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, 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. 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 .
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 on an initial estimate. The initial estimate exploits the latest advances in mixed integer linear programming and can be computed within seconds. One theoretical contribution is that such initial estimators and Jacobian of the moment condition used in the k-step correction need not be even consistent and merely k=4\log n fast iterations are needed to obtain an efficient estimator. The overall proposal scales well to handle extremely large sample sizes because lack of consistency requirement allows one to use a very small subsample to obtain the initial estimate and the k-step iterations on the full sample can be implemented efficiently. Another contribution that is of independent interest is to propose a tuning-free estimation for the Jacobian matrix, whose definition nvolves conditional densities. This Jacobian estimator generalizes bootstrap quantile standard errors and can be efficiently computed via closed-end solutions. We evaluate the performance of the proposal in simulations and an empirical example on the heterogeneous treatment effect of Job Training Partnership Act .
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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
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
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framework, 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
on an initial estimate. The initial estimate exploits the latest advances in mixed integer linear programming and can be computed within seconds. One theoretical contribution is that such initial estimators and Jacobian of the moment condition used in the k-step correction need not be even consistent and merely k=4\log n fast
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to obtain an efficient estimator. The overall proposal scales well to handle extremely large sample sizes because lack of consistency requirement allows one to use a very small subsample to obtain the initial estimate and the
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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
iterations
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. As a result, we can handle massive datasets of millions of observations within seconds. As an empirical illustration, we examine
can be implemented efficiently. Another contribution that is of independent interest is to propose a tuning-free estimation for the Jacobian matrix, whose definition nvolves conditional densities. This Jacobian estimator generalizes bootstrap quantile standard errors and can be efficiently computed via closed-end solutions. We evaluate the performance of the proposal in simulations and an empirical example on
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(JTPA) using a regression with 13 interaction terms of the treatment variable