Supplementary Material A Proof of identification (3)

We state it here for clarity and completeness. The data generating mechanism for ( X, A,Z, W, U) is summarized in Table 1, and the setups of varying parameters in each scenario are summarized in Table 2. Table 1: Data generating mechanism and setup for fixed parameters across scenarios.21 X)null + ωW, (20) where the first equality is due to Assumption 1. Furthermore, note that E[h( W, 1, X)|X, Z,U ] = E[h( W, 1, X)| X,U ] = E[Y | X,A = 1, U] = E[Y | X,A = 1, Z,U ] = b X)null, 22 where the first and third equality is due to Assumption 1, the second equality follows from Theorem 1 of Miao et al. (2018a) under Assumptions 2 and 3, and the last equality is by (19). X) null + ωW, where the second equality is due to Assumption 1, and the third equality is due to Theorem 2.2 of Cui et al. (2023) under Assumptions 4 and 5, and the last equality is due to (20). Step (i) The method we adopt is neural maximum moment restriction (NMMR), which employs multilayer perceptron (MLP) to estimate the confounding bridges (Kompa et al., 2022).