Imputation of Counterfactual Outcomes when the Errors are Predictable

TL;DR

This paper introduces a new method for imputing counterfactual outcomes in causal inference that leverages the predictability of errors, especially when errors are correlated, improving accuracy over traditional methods.

Contribution

It proposes the extit{Predictive Unbiased Predictor} ( extit{PUP}), a practical, non-linear approach that enhances counterfactual prediction by utilizing correlated error structures.

Findings

PUP improves mean-squared error in simulations.

The method is applicable with existing estimators.

Ignoring error predictability can bias conditional inference.

Abstract

A crucial input into causal inference is the imputed counterfactual outcome. Imputation error can arise because of sampling uncertainty from estimating the prediction model using the untreated observations, or from out-of-sample information not captured by the model. While the literature has focused on sampling uncertainty, it vanishes with the sample size. Often overlooked is the possibility that the out-of-sample error can be informative about the missing counterfactual outcome if it is mutually or serially correlated. Motivated by the best linear unbiased predictor (\blup) of \citet{goldberger:62} in a time series setting, we propose an improved predictor of potential outcome when the errors are correlated. The proposed \pup\; is practical as it is not restricted to linear models, can be used with consistent estimators already developed, and improves mean-squared error for a large class of strong mixing error processes. Ignoring predictability in the errors can distort conditional inference. However, the precise impact will depend on the choice of estimator as well as the realized values of the residuals.