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Structural vector autoregressions that are set identified (e.g. with sign restrictions) are typically used to analyse the effects of standard deviation shocks. However, answering questions of economic interest often requires knowing the effects of a 'unit' shock. For example, central bankers want to answer questions like 'what are the effects of a 100 basis point increase in the policy rate?' The problem is that set-identifying restrictions do not always rule out the possibility that a variable does not react contemporaneously to its own shock. As a consequence, identified sets for the impulse responses to unit shocks may be unbounded, which implies that set-identifying restrictions may be extremely uninformative. Simply assuming that responses are non-zero turns out to be an arbitrary and unsatisfactory solution. I argue that it is therefore important to communicate about the extent to which the identified set may be unbounded, since this tells us about the informativeness of the identifying restrictions, and I develop tools to facilitate this. I explain how to draw useful posterior inferences about impulse responses when identified sets are unbounded with positive probability. I illustrate the empirical relevance of these issues by estimating the response of US output to a 100 basis point federal funds rate shock under different sets of identifying restrictions. Some restrictions are very uninformative about the effects of a 100 basis point shock. The output responses I obtain under a rich set of identifying restrictions lie towards the smaller end of the range of existing estimates.

We compare two approaches to using information about the signs of structural shocks at specific dates within a structural vector autoregression (SVAR): imposing 'narrative restrictions' (NR) on the shock signs in an otherwise setidentified SVAR; and casting the information about the shock signs as a discretevalued 'narrative proxy' (NP) to point-identify the impulse responses. The NP is likely to be 'weak' given that the sign of the shock is typically known in a small number of periods, in which case the weak-proxy robust confidence intervals in Montiel-Olea et al. (2021) are the natural approach to conducting inference. However, we show both theoretically and via Monte Carlo simulations that these confidence intervals have distorted coverage -which may be higher or lower than the nominal level -unless the sign of the shock is known in a large number of periods. Regarding the NR approach, we show that the prior-robust Bayesian credible intervals from Giacomini et al. (2021a) deliver coverage exceeding the nominal level, but which converges towards the nominal level as the number of NR increases. * This paper was prepared for the Journal of Business and Economic Statistics Session of the 2022 North American Winter Meeting of the Econometric Society. We thank Lutz Killian, Mikkel Plagborg-Møller and Juan Rubio-Ramírez for their illuminating discussions of our paper, and Chris Hansen for organising the session. We also thank James Stock, Christian Wolf and seminar participants at various venues for discussions that motivated this work. 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). The views in this paper 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 review the literature on robust Bayesian analysis as a tool for global sensitivity analysis and for statistical decision-making under ambiguity. We discuss the methods proposed in the literature, including the different ways of constructing the set of priors that are the key input of the robust Bayesian analysis. We consider both a general set-up for Bayesian statistical decisions and inference and the special case of set-identified structural models. We provide new results that can be used to derive and compute the set of posterior moments for sensitivity analysis and to compute the optimal statistical decision under multiple priors. The paper ends with a self-contained discussion of three different approaches to robust Bayesian inference for setidentified structural vector autoregressions, including details about numerical implementation and an empirical illustration.