Generated outcomes as generated regressors: Equivalences in recursive causal estimation

Abstract

Time-varying treatment effects, surrogate-identified treatment effects, and mediation effects can all be written as recursive regressions, in which each regression's predicted values become generated outcomes for the next regression. We study how standard causal estimators behave in this setting. Formally, we compare the recursive plug-in, recursive balancing weight, and recursive doubly robust estimators. When every stage is fitted by ordinary least squares (OLS), the three recursive estimators coincide in any finite sample, whether or not the models are correctly specified. As such, estimation by recursively regressing generated outcomes is numerically equivalent to estimation by recursively balancing generated regressors. Under ridge penalisation for the balancing weights, the doubly robust estimator is a backward recursion of stage-wise blends of penalised and OLS regressions. The weight on the recursive OLS regression decays geometrically in the number of time periods. Therefore, the intuition from the cross-sectional setting, where the bias correction moves the estimator towards OLS, applies less and less as the number of time periods increases. For general convex penalties, we derive an identity at each stage.