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Important advances have recently been achieved in developing procedures yielding uniformly valid inference for a low dimensional causal parameter when high-dimensional nuisance models must be estimated. In this paper, we review the literature on uniformly valid causal inference and discuss the costs and benefits of using uniformly valid inference procedures. Naive estimation strategies based on regularisation, machine learning, or a preliminary model selection stage for the nuisance models have finite sample distributions which are badly approximated by their asymptotic distributions. To solve this serious problem, estimators which converge uniformly in distribution over a class of data generating mechanisms have been proposed in the literature. In order to obtain uniformly valid results in high-dimensional situations, sparsity conditions for the nuisance models need typically to be made, although a double robustness property holds, whereby if one of the nuisance model is more sparse, the other nuisance model is allowed to be less sparse. While uniformly valid inference is a highly desirable property, uniformly valid procedures pay a high price in terms of inflated variability. Our discussion of this dilemma is illustrated by the study of a double-selection outcome regression estimator, which we show is uniformly asymptotically unbiased, but is less variable than uniformly valid estimators in the numerical experiments conducted. 

Recently, various methods have been proposed to estimate causal effects with confidence intervals that are uniformly valid over a set of data generating processes, when high-dimensional nuisance models are estimated by post-model selection or machine learning estimators. These methods typically require that all the confounders are observed to ensure identification of the effects. We contribute by showing how valid semiparametric inference can be obtained in the presence of unobserved confounders and high-dimensional nuisance models. We propose uncertainty intervals which allow for unobserved confounding, and show that the resulting inference is valid when the amount of unobserved confounding is small relative to the sample size; the latter is formalized in terms of convergence rates. Simulation experiments illustrate the finite sample properties of the proposed intervals and investigate an alternative procedure that improves the empirical coverage of the intervals when the amount of unobserved confounding is large.

Convolutional neural networks (CNN) have been successful in machine learning applications including image classification. When it comes to images, their success relies on their ability to consider the space invariant local features in the data. Here, we consider the use of CNN to fit nuisance models in semiparametric estimation of a one dimensional causal parameter: the average causal effect of a binary treatment. In this setting, nuisance models are functions of pre-treatment covariates that need to be controlled for. In an application where we want to estimate the effect of early retirement on a health outcome, we propose to use CNN to control for time-structured covariates. Thus, CNN is used when fitting nuisance models explaining the treatment assignment and the outcome. These fits are then combined into an augmented inverse probability weighting estimator yielding efficient and uniformly valid inference. Theoretically, we contribute by providing rates of convergence for CNN equipped with the rectified linear unit activation function and compare it to an existing result for feedforward neural networks. We also show when those rates guarantee uniformly valid inference for the proposed estimator. A Monte Carlo study is provided where the performance of the proposed estimator is evaluated and compared with other strategies. Finally, we give results on a study of the effect of early retirement on later hospitalization using a database covering the whole Swedish population.

In Moosavi et al. (2022) a sensitivity analysis method to unobserved confounding was proposed when estimating an average causal effect with a double robust estimator in high dimensional situations. For this purpose, it was assumed that linear models could sparselyapproximate the nuisance functions (treatment assignment and outcome models). In this note, we relax these assumptions making the sensitivity analysis more generally applicable, for instance when nuisance functions are (weakly) consistently estimated with machine learning algorithms. Simulations and a case study illustrate the performance and use of the method.