would be used for ̂. Then
̂ A T T
̂ −1N−1 ∑
Wi − p̂XiΔYi
1 − p̂Xi
is consistent and N -asymptotically normal. HIR discuss variance estimation. Imbens and Wooldridge (2007) provide a simple adjustment available in the case that p̂ is treated as a parametric model. ∙ Similar approach works for ATE ∙ Regression version: .
ΔYi on 1, Wi, p̂Xi, Wi − ̂ p̂Xi, i 1, . . . , N.
The coefficient on Wi is the estimated ATE. Requires some functional form restrictions. Certainly preferred to running the regression Yit on 1, d1t, d1t Wi, p̂Xi. This latter regression