J.Q. Smith and Antonio Santos
the following equations,
f (αt,klDt) α f (yt∖at) f (αt∖αt-1,k) (13)
≤ g (yt↑αt,μt,k)f (at∖at-1,k) (14)
= g yytμ^k,k) g a*t∖at-1,k, yt, μt,k) (15)
a g (at,k∖Dt) (16)
To implement rejection sampling, k is first sampled from a density proportional
to g (yt∣μt,k}, then at is drawn from g (at∖at-1 ,k,yt,μt,k), which is equivalent of
sampling from g (αt, k|Dt).
As this is an approximating density, the pair (k, αt) is
accepted with probability
f (at,k∣Dt)
g (at, k∖Dt)
(17)
which can be rewritten as
f (yt∖at)
g yyt∖at,μkk)
(18)
Pitt and Shephard (1999, 2001) developed these results and applied them to the
SV model in (1)-(2). They used μt,k = φαt-1 ,k and the first order approximation
log g (yt∖at,μt,k) = const
at
2
yt2
2β2 exp μkttk)
μt,k (19)
which, combined with the Gaussian transition density, gives rise to an approximating
density that can be factorized into two densities
g (αt∖αL1,yt,μl,k) = N μ'k∣-σ.2∣)
(20)
and
g yt ∖μtkk a exp
μμt,k μt,k∖ f yt (1 + μt,k)
tʒfe exp 2β2 exp (μtkk)
where
„.._,,.+ ⅛(____y2__1
μtk μtk +2 ' 2 exp (μt,k)
(21)
(22)
G.E.M.F - F.E.U.C.
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