  Propensity score methods have been shown to be powerful in obtaining
efficient estimators of average treatment effect (ATE) from observational data,
especially under the existence of confounding factors. When estimating,
deciding which type of covariates need to be included in the propensity score
function is important, since incorporating some unnecessary covariates may
amplify both bias and variance of estimators of ATE. In this paper, we show
that including additional instrumental variables that satisfy the exclusion
restriction for outcome will do harm to the statistical efficiency. Also, we
prove that, controlling for covariates that appear as outcome predictors, i.e.
predict the outcomes and are irrelevant to the exposures, can help reduce the
asymptotic variance of ATE estimation. We also note that, efficiently
estimating the ATE by non-parametric or semi-parametric methods require the
estimated propensity score function, as described in Hirano et al.
(2003)\cite{Hirano2003}. Such estimation procedure usually asks for many
regularity conditions, Rothe (2016)\cite{Rothe2016} also illustrated this point
and proposed a known propensity score (KPS) estimator that requires mild
regularity conditions and is still fully efficient. In addition, we introduce a
linearly modified (LM) estimator that is nearly efficient in most general
settings and need not estimation of the propensity score function, hence
convenient to calculate. The construction of this estimator borrows idea from
the interaction estimator of Lin (2013)\cite{Lin2013}, in which regression
adjustment with interaction terms are applied to deal with data arising from a
completely randomized experiment. As its name suggests, the LM estimator can be
viewed as a linear modification on the IPW estimator using known propensity
scores. We will also investigate its statistical properties both analytically
and numerically.
