Causal effect estimation for multivariate continuous treatments

Causal inference is widely used in various fields, such as biology, psychology, and economics, etc. In observational studies, balancing the covariates is an important step in estimating the causal effect. This study extends the one-dimensional entropy balancing method to multiple dimensions to balance the covariates. Both parametric and nonparametric methods are proposed to estimate the causal effect of multivariate continuous treatments and theoretical properties of the two estimations are provided. Furthermore, the simulation results show that the proposed method is better than other methods in various cases. Finally, the proposed method is applied to analyze the impact of the duration and frequency of smoking on medical expenditure. The results from the parametric method indicate that the frequency of smoking increases medical expenditure while the duration of smoking does not. The results from the nonparametric method indicate that there is a short-term downward trend and then a long-term upward trend as the duration and frequency of smoking increase.