Error bands for impulse responses

We show how correctly to extend known methods for generating error bands in reduced form VAR's to overidentified models. We argue that the conventiona l pointwise bands common in the literature should be supplemented with meas ures of shape uncertainty, and we show how to generate such measures. We fo cus an bands that characterize the shape of the likelihood. Such bands are not classical confidence regions. We explain that classical confidence regi ons mix information about parameter location with information about model f it, and hence can be misleading as summaries of the implications of the dat a for the location of parameters. Because classical confidence regions also present conceptual and computational problems in multivariate time series models, we suggest that likelihood-based bands, rather than approximate con fidence bands based on asymptotic theory, be standard in reporting results for this type of model.