  In this paper, we apply doubly robust approach to estimate, when some
covariates are given, the conditional average treatment effect under
parametric, semiparametric and nonparametric structure of the nuisance
propensity score and outcome regression models. We then conduct a systematic
study on the asymptotic distributions of nine estimators with different
combinations of estimated propensity score and outcome regressions. The study
covers the asymptotic properties with all models correctly specified; with
either propensity score or outcome regressions locally / globally misspecified;
and with all models locally / globally misspecified. The asymptotic variances
are compared and the asymptotic bias correction under model-misspecification is
discussed. The phenomenon that the asymptotic variance, with
model-misspecification, could sometimes be even smaller than that with all
models correctly specified is explored. We also conduct a numerical study to
examine the theoretical results.
