In this paper, we derive and analyze the implied weights of linear regression methods for causal inference. We obtain new closed-form, finite-sample expressions of the weights for various types of estimators based on multivariate linear regression models. In finite samples, we show that the implied weights have minimum variance, exactly balance the means of the covariates (or transformations thereof) included in the model, and produce estimators that may not be sample bounded. Furthermore, depending on the specification of the regression model, we show that the implied weights may distort the structure of the sample in such a way that the resulting estimator is biased for the average treatment effect for a given target population. In large samples, we demonstrate that, under certain functional form assumptions, the implied weights are consistent estimators of the true inverse probability weights. We examine doubly robust properties of regression estimators from the perspective of their implied weights. We also derive and analyze the implied weights of weighted least squares regression. The equivalence between minimizing regression residuals and optimizing for certain weights allows us to bridge ideas from the regression modeling and causal inference literatures. As a result, we propose a set of regression diagnostics for causal inference. We discuss the connection of the implied weights to existing matching and weighting approaches. As special cases, we analyze the implied weights in common settings such as multi-valued treatments, regression after matching, and two-stage least squares regression with instrumental variables.
翻译:在本文中,我们得出并分析线性回归方法的隐含因果推算权重。 我们获得了基于多变量线性回归模型的新型封闭式、 有限样本式的各类估计值权重表达方式。 在有限的样本中,我们显示隐含的权重有最小差异,精确平衡模型中包含的共变(或变换)手段,并产生可能不受抽样约束的估测器。 此外,根据回归模型的规格,我们显示隐含的权重可能会扭曲样本的结构,从而导致的估算值偏向于特定目标人群的平均治疗效果。 在大样本中,我们显示隐含的权重是最小的。 在某种功能性假设中,隐含的权重是最小的,我们从其隐含的重量的角度来研究回归估测算器的较强性特性。 我们还根据回归模型的隐含的权重来计算和分析加权最低方回归值回归的隐含的权重。 将回归余残留值与某些普通权重的等值的等值与最不相等值的等值的等值的等值, 使得我们能够用某种普通的比重的比重来连接,我们从某些的回归法性分析法的模型中, 讨论现有的因果关系, 和正值分析法系的正值的因果关系,我们从目前的因果关系,我们从现有的因果关系,我们用一个正值分析法系的正值分析法系中, 讨论。