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

J.Q. Smith and Antonio Santos

simplified,

2       2   7           2 ∖        ..2 2       ..2 2 __ ..2 X

/ I *∖           ^{α   α}t  ( ₁  I ^αt ʌ  , ^μt,k  , ^μt,k    ^μt,k ʌ                 ∕_qo∖

g(^y⅛) = ^exp (j-2- (₄¹ + y_y) +   .        .                  ^{( )}

After sampling k from a distribution proportional to (32), the particle αt-1,k is
chosen, and the density, assuming the role of prior density, assumes a Gaussian form
with mean μ_t,_k = φα_t- 1 ,_k and variance σⁱ_η. This is combined with a Gaussian density
with mean α_t² and variance σ² =2. The approximating density thus becomes:

g (^αt^lαt- 1 ,k^{, α2)} = Я ^μk_k^,σt,k}

where

and

(35)

After the particles have been sampled, they must be resampled in order to take into

account the target density. They are resampled using the second stage weights

Following the resampling stage, an approximation of the target posterior distribution
of the states at t is available, which will be used as a prior distribution to update
the states at t + 1.

To summarize, the particles at t - 1 propagated to update the distribution of
the states at t are chosen randomly according to the weights defined in (32). These
weights are influenced by the information contained in yt . By conditioning on each
particle chosen through the first stage weights, new particles are sampled. As these
come from an approximating distribution, a second step is necessary. The particles
are resampled using the weights defined in (36)-(37). Our modification, outlined
above, makes this second order APF straightforward and quick to implement.

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

<<   11   12   13   14   15   16   17   >>

More intriguing information