Simultaneous confidence bands: Theory, implementation, and an application to SVARs

Simultaneous confidence bands are versatile tools for visualizing estimation uncertainty for parameter vectors, such as impulse response functions. In linear models, it is known that that the sup-t confidence band is narrower than commonly used alternatives—for example, Bonferroni and projection bands. We show that the same ranking applies asymptotically even in general nonlinear models, such as vector autoregressions (VARs). Moreover, we provide further justification for the sup-t band by showing that it is the optimal default choice when the researcher does not know the audience's preferences. Complementing existing plug-in and bootstrap implementations, we propose a computationally convenient Bayesian sup-t band with exact finite-sample simultaneous credibility. In an application to structural VAR impulse response function estimation, the sup-t band—which has been surprisingly overlooked in this setting—is at least 35% narrower than other off-the-shelf simultaneous bands.