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This paper presents a simple and efficient estimation of the average treatment effect (ATE) and local average treatment effect (LATE) in models with unmeasured confounders. In contrast to existing studies that estimate some unknown functionals in the influence function either parametrically or semiparametrically, we do not model the influence function nonparametrically. Instead, we apply the calibration method to a growing number of moment restrictions to estimate the weighting functions nonparametrically and then estimate the ATE and LATE by plugging in. The calibration method is similar to the covariatebalancing method in that both methods exploit the moment restrictions. The difference is that the calibration method imposes the sample analogue of the moment restrictions, while the covariate-balancing method does not. A simulation study reveals that our estimators have good finite sample performance and out-ATE WITH UNMEASURED CONFOUNDERS 2 perform existing alternatives. An application to the empirical analysis of return to education illustrates the practical value of the proposed method.

To investigate causal mechanisms, causal mediation analysis decomposes the total treatment effect into the natural direct and indirect effects. This paper examines the estimation of the direct and indirect effects in a general treatment effect model, where the treatment can be binary, multi-valued, continuous, or a mixture. We propose generalized weighting estimators with weights estimated by solving an expanding set of equations. Under some sufficient conditions, we show that the proposed estimators are consistent and asymptotically normal. Specifically, when the treatment is discrete, the proposed estimators attain the semiparametric efficiency bounds. Meanwhile, when the treatment is continuous, the convergence rates of the proposed estimators are slower than O(N −1/2 ); however, they are still more efficient than that constructed from the true weighting function. A simulation study reveals that our estimators exhibit a satisfactory finite-sample performance, while an application shows their practical value.

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