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Florian, Gunsilius. (2021) Distributional synthetic controls.
In: Papers. RePEc:arx:papers:2001.06118.
Full description at Econpapers
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The function W2 2 (P, PYjt ) is continuous in P in the weak topology (with respect to Cq(Yt)) for fixed {PYjt }j=2,...,J+1. This follows from the fact that convergence in the Wasserstein metric on compact sets is equivalent to weak convergence (Santambrogio 2015, Theorem 5.9). Continuous functions on compact sets obtain their optimum, which shows that P(λ) exists for each λ ∈ ∆J−1 . Uniqueness of P(λ) for every λ ∈ ∆J−1 then follows from Proposition 3.5 in Agueh & Carlier (2011) under the assumption that at PYjt is absolutely continuous with respect to Lebesgue measure for j = 2, . . . , J + 1 and t ≤ T0.
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