Second Order Filter Distribution Approximations for Financial Time Series with Extreme Outlier



J.Q. Smith and Antonio Santos

the following equations,


f (αt,klDt)   α  f (ytat) f (αtαt-1,k)                               (13)

≤  g (yt↑αt,μt,k)f (atat-1,k)                         (14)

=  g yytμ^k,k) g a*tat-1,k, yt, μt,k)                  (15)


a g (at,kDt)                                   (16)


To implement rejection sampling, k is first sampled from a density proportional
to
g (ytμt,k}, then at is drawn from g (atat-1 ,k,ytt,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,kDt)
g (at, kDt)


(17)


which can be rewritten as


f (ytat)
g yytat,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 (ytat,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 (αtαL1,ytl,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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