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 (y_t∣α_t) around μ_t,_k,
log g (yt ^αt∙μ_t,k) ^x l (μ_t,k) + l⁰ (μ_t,k) (α^- μ_t,k) +¹ l⁰⁰ (μ_t,k) (α^- μ_t,k¢2 (24)
= const ^{- α}t ^-   2 ^yt ⁽---γ ( (^αt ^- μt,k) + (^αt 2^μt^,k) ^{- 1}!        (25)

2    2^β exp μt_tyk)            ^,           2 y

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

g (a_t∖a_t-I,k^,yt^,μt,k} = ^{n μ}t,k^,σlk}                         (26)

and

Л * 2

l ^μt,k ^-

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 (α_t, k|D_t). The weights used in this resampling step are

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.

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