Fiscal Policy Rules in Practice

For the full likelihood conditional on the regimes we obtain, using (4.3) and (4.6)

T

L(X|St,λ) = l(Xp∣Sp,λ) ∏ l(xt∣St,X-ι,λ).                     (4.7)

t=p+1

Integrating over all possible states gives the unconditional likelihood

T

L(X∣λ) = l(Xp∣λ) ∩ l(xt∣X-ι,λ).                            (4.8)

t=p+1

4.1.2 Generating the Regimes S using Gibbs-Sampling

We generate the regimes S with the help of multi-move Gibbs sampling. The idea is to obtain
the T elements in S within one draw conditional on λ and the observed data X . The starting
point is to make use of the structure of the underlying Markov chain. The density of the regimes
p(S ∣X, λ) can easily be rearranged in a multiplicative relationship as

p(S∣X, λ) = p(S1,... ,ST∣X,λ)

= p(ST ∣X, λ)p(ST -1,... ,S1∣ST,X-1,λ)

= p(ST ∣X, λ)p(ST -1∣ST, X-1, λ)p(ST -2,... ,S1∣ST-1,X-2,λ)

T-1

= p(Sτ∣X,λ) ∏p(St∣St+ι,xt,λ).                                  (4.9)

t=1

Knowing p(ST ∣X, λ) and p(St∣St+1, xt, λ) we could first draw ST. Conditional on ST it would
then possible to obtain ST-1, and again conditional on ST-1 we could draw ST-2 etc.

With some algebra one can show that

pijp(St = i∣xt,λ)

^p(St i|St+1   ^j,^xt,^λ)    k-^k        /a I w'                     (4.1⁰)

_z=1 pzjp(St = z∣xt, λ)

That means that given the matrix of transition probabilities P, it only requires P(St∣xt, λ)
to compute p(St∣St+1, xt, λ). p(St∣xt, λ) can in turn be determined using the filter proposed by
Hamilton (1989). This procedure demands initial values for S₀. We will briefly outline in the
following how these may reasonably be chosen.

4.1.3 Deriving the Initial Probabilities

Using the filter of Hamilton (1989) to compute p(ST ∣X, λ) requires initial values for p(S0∣X). By
assuming that the economy was in a steady state in t = 0, we may use steady-state probabilities
for p(S0 ∣X). The general condition for a steady-state probability is given by

P ∙ p(So∣X)= p(So∣X),
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