For a data-generating process for random variables that can be described with a linear structural equation model, we consider a situation in which (i) a set of covariates satisfying the back-door criterion cannot be observed or (ii) such a set can be observed, but standard statistical estimation methods cannot be applied to estimate causal effects because of multicollinearity/high-dimensional data problems. We propose a novel two-stage penalized regression approach, the penalized covariate-mediator selection operator (PCM Selector), to estimate the causal effects in such scenarios. Unlike existing penalized regression analyses, when a set of intermediate variables is available, PCM Selector provides a consistent or less biased estimator of the causal effect. In addition, PCM Selector provides a variable selection procedure for intermediate variables to obtain better estimation accuracy of the causal effects than does the back-door criterion.

This paper assumes that cause-effect relationships among variables can be described with a linear structural equation model. Then, a situation is considered where a set of observed covariates satisfies the back-door criterion but the ordinary least squares method cannot be applied to estimate linear causal effects because of multicollinearity/high-dimensional data problems. In this situation, we propose a novel regression approach, the “partially adaptive L$_p$-regularized multiple regression analysis” (PAL$_p$MA) method for estimating the total effects. Different from standard regularized regression analysis, PAL$_p$MA provides a consistent or less-biased estimator of the linear causal effect. PAL$_p$MA is also applicable to evaluating direct effects through the single-door criterion. Given space constraints, the proofs, some numerical experiments, and an industrial case study on setting up painting conditions of car bodies are provided in the Supplementary Material.