Introduction

There is a large statistical and econometric literature concerning the topic of unobserved heterogeneity. Observed heterogeneity refers to interindividual differences that are measured by regressors, and unobserved heterogeneity refers to all other differences. Both factors affect survival times. In the presence of unobserved heterogeneity even individuals with the same values of all covariates may have different hazards out of a given state. When unobserved heterogeneity is ignored, its impact is confounded with that of the baseline hazard.

To motivate further study consider a well-known empirical example. The aggregate hazard rate out of unemployment is known to be a declining function of the length of unemployment spell. If all individuals were identical then this would imply negative duration dependence, that is, a falling probability of escaping unemployment the longer an individual has remained unemployed. However, suppose that there are two types of individuals in the unemployed population, type F (fast), who have a constant hazard rate of 0.4, and type S (slow), whose constant hazard rate is 0.1. The population is a 50/50 mixture of the two types. Then for 100 type F people we observe 40 transitions in the first period, 24 transitions in the second period, and 14.4 in the third. For the type S, we observe 10, 9, and 8.1 transitions in the first, second, and third periods, respectively. Hence the aggregate proportion of transitions will be (40 + 10)/200 = 0.25, (24 + 9)/150 = 0.22, and (14.4 + 8.1)/117 = 0.192.

Introduction

In many economic contexts the dependent or response variable of interest is a nonnegative integer or count that we wish to explain or analyze in terms of a set of regressors. Unlike the classical regression model, the response variable is discrete, with a distribution that places probability mass at nonnegative integer values only. Several models discussed earlier in the book, such as the binary outcome model and the duration model, can be shown to be closely related to the count data regression model. Regression models for counts, like other limited or discrete dependent variable models such as the logit and probit, are nonlinear with many properties and special features intimately connected to discreteness and nonlinearity.

Let us consider some examples from microeconometrics, beginning with sample data that are independent cross-section observations. Fertility studies often model the number of live births over a specified age interval of the mother, with interest in analyzing its variation in terms of, say, mother's schooling, age, and household income (Winkelmann, 1995). In some models of family decisions the number of children may appear as an explanatory variable with the acknowledgment that the variable is endogenous. Accident analysis studies model airline safety as measured by the number of accidents experienced by an airline over some period and seek to determine its relationship to airline profitability and other measures of the financial health of the airline (Rose, 1990).

Introduction

Microeconometrics deals with the theory and applications of methods of data analysis developed for microdata pertaining to individuals, households, and firms. A broader definition might also include regional- and state-level data. Microdata are usually either cross sectional, in which case they refer to conditions at the same point in time, or longitudinal (panel) in which case they refer to the same observational units over several periods. Such observations are generated from both nonexperimental setups, such as censuses and surveys, and quasi-experimental or experimental setups, such as social experiments implemented by governments with the participation of volunteers.

A microeconometric model may be a full specification of the probability distribution of a set of microeconomic observations; it may also be a partial specification of some distributional properties, such as moments, of a subset of variables. The mean of a single dependent variable conditional on regressors is of particular interest.

There are several objectives of microeconometrics. They include both data description and causal inference. The first can be defined broadly to include moment properties of response variables, or regression equations that highlight associations rather than causal relations. The second category includes causal relationships that aim at measurement and/or empirical confirmation or refutation of conjectures and propositions regarding microeconomic behavior. The type and style of empirical investigations therefore span a wide spectrum.

Introduction

This chapter deals with several different duration models that can be interpreted broadly as multivariate models, a category that covers both parallel and repeated transitions. Any transition model that involves more than one destination state can be regarded as a multivariate model because the analysis will involve joint distribution of two or more durations. The models we consider arise in a variety of ways and apply to several different types of data. Despite their differences, they are grouped in this chapter for reasons of organizational convenience.

To be concrete consider some examples. A familiar model from labor economics involves a transition from unemployment to employment or out of the labor force. The first transition can be further broken into return to the old job or to a new job. These destinations are mutually exclusive. An unemployment spell may end by a transition to any one of the destinations. A variant of this example considers an unemployed individual who could find either a new full-time or part-time job or remain unemployed. Thus there are three possible states (destinations). The models of Chapters 17 and 18 dealt with transitions between two states. One can still use the two-state methods to handle such data. For example, state 1 could be that of full-time employment and state 0 could be any other state. This would, as before, involve modeling one hazard rate.

Part 4, consisting of chapters 14 to 20, covers the core nonlinear limited dependent variable models for cross-section data, defined by the range of values taken by the dependent variable. Topics covered include models for binary, multinomial, duration and count data. The complications of censoring, truncation and sample selection are also studied. The essential base for Part 4 is least squares and maximum likelihood estimation.

Chapters 14–15 cover models for binary and multinomial data that are standard in the analysis of discrete outcomes and discrete choice. Maximum likelihood methods are dominant. Different parameterizations for the conditional probabilities in these models lead to different models, notably logit and probit models, which are wellestablished. Recent literature has focused on less restrictive modeling with more flexible functional forms for conditional probabilities and on accommodating individual unobserved heterogeneity. These objectives motivate the use of semiparametric methods and simulation-based estimation methods.

