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

J.Q. Smith and Antonio Santos

for example, by using Markov chain Monte Carlo (MCMC) techniques, the main
aim of this paper is to present certain modifications of particle filter methods which
have recently been proposed to predict the process. The predictions made by this
model, in contrast with ARCH family models, are expressed through the posterior
density of the states f ( α_t∣D_t ) and the predictive density of returns f ( y_t+1 ∣D_t ) rather
than through point predictions. Henceforth we assume a closed forecasting system
and we let Dt = {y0, y1, . . . , yt} represent the available information at time t. Our
modifications are straightforward but nevertheless appear to improve the predictive
performance dramatically when these models are applied to stock return series. We
first review current particle filter methods.

2 Particle Filter methods

The Bayes’ rule allows us to assert that the posterior density f (α_t∣D_t) of states is
related to the density f (α_t∣D_t- 1) prior to y_t and the density f (y_t∣α_t) of y_t given α_t
by

^{f (α}tD) α ^{f (}y_t∖a_t) ^{f (α}t^lDt- 1)                          (3)

and the predictive density of yt+1 given Dt is

Instead of numerically estimating these integrals, the particle filter approximates
these densities using a simulated sample.

Particle filters approximate the posterior density of interest, f (α_t∣D_t), through
a set of m “particles” {α_t,1, . . . , α_t,m} and their respective weights {π_t,1, . . . , π_t,m}
where πt,j ≥ 0 and _j^m₌₁ πt,j = 1. To implement these filters, we must first be able
to sample from the nonstandard density f (α_t∣D_t). It is possible to develop simula-
tion procedures to approximate the distribution of interest and to calculate certain

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

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