ACE Bounds; SEMs with Equilibrium Conditions

This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in Statistical Science, 2014, Vol. 29, No. 3, 363–366. This reprint differs from the original in pagination and typographic detail. In other words, they do not involve both Y (x0) and Y (x1), nor X(zi) and X(zj) for i 6= j. where the instrument is randomized, as depicted by the DAG in Figure 1(a). Assumption (iii) may be read (via d-separation) from the Single-World Intervention Graph (SWIG)2 G1(z,x), depicted in Figure 1(b), which represents the factorization of P (Z,X(z), Y (x),U), implied by the IV model. Lastly (iv) consists of only three independence statements, but does assume the existence of an unobserved variable U that is sufficient to control for confounding between X and Y . No assumption is made concerning the existence of counterfactuals X(z); confounding variables (U) between Z and X are permitted (so long as U⊥⊥U ). The DAG G2 and corresponding SWIG G2(x) are shown in Figure 1(c), (d). In Richardson and Robins (2014), we prove the following.