  We expand upon the simulation study of Setodji et al. (2017) which compared
three promising balancing methods when assessing the average treatment effect
on the treated for binary treatments: generalized boosted models (GBM),
covariate-balancing propensity scores (CBPS), and entropy balance (EB). The
study showed that GBM can outperform CBPS and EB when there are likely to be
non-linear associations in both the treatment assignment and outcome models and
CBPS and EB are fine-tuned to obtain balance only on first order moments. We
explore the potential benefit of using higher-order moments in the balancing
conditions for CBPS and EB. Our findings showcase that CBPS and EB should, by
default, include higher order moments and that focusing only on first moments
can result in substantial bias in both CBPS and EB estimated treatment effect
estimates that could be avoided by the use of higher moments.
