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- It is well known that for horizons greater than one period the predictive density of the Bayesian VAR described above is not available in closed form. Simulated draws can, however, be obtained by drawing a sequence of Σ and B (from equations A.3-A.4) and shocks (remembering εt ∼ N ( 0 , Σ )) and then assembling the implied draw of YT+h (see Carriero et al., 2015). In my application, for each draw of a total of 1,000 draws from the posterior of Σ and B, I draw ten paths of shocks (i.e. of εT+1, . . . , εT+h) and thus arrive at a total of 10,000 draws from the predictive density. To be able to apply the results of Appendix A.1 for the SFA, I assume that the predictive density comes from a multivariate normal distribution with mean and variance given by the respective statistics of the simulated draws. Note that the true predictive density is nonGaussian.
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- The approximation of the true predictive density could potentially be improved by mixture-type distributions. Focusing on univariate forecast distributions, Krüger, Lerch, Thorarinsdottir, and Gneiting (2016) show that a mixture-type approximation to a Bayesian forecast distribution outperforms the Gaussian approximation on theoretical and empirical grounds. Empirical evidence by Warne, Coenen, and Christoffel (2017) suggests that similar results may apply in the multivariate case; however, the difference between the mixture versus Gaussian approximations seems rather small in their case.13 diag(x) generates a diagonal matrix with the vector x on its main diagonal (and zeros everywhere else) In my application, I use ν = .01 I wish to thank a referee for making this point.
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Warne, A., G. Coenen, and K. Christoffel (2017). Marginalized predictive likelihood comparisons of linear Gaussian state-space models with applications to DSGE, DSGEVAR, and VAR models. Journal of Applied Econometrics 32(1), 103–119. A Appendix A.1 SFA scenario in the special case of section 2.1 First, note that the general optimization problem of equation (1) can be restated in terms of the natural logarithm of the predictive density: b Yα = argmaxb Y ln fY ( b Y) s.t. Pr h s(Yt+h) > s( b Y) i = α Next, set up the corresponding Lagrangian: L( b Y, λ) = − k h ln 2À + ln |Σ| + b Y − 0 Σ−1 b Y − −λ  ï£Âα − 1 + Φ   a0 b