Tilburg University
Bayesian Vector Autoregressions with Stochastic Volatility
Uhlig, H.F.H.V.S.
Publication date:
1996
Link to publication in Tilburg University Research Portal
Citation for published version (APA):
Uhlig, H. F. H. V. S. (1996).
Bayesian Vector Autoregressions with Stochastic Volatility
. (CentER Discussion
Paper; Vol. 1996-09). Macroeconomics.
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Download date: 10. aug.. 2022
BAYESIAN VECTOR AUTOREGRESSIONS
WITH STOCHASTIC VOLATILITY
1
By Harald Uhlig
February 2, 1996
Abstract
This paper proposes a Bayesian approachtoavector autoregression with
stochastic volatility, where the multiplicativeevolution of the precision matrix
is driven byamultivariate b eta variate. Exact updating formulas are given to
the nonlinear ltering of the precision matrix. Estimation of the autoregressive
parameters requires numerical metho ds: an importance-sampling based approach
is explained here.
i
1
1 Intro duction
This paper introduces Bayesian vector autoregressions with sto chastic volatility. In con-
trast to multivariate autoregressive conditional heteroskedasticity (ARCH), the sto chas-
tic volatility setup here models the error precision matrix as an unobserved component
with sho cks drawn from a multivariate b eta distribution. This allows the interpretation
of a sudden large movement in the data as the result of a draw from a distribution
with a randomly increased but unobserved variance. Exploiting a conjugacy between
Wishart distributions and multivariate singular beta distributions, the integration over
the unobserved shock to the precision matrix can be p erformed in closed form, lead-
ing to a generalization of the standard Kalman-Filter formulas to the nonlinear ltering
problem at hand. Estimating the autoregressive parameters requires numerical methods,
however. The paper focusses on an importance-sampling based approach.
Bayesian vector autoregressions have been studied and popularized by e.g. Litter-
man (1979), Doan, Litterman and Sims (1984) and Doan's RATS Manual (1990). ARCH
models have been introduced by Engle (1982), see the review in Bollerslev, Chou and
Kroner (1992). Sto chastic volatility mo dels provide an alternative approach to model
time variation in the size of uctuations. The stochastic volatility mo del used here
is similar to Shephard (1994), whose model is a univariate, non-Bayesian and non-
autoregressive special case of the model proposed here. In contrast to other Bayesian
approaches to stochastic volatility, see Jacquier, Polson and Rossi (1994), the method
here results in exact up dating formulas for the posterior in the sense that the integra-
tion over the unobserved shocks to the precision matrices is done in closed form. The
conjugacy result needed for this step is established in Uhlig (1994b).
For simplicity, the main ideas are explained in section 2 for the univariate case with
the general case presented in section 3. Section 4 discusses how to analyze the posterior
numerically. Section 5 concludes. Appendix A lists some of the distributions used and
xes notation. Appendix B contains the proofs and one additional theorem. Appendix C
proposes a prior.
2 A Simple Case
Consider the following simple version of the mo del studied in this pap er:
y
t
=
y
t
1
+
h
1
=
2
t
t
;
with
t
N
(0
;
1)
;
(1)
h
t
+1
=
h
t
#
t
=;
with
#
t
B
1
((
+1)
=
2
;
1
=
2)
;
(2)
where all
#
t
's and
t
's drawn independently, where
t
=1
;:::;T
denotes time,
y
t
2
IR
;t
=
0
;:::;T
is data and observable,
>
0,
>
0 are parameters and
B
1
(
p; q
) denotes the
(one-dimensional) b eta-distribution on the interval [0,1].
2
Equation (2) sp ecies the unobserved precision
h
t
of the innovation
h
1
=
2
t
t
to b e
stochastic. The model thus belongs to the family of sto chastic volatility mo dels, see e.g.
Jacquier-Polson and Rossie (1994). The model captures auto correlated heteroskedastic-
ity, a feature often found especially in nancial data series. Another popular specica-
tion whichdoessoistheARCH-family of models. A GARCH(1,1) mo del, for example,
replaces
h
1
=
2
t
in (1) with
t
and replaces (2) with
2
t
+1
=
+
2
t
+
2
t
2
t
;
(3)
where
,
and
are parameters. It thus ties the innovation in the variance to the
size of the current innovation
t
t
. Given
t
1
and
h
t
1
or
t
1
,anunusually large
innovation in (2) can result from a randomly decreased
h
t
as well as a large
t
, whereas
the GARCH-model (3) only allows for an unusually large draw
t
.
