Fiscal Policy Rules in Practice

l(xt∣St,X-ι,λ) = (2π)^-m∕2∣Ω^-1 (St)|^1/2 ∙ exp {-1εt(St)Ω^-1 (St)εt(St)}           (4.3)

with εt(St) = xt - μ(St) - ∑^p=₁ Bh^xt-h.

As before, m denotes the number of variables in x_t and X_-1 = {x₁ , . . . , x_t-1}, i.e. all observa-
tions up to period t - 1. Next, we exploit the recursiveness of (4.2) for the first p observations
by substituting for Xt-1. This yields

Xp = μ + BXp—1 + εp

= μ + B(μ + BXp—2 + ε⁷p-ι) + εp

= μ + Bμ + Bεt-1 + it + B²Xp-2

∞∞

= ∑ B^τ μ + ∑ B^τ εt-τ

τ=0        τ=0

(4.4)

under the assumption that there is no regime shift prior to p. This enables us to write the
unconditional mean of Xp as

∞

E [Xp] =    B^τμi.

τ=0

For the existence of E [Xp] it requires that all eigenvalues of B have absolute values less than
one. For the variance of Xp it follows

V ar[Xp]

E(Xp - E(Xp))(Xp - E(Xp))'

E (∑ B^Ti-τ) (∑ B^T !<-τ) '

E (∑B^τε<-τεt-τ(B^τ)J

E f∑ B^τΩ(B^τ)J

V (Ω,B).

(4.5)

We are now able to approximate l(X_p |S_p , λ), which is the contribution of the first p data
vectors to the likelihood, by

l(Xp∣Sp, λ) = (2n)^-(mp)/2| V(Ω, B)^-1(Sp)|^1/2 ∙ exp ∣-1 Xp(St)V(Ω, B)^-1(Sp)Xp} .      (4.6)

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