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Empirical specification description
"We use the inverse of the propensity scores as weights."
For ATT, control units should be weighted by ωᵢ / (1 - ωᵢ), not 1 / ωᵢ, where ωᵢ is the propensity score of unit i. Suggestions: if the nonstandard weights are appropriate in this context, explain them when the weights are discussed.
Address case when x is negative in Proposition 2.
"Lemma 2. The optimal level y(x) satisfies..."
The proof of the lemma implicitly assumes the argument x is nonnegative, but the maintained assumptions do not require this. Suggestions: either add an argument to deal with the negative case, or make an assumption to rule out negative x.
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