almost surely, where b is the number of realizations discarded in a burn-in period. See, e.g., Robert and Casella
2004 for an
overview on MCMC methods from both theoretical and practi- cal standpoints.
First, we need to specify the expressions of the prior distri- bution, p. We use independent uniform prior distributions
in the domain of all the bounded parameters of the vector , that is, the parameters α
i, 1
, . . . , α
i,p
i
′
of the EGARCH mod- els, the parameters of the correlation matrixes of the vector ,
and the parameters of the Gaussian mixture of the vector . On the other hand, we use independent normal distribution
with mean 0 and a very large variance e.g., 100 for all of the unbounded parameters of the vector , that is, the parame-
ters μ
i
, ω
i
, β
i, 2
, . . . , β
i,q
i
, δ
i
, γ
i ′
of the EGARCH models. The variance of this Gaussian prior is much larger than the variance
of the posterior distribution obtained, so it ensures that we are using noninformative, but proper priors for all of the model pa-
rameters.
Now we construct an MCMC algorithm to sample from the
posterior distribution p|y, z. As noted by Vrontos, Della- portas, and Politis
2003a ,
2003b , the convergence of this type
of algorithm may be accelerated by updating the highly cor- related parameters simultaneously using a block-sampling ap-
proach. Thus we define the following algorithm scheme whose main steps are elaborated as follows:
1. Set n = 0 and initial values =
0′ 1
, . . . ,
0′ K
,
0′
,
0′ ′
.
2. Obtain a sample, z
n+1
, of the distribution of z|y,
n
. 3. Obtain a sample,
n+1 i
, of the distribution of
i
|
n+1 1
, . . . ,
n+1 i−1
,
n i+1
, . . . ,
n K
,
n
,
n
, y, z
n+1
, for i = 1, . . . , K.
4. Obtain a sample,
n+1
, of the distribution of |
n+1 1
, . . . ,
n+1 K
,
n
, y, z
n+1
. 5. Obtain a sample,
n+1
, of the distribution of |
n+1 1
, . . . ,
n+1 K
,
n+1
, y, z
n+1
. 6. Define n = n + 1 and go to 2, until n = N, for a large N.
In step 2 we sample from the conditional posterior proba-
bilities that each multivariate return y
t
, for t = 1, . . . , T, has been generated from the sth component. These probabilities are
given by pz
t
= s | y,
= ρ
s
σ
K s
exp−U
s
2σ
2 s
S v=1
ρ
v
σ
K v
exp−U
v
2σ
2 v
, s = 1, . . . , S,
where U
s
=
t:z
t
=s
y
t
− μ
′
H
−1 t
y
t
− μ, s = 1, . . . , S. In step 3 we sample from the conditional posterior probability of
i
whose kernel is given by κ
i
|
1
, . . . ,
i−1
,
i+1
, . . . ,
K
, , , y, z
= p
i
×
T t=1
H
−12 iit
|R
t
|
−12
× exp −
1 2
S s=1
U
s
σ
2 s
, 9
where p
i
is the prior probability of
i
. To do that, we make use of the random-walk Metropolis–Hastings method RWMH
see, e.g., Robert and Casella 2004
using the following steps: 3.1. Generate a candidate vector
i
from the multivariate normal distribution N
i n
, c
i
, where c is a con- stant and
i
is the covariance matrix of the ML esti- mate of
i
. Let τ
n
i
= min 1,
κ
i
|
n+1 1
, . . . ,
n+1 i−1
,
n i+1
, . . . ,
n K
,
n
,
n
, y, z
n+1
κ
n i
|
n+1 1
, . . . ,
n+1 i−1
,
n i+1
, . . . ,
n K
,
n
,
n
, y, z
n+1
, where κ
i
|
n+1 1
, . . . ,
n+1 i−1
,
n i+1
, . . . ,
n K
,
n
,
n
, y, z
n+1
is given in eq. 9
. 3.2. Define
n+1 i
=
i
, with probability τ
n
i
n i
, with probability 1 − τ
n
i
. The constant c is taken by tuning the acceptance rate to
achieve fast convergence. Usually, an acceptance rate lying be- tween 0.2 to 0.5 is plausible and practical for good convergence.
Finally, in steps 4 and 5 we sample from the conditional pos- terior distributions of and , respectively, whose kernels are
given by κ
|
1
, . . . ,
K
, , y, z
= p ×
T t=1
K i=1
H
−12 iit
|R
t
|
−12
× exp −
1 2
S s=1
U
s
σ
2 s
and κ
|
1
, . . . ,
K
, , y, z
= p ×
S s=1
ρ
s
σ
K s
T
s
×
T t=1
K i=1
H
−12 iit
× exp −
1 2
S s=1
U
s
σ
2 s
, respectively, where T
s
=
T t=1
Iz
t
= s, with
S s=1
T
s
= T, and p and p are the prior probabilities of and , re-
spectively, which can be performed using a similar RWMH as that described in step 3.
Besides making inference on the parameters of the GMDCC model, we may use the Markov chain to estimate in-sample
volatilities and correlations and to predict future volatilities and correlations. First, a sample from the posterior distribu-
tion of each conditional variance, H
iit
, for i = 1, . . . , K and t = 1, . . . , T, can be obtained by calculating the value of each
conditional variance for each draw,
n
, which is denoted by H
n iit
, for n = b + 1, . . . , T. Then the posterior expected value of H
iit
, E[H
iit
| y], can be approached by the mean of the posterior sample of conditional volatilities, that is,
1 N − b
N n=b+1
H
n iit
.
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:34 12 January 2016
In addition, 95 Bayesian confidence intervals can be obtained by just calculating 0.025 and 0.975 quantiles of each poste-
rior sample, respectively. Similarly, we can estimate in-sample correlations R
ijt
, using the draws R
n ijt
. A sample from the pre- dictive distribution of H
ii,T+1
and R
ij,T+1
and 95 predictive intervals can be obtained similarly to the case of in-sample es-
timation. On the other hand, the predictive density of y
T+1
is given by
py
T+1
|y = py
T+1
|y, p|y d, 10
where py
T+1
|y, is a mixture of S multivariate Gaussian
distributions with mean μ and covariances σ
2 s
H
T+1
, for s =
1, . . . , S. Thus the predictive density py
