Fiscal Policy Rules in Practice

2.1 Bayesian Analysis of Markov-Switching Models

The Bayesian analysis of Markov-switching models goes back to McCulloch and Tsay (1994).
They show that Bayesian estimation of Markov-switching models is kept relatively simple when
using the Gibbs sampler, as it solves the problem of drawing samples from a multivariate density
function by drawing successive samples from the corresponding univariate density functions. The
exposition given in the following is based on Harris (1999) and Krolzig (1997).

We consider the following simple univariate model, where the parameters can take on k different
states S,

yt = μ(St)+ XtB(St)+ εt(St),                              (2.2)

where X_t is the vector of explanatory variables and ε_t(S_t) a 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(St) is the vector of coefficients in state St. Furthermore, we define the transition proba-
bilities for a switch from regime i to regime j as pij = p(St = j|St-1 = i). We summarize these
probabilities in the transition matrix P with size (k × k).

Let λ denote the set of all unknown parameters, i.e.

λ = [μ(1),..., μ(k), B(1),..., B(k), Ω(1),..., Ω(k), P].

In partitioned notation this boils down to λ = [Θ, P]. Inference on λ depends on the posterior
distribution

p(^λ Y) α π(^λ)ρ(YI^λ),

where Y' = (y₁,... ,yT) is the vector of observations and π(λ) the prior for the parameter
vector. As we are in a Markov-regime switching environment, we have additional unknown
parameters given by the unobservable states. Therefore, the posterior density (2.3) is obtained
by the integration of the joint probability distribution with respect to the state vector S, i.e.

The problem arising from (2.4) is that the posterior distribution of λ depends on an unknown
multivariate distribution p(λ, SIY). The Gibbs sampler offers a solution to this problem, as it
allows to draw successive samples from univariate distributions for λ and S, namely p(SIY, λ)
and p(λIY, S), instead of the multivariate distribution p(λ, SIY). The Gibbs sampler constructs
a Markov chain on (λ, S) such that the limiting distribution of the chain is the joint distribution
of p(λ, SIY). There are two types of Gibbs sampler, single-move and multi-move, which differ in
the way the states S are generated. We apply multi-move sampling as it - according to Liu et
al. (1994) - will lead to a faster convergence than single-move sampling.

The idea of multi-move Gibbs sampling is to draw all states in S at once conditional on the
observations. The starting point is to make use of the structure of the underlying Markov chain,
i.e.

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