Fiscal Policy Rules in Practice



l(xtSt,X-ι,λ) = (2π)-m∕2Ω-1 (St)|1/2 exp {-1εt(St-1 (Stt(St)}           (4.3)

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


As before, m denotes the number of variables in xt and X-1 = {x1 , . . . , xt-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 X
t-1. This yields


Xp = μ + BXp1 + εp

= μ + B(μ + BXp2 + ε7p-ι) + εp

= μ + + Bεt-1 + it + B2Xp-2


∞∞

= Bτ μ + Bτ 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 X
p as


E [Xp] =    Bτμi.

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 X
p it follows


V ar[Xp]


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


E (∑ BTi-τ) (∑ BT !<-τ) '

E (∑Bτε<-τεt-τ(Bτ)J

E f∑ BτΩ(Bτ)J

V (Ω,B).


(4.5)


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


l(XpSp, 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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