Inference in high-dimensional set-identified affine models

Abstract

This paper proposes both point-wise and uniform confidence sets (CS) for an element θ 1 of a parameter vector θ ∈ 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 half-median unbiased and has an 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 computation time and 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 θ. Inference for coefficients in the linear IV regression model with interval outcome is used as an illustrative example.

Simple Inference on Functionals of Set-Identified Parameters Defined by Linear Moments

“…However, we allow for the linear functional in the bounding problem to be data-dependent, and both our bootstrap procedure and our proof of uniform validity are very different. Overall, we believe our contribution is both practical and theoretical, and complements this recent work by Gafarov (2019).…”

“…Constraint qualifications in various forms have appeared throughout the recent history of partial identification (e.g. Beresteanu and Molinari (2008), Pakes et al (2011) Kaido and Santos (2014), Freyberger and Horowitz (2015, Kaido et al (2019a), Gafarov et al (2018), andGafarov (2019)). We refer to the recent paper of Kaido et al (2019b) for a full comparison of the constraint qualifications used in partial identification.…”

“…The overall approach to constructing confidence sets in this paper is most closely related to the approach in Gafarov (2019), who also shows how to construct uniformly valid confidence sets for linear functionals of a partially identified parameter in an optimization framework. It is well known from Hirano and Porter (2012) that it is impossible to obtain a locally unbiased estimator of the value function when the value function is nondifferentiable, and to address these problems Gafarov (2019) proposes including a regularization term in the objective function to ensure a unique optimal solution is selected. In contrast we assume the existence of unique optimal solutions.…”

“…In contrast we assume the existence of unique optimal solutions. Similar to Gafarov (2019), we also impose a linear independence constraint qualification to ensure uniqueness of the Lagrange multipliers. However, we allow for the linear functional in the bounding problem to be data-dependent, and both our bootstrap procedure and our proof of uniform validity are very different.…”

Constraint Qualifications in Partial Identification

“…While these assumptions can be dispensed with when the researcher's goal is to obtain a confidence set for the partially identified parameter vector that is -pointwise or uniformlyconsistent in level (e.g., Andrews and Soares (2010)), related assumptions reappear when the aim is to obtain a confidence interval for a smooth function of the partially identified parameter vector that is -pointwise or uniformly-consistent in level (e.g., Pakes, Porter, Ho, and Ishii (2011, PPHI henceforth), Bugni, Canay, and Shi (2017, BCS henceforth)). 1 Some more recent contributions (Cho and Russell, 2019;Gafarov, 2019) observe a connection to stochastic programming and show that inference becomes much more tractable under the so-called Linear Independence Constraint Qualification. Some obvious questions arise: How do all these assumptions relate?…”