Joint Bayesian Inference about Impulse Responses in VAR Models

Structural VAR models are routinely estimated by Bayesian methods. Several recent studies have voiced concerns about the common use of posterior median (or mean) response functions in applied VAR analysis. In this paper, we show that these response functions can be misleading because in empirically relevant settings there need not exist a posterior draw for the impulse response function that matches the posterior median or mean response function, even as the number of posterior draws approaches infinity. As a result, the use of these summary statistics may distort the shape of the impulse response function which is of foremost interest in applied work. The same concern applies to error bands based on the upper and lower quantiles of the marginal posterior distributions of the impulse responses. In addition, these error bands fail to capture the full uncertainty about the estimates of the structural impulse responses. In response to these concerns, we propose new estimators of impulse response functions under quadratic loss, under absolute loss and under Dirac delta loss that are consistent with Bayesian statistical decision theory, that are optimal in the relevant sense, that respect the dynamics of the impulse response functions and that are easy to implement. We also propose joint credible sets for these estimators derived under the same loss function. Our analysis covers a much wider range of structural VAR models than previous proposals in the literature including models that combine short-run and long-run exclusion restrictions and models that combine zero restrictions, sign restrictions and narrative restrictions.