Sparse Models and Methods for Optimal Instruments With an Application to Eminent Domain
Sparse Models and Methods for Optimal Instruments With an Application to Eminent Domain
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DOI：
10.3982/ecta9626
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发表时间：
2012-11-01
期刊：
影响因子：
6.1
通讯作者：
Hansen, C.
中科院分区：
文献类型：
--
作者：
Belloni, A.;Chen, D.;Hansen, C.
我们开发了使用Lasso和后Lasso方法的结果，以形成具有许多工具的线性工具变量（IV）模型中的第一阶段预测和估计最优工具，p。我们的结果即使在p远大于样样量n时也适用。我们表明，当第一阶段近似稀疏时，基于使用Lasso或后Lasso的IV估计量是根n一致的，并且渐近正态，即：当给定工具的内生变量的条件期望可以被一组相对较小的变量很好地近似时，这些变量的身份可能是未知的。我们还证明了当结构误差是等方差时，估计器是半参数有效的。值得注意的是，我们的结果允许不完美的模型选择，并且不依赖于广泛用于建立模型选择后推理有效性的不切实际的β -min条件（另见Belloni， Chernozhukov和Hansen (2011b)）。在模拟实验中，与最近提倡的多仪器鲁棒程序相比，基于lasso的具有数据驱动惩罚的IV估计器表现良好。在处理司法征用权决定对经济结果影响的实证示例中，基于lasso的IV估计器优于直观基准。最优工具是条件预期。在发展IV结果的过程中，我们建立了一系列关于非参数条件期望函数的Lasso和后Lasso估计量的新结果，这些结果具有独立的理论和实践意义。我们构造了一个Lasso的修改，用于处理非高斯的异方差干扰，它使用数据加权的11惩罚函数。通过创新地使用自归一化和的中等偏差理论，我们提供了在logp = o（n /3）的条件下得到的Lasso和后Lasso估计的收敛速率，其收敛速率与均匀高斯情况下的相应速率一样快。我们还提供了一种数据驱动的方法，用于选择在获得Lasso和后Lasso估计时必须指定的惩罚水平，并建立了其在非高斯、异方差干扰下的渐近有效性。
We develop results for the use of Lasso and post-Lasso methods to form first-stage predictions and estimate optimal instruments in linear instrumental variables (IV) models with many instruments, p. Our results apply even when p is much larger than the sample size, n. We show that the IV estimator based on using Lasso or post-Lasso in the first stage is root-n consistent and asymptotically normal when the first stage is approximately sparse, that is, when the conditional expectation of the endogenous variables given the instruments can be well-approximated by a relatively small set of variables whose identities may be unknown. We also show that the estimator is semiparametrically efficient when the structural error is homoscedastic. Notably, our results allow for imperfect model selection, and do not rely upon the unrealistic beta-min conditions that are widely used to establish validity of inference following model selection (see also Belloni, Chernozhukov, and Hansen (2011b)). In simulation experiments, the Lasso-based IV estimator with a data-driven penalty performs well compared to recently advocated many-instrument robust procedures. In an empirical example dealing with the effect of judicial eminent domain decisions on economic outcomes, the Lasso-based IV estimator outperforms an intuitive benchmark.Optimal instruments are conditional expectations. In developing the IV results, we establish a series of new results for Lasso and post-Lasso estimators of nonparametric conditional expectation functions which are of independent theoretical and practical interest. We construct a modification of Lasso designed to deal with non-Gaussian, heteroscedastic disturbances that uses a data-weighted l1-penalty function. By innovatively using moderate deviation theory for self-normalized sums, we provide convergence rates for the resulting Lasso and post-Lasso estimators that are as sharp as the corresponding rates in the homoscedastic Gaussian case under the condition that logp = o(n1/3). We also provide a data-driven method for choosing the penalty level that must be specified in obtaining Lasso and post-Lasso estimates and establish its asymptotic validity under non-Gaussian, heteroscedastic disturbances.