Econometric analysis of count data+

Econometric Analysis of Count Data Rainer Winkelmann Econometric Analysis of Count Data Fifth edition 123 Prof Dr Rainer Winkelmann University of Zurich Socioeconomic Institute Zürichbergstr 14 8032 Zürich Switzerland winkelmann@sts.uzh.ch ISBN 978-3-540-77648-2 e-ISBN 978-3-540-78389-3 DOI 10.1007/978-3-540-78389-3 Library of Congress Control Number: 2008922297 c 2008 Springer-Verlag Berlin Heidelberg 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 Violations are liable to prosecution under the German Copyright Law 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 Production: le-tex Jelonek, Schmidt & Vöckler GbR, Leipzig Cover design: WMX Design GmbH, Heidelberg Printed on acid-free paper 987654321 springer.com Preface The “count data” field has further flourished since the previous edition of this book was published in 2003 The development of new methods has not slowed down by any means, and the application of existing ones in applied work has expanded in many areas of social science research This, in itself, would be reason enough for updating the material in this book, to ensure that it continues to provide a fair representation of the current state of research In addition, however, I have seized the opportunity to undertake some major changes to the organization of the book itself The core material on cross-section models for count data is now presented in four chapters, rather than in two as previously The first of these four chapters introduces the Poisson regression model, and its estimation by maximum likelihood or pseudo maximum likelihood The second focuses on unobserved heterogeneity, the third on endogeneity and non-random sample selection The fourth chapter provides an extended and unified discussion of zeros in count data models This topic deserves, in my view, special emphasis, as it relates to aspects of modeling and estimation that are specific to counts, as opposed to general exponential regression models for non-negative dependent variables Count distributions put positive probability mass on single outcomes, and thus offer a richer set of interesting inferences “Marginal probability effects” for zeros – at the “extensive margin” – as well as for any positive outcome – at the “intensive margin” – can be computed, in order to trace the response of the entire count distribution to changes in an explanatory variable The fourth chapter addresses specific methods for flexible modeling and estimation of such distribution responses, relative to the benchmark case of the Poisson distribution The organizational changes are accompanied by extensive changes to the presentation of the existing material Many sections of the book have been entirely re-written, or at least revised to correct for typos and inaccuracies that had slipped through Hopefully, these changes to presentation and organization have made the book more accessible, and thus more useful also as a reference for graduate level courses on the subject The list of newly in- VI Preface cluded topics includes: Poisson polynomial and double Poisson distribution; the significance of Poisson regression for estimating log-linear models with continuous dependent variable; marginal effects at the extensive margin; additional semi-parametric methods for endogenous regressors; new developments in discrete factor modeling, including a more detailed presentation of the EM algorithm; and copula functions I acknowledge my gratitude to those who contributed in various ways, and at various stages, to this book, including Tim Barmby, Kurt Bră annă as, Siddharta Chib, Malcolm Faddy, Bill Greene, Edward Greenberg, James Heckman, Robert Jung, Tom Kniesner, Gary King, Nikolai Kolev, Jochen Mayer, Daniel Miles, Andreas Million, Hans