3 The selection problem in econometrics and statistics

Abstract

This chapter discusses the selection problem in econometrics and statistics. Because censored data are so common, econometricians and statisticians have denoted much effort to their analysis. In particular, the following selection problem has drawn substantial attention: each member of a population is characterized by a triple (y, z, x), where y lies in a finite dimensional real space Y, z = 0 or 1, and x lies in a finite dimensional real space X. The selection problem is the failure of the censored-sampling process to identify P(y│x). The sampling process does identify the selection probability P(z = 1Ix), the censoring probability P(z = 0│x), and the measure of y conditional on selection, P(y │x, z = 1). It is uninformative regarding the measure of y conditional on censoring, P(y│x, z = 0). Although the econometrics and statistics literatures on the selection problem differ in important respects, they both focus primarily on situations in which one has strong prior information on the distribution of (y, z) conditional on x. The chapter discusses the selection problem in the absence of prior information and the selection problem with prior information. Censoring creates an identification problem. Identification depends on the prior knowledge a researcher is willing to assert in the application of interest. As researchers are heterogeneous in their applications and in their prior beliefs, so must be their perspectives on the selection problem.