Preface

The availability of microdata has increased rapidly over the last decades,
and standard statistical and econometric software packages for data analysis
include ever more sophisticated modeling options. The goal of this book is to
familiarize readers with a wide range of commonly used models, and thereby
to enable them to become critical consumers of current empirical research,
and to conduct their own empirical analyses.
The focus of the book is on regression-type models in the context of large
cross-section samples. In microdata applications, dependent variables often
are qualitative and discrete, while in other cases, the sample is not randomly
drawn from the population of interest and the dependent variable is censored
or truncated. Hence, models and methods are required that go beyond the
standard linear regression model and ordinary least squares. Maximum likelihood
estimation of conditional probability models and marginal probability
effects are introduced here as the unifying principle for modeling, estimating
and interpreting microdata relationships. We consider the limitation to maximum
likelihood sensible, from a pedagogical point of view if the book is to
be used in a semester-long advanced undergraduate or graduate course, and
from a practical point of view because maximum likelihood estimation is used
in the overwhelming majority of current microdata research.
In order to introduce and explain the models and methods, we refer to a
number of illustrative applications. The main examples include the determinants
of individual fertility, the intergenerational transmission of secondary
school choices, and the wage elasticity of female labor supply. The models presented,
while chosen with economic applications in mind, should be equally
relevant for other social sciences, for example, quantitative political science
and sociology, and for empirical disciplines outside of the social sciences.
The book can be used as a textbook for an advanced undergraduate, a
Master’s or a first-year Ph.D. course on the topic of microdata analysis. In
economics and related disciplines, such a course is typically offered after a
first course on linear regression analysis. Alternatively, the book can also serve
as a supplementary text to an applied microeconomics field course, such asthose offered in the areas of labor economics, health economics, and the like.
Finally, it is intended as a reference for graduate students, researchers as well
as practitioners who encounter microdata in their work. The mathematical
prerequisites are not very high. In particular, the use of linear algebra is
minimal. On the other hand, some background in mathematical statistics is
useful although not absolutely necessary.
The book includes numerous exercises. Most of the exercises do not require
the use of a computer. Rather, they typically present specific empirical
results, and the task is to assess the validity of the procedure in that particular
context and to provide a correct interpretation of the estimated parameters.
In addition, we encourage the reader to develop practical skills in applied
data analysis by re-estimating the examples we discuss, using a software of
choice. For this purpose, we have made the datasets employed available at our
homepage www.unizh.ch/sts/, both in ASCII format and in Stata 7 format.
An earlier version of the manuscript was used in a course of the same name
taught by us for several years at the economics department of the University of
Zurich. We thank the participants for numerous suggestions for improvement.
We are heavily indebted to Markus Lipp and Adrian Bruhin for careful proofreading,
to Markus in addition for creating all the figures, and to Deborah
Bowen for improving our English.

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 What Are Microdata?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Types of Microdata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Qualitative Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.2 Quantitative Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Why Not Linear Regression? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Common Elements of Microdata Models . . . . . . . . . . . . . . . . . . . . 10
1.5 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5.1 Determinants of Fertility . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5.2 Secondary School Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.5.3 Female Hours of Work and Wages . . . . . . . . . . . . . . . . . . . 17
1.6 Overview of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2 From Regression to Probability Models . . . . . . . . . . . . . . . . . . . . 21
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 Conditional Probability Functions . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2.2 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2.3 Interpretation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3 Probability and Probability Distributions . . . . . . . . . . . . . . . . . . . 29
2.3.1 Axioms of Probability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.3.2 Univariate Random Variables . . . . . . . . . . . . . . . . . . . . . . . 30
2.3.3 Multivariate Random Variables . . . . . . . . . . . . . . . . . . . . . 31
2.3.4 Conditional Probability Models . . . . . . . . . . . . . . . . . . . . . 34
2.4 Further Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3 Maximum Likelihood Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2 Likelihood Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.2.1 Score Function and Hessian Matrix . . . . . . . . . . . . . . . . . . 48
3.2.2 Conditional Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.2.3 Maximization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.3 Properties of the Maximum Likelihood Estimator . . . . . . . . . . . . 53
3.3.1 Expected Score . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.3.2 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.3.3 Information Matrix Equality . . . . . . . . . . . . . . . . . . . . . . . . 56
3.3.4 Asymptotic Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.3.5 Covariance Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.4 Normal Linear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.5 Further Aspects of Maximum Likelihood Estimation . . . . . . . . . 67
3.5.1 Invariance and Delta Method . . . . . . . . . . . . . . . . . . . . . . . 67
3.5.2 Numerical Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.5.3 Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.5.4 Quasi Maximum Likelihood . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.6 Testing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.6.2 Restricted Maximum Likelihood. . . . . . . . . . . . . . . . . . . . . 79
3.6.3 Wald Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.6.4 Likelihood Ratio Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.6.5 Score Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.6.6 Model Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.6.7 Goodness-of-Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.7 Pros and Cons of Maximum Likelihood. . . . . . . . . . . . . . . . . . . . . 89
3.8 Further Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4 Binary Response Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.2 Models for Binary Response Variables . . . . . . . . . . . . . . . . . . . . . . 97
4.2.1 General Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.2.2 Linear Probability Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.2.3 Probit Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.2.4 Logit Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.2.5 Interpretation of Parameters . . . . . . . . . . . . . . . . . . . . . . . . 104
4.3 Discrete Choice Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.4 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.4.1 Maximum Likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.4.2 Perfect Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.4.3 Properties of the Estimator . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.4.4 Endogenous Regressors in Binary Response Models . . . . 116
4.4.5 Estimation of Marginal Effects . . . . . . . . . . . . . . . . . . . . . . 118
4.5 Goodness-of-Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.6 Non-Standard Sampling Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . 127
4.6.1 Stratified Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
4.6.2 Exogenous Stratification . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
4.6.3 Endogenous Stratification . . . . . . . . . . . . . . . . . . . . . . . . . . 128
4.7 Further Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

5 Multinomial Response Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.2 Multinomial Logit Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.2.1 Basic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.2.2 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
5.2.3 Interpretation of Parameters . . . . . . . . . . . . . . . . . . . . . . . . 144
5.3 Conditional Logit Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
5.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
5.3.2 General Model of Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
5.3.3 Modeling Conditional Logits . . . . . . . . . . . . . . . . . . . . . . . . 152
5.3.4 Interpretation of Parameters . . . . . . . . . . . . . . . . . . . . . . . . 155
5.3.5 Independence of Irrelevant Alternatives . . . . . . . . . . . . . . 159
5.4 Generalized Multinomial Response Models . . . . . . . . . . . . . . . . . . 160
5.4.1 Multinomial Probit Model . . . . . . . . . . . . . . . . . . . . . . . . . . 161
5.4.2 Mixed Logit Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
5.4.3 Nested Logit Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
5.5 Further Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
6 Ordered Response Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
6.2 Standard Ordered Response Models. . . . . . . . . . . . . . . . . . . . . . . . 174
6.2.1 General Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
6.2.2 Ordered Probit Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
6.2.3 Ordered Logit Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
6.2.4 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
6.2.5 Interpretation of Parameters . . . . . . . . . . . . . . . . . . . . . . . . 179
6.2.6 Single Indices and Parallel Regression . . . . . . . . . . . . . . . . 186
6.3 Generalized Threshold Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
6.3.1 Generalized Ordered Logit and Probit Models . . . . . . . . . 188
6.3.2 Interpretation of Parameters . . . . . . . . . . . . . . . . . . . . . . . . 189
6.4 Sequential Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
6.4.1 Modeling Conditional Transitions . . . . . . . . . . . . . . . . . . . 194
6.4.2 Generalized Conditional Transition Probabilities . . . . . . 197
6.4.3 Marginal Effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
6.4.4 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
6.5 Interval Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
6.6 Further Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
7 Limited Dependent Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
7.1.1 Corner Solution Outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . 208
7.1.2 Sample Selection Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
7.1.3 Treatment Effect Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
7.2 Tobin’s Corner Solution Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
7.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

7.2.3 Truncated Normal Distribution . . . . . . . . . . . . . . . . . . . . . 214
7.2.4 Inverse Mills Ratio and its Properties . . . . . . . . . . . . . . . . 215
7.2.5 Interpretation of the Tobit Model . . . . . . . . . . . . . . . . . . . 218
7.2.6 Comparing Tobit and OLS . . . . . . . . . . . . . . . . . . . . . . . . . 221
7.2.7 Further Specification Issues . . . . . . . . . . . . . . . . . . . . . . . . . 223
7.3 Sample Selection Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
7.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
7.3.2 Censored Regression Model . . . . . . . . . . . . . . . . . . . . . . . . . 226
7.3.3 Estimation of the Censored Regression Model . . . . . . . . . 228
7.3.4 Truncated Regression Model . . . . . . . . . . . . . . . . . . . . . . . . 230
7.3.5 Incidental Censoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
7.3.6 Example: Estimating a Labor Supply Model . . . . . . . . . . 237
7.4 Treatment Effect Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
7.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
7.4.2 Endogenous Binary Variable . . . . . . . . . . . . . . . . . . . . . . . . 242
7.4.3 Switching Regression Model . . . . . . . . . . . . . . . . . . . . . . . . 243
7.5 Appendix: Bivariate Normal Distribution . . . . . . . . . . . . . . . . . . . 246
7.6 Further Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
8 Event History Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
8.2 Duration Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
8.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
8.2.2 Basic Concepts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
8.2.3 Discrete Time Duration Models . . . . . . . . . . . . . . . . . . . . . 259
8.2.4 Continuous Time Duration Models . . . . . . . . . . . . . . . . . . 262
8.2.5 Key Element: Hazard Function . . . . . . . . . . . . . . . . . . . . . . 265
8.2.6 Duration Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
8.2.7 Unobserved Heterogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . 271
8.3 Count Data Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
8.3.1 The Poisson Regression Model . . . . . . . . . . . . . . . . . . . . . . 279
8.3.2 Unobserved Heterogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . 284
8.3.3 Efficient versus Robust Estimation . . . . . . . . . . . . . . . . . . 289
8.3.4 Censoring and Truncation . . . . . . . . . . . . . . . . . . . . . . . . . . 289
8.3.5 Hurdle and Zero-Inflated Count Data Models . . . . . . . . . 291
8.4 Further Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309