Joint Bayesian inference about impulse responses in VAR models

This 2022 Journal of Econometrics paper by Atsushi Inoue and Lutz Kilian develops a Bayesian decision-theoretic framework for reporting inference about an entire vector of structural VAR impulse responses jointly, rather than horizon by horizon, and shows that the conventional practice of reporting the posterior median or mean impulse response at each horizon together with pointwise credible intervals is not in general the Bayes estimator and can be economically misleading. Formally, letting theta = g(lambda) denote the n_irf-dimensional vector of structural impulse responses implied by the underlying VAR parameters lambda, the Bayes estimator minimizes posterior expected loss over the feasible response space Theta under a chosen loss function – quadratic, absolute, Dirac delta (which yields the posterior mode), or an angular loss invariant to the scaling of the data and structural shocks – and the corresponding joint credible set is constructed as the region of lowest posterior risk (for quadratic or absolute loss) or highest posterior density (for Dirac delta loss), visualized as a “shot-gun” trajectory plot of complete retained impulse-response paths rather than as a separate interval at each horizon. The key theoretical result is that when the number of estimated impulse responses n_irf exceeds the number of underlying structural parameters n_p – the empirically typical case – the space of feasible response vectors Theta is a lower-dimensional manifold embedded in R^{n_irf}, so the vector of pointwise posterior medians or means need not lie in Theta at all; a stylized AR(1) illustration with posterior rho ~ N(0.7, 1/5) yields a mean-response vector across horizons 1-4 of [0.70, 0.69, 0.76, 0.95], a shape that first falls and then rises and is infeasible for any AR(1) coefficient. In two empirical illustrations – a quarterly VAR(4) identifying aggregate demand, aggregate supply, and monetary policy shocks for U.S. real GNP growth, the federal funds rate, and GNP deflator inflation over 1954:IV-2007:IV, exactly identified via short- and long-run exclusion restrictions, and a monthly VAR(24) for the global crude oil market over 1973:2-2018:6 with flow supply, flow demand, and storage demand shocks set-identified via narrative, sign, and impact-elasticity bound restrictions – the paper finds that conventional pointwise 68% error bands “grossly understate” joint estimation uncertainty about the response vector, with the range spanned by the responses in the joint credible set in some cases three times as wide as the conventional bands. Across both examples the absolute, quadratic, and Dirac delta loss functions produce Bayes estimates that are quite similar to one another, and angular loss gives similar answers too, though at greater computational cost, so the paper concludes there is “little to choose” between the loss functions in practice and recommends absolute or quadratic loss as computationally cheap defaults.

Lutz Kilian | Macro Paper Warehouse

Joint Bayesian inference about impulse responses in VAR models