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.
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