学习非光滑模型：工具变量分位数回归 及相关问题

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摘要翻译：
本文提出了计算效率高的工具变量分位数回归(IVQR)方法和有统计保证的相关方法。当我们研究异质性处理效应时，这是非常必要的，因为内源处理和控制变量之间的相互作用导致了内源协变量的增加。我们证明了IVQR的GMM公式是NP难的，求近似解也是NP难的。因此，从纯粹的计算角度来解决这个问题似乎是不太可能的。相反，我们的目标是得到一个具有良好统计性质的估计，并且不一定是任何优化问题的全局解。该建议包括对初始估计采用$k$-步修正。初始估计利用了混合整数线性规划的最新进展，可以在几秒钟内计算出来。一个理论上的贡献是，这种初始估计量与K步修正中所用的矩条件的雅可比量甚至不需要一致，只需要$k=4\logn$快速迭代就可以得到一个有效的估计量。由于缺乏一致性要求，可以使用很小的子样本来获得初始估计值，并且可以有效地实现对全样本的k步迭代，因此整个方案能够很好地处理超大样本。另一个独立的贡献是对雅可比矩阵提出了一个无调谐估计，它的定义是条件密度。该雅可比估计推广了bootstrap分位数标准误差，并可通过闭端解有效地计算。我们在模拟和一个关于工作培训伙伴关系法案异质待遇效应的实证例子中评估了该建议的绩效。
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英文标题：
《Learning non-smooth models: instrumental variable quantile regressions
  and related problems》
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作者：
Yinchu Zhu
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最新提交年份：
2019
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分类信息：

一级分类：Economics        经济学
二级分类：Econometrics        计量经济学
分类描述：Econometric Theory, Micro-Econometrics, Macro-Econometrics, Empirical Content of Economic Relations discovered via New Methods, Methodological Aspects of the Application of Statistical Inference to Economic Data.
计量经济学理论，微观计量经济学，宏观计量经济学，通过新方法发现的经济关系的实证内容，统计推论应用于经济数据的方法论方面。
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一级分类：Statistics        统计学
二级分类：Computation        计算
分类描述：Algorithms, Simulation, Visualization
算法、模拟、可视化
--
一级分类：Statistics        统计学
二级分类：Methodology        方法论
分类描述：Design, Surveys, Model Selection, Multiple Testing, Multivariate Methods, Signal and Image Processing, Time Series, Smoothing, Spatial Statistics, Survival Analysis, Nonparametric and Semiparametric Methods
设计，调查，模型选择，多重检验，多元方法，信号和图像处理，时间序列，平滑，空间统计，生存分析，非参数和半参数方法
--

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英文摘要：
  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.
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PDF链接：
https://arxiv.org/pdf/1805.06855

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关键词：分位数回归 工具变量 分位数 econometrics Contribution 一致性 instrumental variable 可比