The estimated causal effect on the variance based on the front-door criterion in Gaussian linear structural equation models: an unbiased estimator with the exact variance

In this paper, we assume that cause-effect relationships between random variables can be represented by a Gaussian linear structural equation model and the corresponding directed acyclic graph. Then, we consider a situation where a set of random variables that satisfies the front-door criterion is observed to estimate a total effect. In this situation, when the ordinary least squares method is utilized to estimate the total effect, we formulate the unbiased estimator of the causal effect on the variance of the outcome variable. In addition, we provide the exact variance formula of the proposed unbiased estimator.