Econometric Analysis of Count Data

Transcription

1 Econometric Analysis of Count Data

2 Springer-Verlag Berlin Heidelberg GmbH

3 Rainer Winkelmann Econometric Analysis of Count Data Third, Revised and Enlarged Edition With l3 Figures and 20 Tables, Springer

4 Dr. Rainer Winkelmann Institute for the Study of Labor (IZA) P.O. Box Bonn Germany ISBN ISBN (ebook) DOl Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Winkelmann, Rainer: Econometric analysis of count data: with 200 tables 1 Rainer Winkelmann. - 3., rev. and en!. ed. - Berlin; Heidelberg; New York; Barcelona; Hong Kong; London; Milan; Paris; Singapore; Tokyo: Springer, 2000 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg 1994, 1997, 2000 Originally published by Springer-Verlag Berlin Heidelberg New York in 2000 Softcover reprint of the hardcover 3rd edition 2000 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. SPIN Printed on acid-free paper

5 To S.D. and Z.

6 Preface The primary objective of this book is to provide an introduction to the econometric modeling of count data for graduate students and researchers. It should serve anyone whose interest lies either in developing the field further, or in applying existing methods to empirical questions. Much of the material included in this book is not specific to economics, or to quantitative social sciences more generally, but rather extends to disciplines such as biometrics and technometrics. Applications are as diverse as the number of congressional budget vetoes, the number of children in a household, and the number of mechanical defects in a production line. The unifying theme is a focus on regression models in which a dependent count variable is modeled as a function of independent variables which mayor may not be counts as well. The modeling of count data has come of age. Inclusion of some of the fundamental models in basic textbooks, and implementation on standard computer software programs bear witness to that. Based on the standard Poisson regression model, numerous extensions and alternatives have been developed to address the common challenges faced in empirical modeling (unobserved heterogeneity, selectivity, endogeneity, measurement error, and dependent observations in the context of panel data or multivariate data, to name but a few) as well as the challenges that are specific to count data (e.g., over dispersion and underdispersion). Parallely, the literature has engaged in an ongoing argument on the relative merits of fully parametric modeling relative to semi-parametric modeling based on weaker identifying assumptions. Much progress has been made on all of these issues over the last few years which justifies a new edition after the two previous editions of 1994 and A substantial amount of new material is added. One indicator is the number of referenced articles and books that has increased by about 50 percent. The new material includes methodological advances as well as an extended application section with examples from accident research, health studies, demography and marketing. In addition, part ofthe existing material

7 VIII Preface has been reorganized, errors have been corrected, and notation has been adjusted, in an effort to improve on exposition and readability. Many of these improvements are due to comments and suggestions that I received during the process of revising the book. In particular, I would like to thank Kurt Briinniis, Malcolm Faddy, Bill Greene, Nikolai Kolev, Jochen Mayer, Daniel Miles, Andreas Million, Hans van Ophem, Joao Santos Silva and Frank Windmeijer for very helpful comments. My own research in this area has benefited at various stages from discussions or collaborations with Tim Barmby, Siddharta Chib, Ed Greenberg, James Heckman, Robert Jung, Gary King and Pravin Trivedi. Finally lowe a special debt to Klaus F. Zimmermann who inititated my interest in count data models and provided continuing advice and support.

8 Table of Contents Preface... VII 1. Introduction Poisson Regression Model Some Thoughts on Methodology Examples Organization of the Book Probability Models for Count Data Introduction Poisson Distribution Definitions and Properties Genesis of the Poisson Distribution Poisson Process Generalizations of the Poisson Process Poisson Distribution as a Binomial Limit Exponential Interarrival Times Non-Poissonness Further Distributions for Count Data Negative Binomial Distribution Binomial Distribution Logarithmic Distribution Summary Modified Count Data Distributions Truncation Censoring and Grouping Altered Distributions Generalizations Mixture Distributions Compound Distributions Birth Process Generalizations

