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would be used for ̂. Then

  • ̂ A T T

N

  • ̂ 1N1

i1

  • Wi p̂XiΔYi

  • 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

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