Before revision: 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 framework,
After revision: This paper proposes computationally efficient methods that can be used for instrumental variable quantile regressions (IVQR) and related methods with statistical guarantees. This is much needed when we investigate heterogenous treatment effects since interactions between the endogenous treatment and control variables lead to an increased number of endogenous covariates. We prove that the GMM formulation of IVQR is NP-hard. Hence, solving the problem from a purely computational perspective seems unlikely. Instead, we aim to obtain an estimate that has good statistical properties and is not necessarily the global solution of any optimization problem. The proposal consists of employing k-step correction framework,