Censoring, truncation, or sample selection generate several important classes of models that are analyzed in Chapter 16. The long-established Tobit model is central to this literature, but its estimation and inference rely on strong distributional assumptions to permit consistent estimation. We also examine the newer semiparametric methods that rely on weaker assumptions.

Chapters 17–19 consider duration models in which the focus is on either the determinants of spell lengths, such as length of an unemployment spell, or on modeling the hazard rate of transitions from one initial state to another.

Introduction

The preceding chapter considered models for discrete outcome variables that can take one of two possible values. Here we consider several possible outcomes, usually mutually exclusive. Examples include different ways to commute to work (by bus, car, or walking), various types of health insurance (fee-for-service, managed care, or none), different employment status (full-time, part-time, or none), choice of recreational site, occupational choice, and product choice.

Statistical inference is relatively straight forward in principle, as the data must be multinomial distributed, just as binary data must be Bernoulli or binomial distributed. Estimation is most often by maximum likelihood because the data are clearly multinomial distributed. For some complications, however, moment-based estimation is used instead.

Different multinomial models arise owing to different functional forms for the probabilities of the multinomial distribution, similar to the differences between probit and logit in the binary case. A distinction is also made between models where regressors vary across alternatives for a given individual and models where regressors are constant across alternatives. For example, in transportation mode choice some regressors, such as travel time or cost, will vary with choices whereas others, such as age, are choice invariant.

The simplest multinomial model, the conditional or multinomial logit model, is quite straightforward to use but is viewed as too restrictive in practice, especially if the multinomial outcome data arise from individual choice. For unordered outcomes less restrictive models can be obtained using the random utility model.

Introduction

Discrete outcome or qualitative response models are models for a dependent variable that indicates in which one of m mutually exclusive categories the outcome of interest falls. Often there is no natural ordering of the categories. For example, categorization may be on the occupation of a worker.

This chapter considers the simplest case of binary outcomes, where there are two possible outcomes. Examples include whether or not an individual is employed and whether or not a consumer makes a purchase. Binary outcomes are simple to model and estimation is usually by maximum likelihood because the distribution of the data is necessarily defined by the Bernoulli model. If the probability of one outcome equals p, then the probability of the other outcome must be (1 − p). For regression applications the probability p will vary across individuals as a function of regressors. The two standard binary outcome models, the logit and the probit models, specify different functional forms for this probability as a function of regressors. The difference between these estimators is qualitatively similar to use of different functional forms for the conditional mean in least-squares regression.

Section 14.2 provides a data example. Section 14.3 presents a summary of statistical results for standard models including logit and probit models. In Section 14.4 binary outcome models are presented as arising from an underlying latent variable. This formulation is useful as it extends readily to multinomial models (see Chapter 15) and models for censored and selected samples (see Chapter 16).

Introduction

The previous chapter presented variants of the linear panel data model with a fixed or random intercept and regressors that are strongly exogenous. Now we move on to various extensions for linear models, with focus on relaxation of the strong exogeneity assumption to permit consistent estimation of models with endogenous variables and/or lagged dependent variables as regressors.

The use of instrumental variables is a standard method to handle endogenous regressors. It is much easier to obtain instruments with panel data than with cross-section data, since exogenous regressors in other time periods can be used as instruments for endogenous regressors in the current time period. The only complication is to first control for any fixed or random effects.

Panel data permit regressors to additionally include lagged dependent variables, data unavailable with a single cross section. This permits estimation of dynamic models that distinguish between persistence of earnings, for example, as the result of variation around an unobserved individual-specific effect, as in Chapter 21, and persistence caused by the outcomes of previous periods directly determining the outcome of the current period. The estimators of Chapter 21 that control for individual-specific effects become inconsistent, however, if lagged dependent variables are regressors. Instrumental variables estimation using longer lags as instruments leads to consistent estimation.

Panel data provide an excess of moment conditions available for estimation, owing to an abundance of instruments, and panel model errors are usually not iid.

Introduction

In this chapter we present methods for data analysis that require less model specification than the methods of the preceding chapters.

We begin with nonparametric estimation. This makes very minimal assumptions regarding the process that generated the data. One leading example is estimation of a continuous density using a kernel density estimate. This has the attraction of providing a smoother estimate than the familiar histogram. A second leading example is nonparametric regression, such as kernel regression, on a scalar regressor. This places a flexible curve on an (x, y) scatterplot with no parametric restrictions on the form of the curve. Nonparametric estimates have numerous uses, including data description, exploratory analysis of data and of fitted residuals from a regression model, and summary across simulations of parameter estimates obtained from a Monte Carlo study.

Econometric analysis emphasizes multivariate regression of a scalar y on a vector of regressors x. However, nonparametric methods, although theoretically possible with an infinitely large sample, break down in practice because the data need to be sliced in several dimensions, leading to too few data points in each slice.

