Econometrics has traditionally revolved around point identication. Much effort has been devoted to finding the weakest set of assumptions that, together with the available data, deliver point identification of population parameters, finite or infinite dimensional that these might be. And point identification has been viewed as a necessary prerequisite for meaningful statistical inference. The research program on partial identification has begun to slowly shift this focus in the early 1990s, gaining momentum over time and developing into a widely researched area of econometrics. Partial identification has forcefully established that much can be learned from the available data and assumptions imposed because of their credibility rather than their ability to yield point identification. Within this paradigm, one obtains a set of values for the parameters of interest which are observationally equivalent given the available data and maintained assumptions. I refer to this set as the parameters’ sharp identification region.
Econometrics with partial identification is concerned with: (1) obtaining a tractable characterization of the parameters’ sharp identification region; (2) providing methods to estimate it; (3) conducting test of hypotheses and making confidence statements about the partially identified parameters. Each of these goals poses challenges that differ from those faced in econometrics with point identification. This chapter discusses these challenges and some of their solution. It reviews advances in partial identification analysis both as applied to learning (functionals of) probability distributions that are well-defined in the absence of models, as well as to learning parameters that are well-defined only in the context of particular models. The chapter highlights a simple organizing principle: the source of the identification problem can often be traced to a collection of random variables that are consistent with the available data and maintained assumptions. This collection may be part of the observed data or be a model implication. In either case, it can be formalized as a random set. Random set theory is then used as a mathematical framework to unify a number of special results and produce a general methodology to conduct econometrics with partial identification.