Partially identified models occur commonly in economic applications. A common problem in this literature is a regression problem with bracketed (interval-censored) outcome variable Y, which creates a set-identified parameter of interest. The recent studies have only considered finite-dimensional linear regression in such context. To incorporate more complex controls into the problem, we consider a partially linear projection of Y on the set functions that are linear in treatment/policy variables and nonlinear in the controls. We characterize the identified set for the linear component of this projection and propose an estimator of its support function. Our estimator converges at parametric rate and has asymptotic normality properties. It may be useful for labor economics applications that involve bracketed salaries and rich, high-dimensional demographic data about the subjects of the study .
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 .
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Partially identified models occur commonly in economic applications. A common problem in this literature is a regression problem with bracketed (interval-censored) outcome variable Y, which creates a
Set-identified models often restrict the number of covariates leading to wide identified sets in practice. This paper provides estimation and inference methods for
R
parameter of interest. The recent studies have only considered finite-dimensional linear regression in such context. To incorporate more complex controls into the problem, we consider a partially linear projection of Y on the set functions that are linear in treatment/policy variables and nonlinear in the controls. We characterize the identified set for
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
R
linear component of this projection and propose an estimator of its
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
R
Our estimator converges at parametric rate and has asymptotic normality properties. It may be useful for labor economics applications that involve bracketed salaries and rich, high-dimensional demographic data about the subjects of the study
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