How to conduct joint Bayesian inference in VAR models?

Standard methods for constructing error bands around impulse response functions consider them in isolation, neglecting the estimation uncertainty that arises across variables and time horizons due to the joint nature of the underlying structural parameters. For example, one approach to assessing the impact of government expenditure on economic activity-known as the fiscal multiplier-is to estimate the ratio between the cumulative responses of gross domestic product and government expenditure. By convention, practitioners use marginal error bands to address this and similar economic questions, which leads to an underestimation of uncertainty. This paper quantifies the extent to which conventional error bands understate uncertainty and focuses on methods for conducting joint inference to support more robust economic analysis. It conducts a series of simulation experiments using widely adopted Bayesian vector autore- gressions to evaluate the performance of several estimators, discussed by Montiel Olea and Plagborg-Møller (2019) and Inoue and Kilian (2022), that account for the joint uncertainty of impulse response functions. In addition, it proposes strategies to improve upon those joint inference methods, which tend to overstate the uncertainty. When economic analysis requires simultaneous inference across multiple variables and time horizons, this paper shows that conventional pointwise quantiles in Bayesian structural vector autoregressions significantly understate the uncertainty of impulse responses. The performance of recently proposed joint inference methods, which produce noticeably different error band estimates, is evaluated, and calibration routines are suggested to ensure that they achieve the intended nominal probability coverage. Two practical applications illustrate the implications of these findings: (i) within a structural vector autoregression, the fiscal multiplier exhibits error bands that are 51% to 91% wider than previous estimates, and (ii) a pseudo-out-of-sample projection exercise for inflation and gross domestic product shows that joint inference methods could effectively summarize uncertainty for forecasts as well. These results underscore the importance of using joint inference methods for more robust econometric analysis.