Students in both the natural and social sciences often seek regression models to explain the frequency of events, such as visits to a doctor, auto accidents or job hiring. This analysis provides the most comprehensive and up-to-date account of models and methods to interpret such data. The authors combine theory and practice to make sophisticated methods of analysis accessible to practitioners working with widely different types of data and software. The treatment will be useful to researchers in areas such as applied statistics, econometrics, operations research, actuarial studies, demography, biostatistics, and quantitatively-oriented sociology and political science. The book may be used as a reference work on count models or by students seeking an authoritative overview. The analysis is complemented by template programs available on the Internet through the authors' homepages.
Author(s): A. Colin Cameron, Pravin K. Trivedi
Edition: 1st
Year: 1998
Language: English
Pages: 436
Cover......Page 1
Half-title......Page 3
Series-title......Page 5
Title......Page 6
Copyright......Page 8
Dedication......Page 9
Contents......Page 11
List of Figures......Page 15
List of Tables......Page 16
Preface......Page 19
CHAPTER 1 Introduction......Page 23
1.1 Poisson Distribution......Page 25
1.1.2 Poisson Process......Page 27
1.1.3 Waiting Time Distributions......Page 28
1.2 Poisson Regression......Page 30
1.2.3 Regression Specification......Page 31
1.3 Examples......Page 32
1.3.4 Takeover Bids......Page 34
1.3.7 Credit Rating......Page 35
1.3.11 Manufacturing Defects......Page 36
1.4 Overview of Major Issues......Page 37
1.5 Bibliographic Notes......Page 39
2.1 Introduction......Page 41
2.2.1 Example......Page 42
2.2.2 Definitions......Page 43
2.3 Likelihood-Based Models......Page 44
2.3.1 Regularity Conditions......Page 45
2.3.2 Maximum Likelihood......Page 46
2.3.3 Profile Likelihood......Page 47
2.3.4 Misspecified Density......Page 48
2.4 Generalized Linear Models......Page 49
2.4.1 Weighted Linear Least Squares......Page 50
2.4.2 Linear Exponential Family Models......Page 51
2.4.3 LEF with Nuisance Parameter......Page 54
2.4.4 Generalized Linear Models......Page 55
2.4.5 Extensions......Page 58
2.5 Moment-Based Models......Page 59
2.5.1 Estimating Equations......Page 60
2.5.2 Generalized Methods of Moments......Page 61
2.5.3 Optimal GMM......Page 64
2.5.4 Sequential Two-Step Estimators......Page 65
2.6.1 Likelihood-Based Models......Page 66
2.6.2 General Models......Page 68
2.6.3 Conditional Moment Tests......Page 69
2.7 Derivations......Page 72
2.7.1 General Framework......Page 73
2.7.3 Generalized Linear Models......Page 75
2.7.4 Moment-Based Models......Page 76
2.7.5 Conditional Moment Tests......Page 77
2.9 Exercises......Page 79
3.1 Introduction......Page 81
3.2.1 Poisson MLE......Page 83
3.2.2 NB1 and NB2 Variance Functions......Page 84
3.2.3 Poisson PMLE......Page 85
Poisson PMLE with NB1 Variance Function......Page 86
Poisson PMLE with Unspecified Variance Function......Page 87
3.2.4 Poisson GLM......Page 88
3.2.6 Example: Doctor Visits......Page 89
3.3 Negative Binomial MLE and QGPMLE......Page 92
3.3.1 NB2 Model and MLE......Page 93
3.3.3 NB1 Model and MLE......Page 95
3.3.4 Discussion......Page 96
3.4 Overdispersion Tests......Page 99
3.5 Use of Regression Results......Page 101
3.5.1 Interpretation of Coefficients......Page 102
3.5.2 Prediction......Page 106
3.6 Ordered and Other Discrete-Choice Models......Page 107
3.6.1 Binary-Choice Models......Page 108
3.6.2 Ordered Probit......Page 109
3.7 Other Models......Page 110
3.7.2 OLS with Transformation......Page 111
3.7.3 Nonlinear Least Squares......Page 112
3.8 Iteratively Reweighted Least Squares......Page 115
3.9 Bibliographic Notes......Page 116
3.10 Exercises......Page 117
4.1 Introduction......Page 118
4.2 Mixture Models for Unobserved Heterogeneity......Page 119
