Narrative Restrictions and Proxies

Figures & data

Fig. 1 Cumulative (null) distribution function of Wald statistic under narrative proxy.

NOTE: Dashed lines represent 68th and 95th percentiles; data-generating process assumes T = 1000, ${\eta }_{21}=0.4$ and $\text{vech}({\mathbf{\Sigma }}_{tr})\approx (0.7,-1.1,1.4)\prime$; null distribution approximated using 100,000 Monte Carlo replications.

Table 1 Weak-proxy robust inference—Monte Carlo results for $\alpha =0.95$.

K	Coverage prob.	Unbounded	Median width	Average W
1	1.000	1.000	$\infty$	1.001
2	1.000	1.000	$\infty$	1.194
3	1.000	1.000	$\infty$	1.452
4	0.996	0.985	$\infty$	1.761
5	0.978	0.887	$\infty$	2.043
10	0.958	0.549	$\infty$	3.623
20	0.952	0.180	8.222	6.800
30	0.951	0.047	5.067	10.009
40	0.954	0.011	3.975	13.256
50	0.951	0.001	3.371	16.550
100	0.947	0.000	2.133	33.362
500	0.950	0.000	0.894	189.693
1000	0.950	0.000	0.625	468.572

NOTE: “Coverage prob.” is the coverage probability of the 95% weak-proxy robust confidence interval; “Unbounded” is the proportion of Monte Carlo samples in which the confidence interval is unbounded; W is the Wald statistic for testing the null hypothesis that the covariance between the proxy and $y_{1t}$ is zero.

Table 2 Weak-proxy robust inference—Monte Carlo results for $\alpha =0.68$.

K	Coverage prob.	Unbounded	Median width	Average W
1	0.000	0.000	0.000	1.001
2	0.493	0.376	7.824	1.194
3	0.581	0.360	8.549	1.452
4	0.603	0.311	7.008	1.761
5	0.627	0.276	6.075	2.043
10	0.654	0.109	3.634	3.623
20	0.654	0.020	2.386	6.800
30	0.667	0.003	1.895	10.009
40	0.674	0.000	1.640	13.256
50	0.674	0.000	1.457	16.550
100	0.682	0.000	1.008	33.362
500	0.687	0.000	0.448	189.693
1000	0.683	0.000	0.316	468.572

NOTE: “Coverage prob.” is the coverage probability of the 68% weak-proxy robust confidence interval; “Unbounded” is the proportion of Monte Carlo samples in which the confidence interval is unbounded; W is the Wald statistic for testing the null hypothesis that the covariance between the proxy and $y_{1t}$ is zero.

Table 3 Robust Bayesian inference—Monte Carlo results for $\alpha =0.95$.

K	Coverage prob.	Unbounded	Median width
1	1.000	0.766	$\infty$
2	1.000	0.596	$\infty$
3	1.000	0.461	28.343
4	1.000	0.345	8.725
5	0.999	0.271	6.083
10	0.998	0.075	2.742
20	0.996	0.005	1.571
30	0.994	0.000	1.226
40	0.993	0.000	1.034
50	0.993	0.000	0.932
100	0.986	0.000	0.717
500	0.966	0.000	0.545
1000	0.955	0.000	0.522

NOTE: “Coverage prob.” is the coverage probability of the 95% robust credible interval; “Unbounded” is the proportion of Monte Carlo samples in which the robust credible interval is unbounded.

Table 4 Robust Bayesian inference—Monte Carlo results for $\alpha =0.68$.

K	Coverage prob.	Unbounded	Median width
1	1.000	0.757	$\infty$
2	0.998	0.581	$\infty$
3	0.997	0.442	19.893
4	0.995	0.330	7.615
5	0.994	0.256	5.422
10	0.985	0.066	2.436
20	0.970	0.004	1.302
30	0.955	0.000	0.969
40	0.947	0.000	0.778
50	0.932	0.000	0.678
100	0.882	0.000	0.466
500	0.757	0.000	0.296
1000	0.718	0.000	0.275

NOTE: “Coverage prob.” is the coverage probability of the 68% robust credible interval; “Unbounded” is the proportion of Monte Carlo samples in which the robust credible interval is unbounded.