Generalized method of moments with partially missing data

TL;DR

This paper develops a framework for generalized method of moments with partially missing data, providing sharp identified sets, hypothesis testing, and confidence regions, applicable to both continuous and discrete data.

Contribution

It introduces a novel approach to characterize identified sets with missing data using support functions, and proposes valid bootstrap methods for inference.

Findings

The proposed estimator performs well with moderate sample sizes.

The confidence regions are valid and reliable under the proposed method.

The approach is applicable to both continuous and discrete data.

Abstract

We consider a generalized method of moments framework in which a part of the data vector is missing for some units in a completely unrestricted, potentially endogenous way. In this setup, the parameters of interest are usually only partially identified. We characterize the identified set for such parameters using the support function of the convex set of moment predictions consistent with the data. This identified set is sharp, valid for both continuous and discrete data, and straightforward to estimate. We also propose a statistic for testing hypotheses and constructing confidence regions for the true parameter, show that standard nonparametric bootstrap may not be valid, and suggest a fix using the bootstrap for directionally differentiable functionals of Fang and Santos (2019). A set of Monte Carlo simulations demonstrates that both our estimator and the confidence region perform well when samples are moderately large and the data have bounded supports.