4.2.1 Unobserved Heterogeneity and Overdispersion......Page 120
4.2.2 Negative Binomial Model......Page 122
4.2.3 Other Characterizations of NB......Page 124
4.2.4 General Mixture Results......Page 125
4.2.5 Identification......Page 126
4.2.6 Consequences of Misspecified Heterogeneity......Page 127
4.3.1 True and Apparent Contagion......Page 128
4.3.3 Waiting-Time Distribution......Page 129
4.3.4 Gamma Waiting Times......Page 131
4.3.5 Dependence and Dispersion......Page 133
4.4.1 The Katz System......Page 134
4.4.2 Example: Doctor Visits......Page 135
4.4.3 Double Poisson Model......Page 136
4.4.5 Consul’s Generalized Poisson......Page 138
4.5.1 Standard Truncated Models......Page 139
4.5.2 Maximum Likelihood Estimation......Page 142
4.6 Censored Counts......Page 143
4.7.1 With Zeros and Hurdle Models......Page 145
4.7.2 Zero-Inflated Counts......Page 147
4.7.3 Example: Hurdles and Two-Part Decisionmaking......Page 149
4.8.1 Finite Mixtures......Page 150
4.8.3 Latent Class Analysis......Page 152
4.8.4 Estimation......Page 153
4.8.5 Inference on C......Page 154
4.9 Estimation by Simulation......Page 156
4.10 Derivations......Page 157
4.11 Bibliographic Notes......Page 158
4.12 Exercises......Page 159
5.1 Introduction......Page 161
5.2 Residual Analysis......Page 162
5.2.1 Pearson, Deviance, and Anscombe Residuals......Page 163
5.2.2 Generalized Residuals......Page 164
5.2.3 Using Residuals......Page 166
5.2.4 Small Sample Corrections and Influential Observations......Page 167
5.2.5 Example: Takeover Bids......Page 168
5.3.1 Pearson Statistic......Page 173
5.3.2 Deviance Statistic......Page 174
5.3.3 Pseudo R-Squared Measures......Page 175
5.3.4 Chi-Square Goodness of Fit Test......Page 177
5.4 Hypothesis Tests......Page 180
5.4.1 LM Test for Overdispersion against Katz System......Page 181
5.4.2 Auxiliary Regressions for LM Test......Page 182
5.4.3 LM Test against Local Alternatives......Page 184
5.5 Inference with Finite Sample Corrections......Page 185
5.5.1 Bootstrap......Page 186
5.5.2 Other Corrections......Page 189
5.6 Conditional Moment Specification Tests......Page 190
5.6.1 Introduction......Page 191
5.6.2 Regression-Based Tests for Overdispersion......Page 192
5.6.3 Regression-Based CM Tests......Page 196
5.6.4 Orthogonal Polynomial Tests......Page 198
5.6.5 Information Matrix Tests......Page 200
5.6.6 Hausman Tests......Page 202
5.7.1 Information Criteria......Page 204
5.7.2 Tests of Nonnested Models......Page 205
5.8.1 Test of Poisson Against Katz System......Page 207
5.8.2 LM Test Against Local Alternatives......Page 208
5.9 Bibliographic Notes......Page 209
5.10 Exercises......Page 210
6.1 Introduction......Page 211
6.2.1 Fully Parametric Estimation......Page 212
6.2.2 Repeated Events......Page 213
6.3.1 Health Service Data......Page 214
6.3.2 Demand for Medical Care......Page 215
6.3.3 Competing Models......Page 217
6.3.4 Is There a Mixture?......Page 218
6.3.5 Model Comparison and Selection......Page 219
6.3.6 Evaluation of Fitted Models......Page 221
6.3.7 Assessing the Preferred Model......Page 223
6.3.8 Interpreting the Coefficients......Page 225
6.3.9 Economic Significance......Page 227
6.4.1 Recreational Trips Data......Page 229
6.4.2 Initial Specifications......Page 230
6.4.3 Modified Poisson Models......Page 233
Simulation Design......Page 238
Simulation Outcomes......Page 239
6.6 Concluding Remarks......Page 240
6.7 Bibliographic Notes......Page 241
6.8 Exercises......Page 242
7.1 Introduction......Page 243
7.2.1 Linear Models......Page 244
7.2.2 Count Models......Page 246
7.3.1 Estimation......Page 248
7.3.2 Tests of Serial Correlation......Page 249
7.3.4 Example: Strikes......Page 252
7.4.1 Pure Time Series Models......Page 256
7.4.2 Regression Models......Page 259
7.5 Autoregressive Models......Page 260
7.6 Serially Correlated Error Models......Page 262
7.7 State-Space Models......Page 264
7.7.1 Conjugate Distributed Mean......Page 265