To analyze the system (1) to (2) in a Bayesian fashion, one needs to choose a prior
density
0
(
; h
1
) for
and
h
1
, given
y
0
. The goal is to nd the p osterior density
T
(
; h
T
+1
) given data
y
0
;:::;y
T
.We restrict the choice of priors to b e of the following
form. Fix
>
0 and
>
0 (for a more general treatment, see section 3). Cho ose
b
0
2
IR
,
n
0
>
0,
s
2
0
>
0 and a function
g
0
(
)
0 to describ e a prior density prop ortional to
0
(
; h
1
)
/
g
0
(
)
f
NG
(
; h
1
j
b
0
;n
0
;s
0
;
)
;
where
f
NG
denotes the Normal-gamma density, see app endix A. The form of the prior
allows for a exible treatment of a root near or above unity via the function
g
0
(
), see
Uhlig (1994a).
Adapting the Bayesian updating formulas (12), (13), (14) and (15) derived b elowin
section 3 to the simple mo del ab ove results in
n
t
=
n
t
1
+
y
2
t
1
(4)
b
t
=
b
t
1
n
t
1
+
y
t
y
t
1
=n
t
(5)
s
t
=
s
t
1
+
e
2
t
1
y
2
t
1
=n
t
;
(6)
where
e
t
=
y
t
b
t
1
y
t
1
;
and
g
t
(
)=
g
t
1
(
)
(
b
t
)
2
n
t
+
s
t
1
=
2
(7)
for
t
=1
;:::;T
. These deliver the posterior density
T
(
; h
T
+1
)
/
g
T
(
)
f
NG
(
; h
T
+1
j
b
T
;n
T
;s
T
;
)
:
3
Equations (4) and (5) are the recursion formulas or Kalman Filter formulas for geomet-
rically weighted least squares. Dierent observations receive dierentweights according
to the size of
s
t
via equation (7). Equation (6) prescribes to nd the \estimate"
s
t
of
h
t
+1
essentially via a geometric lag on past squared residuals. Notice the formal similarityto
GARCH: ignoring the term (1
y
2
t
1
=n
t
), equation (6) resembles equation (3) rewritten
in terms of observables, using
= 0 and
=
=
.
The key for proving the validity of these updating formulas here or in the next section
is theorem 2 and its pro of (see appendix B): as the unobserved shock
#
t
occurs, one needs
to do a \change of variable" from
d dh
t
d#
t
to
d dh
t
+1
dz
t
for some suitably dened
z
t
. Thanks to the conjugacy b etween the b eta and the gamma distribution, integration
over
dz
t
can be p erformed in closed form, resulting in an integration constant depending
on
and the data. This constant is captured by the function
g
t
(
).
Shephard (1994) nds similar formulas with a classical interpretation for (1) to (2)
without the autoregressive term
y
t
1
.To include autoregressive terms, Shephard (1994)
suggests approximate ltering formulas. In contrast, the Bayesian formulas here are
exact. They do, however, require numerical techniques such as importance-sampling for
the estimation of
. There is no treatment of the multivariate case in Shephard (1994).
For
=
=
(
+1) wehave
=
=1
in equation (6). For
=(
+1)
=
(
+2)
the precision
h
t
is a martingale
E
[
h
t
+1
j
h
t
]=
h
t
on the positive part of the real axis.
Shephard (1994) suggests setting
=
e
r
, where
r
=
E
[log
#
t
]. This avoids the problem,
that otherwise
h
t
!1
a.s. or
h
t
!
0 a.s. (see Nelson (1990)) and makes log
h
t
a
random walk.
For
!1
one obtains a mo del where
h
1
is known a priori,
h
1
=
s
1
0
, and where
h
t
+1
=
h
t
(
+1)
=
(
(
+ 2)). In other words the mo del allows for the greater time
variation in the precision, the smaller the parameter
.
Figure 1 shows parts of the densities for
=#
t
, which are the multiplicative distur-
bances of the variance
2
t
h
1
t
. It shows that (2) typically leads to a slight decrease in
the innovation variance except for occasional and potentially large increases.
3 The General Model
Consider the VAR(k)-mo del with time-varying error precision matrices
Y
t
=
B
(0)
C
t
+
B
(1)
Y
t
1
+
B
(2)
Y
t
2
+
:::
+
B
(
k
)
Y
t
k
+
U
(
H
1
t
)
0
t
;
with
t
N
(0
;I
m
)
;
(8)
H
t
+1
=
U
(
H
t
)
0
t
U
(
H
t
)
=;
with
t
B
m
((
+
c
+
km
)
=
2
;
1
=
2)
;
(9)
where
t
=1
;:::;T
denotes time,
Y
t
;t
=1
k;:::;T
, size
m
1, is observable data,
and
C
t
, size
c
1, denotes deterministic regressors such as a constant and a time trend.
The co ecient matrix
B
(0)
is of size
m
c
, the co ecient matrices
B
(
i
)
;i
=1
;:::;k
are