Table 2.

Simulation 2 with continuous outcome to validate bias and type 1 error rate formulae

			Mean	Mean	Relative		Empirical	Expected
α	Mean F	Mean R 2	OLS estimate	IV estimate	bias	(Mean F)−1	Type 1 error	Type 1 error
0.01	1.2	0.2%	0.999	0.829	0.829	0.825	84.8%	89.5%
0.02	1.8	0.4%	0.998	0.531	0.532	0.545	65.5%	69.8%
0.03	2.9	0.6%	0.996	0.333	0.334	0.346	46.9%	46.1%
0.04	4.4	0.9%	0.993	0.216	0.218	0.229	32.5%	30.6%
0.05	6.3	1.2%	0.990	0.149	0.151	0.160	24.1%	21.8%
0.06	8.6	1.7%	0.985	0.109	0.111	0.117	19.2%	16.8%
0.07	11.3	2.2%	0.980	0.082	0.083	0.089	15.6%	13.6%
0.08	14.4	2.8%	0.974	0.062	0.064	0.069	12.4%	11.5%
0.09	18.0	3.5%	0.967	0.050	0.052	0.056	11.0%	10.1%
0.10	22.0	4.2%	0.960	0.040	0.042	0.046	9.8%	9.0%
0.15	48.3	8.8%	0.914	0.019	0.021	0.021	7.5%	6.6%
0.20	85.0	14.5%	0.856	0.010	0.012	0.012	5.9%	5.8%

Notes: Simulation results with null causal effect ${\beta }_X=0$, and confounder effect ${\beta }_U=2$ to estimate the relative bias and empirical Type 1 error rate (5% nominal significance level) of the two‐stage least squares (or equivalently, inverse‐variance weighted) instrumental variable (IV) estimate; the relative bias is the bias of the IV estimate divided by the bias of the ordinary least squares (OLS) estimate. The relative bias is theoretically predicted to be close to the reciprocal of the mean value of the F statistic, (Mean F)⁻¹.