Intersection Bounds: Estimation and Inference

TL;DR

This paper introduces a new method for inference on intersection bounds, applicable to econometric models with complex inequality structures, and provides bias correction and advanced asymptotic theory for valid large-sample inference.

Contribution

It develops a practical inference method for intersection bounds, including bias correction and theory for non-Donsker empirical processes, expanding tools for econometric models with inequalities.

Findings

Median-bias correction improves finite-sample accuracy.

Valid asymptotic inference is achievable with non-Donsker processes.

New adaptive inequality/moment selection methods are proposed.

Abstract

We develop a practical and novel method for inference on intersection bounds, namely bounds defined by either the infimum or supremum of a parametric or nonparametric function, or equivalently, the value of a linear programming problem with a potentially infinite constraint set. We show that many bounds characterizations in econometrics, for instance bounds on parameters under conditional moment inequalities, can be formulated as intersection bounds. Our approach is especially convenient for models comprised of a continuum of inequalities that are separable in parameters, and also applies to models with inequalities that are non-separable in parameters. Since analog estimators for intersection bounds can be severely biased in finite samples, routinely underestimating the size of the identified set, we also offer a median-bias-corrected estimator of such bounds as a by-product of our inferential procedures. We develop theory for large sample inference based on the strong approximation of a sequence of series or kernel-based empirical processes by a sequence of "penultimate" Gaussian processes. These penultimate processes are generally not weakly convergent, and thus non-Donsker. Our theoretical results establish that we can nonetheless perform asymptotically valid inference based on these processes. Our construction also provides new adaptive inequality/moment selection methods. We provide conditions for the use of nonparametric kernel and series estimators, including a novel result that establishes strong approximation for any general series estimator admitting linearization, which may be of independent interest.