Table 1.

Simulation results for our estimator of γ_XY·Z with PSRs derived from linear models (LM), kernel estimation (kernel), and semiparametric cumulative probability models (CPM) with n=200 and 10,000 simulation replicates

Scenarios			truth	est	% bias	est.SE	emp.SE	MSE	coverage
I
	LM
		normality	0.582	0.582	0.01	0.048	0.049	0.0024	0.946
		empirically	0.582	0.580	−0.35	0.048	0.049	0.0024	0.945
	kernel
		Silverman	0.582	0.552	−5.10	—	0.050	0.0034	—
	CPM
		probit	0.582	0.577	−0.92	0.050	0.049	0.0025	0.950
		logit	0.582	0.573	−1.59	0.050	0.050	0.0026	0.948
		loglog	0.582	0.565	−2.85	0.050	0.049	0.0027	0.942
		cloglog	0.582	0.565	−2.88	0.050	0.049	0.0027	0.941
II
	LM
		normality	0.582	0.394	−32.33	0.058	0.063	0.0393	0.057
		empirically	0.582	0.386	−33.68	0.060	0.063	0.0424	0.051
	kernel
		Silverman	0.582	0.552	−5.10	—	0.050	0.0034	—
	CPM
		probit	0.582	0.577	−0.92	0.050	0.049	0.0025	0.950
		logit	0.582	0.573	−1.59	0.050	0.050	0.0026	0.948
		loglog	0.582	0.565	−2.85	0.050	0.049	0.0027	0.942
		cloglog	0.582	0.565	−2.88	0.050	0.049	0.0027	0.941
III
	LM
		normality	0.520	0.499	−3.86	0.054	0.055	0.0034	0.931
		empirically	0.520	0.500	−3.72	0.054	0.055	0.0034	0.930
	kernel
		Silverman	0.520	0.499	−3.92	—	0.053	0.0032	—
	CPM
		probit	0.520	0.517	−0.52	0.053	0.053	0.0028	0.945
		logit	0.520	0.514	−1.02	0.053	0.053	0.0029	0.945
		loglog	0.520	0.505	−2.84	0.053	0.053	0.0030	0.943
		cloglog	0.520	0.504	−2.90	0.053	0.053	0.0030	0.939
IV
	LM
		normality	0.520	0.361	−30.52	0.054	0.056	0.0283	0.144
		empirically	0.520	0.382	−26.50	0.053	0.055	0.0219	0.226
	kernel
		Silverman	0.520	0.499	−3.92	—	0.053	0.0032	—
	CPM
		probit	0.520	0.517	−0.52	0.053	0.053	0.0028	0.945
		logit	0.520	0.514	−1.02	0.053	0.053	0.0029	0.945
		loglog	0.520	0.505	−2.84	0.053	0.053	0.0030	0.943
		cloglog	0.520	0.504	−2.90	0.053	0.053	0.0030	0.939

est is the mean of the point estimates.

est.SE is the mean of the standard error estimates.

emp.SE is the standard deviation of the point estimates