As a result econometricians have focused on semiparametric methods. These combine a parametric component, greatly reducing the dimensionality, with a nonparametric component. One important application is to permit more flexible models of the conditional mean.

Part 1 covers the essential components of microeconometric analysis – an economic specification, a statistical model and a data set.

Chapter 1 discusses the distinctive aspects of microeconometrics, and provides an outline of the book. It emphasizes that discreteness of data, and nonlinearity and heterogeneity of behavioral relationships are key aspects of individual-level microeconometric models. It concludes by presenting the notation and conventions used throughout the book.

Chapters 2 and 3 set the scene for the remainder of the book by introducing the reader to key model and data concepts that shape the analyses of later chapters.

A key distinction in econometrics is between essentially descriptive models and data summaries at various levels of statistical sophistication and models that go beyond associations and attempt to estimate causal parameters. The classic definitions of causality in econometrics derive from the Cowles Commission simultaneous equations models that draw sharp distinctions between exogenous and endogenous variables, and between structural and reduced form parameters. Although reduced form models are very useful for some purposes, knowledge of structural or causal parameters is essential for policy analyses. Identification of structural parameters within the simultaneous equations framework poses numerous conceptual and practical difficulties. An increasingly-used alternative approach based on the potential outcome model, also attempts to identify causal parameters but it does so by posing limited questions within a more manageable framework. Chapter 2 attempts to provide an overview of the fundamental issues that arise in these and other alternative frameworks.

Introduction

Problems of measurement error pervade all econometrics. In microeconometrics, a common source of the measurement error problem comes from incorrect response to a survey question, incorrect coding of a correct response, and the use of a correctly measured variable as a proxy for another theoretically valid but unobserved variable (e.g., using observed income as a proxy for “normal income”). Questions that seek sensitive information may elicit partial or incorrect responses. That is, a measurement error is triggered by unobservables (or latent variables) when such variables are replaced by proxy variables.

Here are some examples. Consider the problem of testing for the presence of gender bias in a study of earnings. The obvious approach is to regress a measure of earnings on a categorical gender variable while controlling for qualifications, age, experience, and so forth. However, the most relevant variable may be an individual's on-the-job productivity, which may not be directly observed and a proxy may be used instead. Therefore, the impact of measurement error on inferences about the gender discrimination is an important issue. Studies of individual demand for goods and services feature concepts such as “economic cost” or “full price of a service.” However, these are rarely directly measured in published data and must be constructed by the econometrician prior to model estimation. Inevitably their measurement is subject to error.

There are virtually no models discussed in this book that are protected from the problem of measurement errors.

Introduction

Exact finite-sample results are unavailable for most microeconometrics estimators and related test statistics. The statistical inference methods presented in preceding chapters rely on asymptotic theory that usually leads to limit normal and chi-square distributions.

An alternative approximation is provided by the bootstrap, due to Efron (1979, 1982). This approximates the distribution of a statistic by a Monte Carlo simulation, with sampling done from the empirical distribution or the fitted distribution of the observed data. The additional computation required is usually feasible given advances in computing power. Like conventional methods, however, bootstrap methods rely on asymptotic theory and are only exact in infinitely large samples.

The wide range of bootstrap methods can be classified into two broad approaches. First, the simplest bootstrap methods can permit statistical inference when conventional methods such as standard error computation are difficult to implement. Second, more complicated bootstraps can have the additional advantage of providing asymptotic refinements that can lead to a better approximation in-finite samples.

Applied researchers are most often interested in the first aspect of the bootstrap. Theoreticians emphasize the second, especially in settings where the usual asymptotic methods work poorly in finite samples.

The econometrics literature focuses on use of the bootstrap in hypothesis testing, which relies on approximation of probabilities in the tails of the distributions of statistics. Other applications are to confidence intervals, estimation of standard errors, and bias reduction.

Cross-section models have certain inherent limitations. They are predominantly equilibrium models that generally do not shed light on intertemporal dependence of events. They also cannot satisfactorily resolve fundamental issues about the sources of persistence in behavior. Such persistence may be behavioral, i.e. arising from true state dependence, or it may be spurious, being an artifact of the inability to control for heterogeneous behavior in the population. Because panel data, also called longitudinal data, contain periodically repeated observations of the same subjects, they have a large potential for resolving issues that cross-section models cannot satisfactorily handle. Chapters 21 through 23 present methods for panel data. We progress systematically from linear models for continuous data in Chapter 21 to nonlinear panel data models for limited dependent variables in Chapter 23. Both fixed effects and random effects models are considered. A persistent theme through these three chapters is the importance of using panel-robust methods of inference.

Chapter 21, which reviews the key general results for linear panel data regression models, can be read easily by those with a good grasp of linear regression; it does not require the material covered in Parts 2 to 4. We recommend that even those who are interested in more advanced material should quickly peruse through the contents of this chapter first to gain familiarity with key concepts and definitions.

Chapter 22 covers important extensions of Chapter 21, especially to dynamic panels which allow for Markovian dependence structure of current variables.