Table 1. Open in new tab A Comparison of...

	Strong IV (R² = 0.025) (Theoretical F = 50)^b	Moderate IV (R² = 0.005) (Theoretical F = 10)	Weak IV (R² = 0.002) (Theoretical F = 5)
Subsample IV approach
Delta method	0.148	0.057	0.037, 0.259	0.152	0.132	−0.108, 0.411	0.081	0.161	−0.234, 0.397
Sequential regression^c		0.055	0.039, 0.256		0.128	−0.099, 0.403		0.159	−0.231, 0.394
Fieller's theorem		N/A	0.040, 0.272		N/A	−0.108, 0.562		N/A	−0.291, 0.602
Bootstrap^d		0.068	0.014, 0.280		0.137	−0.117, 0.421		0.551	−0.999, 1.162
Bayesian	0.143	0.056	0.040, 0.258	0.174	0.289	−0.161, 0.778	0.089	0.443	−0.563, 0.975
2-sample IV approach
Delta method	0.117	0.068	−0.015, 0.250	0.051	0.119	−0.182, 0.284	−0.086	0.163	−0.405, 0.232
Sequential regression^c		0.065	−0.011, 0.245		0.118	−0.181, 0.282		0.160	−0.440, 0.227
Fieller's theorem		N/A	−0.012, 0.267		N/A	−0.201, 0.336		N/A	−0.610, 0.280
Bootstrap^d		0.071	−0.023, 0.257		0.138	−0.221, 0.322		0.997	−2.041, 1.868
Bayesian	0.119	0.072	−0.013, 0.273	0.055	0.169	−0.232, 0.390	−0.100	0.456	−1.012, 0.554

Strong IV (R² = 0.025)
(Theoretical F = 50)^b

Moderate IV (R² = 0.005)
(Theoretical F = 10)

Weak IV (R² = 0.002)
(Theoretical F = 5)

Subsample IV approach

Delta method

0.148

0.057

0.037, 0.259

0.152

0.132

−0.108, 0.411

0.081

0.161

−0.234, 0.397

Sequential regression^c

0.055

0.039, 0.256

0.128

−0.099, 0.403

0.159

−0.231, 0.394

Fieller's theorem

N/A

0.040, 0.272

N/A

−0.108, 0.562

N/A

−0.291, 0.602

Bootstrap^d

0.068

0.014, 0.280

0.137

−0.117, 0.421

0.551

−0.999, 1.162

Bayesian

0.143

0.056

0.040, 0.258

0.174

0.289

−0.161, 0.778

0.089

0.443

−0.563, 0.975

2-sample IV approach

Delta method

0.117

0.068

−0.015, 0.250

0.051

0.119

−0.182, 0.284

−0.086

0.163

−0.405, 0.232

Sequential regression^c

0.065

−0.011, 0.245

0.118

−0.181, 0.282

0.160

−0.440, 0.227

Fieller's theorem

N/A

−0.012, 0.267

N/A

−0.201, 0.336

N/A

−0.610, 0.280

Bootstrap^d

0.071

−0.023, 0.257

0.138

−0.221, 0.322

0.997

−2.041, 1.868

Bayesian

0.119

0.072

−0.013, 0.273

0.055

0.169

−0.232, 0.390

−0.100

0.456

−1.012, 0.554

	Strong IV (R² = 0.025) (Theoretical F = 50)^b	Moderate IV (R² = 0.005) (Theoretical F = 10)	Weak IV (R² = 0.002) (Theoretical F = 5)
Subsample IV approach
Delta method	0.148	0.057	0.037, 0.259	0.152	0.132	−0.108, 0.411	0.081	0.161	−0.234, 0.397
Sequential regression^c		0.055	0.039, 0.256		0.128	−0.099, 0.403		0.159	−0.231, 0.394
Fieller's theorem		N/A	0.040, 0.272		N/A	−0.108, 0.562		N/A	−0.291, 0.602
Bootstrap^d		0.068	0.014, 0.280		0.137	−0.117, 0.421		0.551	−0.999, 1.162
Bayesian	0.143	0.056	0.040, 0.258	0.174	0.289	−0.161, 0.778	0.089	0.443	−0.563, 0.975
2-sample IV approach
Delta method	0.117	0.068	−0.015, 0.250	0.051	0.119	−0.182, 0.284	−0.086	0.163	−0.405, 0.232
Sequential regression^c		0.065	−0.011, 0.245		0.118	−0.181, 0.282		0.160	−0.440, 0.227
Fieller's theorem		N/A	−0.012, 0.267		N/A	−0.201, 0.336		N/A	−0.610, 0.280
Bootstrap^d		0.071	−0.023, 0.257		0.138	−0.221, 0.322		0.997	−2.041, 1.868
Bayesian	0.119	0.072	−0.013, 0.273	0.055	0.169	−0.232, 0.390	−0.100	0.456	−1.012, 0.554

Table 1.

