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,k∣Dt ) a f ( y_t ∣αt ) f ( at |a_t_ 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 (k∣D_t) and then from f (α_t∣k, D_t), obtaining the sample {(α_t,j, kj) ; j = 1, . . . , m}.
The marginal density f (αt∣Dt) 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} yy_t∖μ_u) ^f (at\at_ɪ) ∏k

where μ_t,_k is the mean, mode or a highly probable value associated to f (a_t|a_t_ɪ).
It can easily be seen that

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 (a_t\a_t_ɪ,_k) and
the second stage weights are defined as

f ₌ ^{f (}yt∣^αtj)
w^j f (yt∖μtj)

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

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

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