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Title: Learning non-smooth models: Instrumental variable quantile regressions and related problems Authors:  Yinchu Zhu - Brandeis University (United States) [presenting]
Abstract: Computationally efficient methods are proposed that can be used for instrumental variable quantile regressions (IVQR) and related methods with statistical guarantees. We prove that the GMM formulation of IVQR is NP-hard, and finding an approximate solution is also NP-hard. 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. The 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. We evaluate the performance of the proposal in simulations and an empirical example on the heterogeneous treatment effect of the Job Training Partnership Act.