van Ophem, Joao Santos Silva, Pravin Trivedi, Frank Windmeijer and Klaus Zimmermann Large parts of this fifth edition were read by Stefan Boes, Adrian Bruhin and Kevin Staub, and their insights and comments lead to substantial improvements Part of the revision was completed while I was on leave at the University of California at Los Angeles and at the Center for Economic Studies at the University of Munich I am grateful for the hospitality experienced at both institutions In particular, I owe a great debt to doctoral students at UCLA and in Munich, whose feedback to a count data course I was teaching there led, I trust, to substantial improvements in the presentation of the material Ză urich, January 2008 Rainer Winkelmann Contents Preface V Introduction 1.1 Poisson Regression Model 1.2 Examples 1.3 Organization of the Book 1 Probability Models for Count Data 2.1 Introduction 2.2 Poisson Distribution 2.2.1 Definitions and Properties 2.2.2 Genesis of the Poisson Distribution 2.2.3 Poisson Process 2.2.4 Generalizations of the Poisson Process 2.2.5 Poisson Distribution as a Binomial Limit 2.2.6 Exponential Interarrival Times 2.2.7 Non-Poissonness 2.3 Further Distributions for Count Data 2.3.1 Negative Binomial Distribution 2.3.2 Binomial Distribution 2.3.3 Logarithmic Distribution 2.3.4 Summary 2.4 Modified Count Data Distributions 2.4.1 Truncation 2.4.2 Censoring and Grouping 2.4.3 Altered Distributions 2.5 Generalizations 2.5.1 Mixture Distributions 2.5.2 Compound Distributions 2.5.3 Birth Process Generalizations 2.5.4 Katz Family of Distributions 7 7 10 11 14 15 16 17 20 20 25 27 28 30 30 31 32 33 33 36 39 40 VIII Contents 2.5.5 Additive Log-Differenced Probability Models 2.5.6 Linear Exponential Families 2.5.7 Summary 2.6 Distributions for Over- and Underdispersion 2.6.1 Generalized Event Count Model 2.6.2 Generalized Poisson Distribution 2.6.3 Poisson Polynomial Distribution 2.6.4 Double Poisson Distribution 2.6.5 Summary 2.7 Duration Analysis and Count Data 2.7.1 Distributions for Interarrival Times 2.7.2 Renewal Processes 2.7.3 Gamma Count Distribution 2.7.4 Duration Mixture Models 41 42 44 45 45 46 47 49 49 50 52 54 56 59 Poisson Regression 63 3.1 Specification 63 3.1.1 Introduction 63 3.1.2 Assumptions of the Poisson Regression Model 63 3.1.3 Ordinary Least Squares and Other Alternatives 65 3.1.4 Interpretation of Parameters 70 3.1.5 Period at Risk 74 3.2 Maximum Likelihood Estimation 77 3.2.1 Introduction 77 3.2.2 Likelihood Function and Maximization 77 3.2.3 Newton-Raphson Algorithm 78 3.2.4 Properties of the Maximum Likelihood Estimator 80 3.2.5 Estimation of the Variance Matrix 82 3.2.6 Approximate Distribution of the Poisson Regression Coefficients 83 3.2.7 Bias Reduction Techniques 84 3.3 Pseudo-Maximum Likelihood 87 3.3.1 Linear Exponential Families 89 3.3.2 Biased Poisson Maximum Likelihood Inference 90 3.3.3 Robust Poisson Regression 91 3.3.4 Non-Parametric Variance Estimation 95 3.3.5 Poisson Regression and Log-Linear Models 97 3.3.6 Generalized Method of Moments 98 3.4 Sources of Misspecification 102 3.4.1 Mean Function 102 3.4.2 Unobserved Heterogeneity 103 3.4.3 Measurement Error 105 3.4.4 Dependent Process 107 3.4.5 Selectivity 107 3.4.6 Simultaneity and Endogeneity 108 Contents IX 3.4.7 Underreporting 109 3.4.8 Excess Zeros 109 3.4.9 Variance Function 110 3.5 Testing for Misspecification 112 3.5.1 Classical Specification Tests 112 3.5.2 Regression Based Tests 118 3.5.3 Goodness-of-Fit Tests 118 3.5.4 Tests for Non-Nested Models 120 3.6 Outlook 125 Unobserved Heterogeneity 127 4.1 Introduction 127 4.1.1 Conditional Mean Function 127 4.1.2 Partial Effects with Unobserved Heterogeneity 128 4.1.3 Unobserved Heterogeneity in the Poisson Model 129 4.1.4 Parametric and Semi-Parametric Models 130 4.2 Parametric