P1: KNP/FFX CB733-FM P2: GDZ/FFX QC: GDZ/FFX T1: GDZ CB733-Frees-v4 June 18, 2004 This page intentionally left blank ii 14:23 P1: KNP/FFX CB733-FM P2: GDZ/FFX QC: GDZ/FFX T1: GDZ CB733-Frees-v4 June 18, 2004 Longitudinal and Panel Data This book focuses on models and data that arise from repeated observations of a cross section of individuals, households, or firms These models have found important applications within business, economics, education, political science, and other social science disciplines The author introduces the foundations of longitudinal and panel data analysis at a level suitable for quantitatively oriented graduate social science students as well as individual researchers He emphasizes mathematical and statistical fundamentals but also describes substantive applications from across the social sciences, showing the breadth and scope that these models enjoy The applications are enhanced by real-world data sets and software programs in SAS, Stata, and R EDWARD W FREES is Professor of Business at the University of Wisconsin–Madison and is holder of the Fortis Health Insurance Professorship of Actuarial Science He is a Fellow of both the Society of Actuaries and the American Statistical Association He has served in several editorial capacities including Editor of the North American Actuarial Journal and Associate Editor of Insurance: Mathematics and Economics An award-winning researcher, he has published in the leading refereed academic journals in actuarial science and insurance, other areas of business and economics, and mathematical and applied statistics i 14:23 P1: KNP/FFX CB733-FM P2: GDZ/FFX QC: GDZ/FFX T1: GDZ CB733-Frees-v4 June 18, 2004 ii 14:23 P1: KNP/FFX CB733-FM P2: GDZ/FFX QC: GDZ/FFX T1: GDZ CB733-Frees-v4 June 18, 2004 Longitudinal and Panel Data Analysis and Applications in the Social Sciences EDWARD W FREES University of Wisconsin–Madison iii 14:23 cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge cb2 2ru, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521828284 © Edward W Frees 2004 This publication is in copyright Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press First published in print format 2004 isbn-13 isbn-10 978-0-511-21169-0 eBook (EBL) 0-511-21346-8 eBook (EBL) isbn-13 isbn-10 978-0-521-82828-4 hardback 0-521-82828-7 hardback isbn-13 isbn-10 978-0-521-53538-0 paperback 0-521-53538-7 paperback Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate P1: KNP/FFX CB733-FM P2: GDZ/FFX QC: GDZ/FFX T1: GDZ CB733-Frees-v4 June 18, 2004 Contents Preface page ix Introduction 1.1 What Are Longitudinal and Panel Data? 1.2 Benefits and Drawbacks of Longitudinal Data 1.3 Longitudinal Data Models 1.4 Historical Notes Fixed-Effects Models 2.1 Basic Fixed-Effects Model 2.2 Exploring Longitudinal Data 2.3 Estimation and Inference 2.4 Model Specification and Diagnostics 2.5 Model Extensions Further Reading Appendix 2A Least-Squares Estimation Exercises and Extensions Models with Random Effects 3.1 Error-Components/Random-Intercepts Model 3.2 Example: Income Tax Payments 3.3 Mixed-Effects Models 3.4 Inference for Regression Coefficients 3.5 Variance Components Estimation Further Reading Appendix 3A REML Calculations Exercises and Extensions Prediction and Bayesian Inference 4.1 Estimators versus Predictors 4.2 Predictions for One-Way ANOVA Models v 1 12 15 18 18 24 31 38 46 52 53 57 72 72 81 86 94 100 106 107 113 125 125 126 14:23 P1: KNP/FFX CB733-FM P2: GDZ/FFX QC: GDZ/FFX T1: GDZ CB733-Frees-v4 vi June 18, 2004 Contents 4.3 Best Linear Unbiased Predictors 4.4 Mixed-Model Predictors 4.5 Example: Forecasting Wisconsin Lottery Sales 4.6 Bayesian Inference 4.7 Credibility Theory Further Reading Appendix 4A Linear Unbiased Prediction Exercises and Extensions Multilevel Models 5.1 Cross-Sectional Multilevel Models 5.2 Longitudinal Multilevel Models 5.3 Prediction 