Measuring the Effects of a Shock to Monetary Policy - Humboldt ...

30 Bayesian FAVARs with Agnostic Identification
dimensional models, due to the fact that in case of large cross-sections, highly dimen-
sional likelihoods make irregularities more likely. This can reduce the number of draws
relevant for convergence and hence saves time, which in a computer-intensive statistical
framework is of great relevance. I follow the suggestions of Eliasz [2005] and apply the
first step estimates of PCA to select the starting values. A detailed description how to
obtain the starting values via the first step PCA can be found his paper. Since Gelman
and Rubin [1992] have shown that a single chain of the Gibbs sampler might give a ”false
sense of security ”, it has become common practice to try out different starting values, at
best from a randomly (over)dispersed set of parameters and then check the convergence
verifying that they lead to similar empirical distributions. The Inference part is very close
to BBE [2005] and Eliasz [2005]. This part can also be found in a slightly more elaborate
verion in their papers. But for completenes it is stated here.
Conditional density of the factors
In this subsection we want to draw from
pF ( ˜ FT | ˜ XT , θ)
assuming that the hyperparameters of the parameter space θ are given, hence I describe
Bayesian Inference on the dynamic evolution of the factors Ft conditional on Xt for
t = 1, . . . , T and conditional on θ. The transformations that are required to draw the
factors have been done in the previous section. The conditional distribution, from which
the state vector is generated, can be expressed as the product of conditional distributions
by exploiting the Markov property of state space models in the following way
pF ( ˜ FT | ˜ XT , θ) = pF (FT | ˜ XT , θ)pF (FT −1 | FT , ˜ XT , θ), . . . , pF (F1 | F2, ˜ XT , θ)
pF ( ˜ FT | ˜ XT , θ) = pF (FT | ˜ XT , θ) T −1
t=1 pF (Ft | Ft+1, ˜ XT , θ)
At this point it is important to note that the conditioning is on the first [K + M] rows of
FT only, since otherwise for the case of d > 1 the VCV of the density would be singular.