Mixture Models 130 4.2.1 Gamma Mixture 131 4.2.2 Inverse Gaussian Mixture 131 4.2.3 Log-Normal Mixture 132 4.3 Negative Binomial Models 134 4.3.1 Negbin II Model 135 4.3.2 Negbin I Model 136 4.3.3 Negbink Model 136 4.3.4 NegbinX Model 137 4.4 Semiparametric Mixture Models 138 4.4.1 Series Expansions 138 4.4.2 Finite Mixture Models 139 Sample Selection and Endogeneity 143 5.1 Censoring and Truncation 143 5.1.1 Truncated Count Data Models 144 5.1.2 Endogenous Sampling 144 5.1.3 Censored Count Data Models 146 5.1.4 Grouped Poisson Regression Model 147 5.2 Incidental Censoring and Truncation 148 5.2.1 Outcome and Selection Model 148 5.2.2 Models of Non-Random Selection 149 5.2.3 Bivariate Normal Error Distribution 150 5.2.4 Outcome Distribution 152 5.2.5 Incidental Censoring 153 5.2.6 Incidental Truncation 154 5.3 Endogeneity in Count Data Models 156 5.3.1 Introduction and Examples 156 5.3.2 Parameter Ancillarity 157 X Contents 5.3.3 Endogeneity and Mean Function 159 5.3.4 A Two-Equation Framework 161 5.3.5 Instrumental Variable Estimation 162 5.3.6 Estimation in Stages 165 5.4 Switching Regression 167 5.4.1 Full Information Maximum Likelihood Estimation 168 5.4.2 Moment-Based Estimation 170 5.4.3 Non-Normality 171 5.5 Mixed Discrete-Continuous Models 171 Zeros in Count Data Models 173 6.1 Introduction 173 6.2 Zeros in the Poisson Model 174 6.2.1 Excess Zeros and Overdispersion 174 6.2.2 Two-Crossings Theorem 175 6.2.3 Effects at the Extensive Margin 176 6.2.4 Multi-Index Models 177 6.2.5 A General Decomposition Result 177 6.3 Hurdle Count Data Models 178 6.3.1 Hurdle Poisson Model 181 6.3.2 Marginal Effects 182 6.3.3 Hurdle Negative Binomial Model 183 6.3.4 Non-nested Hurdle Models 183 6.3.5 Unobserved Heterogeneity in Hurdle Models 185 6.3.6 Finite Mixture Versus Hurdle Models 186 6.3.7 Correlated Hurdle Models 187 6.4 Zero-Inflated Count Data Models 188 6.4.1 Introduction 188 6.4.2 Zero-Inflated Poisson Model 189 6.4.3 Zero-Inflated Negative Binomial Model 191 6.4.4 Marginal Effets 191 6.5 Compound Count Data Models 192 6.5.1 Multi-Episode Models 193 6.5.2 Underreporting 193 6.5.3 Count Amount Model 196 6.5.4 Endogenous Underreporting 197 6.6 Quantile Regression for Count Data 199 Correlated Count Data 203 7.1 Multivariate Count Data 203 7.1.1 Multivariate Poisson Distribution 205 7.1.2 Multivariate Negative Binomial Model 210 7.1.3 Multivariate Poisson-Gamma Mixture Model 212 7.1.4 Multivariate Poisson-Log-Normal Model 213 7.1.5 Latent Poisson-Normal Model 216 A Probability Generating Functions 283 Proof: (A.7) follows directly from the definition of the probability generating function: ∞ PT (s) = E(sXT ) = j=1 pj sj − p0 where p0 = P(0) There exists a close relationship between the probability generating function and the moment generating function M(t): M(t) = E(etX ) = P(et ) (A.8) While the moment generating function is a concept that can be used for any distribution with existing moments, the probability generating function is defined for non-negative integers Since s = et = if and only if t = 0, we obtain E(X) = P (1) = M (0) In the same way as in (A.1) one can define a bivariate probability generating function Definition Let X, Y be a pair of integer-valued random variables with joint distribution P (X = j, Y = k) = pjk , j, k ∈ IN0 The bivariate probability generating function is given by: ∞ ∞ Y pjk sj1 sk2 = E(sX s2 ) P(s1 , s2 ) = (A.9) j=0 k=0 Proposition The probability generating functions of the marginal distributions P (X = j) and P (Y = k) are P(s, 1) = E(sX ) and P(1, s) = E(sY ), respectively Proposition The probability generating function of X + Y is given by P(s, s) = E(sX+Y ) Proposition The variables X and Y are independent if and only if P(s1 , s2 ) = P(s1 , 1)P(1, s2 ) for all s1 , s2 Probability