This paper surveys some of the recent literature on inference in partially identified models. After reviewing some basic concepts, including the definition of a partially identified model and the identified set, we turn our attention to the construction of confidence regions in partially identified settings. In our discussion, we emphasize the importance of requiring confidence regions to be uniformly consistent in level over relevant classes of distributions. Due to space limitations, our survey is mainly limited to the class of partially identified models in which the identified set is characterized by a finite number of moment inequalities or the closely related class of partially identified models in which the identified set is a function of a such a set. The latter class of models most commonly arises when interest focuses on a subvector of a vector-valued parameter, whose values are limited by a finite number of moment inequalities. We then rapidly review some important parts of the broader literature on inference in partially identified models and conclude by providing some thoughts on fruitful directions for future research.
A partially identified model is a model in which the parameter of interest is not uniquely determined by the distribution of the observed data. Instead, as we will explain further below, the parameter of interest is only limited by the distribution of the observed data to a set of possible values, commonly referred to as the identified set. Such models have a surprisingly long history: early contributions include the analysis of linear regressions with mismeasured regressors by Frisch (1934) and the analysis of Cobb-Douglas production functions by Marschak and Andrews (1944). Now, partially identified models are common in virtually all parts of economics and econometrics: measurement error (Klepper and Leamer, 1984; Horowitz and Manski, 1995), missing data (Manski, 1989, 1994; Horowitz and Manski, 1998; Manski and Tamer, 2002), industrial organization (Tamer, 2003; Haile and Tamer, 2003; Ho and Pakes, 2014; Pakes et al., 2015), finance (Hansen and Jagannathan, 1991; Hansen et al., 1995), labor economics (Blundell et al., 2007; Kline et al., 2013; Kline and Tartari, 2015), program evaluation (Manski, 1990, 1997; Manski and Pepper, 2000; Heckman and Vytlacil, 2001; Bhattacharya et al., 2008, 2012; Shaikh and Vytlacil, 2011), and macroeconomics (Faust, 1998; Canova and De Nicolo, 2002; Uhlig, 2005).