Open in new tab

A Comparison of Different Methods of Estimating 95% Confidence Intervals for Selected Simulated Data Sets^a

	Strong IV (R² = 0.025) (Theoretical F = 50)^b			Moderate IV (R² = 0.005) (Theoretical F = 10)			Weak IV (R² = 0.002) (Theoretical F = 5)
	β	SE	CI	β	SE	CI	β	SE	CI
Subsample IV approach
Delta method	0.148	0.057	0.037, 0.259	0.152	0.132	−0.108, 0.411	0.081	0.161	−0.234, 0.397
Sequential regression^c		0.055	0.039, 0.256		0.128	−0.099, 0.403		0.159	−0.231, 0.394
Fieller's theorem		N/A	0.040, 0.272		N/A	−0.108, 0.562		N/A	−0.291, 0.602
Bootstrap^d		0.068	0.014, 0.280		0.137	−0.117, 0.421		0.551	−0.999, 1.162
Bayesian	0.143	0.056	0.040, 0.258	0.174	0.289	−0.161, 0.778	0.089	0.443	−0.563, 0.975
2-sample IV approach
Delta method	0.117	0.068	−0.015, 0.250	0.051	0.119	−0.182, 0.284	−0.086	0.163	−0.405, 0.232
Sequential regression^c		0.065	−0.011, 0.245		0.118	−0.181, 0.282		0.160	−0.440, 0.227
Fieller's theorem		N/A	−0.012, 0.267		N/A	−0.201, 0.336		N/A	−0.610, 0.280
Bootstrap^d		0.071	−0.023, 0.257		0.138	−0.221, 0.322		0.997	−2.041, 1.868
Bayesian	0.119	0.072	−0.013, 0.273	0.055	0.169	−0.232, 0.390	−0.100	0.456	−1.012, 0.554

	Strong IV (R² = 0.025) (Theoretical F = 50)^b			Moderate IV (R² = 0.005) (Theoretical F = 10)			Weak IV (R² = 0.002) (Theoretical F = 5)
	β	SE	CI	β	SE	CI	β	SE	CI
Subsample IV approach
Delta method	0.148	0.057	0.037, 0.259	0.152	0.132	−0.108, 0.411	0.081	0.161	−0.234, 0.397
Sequential regression^c		0.055	0.039, 0.256		0.128	−0.099, 0.403		0.159	−0.231, 0.394
Fieller's theorem		N/A	0.040, 0.272		N/A	−0.108, 0.562		N/A	−0.291, 0.602
Bootstrap^d		0.068	0.014, 0.280		0.137	−0.117, 0.421		0.551	−0.999, 1.162
Bayesian	0.143	0.056	0.040, 0.258	0.174	0.289	−0.161, 0.778	0.089	0.443	−0.563, 0.975
2-sample IV approach
Delta method	0.117	0.068	−0.015, 0.250	0.051	0.119	−0.182, 0.284	−0.086	0.163	−0.405, 0.232
Sequential regression^c		0.065	−0.011, 0.245		0.118	−0.181, 0.282		0.160	−0.440, 0.227
Fieller's theorem		N/A	−0.012, 0.267		N/A	−0.201, 0.336		N/A	−0.610, 0.280
Bootstrap^d		0.071	−0.023, 0.257		0.138	−0.221, 0.322		0.997	−2.041, 1.868
Bayesian	0.119	0.072	−0.013, 0.273	0.055	0.169	−0.232, 0.390	−0.100	0.456	−1.012, 0.554

Abbreviations: CI, confidence interval; IV, instrumental variable; N/A, not applicable; SE, standard error.

^a The simulated data sets consisted of 10,000 persons with data on G and Y and 2,000 persons with data on G and X. The true effect of X on Y was set to 0.1, and a confounding variable U had the effect of 0.2 on both X and Y.

^b Theoretical F values were obtained using the following equation: F = R² (n_X − 1)/(1 − R²).

^c For the second-stage regression (of sequential regression), robust SEs are reported.

^d Bootstrapping was conducted using 1,000 replications, with samples of size n_X and n_Y randomly selected (with replacement) from the original samples of size n_X and n_Y.

This Feature Is Available To Subscribers Only