generating functions can be used to establish the distribution of a sum of independent variables This is also called a convolution Using Proposition and Proposition 4, the probability generating function of Z = X + Y is given by: ( ) P (Z) (s) = E(sZ ) = E(sX+Y ) = E(sX sY ) = E(sX )E(sY ) (A.10) 284 A Probability Generating Functions where ( ) follows from the independence assumption Example: Let X have a binomial distribution function with B(1, p) Consider the convolution Z = X + + X Then: n−times P (Z) (s) = (q + ps)n (A.11) Z has a binomial distribution function B(n, p) Conversely, the binomial distribution is obtained by a convolution of identically and independently distributed Bernoulli variables B Gauss-Hermite Quadrature This appendix describes the basic steps required for a numerical evaluation of the likelihood function of count data models with unobserved heterogeneity of the log-normal type The method is illustrated for the Poisson-log-normal model, although a similar algorithm can be used to estimate the models with endogenous selectivity presented in Chap 5.2 Butler and Moffitt (1982) discuss Gauss-Hermite quadrature in the context of a panel probit models Million (1998) points out that the Poisson-log-normal integral can be approximated using Gauss-Laguerre and Gauss-Legendre polynomials as well, and he evaluates the relative performance of the three methods Crouch and Spiegelman (1990) discuss numerical integration in the related logistic-normal model Starting point for Gauss-Hermite quadrature is the integral ∞ f (y|x, β, ε)g(ε|σ )dε (B.1) −∞ that cannot be solved by analytical methods However, assume that by appropriate change of variable, B.1 can be brought into the form ∞ h(ν; y, x, β, σ ) exp(−ν )dν (B.2) −∞ In this case, Gauss-Hermite quadrature can be applied to numerically evaluate the integral (B.1), and thus the marginal likelihood L(y|x) Once the evaluation has been done, the logarithm ln L(y|x) can be passed on to a maximizer that uses numerical derivatives in order to find the maximum likelihood estimators βˆ and σ ˆ2 The Poisson-log-normal model has the following components (see also Chap 4.2): f (y|ε) = exp(− exp(x β + ε)) exp(x β + ε)y y! where ε ∼ N (0, σ ), i.e., 286 B Gauss-Hermite Quadrature f (ε) = √ ε e− ( σ ) 2πσ Change of variable from ε to ν where ε ν=√ 2σ √ √ has inverse ε = ν 2σ and Jacobian df (ν)/dν = 2σ Therefore g(ν) = √ e−ν π and √ √ exp(− exp(x β + ν 2σ)) exp(x β + ν 2σ)y −ν √ f (y|ν)g(ν) = e πy! Let √ √ exp(− exp(xi β + ν 2σ)) exp(xi β + ν 2σ)yi √ hi (ν) = πyi ! where the subscript i reminds us that this function depends on observations yi and xi Then the Gauss-Hermite approximation to the integral B.1 is obtained as Lgh i = ∞ −∞ n ≈ hi (ν) exp(−ν )dν wj hi (νj ) j=1 where wj are weights and νj are the evaluation points The likelihood function for n independent observations is given by n n Lgh = wj hi (νj ) i=1 j=1 Weight factors and abscissas for 20-point quadrature are given in Tab B.1 (Source: Abramowitz and Stegun, 1964, p 924) B Gauss-Hermite Quadrature 287 Table B.1 Abcissas and Weight Factors for 20-point Gauss-Hermite Integration ui wi -5.3874809 -4.6036824 -3.9447640 -3.3478546 -2.7888061 -2.2549740 -1.7385377 -1.2340762 -0.73747373 -0.24534071 0.24534071 0.73747373 1.2340762 1.7385377 2.2549740 2.7888061 3.3478546 3.9447640 4.6036824 5.3874809 2.2939000e-13 4.3993400e-10 1.0860000e-07 7.8025500e-06 0.00022833863 0.0033243773 0.024810521 0.10901721 0.28667551 0.46224367 0.46224367 0.28667551 0.10901721 0.024810521 0.0033243773 0.00022833863 7.8025500e-06 1.0860000e-07 4.3993400e-10 2.2939000e-13 Source: Abramowitz and Stegun, 1964, p 924 C Software Most statistical and econometric software distributions contain built-in procedures for standard count data models, such as