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

26 Bayesian FAVARs with Agnostic Identification
estimated values of the hyperparameters 26 of the model, which are obtained with the
maximum likelihood method. One has to treat them as they were the true values of
the models nonrandom hyperparameters. This is a disadvantage towards the Bayesian
approach where in the vein of Bayesian data analysis either the models hyperparameters
and the unobserved state vector are treated as random variables. The classical approach
does not exploit the fact that Yt is observed and furthermore does not exploit the structure
of the state equation in the estimation of the factors.
5.2 State-Space Representation
This part elaborates on the specific model and the required transformations in order to
estimate the model via Gibbs sampling. For more details the reader is referred to Eliasz
[2005] and Kim and Nelson [1999] for a very good and elaborate survey. In order to prepare
(1) and (2) for the estimation the model has to be cast into the following state-space form:
⎡
⎢
⎣ Xt
Yt
⎤
⎥
⎦ =
⎡
⎢
⎣ Ft
Yt
⎤
⎡
⎢
⎣ Λf Λy 0 I
⎤ ⎡
⎥ ⎢
⎦ ⎣ Ft
Yt
⎡
⎥
⎢
⎦ = Φ(L) ⎣ Ft−1
Yt−1
⎤
⎥
⎦ +
⎤
⎡
⎢
⎣ et
0
⎥
⎦ + vt
⎤
⎥
⎦ , (7)
The respective variables are the same as explained in the preceding sections. The
loadings Λ f and Λ y are restricted and identified against rotational indeterminacies as
it has been implemented by BBE and described in the previous section. According
to BBE the inclusion of the policy instrument Yt in both equations will not change
the model, it merely should serve notational and computational simplification. The
Bayesian econometricians treats the parameters of the model, interested to do inference
on, as random variables. We are interested in doing inference on the parameter space
θ =
Λf , Λy , R, vec(Φ), Q
. Note that vec(Φ) is the vectorized finite order conformable
lag polynomial, i.e. Φ is columnwise stacked to have a vectorized form 27 . To apply the
26 Hyperparameters are the elements of the parameter space to be estimated
27 for more details about the vec operator please refer to Lütkepohl [1993]
(8)