TABLE 1.

Bias, standard deviation (SD), average standard error (SE), and coverage probability (CP) of 95% asymptotic confidence interval based on 1,000 repetitions with $n={10}^5$

Method	Correct model	Estimator	Bias	SD	SE	CP
OLS			2.466	0.024	0.023	0.000
Standard IV			−38.525	3.385	3.316	0.000
iDID-Wald		${\widehat{\beta }}_{\mathrm{S}1,\text{wald}}$	−0.014	0.247	0.251	0.950
		${\widehat{\beta }}_{\mathrm{S}2,\text{wald}}$	0.007	0.253	0.259	0.958
		${\widehat{\beta }}_{\mathrm{S}3,\text{wald}}$	−0.008	0.250	0.259	0.961
		${\widehat{\beta }}_{\mathrm{S}4,\text{wald}}$	0.000	0.289	0.284	0.943
Constant working model $\beta (\mathit{V};\mathit{\psi })=\psi$
iDID-MR	$\left({ℳ}_1,{ℳ}_2,{ℳ}_3\right)$ all correct	${\widehat{\psi }}_{\mathrm{m}\mathrm{r}}$	−0.002	0.111	0.114	0.956
	${ℳ}_1$ correct	${\widehat{\psi }}_{\mathrm{m}\mathrm{r}}$	−0.001	0.110	0.114	0.960
	${ℳ}_2$ correct	${\widehat{\psi }}_{\mathrm{m}\mathrm{r}}$	−0.003	0.136	0.139	0.944
	${ℳ}_3$ correct	${\widehat{\psi }}_{\mathrm{m}\mathrm{r}}$	−0.003	0.137	0.140	0.945
	none	${\widehat{\psi }}_{\mathrm{m}\mathrm{r}}$	−0.355	0.144	0.142	0.293
iDID-Reg	${ℳ}_1$ correct	${\widehat{\psi }}_{\mathrm{r}\mathrm{e}\mathrm{g}}$	−0.002	0.109	0.114	0.960
	${ℳ}_1$ incorrect	${\widehat{\psi }}_{\mathrm{r}\mathrm{e}\mathrm{g}}$	−0.351	0.144	0.149	0.335
iDID-IPW	${ℳ}_2$ correct	${\widehat{\psi }}_{\mathrm{i}\mathrm{p}\mathrm{w}}$	−0.021	0.225	0.225	0.948
	${ℳ}_2$ incorrect	${\widehat{\psi }}_{\mathrm{i}\mathrm{p}\mathrm{w}}$	−0.271	0.234	0.242	0.816
iDID-G	${ℳ}_3$ correct	${\widehat{\psi }}_{\mathrm{g}}$	−0.021	0.225	0.224	0.948
	${ℳ}_3$ incorrect	${\widehat{\psi }}_{\mathrm{g}}$	−0.276	0.235	0.233	0.814
Linear working model $\beta (\mathit{V};\mathit{\psi })={\psi }_1+{\psi }_2{X}_1$
iDID-MR	$\left({ℳ}_1,{ℳ}_2,{ℳ}_3\right)$ all correct	${\widehat{\psi }}_{1,\mathrm{m}\mathrm{r}}$	−0.002	0.110	0.114	0.956
		${\widehat{\psi }}_{2,\mathrm{m}\mathrm{r}}$	0.004	0.113	0.115	0.950
	${ℳ}_1$ correct	${\widehat{\psi }}_{1,\mathrm{m}\mathrm{r}}$	−0.001	0.110	0.114	0.960
		${\widehat{\psi }}_{2,\mathrm{m}\mathrm{r}}$	0.004	0.115	0.118	0.946
	${ℳ}_2$ correct	${\widehat{\psi }}_{1,\mathrm{m}\mathrm{r}}$	−0.003	0.136	0.139	0.944
		${\widehat{\psi }}_{2,\mathrm{m}\mathrm{r}}$	0.003	0.146	0.150	0.960
	${ℳ}_3$ correct	${\widehat{\psi }}_{1,\mathrm{m}\mathrm{r}}$	−0.003	0.137	0.140	0.946
		${\widehat{\psi }}_{2,\mathrm{m}\mathrm{r}}$	0.004	0.144	0.149	0.958
	None	${\hat{\psi }}_{1,\mathrm{m}\mathrm{r}}$	−0.355	0.144	0.142	0.292
		${\widehat{\psi }}_{2,\mathrm{m}\mathrm{r}}$	−0.129	0.221	0.175	0.908
iDID-Reg	${ℳ}_1$ correct	${\widehat{\psi }}_{1,\text{reg}}$	−0.001	0.109	0.114	0.960
		${\widehat{\psi }}_{2,\text{reg}}$	−0.005	0.114	0.118	0.949
	${ℳ}_1$ incorrect	${\widehat{\psi }}_{1,\text{reg}}$	−0.351	0.144	0.149	0.332
		${\widehat{\psi }}_{2,\text{reg}}$	−0.110	0.473	0.478	0.942
iDID-IPW	${ℳ}_2$ correct	${\widehat{\psi }}_{1,\mathrm{i}\mathrm{p}\mathrm{w}}$	−0.021	0.225	0.225	0.948
		${\widehat{\psi }}_{2,\mathrm{i}\mathrm{p}\mathrm{w}}$	−0.010	0.270	0.269	0.957
	${ℳ}_2$ incorrect	${\widehat{\psi }}_{1,\mathrm{i}\mathrm{p}\mathrm{w}}$	−0.271	0.234	0.242	0.813
		${\widehat{\psi }}_{2,\mathrm{i}\mathrm{p}\mathrm{w}}$	−0.072	0.267	0.313	0.957
iDID-G	${ℳ}_3$ correct	${\widehat{\psi }}_{1,\mathrm{g}}$	−0.021	0.225	0.224	0.949
		${\widehat{\psi }}_{2,\mathrm{g}}$	−0.007	0.245	0.247	0.953
	${ℳ}_3$ incorrect	${\widehat{\psi }}_{1,\mathrm{g}}$	−0.276	0.235	0.233	0.812
		${\widehat{\psi }}_{2,\mathrm{g}}$	−0.234	0.246	0.241	0.863

Abbreviations: iDID, instrumented difference-in-differences; OLS, ordinary least squares.