Full Simulation Results for Each Method (nsims = 200)
across different treatment heterogeneity levels
Method	True ATE	Estimated ATE	Bias	Analytic SE	Monte Carlo SE	MSE	Coverage
Setting 1
Oracle Estimator	0.996	0.985	−0.011	0.034	0.035	0.001	0.920
Identification Formula 1	0.996	0.971	−0.026	0.038	0.035	0.002	0.885
Identification Formula 1, Misspecified	0.996	0.968	−0.028	0.034	0.030	0.002	0.900
Identification Formula 2a	0.996	0.983	−0.014	0.051	0.052	0.003	0.950
Identification Formula 2b	0.996	0.956	−0.041	0.046	0.045	0.004	0.835
Setting 2
Oracle Estimator	1.000	0.991	−0.009	0.034	0.033	0.001	0.960
Identification Formula 1	1.000	0.976	−0.024	0.038	0.032	0.002	0.940
Identification Formula 1, Misspecified	1.000	0.983	−0.017	0.034	0.028	0.001	0.970
Identification Formula 2a	1.000	0.992	−0.008	0.050	0.052	0.003	0.935
Identification Formula 2b	1.000	0.955	−0.045	0.047	0.050	0.004	0.825
Setting 3
Oracle Estimator	1.050	1.051	0.001	0.034	0.030	0.001	0.970
Identification Formula 1	1.050	1.035	−0.015	0.038	0.029	0.001	0.965
Identification Formula 1, Misspecified	1.050	1.059	0.008	0.035	0.026	0.001	0.985
Identification Formula 2a	1.050	1.041	−0.009	0.051	0.048	0.002	0.955
Identification Formula 2b	1.050	1.012	−0.037	0.046	0.043	0.003	0.870
Setting 4
Oracle Estimator	1.106	1.098	−0.008	0.034	0.034	0.001	0.950
Identification Formula 1	1.106	1.082	−0.022	0.038	0.033	0.002	0.925
Identification Formula 1, Misspecified	1.106	1.130	0.021	0.036	0.030	0.001	0.930
Identification Formula 2a	1.106	1.102	−0.003	0.050	0.047	0.002	0.965
Identification Formula 2b	1.106	1.070	−0.033	0.047	0.044	0.003	0.880
Setting 5
Oracle Estimator	1.501	1.489	−0.008	0.035	0.031	0.001	0.960
Identification Formula 1	1.501	1.470	−0.020	0.037	0.034	0.002	0.895
Identification Formula 1, Misspecified	1.501	1.720	0.146	0.041	0.029	0.049	0.000
Identification Formula 2a	1.501	1.497	−0.002	0.050	0.053	0.003	0.925
Identification Formula 2b	1.501	1.467	−0.023	0.048	0.047	0.003	0.900