Logit and Probit Models

Abstract

Statistical models in which the endogenous random variables take only discrete values are known as discrete, categorical, qualitative-choice, or quantal response models. 1 This class of models was originally developed by psychologists and later adapted and extended by economists for describing consumers choices. These models have numerous applications because many behavioural responses are qualitative in nature: a commuter decides whether to take public transit; a consumer decides whether to buy a car; a high school student decides whether to attend a college; a housewife decides whether to participate in the labour force, etc. The focus of these models is usually on the individual decision making units. The estimation of these models is also facilitated by the availability of an increasingly large number of survey data on households and individual firms. In Section 16.1 we introduce the pro bit and logit models for panel data analysis. The maximum likelihood estimator (MLE), conditional MLE, and semi-parametric methods for estimating fixed effects binary choice models are reviewed in Section 16.2. The computational issues of estimating random effects models are discussed in Section 16.3. In Section 16.4 a two-step procedure for testing unobserved heterogeneity among cross-sectional units based on the work of Lee [1987] is suggested. We will not discuss the measurement errors issue since this is covered in the chapters on measurement errors (Chapter 10) and nonlinear latent variable models (Chapter 17). The literature on qualitative choice models, both applied and theoretical, is vast (e.g., Amemiya [1981], Maddala [1983], McFadden [1976]' [1981]' [1984], and Train [1985]). To facilitate the discussion on how one may utilize information provided by panel data to control for unobserved characteristics of individual units to avoid specification bias and to improve the efficiency of parameter estimates, we will focus on binary choice models, namely, the dependent variable can assume only two outcomes, the presence or absence of an event. We consider situations in which an analyst has at his disposal a random sample of N individuals having recorded histories indicating the presence or absence of an event in each of T equally spaced discrete time periods.