Mixed integer linear programming: a new approach for instrumental variable quantile regressions and related problems
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This paper proposes a new framework for estimating instrumental variable (IV) quantile models. 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 1000 with 20 endogenous variables in 90 seconds; for high-dimensional problems, our formulation can deliver decent estimates within minutes for problems with 550 endogenous regressors. We also establish asymptotic theory and provide an inference procedure. In our simulations, the asymptotic theory provides an excellent approximation even if we terminate MILP before a certified global solution is found. This suggests that MILP in our setting can quickly approach the global solution. In addition, we show that MILP can also be used for related problems, including censored regression, censored IV quantile regression and high-dimensional IV quantile regression. 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.
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