Linear programming approach to partially identified econometric models

The bounds on a parameter of interest in partially identified settings are often given by the values of linear programs (LP). This paper studies estimation and inference for the value of a LP, where all the problem's parameters are inferred from the data. We develop the first root-n-consistent estimator that does not require additional restrictions. Unlike existing methods, our estimator remains valid under point-identification, over-identifying constraints and solution multiplicity. Exact and computationally simple inference procedure is developed. Turning to uniform properties, we prove that there exists no uniformly consistent estimator absent further conditions. We propose the 'delta-condition', under which our estimator is uniformly consistent. Delta-condition does not rule out economically relevant problematic scenarios, covers the unrestricted set of measures in the limit, and is strictly weaker than all previously proposed restrictions. We complement our estimation approach with a general identification result for models described by affine inequalities over conditional moments (AICM), potentially augmented with relevant almost sure restrictions on the potential outcomes and missing data conditions. Sharp bounds on affine treatment parameters under AICM are shown to take the form of a LP. Our results allow applied work to employ previously intractable conditions, including arbitrary combinations of existing restrictions, and conduct sensitivity analysis. We apply our findings to estimating returns to education. For that, we develop the conditionally monotone IV assumption (cMIV) that tightens classical bounds. We argue that cMIV remains unrestrictive relative to the classical conditions and provide a formal test for it. Under cMIV, university education in Colombia is shown to increase the average wage by at least 5.91%. In contrast, classical conditions fail to produce an informative bound.