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



J.Q. Smith and Antonio Santos

fined through the second order Taylor approximation of log f (ytαt) around μt,k,
log g (yt αt∙μt,k) x l (μt,k) + l0 (μt,k) (α- μt,k) +1 l00 (μt,k) (α- μt,k¢2 (24)
=
const - αt -   2 yt (---γ ( (αt - μt,k) + (αt 2μt,k) - 1!        (25)

2    2β exp μttyk)            ,           2 y

Using this second order approximation, the density g (αt, k|Dt) is factorized as
in equation (15) and the factors are

g (atat-I,k,yt,μt,k} = n μt,klk}                         (26)

and

where


and


g yytμt,k)


exp (2 ɑ + 2β 2 exp ( μt,k )


Л * 2

l μt,k -


× exp


-_ yt (1 + μt,k)
2β2 exp (μt,k)


1j+ + yt ʌ 1 ï yt (1 + μt,k ) + μt,k _ 1
σηση 2β2 exp (μt,k)J y 2β2 exp (μt,k)    ^1 2


2 β 2 σ П

2β2 + exp (-μt,k) σ2У2


(27)

(28)


(29)

(30)


As we sample from g (αt, k|Dt), an approximating sample, the elements in it

must be resampled in order to obtain a sample that gives a better approximation of
the target density
f (αt, k|Dt). The weights used in this resampling step are

log wj

πj


У2

2β2 exp (αt,j )


yt (i_ αt,j (1 _ αj+μt,k ) + ( μt,k+μ2k ´´
2β2 exp μttjβ)


wj
p mwi,


j = 1, . . . ,m


These are the so-called second stage weights that allow the modification of the
approximating distribution towards the target distribution. Obviously, these weights

G.E.M.F - F.E.U.C.

11




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