Joint Bayesian Inference about Impulse Responses in VAR Models

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See all articles by Lutz Kilian

Lutz Kilian

Federal Reserve Banks - Federal Reserve Bank of Dallas; Centre for Economic Policy Research (CEPR)

Atsushi Inoue

Vanderbilt University - College of Arts and Science - Department of Economics

Date Written: November 2, 2020

Abstract

We derive the Bayes estimator of vectors of structural VAR impulse responses under a range of alternative loss functions. We also derive joint credible regions for vectors of impulse responses as the lowest posterior risk region under the same loss functions. We show that conventional impulse response estimators such as the posterior median response function or the posterior mean response function are not in general the Bayes estimator of the impulse response vector obtained by stacking the impulse responses of interest. We show that such pointwise estimators may imply response function shapes that are incompatible with any possible parameterization of the underlying model. Moreover, conventional pointwise quantile error bands are not a valid measure of the estimation uncertainty about the impulse response vector because they ignore the mutual dependence of the responses. In practice, they tend to understate substantially the estimation uncertainty about the impulse response vector.

Keywords: Loss function, joint inference, median response function, mean response function, modal model, posterior risk

JEL Classification: C22, C32, C52

Suggested Citation

Kilian, Lutz and Inoue, Atsushi, Joint Bayesian Inference about Impulse Responses in VAR Models (November 2, 2020). CFS Working Paper, No. 650, 2020, Available at SSRN: https://ssrn.com/abstract=

Lutz Kilian (Contact Author)

Federal Reserve Banks - Federal Reserve Bank of Dallas ( email )

2200 North Pearl Street
PO Box 655906
Dallas, TX 75265-5906
United States

Centre for Economic Policy Research (CEPR)

London
United Kingdom

Atsushi Inoue

Vanderbilt University - College of Arts and Science - Department of Economics ( email )

Box 1819 Station B
Nashville, TN 37235
United States

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