Supplementary Material A Proof of identification (3) zw (X, W, Z) = 1} = I{d zw D } I{d

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. 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). In practice, we use the empirical risk instead, i.e., In addition, we add a penalty term with respect to network weights to avoid overfitting. As for the hyperparameters tuning procedure, we consider employing multilayer perceptrons with 2-8 fully connected layers with a variable number of hidden units.