Simultaneous Confidence Bands: Theory, Implementation, and an Application to SVARs

This 2019 Journal of Applied Econometrics paper by José Luis Montiel Olea and Mikkel Plagborg-Møller addresses a practical problem in applied time-series econometrics: which “simultaneous” (joint, across-parameter) confidence band should researchers use by default when reporting an entire vector of estimated quantities – such as impulse responses across horizons from a structural VAR – rather than one interval per parameter reported separately. The authors set up a general framework in which a possibly nonlinear transformation theta = h(mu) of an asymptotically normal estimator mu-hat is to be covered jointly, and show that a wide class of popular bands (pointwise, Bonferroni, Sidak, projection, and the “sup-t” band) can all be written as members of a single “one-parameter class” that scales every pointwise standard error by one common critical value c. Within this class, the sup-t band – whose critical value is the quantile of the maximum absolute studentized draw from the estimator’s joint asymptotic distribution – is the narrowest band that still achieves exact asymptotic simultaneous coverage, and the paper adds a decision-theoretic result (Proposition 1) showing it uniquely minimizes worst-case regret across all degree-one-homogeneous loss functions, which makes it a defensible default when the researcher does not know which feature of the band matters most to different readers. The paper gives three computationally convenient ways to construct it – a plug-in (delta-method) simulation algorithm, a bootstrap algorithm, and a Bayesian algorithm delivering exact finite-sample simultaneous credibility – and notes all three are first-order asymptotically equivalent. In an empirical application to a monthly U.S. SVAR (July 1979-June 2012, 12 lags; identified two ways, via a recursive/Cholesky scheme and via a Gertler-Karadi (2015)-style external instrument using federal-funds-futures surprises from January 1990) with industrial production, CPI, a one-year bond yield, and the excess bond premium, the sup-t band is substantially narrower than the Bonferroni or Sidak bands – around 35% narrower in the external-instrument specification at 68% confidence – and the narrowing is not merely cosmetic: at the 68% simultaneous level the plug-in sup-t band excludes zero for the industrial-production response at horizons of roughly 13-36 months, letting the authors reject the no-effect null at some horizon in that range, whereas the Bonferroni band does not permit that rejection; conversely, an output response that looks pointwise significant at the 2-month horizon is no longer simultaneously significant once the sup-t multiple-comparison adjustment is applied. A companion Monte Carlo study of bivariate VARs finds the sup-t band 20-25% narrower than Bonferroni/Sidak at 68% confidence and 10-20% narrower at 90% confidence, though for highly persistent data only the Bayesian sup-t implementation is reported to achieve satisfactory finite-sample coverage. The theory is developed for point-identified models with a continuously differentiable transformation h(.); partially identified (e.g., sign-restricted) SVARs require additional considerations, though the authors suggest the Bayesian sup-t band may still be usable there for subjective Bayesian analysis.

Journal of Applied Econometrics | Macro Paper Warehouse

Simultaneous Confidence Bands: Theory, Implementation, and an Application to SVARs