Estimation in Financial Models - RiskLab

The important property of the Gibbs sampling algorithm is that we only sample
from the full conditional distributions. As t tends to innity, ( t 1;:::; t k)
tends in distribution to a random vector with joint density (jx), see e.g.
[29]. In particular, t i tends in distribution to a random quantity with density
( i jx). The above algorithm is assumed to be replicated m times independently,
that means we have m replicates of t =( t 1;:::; t k). Then for large t
the replicates ( t i1;:::; t im) are approximately a random sample from ( i jx).
For m suciently large we obtain an estimate ^( i jx) for ( i jx) from
^( i jx) = 1 m
mX
l=1
For more details we refer to [55], p.226.
The Metropolis{Hastings algorithm
( i jx; t jl;j 6= i):
In order to construct a Markov chain 1 ;:::; t ;::: with state space and
equilibrium distribution (jx) we nd the transition probability from t
to the next state t+1 via the Metropolis{Hastings algorithm as follows:
Denote by q(; 0 ) a transition probability function such that in the case
t = , 0 drawn from q(; 0 ) is considered as a proposed possible value for
t+1 .However, a further randomization takes place. With probability (; 0 )
we accept t+1 = 0 , otherwise we reject the value generated from q(; 0 ) and
set t+1 = .
This construction denes a Markov chain with transition probability given
by
(
(; 0 q(;
)=
0 )(; 0 ); if 0 6= ;
1 , P 00 q(; 00 )(; 00 ); if 0 = :
With the denition
(; 0 )=
8
><
>:
( )
( 0 jx)q( 0 ;)
min
(jx)q(; 0 ) ; 1 ; if (jx) q(; 0 ) > 0;
1; if (jx) q(; 0 )=0;
provided that q(; 0 ) is chosen to be irreducible and aperiodic on a suitable
state space, we have that (jx) is the equilibrium distribution of the
constructed chain. For more details see [71].
Hierarchical models
An hierarchical model forms a structure in the following way: the distribution
of the data x is written conditionally on parameters 1 as p(xj 1 ),
and the distribution of 1 is written conditionally on 'hyperparameters' 2 as
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