We derive general, yet simple, sharp bounds on the size of the omitted variable bias for a broad class of causal parameters that can be identified as linear functionals of the conditional expectation function of the outcome. Such functionals encompass many of the traditional targets of investigation in causal inference studies, such as, for example, (weighted) average of potential outcomes, average treatment effects (including subgroup effects, such as the effect on the treated), (weighted) average derivatives, and policy effects from shifts in covariate distribution -- all for general, nonparametric causal models. Our construction relies on the Riesz-Frechet representation of the target functional. Specifically, we show how the bound on the bias depends only on the additional variation that the latent variables create both in the outcome and in the Riesz representer for the parameter of interest. Moreover, in many important cases (e.g, average treatment effects and avearage derivatives) the bound is shown to depend on easily interpretable quantities that measure the explanatory power of the omitted variables. Therefore, simple plausibility judgments on the maximum explanatory power of omitted variables (in explaining treatment and outcome variation) are sufficient to place overall bounds on the size of the bias. Furthermore, we use debiased machine learning to provide flexible and efficient statistical inference on learnable components of the bounds. Finally, empirical examples demonstrate the usefulness of the approach.

翻译：我们推导出针对一大类可识别为结果条件期望函数线性泛函的因果参数的遗漏变量偏误大小的通用、简洁且精确的界。这类泛函涵盖了因果推断研究中许多传统研究目标，例如（加权）潜在结果均值、平均处理效应（包括子组效应，如处理组效应）、（加权）平均导数，以及协变量分布变化带来的政策效应——所有这些均基于一般的非参数因果模型。我们的构造依赖于目标泛函的Riesz-Fréchet表示。具体而言，我们展示了偏误边界如何仅取决于潜在变量在结果和感兴趣参数的Riesz表示中引入的额外变异。此外，在许多重要情形下（例如平均处理效应和平均导数），该边界被证明依赖于衡量遗漏变量解释力的易于解释的量。因此，关于遗漏变量最大解释力（在解释处理变量和结果变异方面）的简单可信判断足以对偏误大小设置整体边界。进一步，我们利用去偏机器学习对边界的可学习分量提供灵活且高效的统计推断。最后，实证例子展示了该方法的实用性。