论文标题
治疗效应估计，没有观察到异质的混杂变量
Treatment Effect Estimation with Unobserved and Heterogeneous Confounding Variables
论文作者
论文摘要
在存在未观察到的混杂变量的情况下，通常将治疗效果的估计估计通常被称为隐藏变量。尽管最近提出了一些方法来处理隐藏变量的效果，但这些方法通常忽略了观察到的治疗变量与未观察到的协变量之间任何相互作用的可能性。在这项工作中，我们通过研究一个多变量响应回归问题来解决这一缺点，以及表格$ y = y = a^t x+ b^t x+ b^t z+ sum_ {变量，$ x \ in \ mathbb {r}^p $是观察到的协变量（包括处理变量），$ z \ in \ mathbb {r}^k $是$ k $ - 二维的未观察到的混杂因素，$ e \ in \ in \ mathbb {r}^m $是随机噪声。允许$ x_j $和$ z $之间的相互作用引起异质混杂效果。我们的目标是估算未知的矩阵$ a $，观察到的协变量的直接影响或对响应的处理。为此，我们通过SVD提出了一种新的伪造估计方法，以消除未观察到的混杂变量的效果。估计量的收敛速率均在均质和异性噪声下建立。我们还提供了几个模拟实验和一个现实世界数据应用程序，以证实我们的发现。
The estimation of the treatment effect is often biased in the presence of unobserved confounding variables which are commonly referred to as hidden variables. Although a few methods have been recently proposed to handle the effect of hidden variables, these methods often overlook the possibility of any interaction between the observed treatment variable and the unobserved covariates. In this work, we address this shortcoming by studying a multivariate response regression problem with both unobserved and heterogeneous confounding variables of the form $Y=A^T X+ B^T Z+ \sum_{j=1}^{p} C^T_j X_j Z + E$, where $Y \in \mathbb{R}^m$ are $m$-dimensional response variables, $X \in \mathbb{R}^p$ are observed covariates (including the treatment variable), $Z \in \mathbb{R}^K$ are $K$-dimensional unobserved confounders, and $E \in \mathbb{R}^m$ is the random noise. Allowing for the interaction between $X_j$ and $Z$ induces the heterogeneous confounding effect. Our goal is to estimate the unknown matrix $A$, the direct effect of the observed covariates or the treatment on the responses. To this end, we propose a new debiased estimation approach via SVD to remove the effect of unobserved confounding variables. The rate of convergence of the estimator is established under both the homoscedastic and heteroscedastic noises. We also present several simulation experiments and a real-world data application to substantiate our findings.
