Narrative Restrictions and Proxies: Rejoinder

Abstract

This rejoinder addresses the discussants’ specific comments on the article “Narrative Restrictions and Proxies” (Section 2) as well as more general comments on the approach to robust Bayesian inference that we have proposed in previous work (Section 1).

Acknowledgments

The views in this article are the authors’ and do not reflect the views of the Federal Reserve Bank of Chicago, the Federal Reserve System or the Reserve Bank of Australia. We thank our discussants—Lutz Kilian, Mikkel Plagborg-Møller and Juan Rubio-Ramírez—for their insightful discussions of our article.

Notes

1 The indicator function could also be a function of the data separately from the reduced-form parameters, such as when $\mathcal{H}$ is a hypothesis about the values of structural shocks or the historical decomposition in specific periods. We leave this potential dependence on the data implicit.

2 Theorem 1 of Giacomini and Kitagawa (2021) expresses the posterior lower and upper probabilities that some subvector or transformation, $\mathit{\eta }$, of the structural parameters lies within a region D in terms of the posterior for $\mathit{\varphi }$. Replacing $\mathit{\eta }$ in their Theorem 1 with $1(\mathit{\varphi },\mathbf{Q};\mathcal{H})$ and D with {1} yields the expressions in (1) and (2).

3 Methods for checking whether the identified set for Q is empty and for drawing from a uniform distribution over the identified set for Q are described in Giacomini and Kitagawa (2021) and Giacomini et al. (2021b). If multiple hypotheses are of interest, the same draws of Q can be used to evaluate a different indicator function corresponding to each hypothesis.

4 The results are based on 1000 draws of $\mathit{\varphi }$ from its posterior such that the conditional identified set is nonempty and 100,000 draws of Q at each draw of $\mathit{\varphi }$.

5 When specifying a prior for the structural parameters, it remains the case that a component of the prior will not be updated, so it may still be desirable to use robust Bayesian methods to assess posterior sensitivity to the choice of prior. When partially credible prior information about the structural parameters is available, the approach in Giacomini et al. (2019) can be used to assess posterior sensitivity to perturbations of the prior within some neighborhood around the “benchmark” prior.

6 As a stark example, consider imposing a point-mass prior for a single value of $\mathit{\varphi }$ and a conditionally uniform prior for Q given $\mathit{\varphi }$. The implied prior for the impulse responses will in general not be uniform, despite the fact that all impulse responses with positive prior density are observationally equivalent, since they share the same value of $\mathit{\varphi }$.

7 Baumeister and Hamilton (2022) discuss this point. For alternative intuition, consider the model’s orthogonal reduced-form. Given a valid proxy variable for the last structural shock, the last column of Q is point-identified (e.g., Arias et al. 2021; Giacomini et al. 2022). Since ${\mathbf{A}}_0=\mathbf{Q}\prime {\mathbf{\Sigma }}_{tr}^{-1}$. the last row of ${\mathbf{A}}_0$ is a function of ${\mathbf{\Sigma }}_{tr}$ and the last column of Q, so the coefficients in the last structural equation are also point-identified.

Additional information

Funding

We gratefully acknowledge financial support from ERC grants (numbers 536284 and 715940) and the ESRC Centre for Microdata Methods and Practice (CeMMAP) (grant number RES-589-28-0001).