摘要翻译：
我们提出了一个综合控制方法的鲁棒推广比较案例研究。与经典方法一样，我们提出了一种算法来估计治疗单元的不可观测反事实。我们的算法的一个显著特点是通过奇异值阈值化对数据矩阵进行去噪，这使得我们的方法在多个方面具有鲁棒性：它自动识别良好的捐赠者子集，克服了丢失数据的挑战，并在可能无法提供协变量信息的情况下继续工作。首先，我们确定了合成控制方法中的基本假设成立的条件，即当治疗单位和供体池之间的线性关系在干预前和干预后期间都占上风时。我们提供了第一个有限样本分析的更广泛的一类模型，潜变量模型，与以前在文献中考虑的因素模型相比。进一步地，我们证明了我们的去噪过程准确地输入了缺失项，对于一些$\zeta>0$，产生了基础信号矩阵的一致估计量$p=\omega(t^{-1+\zeta}）$；这里，$P$是观察数据的分数，$T$是感兴趣的时间间隔。在相同的条件下，我们证明了预测估计中的均方差(MSE)为$O(\sigma^2/p+1/\sqrt{T}）$,其中$\sigma^2$是噪声方差。利用数据聚合方法，我们证明了对于(0,1/2)$中的任意$\gamma\，MSE可以小到$O(t^{-1/2+\gamma})$,从而得到一致估计量。我们还引入了一个贝叶斯框架，通过后验概率来量化模型的不确定性。我们的实验，使用真实世界和合成数据集，证明了我们的鲁棒泛化产生了一个改进的经典合成控制方法。
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英文标题：
《Robust Synthetic Control》
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作者：
Muhammad Jehangir Amjad, Devavrat Shah, and Dennis Shen
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最新提交年份：
2017
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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：Applications        应用程序
分类描述：Biology, Education, Epidemiology, Engineering, Environmental Sciences, Medical, Physical Sciences, Quality Control, Social Sciences
生物学，教育学，流行病学，工程学，环境科学，医学，物理科学，质量控制，社会科学
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一级分类：Statistics        统计学
二级分类：Machine Learning        机器学习
分类描述：Covers machine learning papers (supervised, unsupervised, semi-supervised learning, graphical models, reinforcement learning, bandits, high dimensional inference, etc.) with a statistical or theoretical grounding
覆盖机器学习论文（监督，无监督，半监督学习，图形模型，强化学习，强盗，高维推理等）与统计或理论基础
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英文摘要：
  We present a robust generalization of the synthetic control method for comparative case studies. Like the classical method, we present an algorithm to estimate the unobservable counterfactual of a treatment unit. A distinguishing feature of our algorithm is that of de-noising the data matrix via singular value thresholding, which renders our approach robust in multiple facets: it automatically identifies a good subset of donors, overcomes the challenges of missing data, and continues to work well in settings where covariate information may not be provided. To begin, we establish the condition under which the fundamental assumption in synthetic control-like approaches holds, i.e. when the linear relationship between the treatment unit and the donor pool prevails in both the pre- and post-intervention periods. We provide the first finite sample analysis for a broader class of models, the Latent Variable Model, in contrast to Factor Models previously considered in the literature. Further, we show that our de-noising procedure accurately imputes missing entries, producing a consistent estimator of the underlying signal matrix provided $p = \Omega( T^{-1 + \zeta})$ for some $\zeta > 0$; here, $p$ is the fraction of observed data and $T$ is the time interval of interest. Under the same setting, we prove that the mean-squared-error (MSE) in our prediction estimation scales as $O(\sigma^2/p + 1/\sqrt{T})$, where $\sigma^2$ is the noise variance. Using a data aggregation method, we show that the MSE can be made as small as $O(T^{-1/2+\gamma})$ for any $\gamma \in (0, 1/2)$, leading to a consistent estimator. We also introduce a Bayesian framework to quantify the model uncertainty through posterior probabilities. Our experiments, using both real-world and synthetic datasets, demonstrate that our robust generalization yields an improvement over the classical synthetic control method.
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PDF链接：
https://arxiv.org/pdf/1711.06940