Table 3.

Results of the simulation study under correct selection of γ and h. Each simulation scenario is described by μ_A (instrument strength, given by average level of engagement) and γ₀ (extent of departure from exclusion restriction); further, results are shown under two different sample sizes (N =50 and N =200). Provided in the table for reference is the implied intention-to-treat effect, Δ_ITT. Depicted are the average point estimates, empirical standard error (ESE), the average bootstrap standard error for ${\widehat{\Delta }}_{\gamma }(1)$, the coverage probability, and power. Note that in all cases, Δγ(1)=−0.80.

			N = 50					N = 200
μ A	γ 0	ΔITT	Est.	ESE	$\widehat{\text{SE}}$	CP	Power	Est.	ESE	$\widehat{\text{SE}}$	CP	Power
0.25	0.00	−0.20	−0.810	1.137	1.378	0.987	0.040	−0.794	0.486	0.503	0.966	0.370
0.25	0.25	−0.35	−0.794	0.556	0.563	0.956	0.293	−0.794	0.265	0.270	0.947	0.850
0.25	0.50	−0.50	−0.799	0.384	0.379	0.943	0.563	−0.798	0.187	0.187	0.951	0.990
0.25	0.75	−0.65	−0.809	0.293	0.290	0.946	0.796	−0.801	0.143	0.144	0.949	1.000
0.25	1.00	−0.80	−0.808	0.239	0.235	0.943	0.921	−0.801	0.120	0.117	0.942	1.000
0.50	0.00	−0.40	−0.806	0.494	0.507	0.958	0.371	−0.802	0.237	0.238	0.951	0.927
0.50	0.25	−0.50	−0.804	0.387	0.386	0.950	0.552	−0.800	0.187	0.188	0.954	0.989
0.50	0.50	−0.60	−0.802	0.320	0.317	0.942	0.717	−0.798	0.159	0.156	0.944	0.999
0.50	0.75	−0.70	−0.797	0.276	0.269	0.937	0.836	−0.798	0.134	0.134	0.951	1.000
0.50	1.00	−0.80	−0.799	0.240	0.235	0.938	0.917	−0.801	0.116	0.117	0.949	1.000
0.75	0.00	−0.60	−0.801	0.323	0.325	0.950	0.703	−0.803	0.158	0.157	0.948	0.999
0.75	0.25	−0.65	−0.801	0.295	0.295	0.948	0.770	−0.803	0.145	0.145	0.948	1.000
0.75	0.50	−0.70	−0.801	0.277	0.271	0.940	0.837	−0.801	0.135	0.134	0.947	1.000
0.75	0.75	−0.75	−0.804	0.253	0.250	0.943	0.891	−0.804	0.124	0.125	0.950	1.000
0.75	1.00	−0.80	−0.798	0.236	0.235	0.945	0.917	−0.800	0.117	0.117	0.951	1.000