Online Appendix Limit properties and variance: A matching estimator based on a bi-level optimization problem

In this document we develop limit properties of a version of the BLOP matching estimator introduced by Dı́az, Rau & Rivera (Forthcoming). We show that this estimator is consistent and asymptotically normal, and that the conditional bias is Op(N −1/K), with K being the dimension of the vector of covariates. We introduce a consistent estimator of its marginal variance. 1 Basic concepts and notation Let Ω = (X, Y,W ) be a random vector whereW ∈ {0, 1} indicates whether a treatment was received (W = 1) or not (W = 0), and X ∈ X represents a vector of individual characteristics or covariates with dimension K ∈ N. For simplicity covariates are assumed to be continuous random variables and X is the supporting set of these variables. Departamento de Economı́a, Universidad de Chile. Diagonal Paraguay 257, Santiago, Chile, email : juadiaz@fen.uchile. cl Instituto de Economı́a, Pontificia Universidad Católica de Chile. Vicuña Mackenna 4860, Santiago, Chile, email : trau@uc.cl Departamento de Economı́a, Universidad de Chile. Diagonal Paraguay 257, Santiago, Chile, email : jrivera@econ.uchile. cl 1 The observed outcome is Y = W Y (1) + (1−W ) Y (0), with Y (1) and Y (0) being the outcomes that the individual would have obtained with or without treatment –the potential outcomes according to Rosenbaum & Rubin (1983)–. For x ∈ X and w ∈ {0, 1}, the conditional expectation and the conditional variance of Y are denoted by μw(x) = E(Y (w) | X = x), σ w(x) = V(Y (w) | X = x), (1) μ(x, w) = E(Y | X = x,W = w), σ(x, w) = V(Y | X = x,W = w), (2) thus the average treatment effect, ATE, and the average treatment effect on the treated, ATT, are, respectively τ = E (μ(x, 1)− μ(x, 0)) , τtre = E (μ(x, 1)− μ(x, 0) | W = 1) . The assumptions below are needed for limit properties, and they will be part of our standing assumptions throughout this work. These are quite standard in the literature. See Rosenbaum & Rubin (1983), Heckman, Ichimura & Todd (1998), Imbens (2004) and Imbens & Wooldridge (2009) for a detailed discussion on them. Assumption 1. X is a compact and convex subset of R and the distribution of X is bounded away from zero. Assumption 2. Unconfounded Treatment Assignment: W ⊥⊥ ((Y (0), Y (1)) | X). Assumption 3. Overlap: there is 0 < c < 1 such that 0 < P(W = 1 | X) < 1− c. In view of the standing assumptions, it is easy to see that μ(x, w) = μw(x), σ (x, w) = σ w(x). For N ∈ N, let Ω = {(Xi, Yi,Wi)}i=1 be a sample of Ω, and N1 and N0 are the 2 number of treated and control units, respectively. We note that for each unit i the number of its opposites regarding the treatment is N1−Wi ∈ {N0, N1}. By reordering, in what follows we assume that control units are indexed by 1, . , N0, thus the treated ones are labeled by N0 + 1, . , N0 +N1 ( = N). Without loss of generality, in this work we use the Euclidean norm, ‖ · ‖, as the matching metric. Given that, using the notation in Abadie & Imbens (2006), for m ∈ N we set jm(i) ∈ {1, . , N} as the index of the first m-th nearest neighbor to unit i ∈ {1, . , N} in the opposite treatment group, and the set of covariates for the first M matches for this unit is denoted by XM(i), that is, 2 XM(i) = { Xj1(i), . , XjM (i) } . For further purposes regarding the study of the variance, we set j m(i) ∈ N as the index of the first m-nearest neighbor to unit i in the same treatment group, leaving the i-th unit out. For integer M , we set X ∗ M(i) = { Xj∗ 1 (i), . , Xj∗ M (i) } ⊆ {Xj : Wj = Wi, j 6= i} . The “BLOP matching estimator” introduced in Dı́az et al. (Forthcoming) rests on the weighting scheme that for a treated unit i solves the following optimization problem: