Partial Identification in Econometrics

Abstract

Suppose that one wants to use sample data to draw conclusions about a population of interest. Econometricians have long found it useful to separately study identification problems and problems of statistical inference. Studies of identification characterize the conclusions that could be drawn if one were able to observe an unlimited number of realizations of the sampling process. Studies of statistical inference characterize the generally weaker conclusions that can be drawn given a sample of positive but finite size. Koopmans (1949, p. 132) put it this way in the article that introduced the term ‘identification’: In our discussion we have used the phrase ‘a parameter that can be determined from a sufficient number of observations.’ We shall now define this concept more sharply, and give it the name identifiability of a parameter. Instead of reasoning, as before, from ‘a sufficiently large number of observations’ we shall base our discussion on a hypothetical knowledge of the probability distribution of the observations, as defined more fully below. It is clear that exact knowledge of this probability distribution cannot be derived from any finite number of observations. Such knowledge is the limit approachable but not attainable by extended observation. By hypothesizing nevertheless the full availability of such knowledge, we obtain a clear separation between problems of statistical inference arising from the variability of finite samples, and problems of identification in which we explore the limits to which inference even from an infinite number of observations is suspect.