On Estimating Causal Effects based on Supplemental Variables

This paper considers the problem of estimating causal effects of a treatment on a response using supplementary variables. Under the assumption that a treatment is associated with a response through a univariate supplementary variable in the framework of linear regression models, Cox (1960) showed that the estimation accuracy of the regression coefficient of the treatment on the response in the single linear regression model can be improved by using the recursive linear regression model based on the supplementary variable from the viewpoint of the asymptotic variance. However, such assumptions may not hold in many practical situations. In this paper, we consider the situation where a treatment is associated with a response through a set of supplementary variables in both linear and discrete models. Then, we show that the estimation accuracy of the causal effect can be improved by using the supplementary variables. Different from Cox (1960), the results of this paper are derived without the assumption of Gaussian error terms in linear models or dichotomous variables in discrete models. The results of this paper help us to obtain the reliable evaluation of causal effects from observed data.