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

J.Q. Smith and Antonio Santos

structure of the model (1)-(2), due to the Gaussian characteristics of the transition
density, f (α_t∣α_t- 1 ), this would be a natural candidate for the approximating density.
However, as stated by Pitt and Shephard (1999, 2001), this is not the most efficient
procedure because it constitutes a blind proposal that does not take into account
the information contained in yt . One way of improving forecasting procedures is to
include this information in the approximating distribution. When this is done, the
nonlinear/non-Gaussian component of the measurement equation starts to play an
important role and certain algebraic manipulations need to be carried out in order
to use a standard approximation.

The design of the samplers must approximate the target distribution well but
another important aspect need to be taken into account. When states are updated,
in the presence of extreme observations, there are many particles with negligible
weight and it is extremely difficult to propagate such particles. More rudimentary
procedures, that treat all previous particles equally, will imply that only a small set
of the new particles have non-negligible weight.

3 Auxiliary Particle Filter procedures

To overcome the problems posed by more rudimentary particle filter procedures,
Pitt and Shephard (1999, 2001) proposed the APF method. The basic idea is that
only part of the particles available at t - 1 are propagated. These particles are chosen
randomly but take into account the information presented in yt . Only particles with
non-negligible likelihood are propagated.

This can be accomplished by sampling from a higher dimensional distribution.
First an index k is sampled, which defines the particles at t - 1 that are propagated

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

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