the Poisson and the negative binomial regression models Development in the software sector is fast, and specific recommendations risk to become outdated very quickly Nevertheless, there are a few general points that should be of help to anyone interested in working with count data and estimating the models presented in this book Within the econometrics research community, Gauss traditionally has been the major development tool Gauss is mostly a programming environment, but specialised procedures are available both as part of the general distribution, and through web sites and mailing lists For example, the “count” module allows the estimation of seemingly unrelated regression models, of various types of negative binomial models as well as hurdle Poisson models Yet, the development of this module has stalled for some time, and the latest models are not available Two alternative programs with a much more ambitious offering in this area are Stata and Limdep This appendix is not intended as a comprehensive review of available software for count data, and there may be other software with similar or even broader scope Yet, the possibilities that these two packages offers should be closely scrutinized by anyone seriously interested in count data applications who wants to apply up-to-date methods without doing the programming for herself In fact, most of the models discussed in this book are easily estimated with Stata or Limdep, providing little support for those who resort to the most basic models in want of available software for the more appropriate ones The following short summary refers to Stata release 7.0 This release includes built-in procedures, apart from the standard Poisson and Negbin models (in its various parameterizations, as Negbin I, Negbin II or with more flexible variance function), for zero-inflated Poisson and zero-inflated negative binomial models, and for fixed and random effects panel count data models Random effects models include the negative binomial panel model (with fixed or random effects) but also the panel Poisson-log-normal model This proce- 290 C Software dure can also be used in cross sections to estimate the standard Poisson-lognormal model that frequently has a better fit than the Negbin model Hurdle Poisson or negative binomial models are not included in the standard distribution However, they can be estimated using routines on truncated-at-zero models authored by Joseph Hilbe and described in the Stata Technical Bulletin Nr 47 Most procedures include options for the computations of robust standard errors (to perform pseudo maximum likelihood estimation) as well as account for clustered sampling The latest version of Limdep is release 8.0 Apart from the standard count data models, its capabilities include the estimation of sample selection models by maximum likelihood, parametric models for underreporting where the observed counts represent only the reported fraction of the total events which have occurred, and maximum likelihood estimation of various types of hurdle models and zero-inflated models Limdep and Stata are both quite versatile in the area of count data modelling D Tables Table D.1 Number of Job Changes: Poisson and Poisson-Log-Normal Constant Education∗10−1 Experience∗10−1 Experience2 ∗ 10−2 Union Single German Qualified White Collar Ordinary White Collar Qualified Blue Collar σ2 Log likelihood Log likelihood (β1 , , β9 = 0) Number of Observations Poisson Poisson-log-normal Mean 0.501** (0.158) -0.138 (0.137) -0.770** (0.111) 0.119** (0.037) -0.292** (0.065) -0.050 (0.108) -0.368** (0.076) 0.067 (0.131) 0.185 (0.147) 0.147 (0.082) 0.072 (0.227) -0.120 (0.187) -0.846** (0.155) 0.127* (0.050) -0.324** (0.088) -0.093 (0.153) -0.390** (0.104) -0.002 (0.179) 0.190 (0.207) 0.112 (0.114) 