Fiscal Policy Rules in Practice



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(S
t)∙ μ(St) denotes a constant and Bh(t) is the matrix of coefficients for the
h
th 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 p
ij 0∙ We define the transition probabilities
for a switch from regime i to regime j as p
ij = 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
xt-1

,μ(St) =

' μ(St) '

0


, B(St) =

1
BS(t)

Im
0

2
BS(t)

0
I
m

...

...

p
BS(t)
0

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
S
t is given by

11The procedure may be directly applied to the univariate case.

14



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