Inference in high-dimensional set-identified affine models

Abstract

This paper proposes both point-wise and uniform confidence sets (CS) for an element $\theta_{1}$ of a parameter vector $\theta\in\mathbb{R}^{d}$ that is partially identified by affine moment equality and inequality conditions. The method is based on an estimator of a regularized support function of the identified set. This estimator is \emph{half-median unbiased} and has an \emph{asymptotic linear representation} which provides closed form standard errors and enables optimization-free multiplier bootstrap. The proposed CS can be computed as a solution to a finite number of linear and convex quadratic programs, which leads to a substantial decrease in \emph{computation time} and \emph{guarantee of global optimum}. As a result, the method provides uniformly valid inference in applications with the dimension of the parameter space, $d$, and the number of inequalities, $k$, that were previously computationally unfeasible ($d,k >100$). The proposed approach is then extended to construct polygon-shaped joint CS for multiple components of $\theta$. Inference for coefficients in the linear IV regression model with interval outcome is used as an illustrative example. Key Words: Affine moment inequalities; Asymptotic linear representation; Delta\textendash Method; Interval data; Intersection bounds; Partial identification; Regularization; Strong approximation; Stochastic Programming; Subvector inference; Uniform inference.Comment: The earlier version of the paper was previously circulated under title "Inference on scalar parameters in set-identified affine models" and was a chapter in my PhD dissertatio