Title: Machine Learning for Set-Identified Linear Models,
Abstract: Set-identified models often restrict the number of covariates leading to wide
identified sets in practice. This paper provides estimation and inference
methods for set-identified linear models with high-dimensional covariates where
the model selection is based on modern machine learning tools. I characterize
the boundary (i.e, support function) of the identified set using a
semiparametric moment condition. Combining Neyman-orthogonality and sample
splitting ideas, I construct a root-N consistent, the uniformly asymptotically
Gaussian estimator of the support function. I also prove the validity of the
Bayesian bootstrap procedure to conduct inference about the identified set. I
provide a general method to construct a Neyman-orthogonal moment condition for
the support function. I apply this result to estimate sharp nonparametric
bounds on the average treatment effect in Lee (2008)'s model of endogenous
selection and substantially tighten the bounds on this parameter in Angrist et
al. (2006)'s empirical setting. I also apply this result to estimate sharp
identified sets for two other parameters - a new parameter, called a partially
linear predictor, and the average partial derivative when the outcome variable
is recorded in intervals.