9 X Table of Contents Katz Family of Distributions Linear Exponential Families Additive Log-Differenced Probability Models Summary Duration Analysis and Count Data Distributions for Interarrival Times Renewal Processes Duration Dependence and Dispersion Gamma Count Distribution Duration Mixture Models Econollletric Modeling - Basic Issues Poisson Regression Model Specification of the Poisson Regression Model Ordinary Least Squares and Other Alternatives Interpretation of the Parameters Period at Risk Estimation Maximum Likelihood Properties of Maximum Likelihood Estimators Pseudo-Maximum Likelihood Generalized Method of Moments Generalized Linear Model Bias Reduction Techniques Sources of Misspecification Mean Function Unobserved Heterogeneity Dependent Process Selectivity Simultaneity and Endogeneity Underreporting Excess Zeros Variance Function Testing for Misspecification Classical Specification Tests Regression Based Tests Goodness-of-Fit Tests Hypothesis Testing for Non-Nested Models

10 Table of Contents XI 4. Econometric Modeling - Extensions Introduction Unobserved Heterogeneity Parametric Mixture Models Negative Binomial Model Semiparametric Mixture Models Dependent Count Process Exogenous Selectivity Truncated Count Data Models Censored Count Data Models Hurdle Count Data Models Zero-Inflated Count Data Models Underreporting Count Amount Model Endogenous Selectivity Endogenous Censoring Endogenous Truncation Endogenous Reporting Endogenous Regressors Instrumental Variable Estimation Switching Regression Simultaneous Equations Generalized Variance Models Generalized Event Count Model Generalized Poisson Regression Robust Poisson Regression Non-Parametric Variance Estimation Correlated Count Data Multivariate Count Data Multivariate Poisson Distribution Multivariate Negative Binomial Model Multivariate Poisson-Gamma Mixture Model Multivariate Poisson-log-normal Model Latent Poisson-Normal Model Moment-Based Methods Panel Data Models Fixed Effects Poisson Model Fixed Effects Negative Binomial Model Random Effects Count Data Models Moment-Based Methods

11 XII Table of Contents Dynamic Panel Count Data Models Time-Series Count Data Models Bayesian Analysis of Count Variables Bayesian Analysis of the Poisson Model A Poisson Model with Underreporting Estimation of the Multivariate Poisson-log-normal Model by MCMC Estimation of a Random Coefficients Model by MCMC Applications Accidents Health Economics Demography Marketing and Management Labor Mobility Conceptual Issues The Economics of Labor Mobility Previous Literature Data and Descriptive Statistics Regression Results Specification Analysis Two Types of Labor Mobility A. Probability Generating Functions..... " B. Gauss-Hermite Quadrature C. Tables References Index

12 List of Figures 2.1 Count Data Distributions (E(X) = 3.5) Negative Binomial Distributions Probability functions for gamma count and Poisson distributions; a = 0.5 (overdispersion): heavily shaded, Gamma count; lightly shaded, Poisson Probability functions for gamma count and Poisson distributions; a = 1.5 (underdispersion): heavily shaded, Gamma count; lightly shaded, Poisson Variance to mean ratio for gamma count distribution; 0 < a < Variance to mean ratio for gamma count distribution; a > Bias in the log-linear model when a constant is added in order to deal with zero counts Mean and variance of exponential blockage model for 0.1 < A < Variance-Mean Relationships for different k's Simulation of d under Fa SimulationofdunderGg Kennan's Strike Data Simulated INAR(I) Time Series for a =

13 List of Tables 1.1 Count Data Frequency Distributions Distributions for Count Data Sub-Models of the Katz System Linear Exponential Families Bias Reduced Poisson Estimates Simulation Study for Poisson-PMLE: N= Simulation Study for Poisson-PMLE: N= Frequency of Direct Changes and Unemployment Mobility Rates by Exogeneous Variables Direct Job Changes: Comparison of Results Predictions from Poisson Regression Predictions from GECk and Hurdle-Poisson Direct Job Changes: Log likelihoods B.1 Abcissas and Weight Factors for 20-point Gauss-Hermite Integration C.1 Direct Job Changes: Poisson and GECk Results C.2 Direct Job Changes: Compound Poisson Models C.3 Direct Job Changes: Robust Poisson Regression C.4 Direct Job Changes: Hurdle Poisson Model and Ordered Probit C.5 Direct Job Changes: Poisson-Logistic Regression C.6 Unemployment: Poisson and GECk Results