840 The Econometrics of Monetary Policy
where:
=∗(θ)+u=u∗(θ)+u.The DSGE restrictions are imposed on the VAR by defining:
XX(θ)=EθD[
XtX′t]XZ(θ)=EθD[
XtZ′t]
,whereEDθ defines the expectation with respect to the distribution generated by the
DSGE model. Such a distribution needs to be well defined. We then have:
∗(θ)=XX(θ)−^1 XZ(θ).Beliefs about the DSGE model parametersθand model misspecification matricesandu are summarized in prior distributions, that, as shown in Del Negro
et al.(2006), can be transformed into priors for the VAR parametersandu.In
particular we have:
u|θ∼IW(
λT∗u(θ),λT−k,n)∣
∣u,θ∼N(
#∗(θ),
1
λT[
−u^1 ⊗XX(θ)]− 1 )
,where the parameterλcontrols the degree of model misspecification with respect
to the VAR: for small values ofλthe discrepancy between the VAR and the DSGE-
VAR is large and a sizeable distance is generated between unrestricted VAR and
DSGE estimators, large values ofλcorrespond to small model misspecification
and, forλ=∞, beliefs about DSGE misspecification degenerate to a point mass
at zero. Bayesian estimation could be interpreted as estimation based on a sample
in which data are augmented by a hypothetical sample in which observations are
generated by the DSGE model; within this frameworkλdetermines the length of
the hypothetical sample.
Given the prior distribution, posteriors are derived by Bayes’ theorem:
u|θ,Z∼IW(
(λ+ 1 )Tˆ
u,b(θ),(λ+ 1 )T−k,n)∣∣
u,θ,Z∼N(
ˆ
b(θ),u⊗[
λTXX(θ)+X′X]− 1
)ˆ
b(θ)=(
λTXX(θ)+X′X)− 1 (
λTXZ(θ)+X′Z)ˆ
u,b(θ)=
1
(λ+ 1 )T[(
λTZZ(θ)+Z′Z)
−(
λTXZ(θ)+X′Z)ˆ
b(θ)]
,