Advanced Econometrics Takeshi Amemiya

Qualitative Response Models

9.1 Introduction

Qualitative response models (henceforth to be abbreviated as QR models) are regression models in which dependent (or endogenous) variables take discrete values. These models have numerous applications in economics because many behavioral responses are qualitative in nature: A consumer decides whether to buy a car or not; a commuter chooses a particular mode of trans­portation from several available ones; a worker decides whether to take a job offer or not; and so on. A long list of empirical examples of QR models can be found in my recent survey (Amemiya, 1981).

Qualitative response models, also known as quantal, categorical, or discrete models, have been used in biometric applications longer than they have been used in economics. Biometricians use the models to study, for example, the effect of an insecticide on the survival or death of an insect, or the effect of a drug on a patient. The kind of QR model used by biometridans is usually the simplest kind—univariate binary (or dichotomous) dependent variable (sur­vival or death) and a single independent variable (dosage).

Economists (and sodologists to a certain extent), on the other hand, must deal with more complex models, such as models in which a single dependent variable takes more than two discrete values (multinomial models) or models that involve more than one discrete dependent variable (multivariate models), as well as considering a larger number of independent variables. The estima­tion of the parameters of these complex models requires more elaborate techniques, many of which have been recently developed by econometricians.

This chapter begins with a discussion of the simplest model—the model for a univariate binary dependent variable (Section 9.2), and then moves on to multinomial and multivariate models (Sections 9.3 and 9.4). The emphasis here is on the theory of estimation (and hypothesis testing to a lesser extent) and therefore complementary to Amemiya’s survey mentioned earlier, which discussed many empirical examples and contained only fundamental results on the theory of statistical inference. We shall also discuss important topics

omitted by the survey—choice-based sampling (Section 9.5), distribution - free methods (Section 9.6), and panel data QR models (Section 9.7).