Econometrics with Partial Identification

Econometrics has traditionally revolved around point identi cation. Much effort has been devoted to finding the weakest set of assumptions that, together with the available data, deliver point identifi cation of population parameters, finite or infi nite dimensional that these might be. And point identifi cation has been viewed as a necessary prerequisite for meaningful statistical inference. The research program on partial identifi cation 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 identifi cation 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 identifi cation. 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 identifi cation region. Econometrics with partial identi fication is concerned with: (1) obtaining a tractable characterization of the parameters' sharp identi fication region; (2) providing methods to estimate it; (3) conducting test of hypotheses and making con fidence statements about the partially identi fied parameters. Each of these goals poses challenges that differ from those faced in econometrics with point identifi cation. This chapter discusses these challenges and some of their solution. It reviews advances in partial identifi cation analysis both as applied to learning (functionals of) probability distributions that are well-defi ned in the absence of models, as well as to learning parameters that are well-defi ned only in the context of particular models. The chapter highlights a simple organizing principle: the source of the identi fication 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 identi fication.