Generalized Kernel Ridge Regression for Causal Inference with Missing-at-Random Sample Selection

Abstract

I propose kernel ridge regression estimators for nonparametric dose response curves and semiparametric treatment effects in the setting where an analyst has access to a selected sample rather than a random sample; only for select observations, the outcome is observed. I assume selection is as good as random conditional on treatment and a sufficiently rich set of observed covariates, where the covariates are allowed to cause treatment or be caused by treatment-an extension of missingness-at-random (MAR). I propose estimators of means, increments, and distributions of counterfactual outcomes with closed form solutions in terms of kernel matrix operations, allowing treatment and covariates to be discrete or continuous, and low, high, or infinite dimensional. For the continuous treatment case, I prove uniform consistency with finite sample rates. For the discrete treatment case, I prove √ n consistency, Gaussian approximation, and semiparametric efficiency.

Generalized Kernel Ridge Regression for Long Term Causal Inference: Treatment Effects, Dose Responses, and Counterfactual Distributions

“…[24] present an RKHS estimator for distributional ATT and ATE with binary treatment. The present work more closely relates to works that incorporate kernel ridge regression into treatment effect, dose response, and counterfactual distribution estimation in static [36], proximal [32], dynamic [37], and missing-at-random settings [34]. I prove equally strong results despite additional complexity in the chain of causal influence and in the scope for nonlinearity due to data fusion.…”

A Finite Sample Theorem for Longitudinal Causal Inference with Machine Learning: Long Term, Dynamic, and Mediated Effects

“…The techniques developed in this paper also apply to other localized longitudinal settings, e.g. heterogeneous sample selection and heterogeneous mediation analysis or selected, dynamic, and mediated dose response curves [45,39,86,82].…”