7.8 Hidden Markov Models......Page 266
7.9 Discrete ARMA Models......Page 267
7.10 Application......Page 268
7.11 Derivations......Page 270
7.13 Exercises......Page 272
8.1 Introduction......Page 273
8.2.1 Some Approaches......Page 274
8.2.2 Example......Page 276
8.3.1 Bivariate Poisson......Page 278
8.3.2 Other Fully Parametric Models......Page 280
8.4.1 Bivariate Poisson with Heterogeneity......Page 282
8.4.2 Seemingly Unrelated Regressions......Page 284
8.5.1 Definitions and Conditions......Page 285
8.5.2 Univariate Expansion......Page 286
8.5.3 Multivariate Expansions......Page 287
8.5.4 Tests of Independence......Page 288
8.5.5 Example: Medical Services......Page 290
8.6.1 Discrete Choice and Counts......Page 291
8.6.2 Counts and Continuous Variables......Page 293
8.7 Derivations......Page 294
8.8 Bibliographic Notes......Page 295
9.1 Introduction......Page 297
9.2 Models for Longitudinal Data......Page 298
9.2.1 Linear Models......Page 299
9.2.2 Count Models......Page 301
9.3.1 Maximum Likelihood......Page 302
9.3.2 Conditional Maximum Likelihood......Page 304
9.3.3 Moment-Based Methods......Page 306
9.3.4 Example: Patents......Page 308
9.4.1 Conjugate-Distributed Random Effects......Page 309
9.4.3 Moment-Based Methods......Page 311
9.5 Discussion......Page 312
9.6.2 Tests for Serial Correlation......Page 315
9.7.1 Some Approaches......Page 316
9.7.2 Fixed Effects Models......Page 317
9.8.2 Density for Poisson with Gamma Random Effects......Page 321
9.10 Exercises......Page 322
10.1 Introduction......Page 323
10.2 Measurement Errors in Exposure......Page 324
10.2.1 Correctly Observed Exposure......Page 325
10.2.2 Multiplicative Error in Exposure......Page 326
10.2.3 Proxy Variables for Exposure......Page 328
10.3.2 Multiplicative Error in Regressors......Page 329
10.4.1 Additive Measurement Errors in Counts......Page 331
10.4.2 Misclassified Counts......Page 332
10.4.3 Outlying Counts......Page 334
10.5.1 Mechanism and Examples......Page 335
10.5.2 Dependence Between Events and Recording......Page 336
Poisson Distribution......Page 338
Poisson Case......Page 339
Example: Safety Violations......Page 340
10.5.5 Underreported-Count Regressions under Dependence......Page 341
10.5.6 MLE MLE under Independence......Page 342
10.5.8 Example: Self-Reported Doctor Visits......Page 344
10.6 Derivations......Page 345
10.7 Bibliographic Notes......Page 346
10.8 Exercises......Page 347
11.2 Alternative Sampling Frames......Page 348
11.2.3 Endogenous or Choice-Based Sampling......Page 349
11.2.4 Counts with Endogenous Stratification......Page 351
11.3 Simultaneity......Page 353
11.3.2 Additive Errors......Page 354
11.3.3 Multiplicative Errors......Page 356
11.4 Sample Selection......Page 358
11.4.1 Normal Linear Case......Page 359
11.4.2 Selection Effect in a Count Model......Page 360
11.4.4 Two-Part Models......Page 364
11.5 Bibliographic Notes......Page 365
12.1 Introduction......Page 366
12.2.1 Estimating Equations and Quasilikelihood......Page 367
12.2.2 Generalized Method of Moments......Page 370
12.3.1 Seminonparametric Maximum Likelihood......Page 372
12.3.2 General Results......Page 373
12.3.3 Poisson Polynomial Model......Page 375
12.3.4 Modified Power Series Distributions......Page 377
12.4 Flexible Models of Conditional Mean......Page 378
12.5.1 Mixture Models for Unobserved Heterogeneity......Page 380
12.5.2 Series Expansions for Unobserved Heterogeneity......Page 381
12.5.3 Nonparametric Estimation of Variance......Page 384
12.6 Example and Model Comparison......Page 386
12.8 Count Models: Retrospect and Prospect......Page 389
12.9 Bibliographic Notes......Page 391
APPENDIX A Notation and Acronyms......Page 393
B.1 Gamma Function......Page 396
B.2.3 Negative Binomial......Page 397
B.3 Moments of Truncated Poisson......Page 398
APPENDIX C Software......Page 400
References......Page 401
Author Index......Page 421
Subject Index......Page 426
