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



J.Q. Smith and Antonio Santos

to t. This corresponds to sampling from

f ( αt,kDt ) a f ( yt αt ) f ( at |at_ ɪ) k, k =1 ,...,m              (7)

where π k represents the weight given to each particle. The aim is then to sample first
from
f (kDt) and then from f (αtk, Dt), obtaining the sample {(αt,j, kj) ; j = 1, . . . , m}.
The marginal density
f (αtDt) is obtained by dropping the index k.

This resolves the problem of too many states with negligible weight being carried
forward. However, the problem of defining a good approximation to the target
distribution still remains. One of the simplest approaches is to define

g (αι,k Dt) a f yytμu) f (at\at_ɪ) k

(8)


where μt,k is the mean, mode or a highly probable value associated to f (at|at_ɪ).
It can easily be seen that

g (k|Dt)


a f f (ytμt,k) f (at\a_ɪ) πk dαt

= f {ytμt,k)


(9)

(10)


This density is used to define the first stage weights. These are the ones used to
sample the index that tell us which particles at
t - 1 are used to define the posterior
distribution at
t. Given a set of indexes, the states are drawn from f (at\at_ɪ,k) and
the second stage weights are defined as

f = f (ytαtj)
wj f (ytμtj)

(11)


The information contained in yt is carried forward through first stage weights. After
the particles
at_ɪ,k, k = 1,... ,m are chosen, the densities used, f (at\at_ɪ,k), k =
1
, . . . , m do not depend any further on yt .

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



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