Double Robustness in Estimation of Causal Treatment Effects

E
5. Doubly Robust Estimator
� �
{Z − e(X, β)}
{Y1 − E(Y |Z = 1, X)}
e(X, β)
�� � ��
{Z − e(X, β)}
�
= E
{Y1 − E(Y |Z = 1, X)} �
e(X, β)
� Z, X
� �
{Z − e(X, β)}
= E
E [{Y1 − E(Y |Z = 1, X)}| Z, X]
e(X, β)
� �
{Z − e(X, β)}
= E
{E(Y1|Z, X) − E(Y |Z = 1, X)}
e(X, β)
� �
{Z − e(X, β)}
= E
{E(Y1|X) − E(Y1|X)} = 0 (1)
e(X, β)
(1) uses no unmeasured confounders, which says
E(Y |Z = 1, X) = E(Y1|Z = 1, X) = E(Y1|X) = E(Y1|Z, X)
Double Robustness, EPID 369, Spring 2007 33