Double Robustness in Estimation of Causal Treatment Effects

2. Counterfactual Model
Conditional expectation: For two random variables Y and Z,
E(Y |Z = z)
is the “average ” across all values of Y for which the corresponding value
of Z is equal to z
• E(Y |Z) is a function of Z taking on values E(Y |Z = z) as Z takes
on values z
• E{ E(Y |Z) } = E(Y ) (“average the averages ” across all possible
values of Z)
• E{ Y f(Z)|Z } = f(Z)E(Y |Z) – f(Z) is constant for any value of
Z so factors out of the average
• If Y � Z, then E(Y |Z = z) = E(Y ) for any z
Double Robustness, EPID 369, Spring 2007 9