4 Appendix
4.1 Bayesian Analysis of Markov-Regime Switching Models
We consider the more general case of a VAR11 of order p given by
p
xt = μ(St ) + BS(t)xt-h + εt(St), (4∙1)
h=1
where εt(St) is an normally distributed i.i.d. error term with mean zero and regime-dependent
covariance matrix Ω(St)∙ μ(St) denotes a constant and Bh(t) is the matrix of coefficients for the
hth lag included∙ As indicated by St we assume that both the parameters included in B as well as
the covariance Ω can adopt k different states. In any period parameters and covariance matrix
may switch to a new state with a probability pij ≥ 0∙ We define the transition probabilities
for a switch from regime i to regime j as pij = p(St = j|St-1 = i)∙ We then summarize these
probabilities in the transition matrix P with size (k × k)∙
The aim is then to estimate the set of unknown parameters given by
λ ≡{μι ,...,μk ,Bι,...Bk, Ωυ..., Ωk ,P} .
In partitioned notation this expression reduces to λ ≡ {Θ, P}∙ Furthermore, it is convenient
to rewrite (4∙1) in stacked form as a VAR(1) model, i∙e∙
(4∙2)
Xt = μ(St) + B (St )Xt-ι + εt(St),
where
Xt = |
xt ∙ ∙ |
,μ(St) = |
' μ(St) ' 0 ∙ |
, B(St) = |
1 Im |
2 0 |
... ... ∙ ∙ ∙ |
p 0 |
,ε(St) = |
' εt(St) 0 ∙ |
xt-p+1 |
0 |
∙ ∙ 0 |
∙ ∙ ∙ 0 |
∙ ∙ ∙ 0 |
∙ ∙ 0 |
0 |
Furthermore, let X = (x1 , . . . , xT) be the vector of all observations∙
4.1.1 The Likelihood Function
The contribution of the tth vector of observations xt to the likelihood conditional on the regime
St is given by
11The procedure may be directly applied to the univariate case.
14