Data Models 207
6.5.1 Multi-Episode Models 208
6.5.2 Underreporting 208
6.5.3 Count Amount Model 211
6.5.4 Endogenous Underreporting 212
6.6 Quantile Regression for Count Data 214
7 Correlated Count Data 218
7.1 Multivariate Count Data 218
7.1.1 Multivariate Poisson Distribution 220
7.1.2 Multivariate Negative Binomial Model 225
7.1.3 Multivariate Poisson-Gamma Mixture Model 227
7.1.4 Multivariate Poisson-Log-Normal Model 228
7.1.5 Latent Poisson-Normal Model 231
7.1.6 Moment-Based Methods 232
7.1.7 Copula Functions 234
7.2 Panel Data Models 235
7.2.1 Fixed Effects Poisson Model 237
7.2.2 Moment-based Estimation of the Fixed Effects Model 240
7.2.3 Fixed Effects Negative Binomial Model 242
7.2.4 Random Effects Count Data Models 243
7.2.5 Dynamic Panel Count Data Models 245
7.3 Time-Series Count Data Models 247
8 Bayesian Analysis of Count Data 256
8.1 Bayesian Analysis of the Poisson Model 257
8.2 A Poisson Model with Underreporting 260
8.3 Estimation of the Multivariate Poisson-Log-Normal Model by MCMC 262
8.4 Estimation of a Random Coefficients Model by MCMC 263
9 Applications 266
9.1 Accidents 266
9.2 Crime 267
9.3 Trip Frequency 267
9.4 Health Economics 269
9.5 Demography 272
9.6 Marketing and Management 275
9.7 Labor Mobility 276
9.7.1 Economics Models of Labor Mobility 277
9.7.2 Previous Literature 278
9.7.3 Data and Descriptive Statistics 280
direct 280
job changes 280
unemployment spells 280
direct job change 280
unemployment spell 280
unemployment spells 280
direct job changes. 281
direct job changes 281
unemployment 281
spells 281
direct job changes 281
unemployment spells, 281
direct job changes 281
unemployment 281
spells 281
Education 281
experience. 281
Quali.ed White Collar 281
Ordinary White Collar 281
Quali.ed Blue Collar 281
Ordinary Blue Collar 281
German 281
Single 281
Union 281
direct job changes 281
unemployment 281
spells) 281
experience. 281
direct 282
job changes, 282
unemployment spells 282
education 283
direct job 283
changes. 283
unemployment spells, 283
direct job changes 283
unemployment spells 283
9.7.4 Regression Results 284
number of job changes. 284
education, experience, squared experience, union, 284
single, German, quali.ed white collar, ordinary white collar, 284
quali.ed blue collar 284
number of job changes 284
education 285
direct job changes, 285
9.7.5 Model Performance 287
9.7.6 Marginal Probability E.ects 289
9.7.7 Structural Inferences 293
A Probability Generating Functions 296
B Gauss-Hermite Quadrature 300
C Software 304
D Tables 306
References 314
Author’s Index 336
Subject Index 342