1.048** (0.048) -2044.47 -2155.40 1962 -1866.80 -1934.53 Source: German Socio-Economic Panel, Wave A/1984; own calculations Note: Asymptotic standard errors in parentheses 1.216 1.460 2.943 0.429 0.077 0.668 0.137 0.058 0.501 292 D Tables Table D.2 Number of Job Changes: Negative Binomial Models Constant Education∗10−1 Experience∗10−1 Experience2 ∗ 10−2 Union Single German Qualified White Collar Ordinary White Collar Qualified Blue Collar σ2 Negbin I Negbin II GECk 0.341 (0.191) 0.008 (0.162) -0.762** (0.139) 0.113* (0.046) -0.274** (0.080) -0.114 (0.139) -0.316** (0.097) -0.022 (0.163) 0.213 (0.176) 0.086 (0.103) 0.823** (0.088) 0.616** (0.224) -0.179 (0.187) -0.786** (0.152) 0.118* (0.048) -0.308** (0.087) -0.054 (0.152) -0.404** (0.102) 0.043 (0.174) 0.188 (0.201) 0.132 (0.111) 1.378** (0.137) 0.380 (0.212) -0.011 (0.180) -0.775** (0.144) 0.115* (0.047) -0.283** (0.084) -0.108 (0.141) -0.331** (0.103) -0.013 (0.173) 0.214 (0.181) 0.094 (0.107) 0.892** (0.080) 0.139 (0.281) -1873.28 1962 -1878.63 -1873.17 k Log likelihood Number of Observations Source: German Socio-Economic Panel, Wave A/1984; own calculations Notes: Asymptotic standard errors in parentheses For σ > and k = 0, the GECk model coincides with the Negbin I model For σ > and k = 1, the GECk model coincides with the Negbin II model D Tables 293 Table D.3 Number of Job Changes: Robust Poisson Regression Coefficient Constant Education∗10−1 Experience∗10−1 Experience2 ∗ 10−2 Union Single German Qualified White Collar Ordinary White Collar Qualified Blue Collar Log likelihood Number of Observations 0.501 -0.138 -0.770 0.119 -0.292 -0.050 -0.368 0.067 0.185 0.147 tPoisson 3.167 -1.006 -6.929 3.269 -4.499 -0.460 -4.843 0.514 1.255 1.794 Robust t-Values tWHITE tLVF tQVF 2.617 -0.823 -4.830 2.385 -3.115 -0.309 -2.892 0.343 0.964 1.261 2.229 -0.707 -4.877 2.301 -3.167 -0.323 -3.409 0.361 0.883 1.263 2.304 -0.749 -5.055 2.486 -3.385 -0.326 -3.503 0.384 0.917 1.308 -2044.47 1962 Notes: Three alternative methods to calculate robust standard errors (and thus robust t-values) were given in Chap 3.3.3 tLV F and tQV F are based on the assumption of a quadratic and linear variance function, respectively, while the White method makes no explicit assumption 294 D Tables Table D.4 Number of Job Changes: Poisson-Logistic Regression Variable Constant Education∗10−1 Experience∗10−1 Experience2 ∗ 10−2 Union a) Overlapping b) Non Overlapping Offers Acceptance Offers Acceptance 0.812 ( 3.746) -0.322 -2.073) -0.668 (-4.804) 0.071 ( 1.382) -0.291 (-4.477) Single 1.151 ( 9.740) 3.732 ( 1.582) -6.044 (-1.221) 3.321 ( 1.132) -0.260 (-1.633) -1.068 (-7.678) 0.175 ( 3.920) -0.290 (-4.470) 0.379 ( 0.153) German -0.068 (-0.460) -0.397 (-5.112) Qualified White Collar 0.069 ( 0.452) Ordinary White Collar 0.178 ( 1.125) Qualified Blue Collar 0.132 ( 1.389) -0.355 (-4.708) 0.088 ( 0.684) 0.195 ( 1.328) 0.156 ( 1.919) Log likelihood Observations -2043.88 -2039.35 1962 Notes: Asymptotic t-values in parentheses D Tables Table D.5 Number of Job Changes: Hurdle Count Data Models Hurdle Poisson Probit-Poisson-log-normal Variable 1+/0 1+ 1+/0 1+ Constant -0.069 (0.202) 0.133 (0.170) -0.758** (0.148) 0.107** (0.048) -0.268** (0.084) -0.194 (0.149) -0.330** (0.101) -0.071 (0.170) 0.239 (0.185) 0.069 (0.109) 1.163 (0.245) -0.600** (0.218) -0.403** (0.156) 0.085 (0.050) -0.167* (0.097) 0.192 (0.147) -0.206** (0.108) 0.271 (0.196) -0.039 (0.236) 0.184 (0.117) 0.269 (0.157) 0.094 (0.128) -0.629** (0.111) 0.098** (0.034) -0.205** (0.061) -0.149 (0.114) -0.254** (0.076) -0.076 (0.125) 0.200 (0.143) 0.042 (0.081) 0.932** (0.156) 0.212 (0.893) -1856.70 0.799 (0.666) -0.764** (0.324) -0.544 (0.405) 0.103 (0.088) -0.230 (0.189) 0.195 (0.249) -0.223 (0.208) 0.283 (0.285) -0.057 (0.336) 0.167 (0.172) Education∗10−1 