Model selection and inference for estimation of causal parameters

In causal inference there are often multiple reasonable estimators for a given target quantity. For example, one may reasonably use inverse probability weighting, an instrumental variables approach, or construct an estimate based on proxy outcomes if the actual outcome is difficult to measure. Ideally, the practitioner decides on an estimator before looking at the data. However, this might be challenging in practice since a priori it might not be clear to a practitioner how to choose the method. If the final model is chosen after peeking at the data, naive inferential procedures may fail. This raises the need for a model selection tool, with rigorous asymptotic guarantees. Since there is usually no loss function available in causal inference, standard model selection techniques do not apply. We propose a model selection procedure that estimates the squared pound 2- deviation of a finite-dimensional estimator from its target. The procedure relies on knowing an asymptotically unbiased (potentially highly variable) estimate of the parameter of interest. The resulting estimator is discontinuous and does not have a Gaussian limit distribution. Thus, standard asymptotic expansions do not apply. We derive asymptotically valid confidence intervals for low-dimensional settings that take into account the model selection step. The performance of the approach for estimation and inference for average treatment effects is evaluated on simulated data sets in low-dimensional settings, including experimental data, instrumental variables settings and observational data with selection on observables.