[Paper Review] Learning non-smooth models: instrumental variable quantile regressions and related problems

Yinchu Zhu|arXiv (Cornell University)|May 17, 2018

Statistical Methods and Inference62 references4 citations

TL;DR

This paper proposes a computationally efficient k-step correction method for instrumental variable quantile regression (IVQR) that achieves asymptotically efficient estimation without solving the NP-hard GMM formulation. By using a fast, inconsistent initial estimator from mixed-integer linear programming and a tuning-free Jacobian estimator, the method enables scalable inference for heterogeneous treatment effects even with many endogenous regressors.

ABSTRACT

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 and finding an approximate solution is also 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.

Motivation & Objective

• Address the computational intractability of standard GMM-based IVQR estimation when multiple endogenous regressors arise from interaction terms.
• Overcome the NP-hard nature of IVQR GMM optimization, which precludes efficient global solution-finding.
• Develop a method that prioritizes statistical efficiency over computational optimality, enabling scalable inference in large samples.
• Provide a tuning-free estimator for the Jacobian matrix in non-smooth GMM models involving conditional densities.
• Enable practical estimation and inference for heterogeneous treatment effects in large-scale econometric applications, such as policy evaluation.

Proposed method

• Use a k-step correction procedure starting from an initial estimator obtained via mixed-integer linear programming (MILP), which can be computed in seconds.
• The initial estimator does not require consistency, allowing the use of very small subsamples to speed up computation.
• Apply k = 4 log n iterations of correction on the full sample to achieve asymptotic efficiency, even when the initial estimator is inconsistent.
• Propose a tuning-free Jacobian estimator that generalizes bootstrap quantile standard errors and is computed via closed-form solutions.
• Formulate the IVQR problem using moment conditions involving conditional quantiles and apply GMM-style inference without solving the original NP-hard optimization.
• Use Wald statistics and t-statistics for inference on model parameters, with variance-covariance matrices estimated via the proposed Jacobian estimator.

Experimental results

Research questions

• RQ1 Is the GMM formulation of IVQR computationally tractable, or is it NP-hard?
• RQ2 Can we achieve asymptotically efficient estimation in IVQR without solving the NP-hard GMM optimization problem?
• RQ3 Can we construct a tuning-free estimator for the Jacobian matrix in non-smooth GMM models involving conditional densities?
• RQ4 How does the proposed k-step correction method perform in terms of statistical efficiency and computational scalability?
• RQ5 To what extent do interaction terms between treatment and covariates reveal heterogeneous treatment effects in real-world policy evaluation?

Key findings

• The GMM formulation of IVQR is proven to be NP-hard, and finding an approximate solution within a constant factor is also NP-hard, establishing inherent computational intractability.
• The proposed k-step correction method achieves asymptotic efficiency with only k = 4 log n iterations, even when the initial estimator is inconsistent.
• The method scales efficiently to large samples because the initial estimator can be computed on a tiny subsample, and full-sample iterations are computationally lightweight.
• The proposed tuning-free Jacobian estimator generalizes bootstrap standard errors and enables efficient computation via closed-end solutions.
• In the JTPA empirical application, the null hypothesis of no heterogeneity in treatment effects (θ_j(τ) = 0) is strongly rejected at all quantiles, with Wald statistics exceeding 90 for τ = 0.85.
• The results show significant heterogeneity: married individuals benefit more from JTPA at higher income levels, and prior employment status affects treatment effects only among higher earners.