Experience∗10−1 Experience2 ∗ 10−2 Union Single German Qualified White Collar Ordinary White Collar Qualified Blue Collar σ2 ρ Log-likelihood Observations -1928.00 1962 Notes: Asymptotic standard errors in parentheses Hurdle negbin results are nor displayed because of convergence problems 295 296 D Tables Table D.6 Number of Job Changes: Finite Mixture Models 2-components Poisson 2-components Negbin II Variable group group group group Constant -0.000 (0.226) 0.078 (0.184) -0.857** (0.172) 0.104* (0.060) -0.309** (0.095) -0.156 (0.158) -0.351** (0.114) -0.037 (0.192) 0.253 (0.202) 0.082 (0.124) 2.229** (0.433) -0.368 (0.374) -0.541** (0.231) 0.081 (0.074) -0.207 (0.141) 0.057 (0.229) -0.478** (0.158) 0.168 (0.288) 0.103 (0.358) 0.317* (0.173) 1.047* (0.630) -0.648 (0.425) -0.371 (0.346) 0.050 (0.099) -0.259 (0.181) 0.200 (0.325) -0.609** (0.236) 0.320 (0.386) -1.037 (0.945) 0.393 (0.263) 2.096** (0.949) 0.395** (0.158) -1856.05 0.154 (0.458) 0.243 (0.307) -1.140** (0.296) 0.138 (0.105) -0.328** (0.152) -0.274 (0.267) -0.101 (0.229) -0.263 (0.327) 0.609* (0.353) -0.130 (0.224) 0.146 (0.281) Education∗10−1 Experience∗10−1 Experience2 ∗ 10−2 Union Single German Qualified White Collar Ordinary White Collar Qualified Blue Collar σ2 π1 Log-likelihood Observations 0.930** (0.013) -1868.16 1962 Notes: Asymptotic standard errors in parentheses Hurdle Negbin I results are nor displayed because of convergence problems D Tables Table D.7 Number of Job Changes: Zero Inflated Count Data Models zero-inflated Poisson zero-inflated Negbin II Variable logit Poisson logit Negbin II Constant 1.132 (0.245) -0.583** (0.203) -0.373** (0.153) 0.072 (0.049) -0.158 (0.097) 0.151 (0.154) -0.173 (0.106) 0.272 (0.189) -0.123 (0.243) 0.166 (0.115) -0.303 (0.529) 1.016** (0.455) 1.035** (0.312) -0.157 (0.091) 0.293 (0.179) 0.461 (0.297) 0.435** (0.211) 0.519 (0.354) -0.870 (0.711) 0.094 (0.219) 0.483* (0.255) -0.262 (0.216) -0.613** (0.192) 0.129** (0.062) -0.253** (0.102) 0.066 (0.169) -0.236** (0.118) 0.178 (0.206) 0.025 (0.203) 0.151 (0.129) 1.103 (0.146) -7.390** (2.777) -0.746 (1.152) 4.535** (1.715) -.759** (0.346) 0.351 (0.465) 1.368 (0.962) 1.306 (0.788) 1.228 (0.843) 11.967 (349.869) 0.193 (0.614) Education∗10−1 Experience∗10−1 Experience2 ∗ 10−2 Union Single German Qualified White Collar Ordinary White Collar Qualified Blue Collar σ2 Log-likelihood Observations -1926.28 1962 Notes: Asymptotic standard errors in parentheses -1866.73 297 298 D Tables Table D.8 Number of Job Changes: Quantile Regressions Qz (0.5, x) Qz (0.75, x) Qz (0.9, x) Constant Education∗10−1 Experience∗10−1 Experience2 ∗ 10−2 Union Single German Qualified White Collar Ordinary White Collar Qualified Blue Collar Observations -0.181 (0.468) 0.319 (0.373) -1.346 (0.249) 0.288 (0.069) -0.388 (0.193) -0.469 (0.324) -0.479 (0.246) -0.144 (0.304) 0.240 (0.319) -0.142 (0.212) 1962 1.138 (0.475) 0.026 (0.343) -1.413 (0.258) 0.220 (0.083) -0.395 (0.187) -0.191 (0.248) -0.522 (0.213) -0.063 (0.302) 0.312 (0.340) -0.078 (0.179) 1.768 (0.272) -0.343 (0.207) -0.721 (0.196) 0.066 (0.054) -0.336 (0.117) -0.128 (0.220) -0.209 (0.162) -0.020 (0.237) -0.061 (0.188) -0.082 (0.137) Notes: Bootstrap standard errors in parentheses (50 replications) .. .Econometric Analysis of Count Data Rainer Winkelmann Econometric Analysis of Count Data Fifth edition 123 Prof Dr Rainer Winkelmann University of Zurich Socioeconomic... D.1 D.2 D.3 D.4 D.5 D.6 D.7 D.8 Number Number Number Number Number Number Number Number of of of of of of of of Job Job Job Job Job Job Job Job Changes: Changes: Changes: Changes: Changes: Changes:... count process, while for fixed T , N (T ) is a count variable The stochastic properties of the count process (and thus of the count) are fully determined once the joint distribution function of