Regression-Based Estimation of Causal Effects in the Presence of Selection Bias and Confounding

We consider the problem of estimating the expected causal effect $E[Y|do(X)]$ for a target variable $Y$ when treatment $X$ is set by intervention, focusing on continuous random variables. In settings without selection bias or confounding, $E[Y|do(X)] = E[Y|X]$, which can be estimated using standard regression methods. However, regression fails when systematic missingness induced by selection bias, or confounding distorts the data. Boeken et al. [2023] show that when training data is subject to selection, proxy variables unaffected by this process can, under certain constraints, be used to correct for selection bias to estimate $E[Y|X]$, and hence $E[Y|do(X)]$, reliably. When data is additionally affected by confounding, however, this equality is no longer valid. Building on these results, we consider a more general setting and propose a framework that incorporates both selection bias and confounding. Specifically, we derive theoretical conditions ensuring identifiability and recoverability of causal effects under access to external data and proxy variables. We further introduce a two-step regression estimator (TSR), capable of exploiting proxy variables to adjust for selection bias while accounting for confounding. We show that TSR coincides with prior work if confounding is absent, but achieves a lower variance. Extensive simulation studies validate TSR's correctness for scenarios which may include both selection bias and confounding with proxy variables.

A Cautionary Tale About "Neutrally" Informative AI Tools Ahead of the 2025 Federal Elections in Germany

In this study, we examine the reliability of AI-based Voting Advice Applications (VAAs) and large language models (LLMs) in providing objective political information. Our analysis is based upon a comparison with party responses to 38 statements of the Wahl-O-Mat, a well-established German online tool that helps inform voters by comparing their views with political party positions. For the LLMs, we identify significant biases. They exhibit a strong alignment (over 75% on average) with left-wing parties and a substantially lower alignment with center-right (smaller 50%) and right-wing parties (around 30%). Furthermore, for the VAAs, intended to objectively inform voters, we found substantial deviations from the parties' stated positions in Wahl-O-Mat: While one VAA deviated in 25% of cases, another VAA showed deviations in more than 50% of cases. For the latter, we even observed that simple prompt injections led to severe hallucinations, including false claims such as non-existent connections between political parties and right-wing extremist ties.