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

Abstract

This paper considers uniformly valid (over a class of data generating processes) inference for linear functionals of partially identified parameters in cases where the identified set is defined by linear (in the parameter) moment inequalities. We propose a bootstrap procedure for constructing uniformly valid confidence sets for a linear functional of a partially identified parameter. The proposed method amounts to bootstrapping the value functions of a linear optimization problem, and subsumes subvector inference as a special case. In other words, this paper shows the conditions under which "naively" bootstrapping a linear program can be used to construct a confidence set with uniform correct coverage for a partially identified linear functional. Unlike other proposed subvector inference procedures, our procedure does not require the researcher to repeatedly invert a hypothesis test, and is extremely computationally efficient. In addition to the new procedure, the paper also discusses connections between the literature on optimization and the literature on subvector inference in partially identified models.

Constraint Qualifications in Partial Identification

“…Also, if J = d, then the PPHI assumptions imply LICQ. This clarifies relation to recent work by Cho and Russell (2019) and Gafarov (2019): Both effectively impose LICQ and benefit from this by being able to propose relatively simple inference. However, while stronger than assumptions in CHT, BCS, and certainly Kaido, Molinari, and Stoye (2019), the assumptions exceed those in PPHI only in the sense of excluding "overidentified" support points, i.e.…”

“…The literature on partial identification uses constraint qualifications in many ways: To ensure Hausdorff consistency or rates of convergence for simple estimators of identified sets (Chernozhukov, Hong, and Tamer, 2007;Yildiz, 2012), to justify inference for the full parameter vector θ (Chernozhukov, Hong, and Tamer, 2007) or subvectors (Cho and Russell, 2019;Gafarov, 2019;Pakes, Porter, Ho, and Ishii, 2011), or to justify efficiency bounds (Kaido and Santos, 2014). However, only some of these uses are explicit, making it difficult even for expert readers to compare assumptions.…”

“…These conditions are frequently invoked in the statistical literature; e.g., LICQ or uniqueness of Lagrange multipliers (which are essentially the same assumption; see Wachsmuth ( 2013)) in Shapiro (1990Shapiro ( , 1991Shapiro ( , 1993. In econometrics, Cho andRussell (2019), Gafarov (2019), and Kaido and Santos (2014) recently invoke LICQ in order to achieve uniqueness of Lagrange multipliers respectively a Lagrangian representation of the support function. Recall also that with linear constraints (as in Gafarov ( 2019)), ACQ will always hold.…”

“…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?…”

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 ispointwise or uniformly-consistent 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 et al, 2011, PPHI henceforth Bugni, Canay, andShi, 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?…”