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 ε_t(S_t) is an normally distributed i.i.d. error term with mean zero and regime-dependent
covariance matrix Ω(S_t)∙ μ(S_t) denotes a constant and B^h(_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 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 ,...,μ_k ,Bι,...Bk, Ω_υ..., 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∙

Xt = μ(St) + B (St )Xt-ι + ^εt(St),

where

Furthermore, let X = (x₁ , . . . , x_T) be the vector of all observations∙

4.1.1 The Likelihood Function

The contribution of the t^th 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.

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