Model selection for estimation of causal parameters

Abstract: A popular technique for selecting and tuning machine learning estimators is cross-validation. Cross-validation evaluates overall model fit, usually in terms of predictive accuracy. In causal inference, the optimal choice of estimator depends not only on the fitted models but also on assumptions the statistician is willing to make. In this case, the performance of different (potentially biased) estimators cannot be evaluated by checking overall model fit. We propose a model selection procedure that estimates the squared l2-deviation of a finite-dimensional estimator from its target. The procedure relies on knowing an asymptotically unbiased "benchmark estimator" of the parameter of interest. Under regularity conditions, we investigate bias and variance of the proposed criterion compared to competing procedures and derive a finite-sample bound for the excess risk compared to an oracle procedure. 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 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, including experimental data, instrumental variables settings, and observational data with selection on observables.