Asymptotic Properties for a Class of Partially Identified Models

Abstract

We propose inference procedures for partially identified population features for which the population identification region can be written as a transformation of the Aumann expectation of a properly defined set valued random variable (SVRV). An SVRV is a mapping that associates a set (rather than a real number) with each element of the sample space. Examples of population features in this class include interval-identified scalar parameters, best linear predictors with interval outcome data, and parameters of semiparametric binary models with interval regressor data. We extend the analogy principle to SVRVs and show that the sample analog estimator of the population identification region is given by a transformation of a Minkowski average of SVRVs. Using the results of the mathematics literature on SVRVs, we show that this estimator converges in probability to the population identification region with respect to the Hausdorff distance. We then show that the Hausdorff distance and the directed Hausdorff distance between the population identification region and the estimator, when properly normalized by √ n converge in distribution to functions of a Gaussian process whose covariance kernel depends on parameters of the population identification region. We provide consistent bootstrap procedures to approximate these limiting distributions. Using similar arguments as those applied for vector valued random variables, we develop a methodology to test assumptions about the true identification region and its subsets. We show that these results can be used to construct a confidence collection and a directed confidence collection. Those are (respectively) collection of sets that, when specified as a null hypothesis for the true value (a subset of values) of the population identification region, cannot be rejected by our tests.

Rounding Probabilistic Expectations in Surveys

“…Beresteanu and Molinari (2008) show that when v is only known to lie in the random interval [ v L , v U ], the identification region for β , denoted H [ β ], is the set of parameter values that solve a set of equations similar to (2). Formally,…”

“…Beresteanu and Molinari (2008) show that the identification region (3) for the BLP parameter vector can be estimated as a Minkowski average of properly defined set-valued random variables. The identification region for each component of the BLP parameter vector can be estimated by simple linear projections.…”

“…We focus on familiar problems of regression and best linear prediction, where the objective is to predict the quantity of interest conditional on specified covariates. Manski and Tamer (2002), Manski (2003), Chernozhukov, Hong, and Tamer (2007), and Beresteanu and Molinari (2008) have studied identification of and statistical inference on regressions and best linear predictors with interval data. We draw on their work and use the HRS data to illustrate.…”

Identification partielle : méthodes et conséquences pour les applications empiriques

“…Les techniques générales d'inférence reposent soit sur la construction d'une fonction critère et l'inversion d'un test basé sur cette fonction (Chernozhukov, Hong et Tamer, 2007) soit sur la caractérisation directe de la frontière (Beresteanu et Molinari, 2008;Bontemps et al, 2012). On peut résumer ces deux méthodes par :…”

“…Pour les extensions au modèle de prédiction linéaire à plusieurs variables, on pourra se référer à la section 2.3 où les résultats de Stoye (2007), Beresteanu et Molinari (2008) et Bontemps, Magnac et Maurin (2012) sont brièvement développés.…”