k-Step Correction for Mixed Integer Linear Programming: A New Approach for Instrumental Variable Quantile Regressions and Related Problems

Abstract

This paper proposes a new framework for estimating instrumental variable (IV) quantile models. The first part of our proposal can be cast as a mixed integer linear program (MILP), which allows us to capitalize on recent progress in mixed integer optimization. The computational advantage of the proposed method makes it an attractive alternative to existing estimators in the presence of multiple endogenous regressors. This is a situation that arises naturally when one endogenous variable is interacted with several other variables in a regression equation. In our simulations, the proposed method using MILP with a random starting point can reliably estimate regressions for a sample size of 500 with 20 endogenous variables in 5 seconds. Theoretical results for early termination of MILP are also provided. The second part of our proposal is a k-step correction frameowork, which is proved to be able to convert any point within a small but fixed neighborhood of the true parameter value into an estimate that is asymptotically equivalent to GMM. Our result does not require the initial estimate to be consistent and only 2 log(n) iterations are needed. Since the k-step correction does not require any optimization, applying the k-step correction to MILP estimate provides a computationally attractive way of obtaining efficient estimators. When dealing with very large data sets, we can run the MILP algorithm on only a small subsample and our theoretical results guarantee that the resulting estimator from the k-step correction is equivalent to computing GMM on the full sample. As a result, we can handle massive datasets of millions of observations within seconds. In Monte Carlo simulations, we observe decent performance of confidence intervals even if MILP uses only 0.01% of samples of size 5 million. As an empirical illustration, we examine the heterogeneous treatment effect of Job Training Partnership Act (JTPA) using a regression with 13 interaction terms of the treatment variable.