Inference for best linear approximations to set identified functions

TL;DR

This paper develops inference methods for best linear approximations to functions within a band, extending partial identification to include parametric and non-parametric bounds, with applications in various econometric models.

Contribution

It introduces a versatile framework for inference on linear approximations to band-defined functions, accommodating indexed bounds and providing limit theory and bootstrap validity.

Findings

Support function converges to a Gaussian process

Bayesian bootstrap is valid for inference

Framework applies to multiple econometric problems

Abstract

This paper provides inference methods for best linear approximations to functions which are known to lie within a band. It extends the partial identification literature by allowing the upper and lower functions defining the band to be any functions, including ones carrying an index, which can be estimated parametrically or non-parametrically. The identification region of the parameters of the best linear approximation is characterized via its support function, and limit theory is developed for the latter. We prove that the support function approximately converges to a Gaussian process and establish validity of the Bayesian bootstrap. The paper nests as special cases the canonical examples in the literature: mean regression with interval valued outcome data and interval valued regressor data. Because the bounds may carry an index, the paper covers problems beyond mean regression; the framework is extremely versatile. Applications include quantile and distribution regression with interval valued data, sample selection problems, as well as mean, quantile, and distribution treatment effects. Moreover, the framework can account for the availability of instruments. An application is carried out, studying female labor force participation along the lines of Mulligan and Rubinstein (2008).