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 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 Xt-1. This yields
Xp = μ + BXp—1 + εp
= μ + B(μ + BXp—2 + ε7p-ι) + εp
= μ + Bμ + 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 Xp 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 Xp 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(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)
15
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