Then the average long-term treatment effect is identifiable:
τ = a∈{0,1} (−1) 1−a E h E (a, X) | G = O + E P (G = E) P (G = O | X) P (G = O) P (G = E | X) I [A = a] P (A = a | X, G = E) h (S 3 , S 2 , A, X) −h E (A, X) | G = E + E P (G = E | A = a) P (G = O | X) P (G = O | A = a) P (G = E | X) I [A = a] P (A = a | X, G = E) q (S 2 , S 1 , A, X) (Y − h (S 3 , S 2 , A, X)) | G = O
Theorem 7 shows that even under weaker Assumptions 10 and 11, outcome and selection bridge functions can be still used to identify the average long-term treatment effect. This again has the doubly robust property in that it only requires one of the bridge functions to be correct rather than both. Compared to Theorem 3, Theorem 7 additionally incorporates the ratios P (G = O | X) /P (G = E | X) to adjust for the covariate distribution discrepancy in the two types of **data** (Assumption 10). It also uses the propensity score P (A = a | X, G = E) to account for the dependence between treatment A and covariates X in the experimental **data** (Assumption 11).