Two Layers of Instability in Causal Estimation
For researchers and practitioners in causal inference, this work provides a taxonomy of estimator stability with decision-theoretic implications.
The paper identifies two layers of instability in causal estimation: one inherent to the causal effect itself (discontinuous in data distribution) and another dependent on the estimator choice. It shows that standard estimators like IPW and regression can be discontinuous, while posterior means and medians are continuous.
There is a precise sense in which drawing causal inferences from observational data is hard, even when identifiability is assumed. In particular, Robins and Ritov (1997) and Robins et al. (2003) showed that causal effects can be discontinuous as a function of the data distribution: two arbitrarily close data distributions might correspond to different causal effects. This is a fact independent of the choice of estimator; however, not all estimators are equally unstable. Our contribution is to surface a second layer of instability that depends on the choice of estimator. We show that many standard point estimates can be read as point summaries of multimodal distributions over the space of structural causal models. As such, estimators can jump discontinuously in the data distribution. This defines a taxonomy of estimators that admits a decision-theoretic reading: stability depends on whether the implicit loss function an estimator optimizes is aligned with the causal effect itself. Specifically, inverse propensity weighted estimators and regression estimators are examples of discontinuous summaries, while explicit posterior means and medians are shown to be continuous.