5.4 Testing Variance Components Further Reading Appendix 5A High-Order Multilevel Models Exercises and Extensions Stochastic Regressors 6.1 Stochastic Regressors in Nonlongitudinal Settings 6.2 Stochastic Regressors in Longitudinal Settings 6.3 Longitudinal Data Models with Heterogeneity Terms and Sequentially Exogenous Regressors 6.4 Multivariate Responses 6.5 Simultaneous-Equations Models with Latent Variables Further Reading Appendix 6A Linear Projections Modeling Issues 7.1 Heterogeneity 7.2 Comparing Fixed- and Random-Effects Estimators 7.3 Omitted Variables 7.4 Sampling, Selectivity Bias, and Attrition Exercises and Extensions Dynamic Models 8.1 Introduction 8.2 Serial Correlation Models 8.3 Cross-Sectional Correlations and Time-Series Cross-Section Models 8.4 Time-Varying Coefficients 8.5 Kalman Filter Approach 129 133 138 147 152 156 157 159 166 166 174 180 184 187 187 191 199 199 208 213 221 231 240 240 242 242 247 256 263 272 277 277 280 286 288 295 14:23 P1: KNP/FFX CB733-FM P2: GDZ/FFX QC: GDZ/FFX T1: GDZ CB733-Frees-v4 June 18, 2004 Contents 8.6 Example: Capital Asset Pricing Model Appendix 8A Inference for the Time-Varying Coefficient Model Binary Dependent Variables 9.1 Homogeneous Models 9.2 Random-Effects Models 9.3 Fixed-Effects Models 9.4 Marginal Models and GEE Further Reading Appendix 9A Likelihood Calculations Exercises and Extensions 10 Generalized Linear Models 10.1 Homogeneous Models 10.2 Example: Tort Filings 10.3 Marginal Models and GEE 10.4 Random-Effects Models 10.5 Fixed-Effects Models 10.6 Bayesian Inference Further Reading Appendix 10A Exponential Families of Distributions Exercises and Extensions 11 Categorical Dependent Variables and Survival Models 11.1 Homogeneous Models 11.2 Multinomial Logit Models with Random Effects 11.3 Transition (Markov) Models 11.4 Survival Models Appendix 11A Conditional Likelihood Estimation for Multinomial Logit Models with Heterogeneity Terms Appendix A Elements of Matrix Algebra A.1 Basic Terminology A.2 Basic Operations A.3 Additional Definitions A.4 Matrix Decompositions A.5 Partitioned Matrices A.6 Kronecker (Direct) Product Appendix B Normal Distribution B.1 Univariate Normal Distribution B.2 Multivariate Normal Distribution B.3 Normal Likelihood B.4 Conditional Distributions vii 302 312 318 318 329 335 339 343 344 347 350 350 356 360 366 371 376 380 380 386 387 387 398 400 411 415 417 417 418 418 419 420 421 422 422 422 423 423 14:23 P1: KNP/FFX CB733-FM P2: GDZ/FFX QC: GDZ/FFX T1: GDZ CB733-Frees-v4 viii June 18, 2004 Contents Appendix C Likelihood-Based Inference C.1 Characteristics of Likelihood Functions C.2 Maximum Likelihood Estimators C.3 Iterated Reweighted Least Squares C.4 Profile Likelihood C.5 Quasi-Likelihood C.6 Estimating Equations C.7 Hypothesis Tests C.8 Goodness-of-Fit Statistics C.9 Information Criteria Appendix D State Space Model and the Kalman Filter D.1 Basic State Space Model D.2 Kalman Filter Algorithm D.3 Likelihood Equations D.4 Extended State Space Model and Mixed Linear Models D.5 Likelihood Equations for Mixed Linear Models Appendix E Symbols and Notation Appendix F Selected Longitudinal and Panel Data Sets References Index 424 424 425 426 427 427 428 431 432 433 434 434 435 436 436 437 439 445 451 463 14:23 ... introduces the many key features of the data and models used in the analysis of longitudinal and panel data Here, longitudinal and panel data are defined and an indication of their widespread usage... time Defining Longitudinal and Panel Data Longitudinal data analysis represents a marriage of regression and time-series analysis As with many regression data sets, longitudinal data are composed... consider the more broadly applicable unbalanced data case Prevalence of Longitudinal and Panel Data Analysis Longitudinal and panel databases and models have taken on important roles in the literature