Palgrave Handbook of Econometrics: Applied Econometrics

Stephen G. Hall and James Mitchell 207

VAR (BVAR) model, which incorporates both drifting coefficients and stochastic
volatility in the errors. Earlier work by Sims and Zha (1998) considered the use of
Bayesian methods to compute fan charts from VAR models. Geweke and White-
man (2006) review Bayesian methods for the construction of density forecasts, or
posteriorpredictive densities. Adolfson, Linde and Villani (2007), in a forecasting
application to the Euro-area, use Bayesian methods to produce density forecasts
from both DSGE and VAR models.
Cogley, Morosov and Sargent (2005) construct a second-order VAR for RPIX infla-
tion, the output gap and the nominal three-month treasury bill rate, denoted by
the vectorYt:
Yt=X′tθt+εt, (5.7)

where the vectorXtincludes lags ofYtand a constant, andεt=R0.5t ξtis a vector
of measurement innovations, whereξt∼N(0, 1)andRtis a stochastic volatility
matrix, discussed below.
This model differs from a standard VAR in two important ways. The error process
has a stochastic volatility structure, to be described below, and the parameters of
the VAR are time-varying and follow a random walk, which is represented by the
following joint prior:

f(θT,Q)=f(θT|Q)f(Q)=f(Q)

T∏− 1

s= 0

f(θs+ 1 |θs,Q), (5.8)

where θT =[θ 1 ′,...,θT′]′ represents the history of the drifting parameters,

θT+1,T+F=[θT′+ 1 ,...,θT′+F]′their potential future paths, and:

f(θt+ 1 |θt,Q)∼N(θt,Q). (5.9)

Thus the parameters are effectively random walks without drift and with a constant
covariance structure.
In addition to this, a prior belief is imposed on the VAR that the roots of the lag
polynomial must lie inside the unit circle, to ensure that the VAR is stable. This is
done by creating a reflecting barrier using an indicator function:

I(θT)=

∏T

s= 1

I(θs), (5.10)

where the functionI(θs)=0 when the roots of the VAR are stable andI(θs)= 1
when they are unstable. This reflecting barrier then modifies the random walk prior
so that we then have:
p(θT,Q)∝I(θT)f(θT,Q). (5.11)
Hence the conditional prior is:

p(θT|Q)=

I(θT)f(θT|Q)

∫

I(θT)f(θT|Q)dθT

. (5.12)