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

J.Q. Smith and Antônio Santos

The probabilities of acceptance referred to in (17)-(18) can be rewritten as

However, in the presence of outliers, this first order approximation turns out to
be a poor approximation of the target density. It is very difficult to accept any
candidate and the procedure can take an excessive amount of time to update the
posterior density. On the other hand, if instead of probabilities, (21) and (23) are
used to define first and second stage weights in an SIR procedure, then only a small
number of weights are non-negligible and so a continuous target is approximated by
a small number of distinct particles.

A common characteristic associated with financial time series like stock market
series is the presence of extreme observations. It is sometimes difficult to update
the information contained in yt when this represents an extreme observation. We
demonstrate below that the procedure based on a first order approximation cannot
cope with extreme observations. The approximating distribution is not close enough
to the target distribution and the approximation of the posterior distribution is very
poor.

Pitt and Shephard (1999, 2001) suggested the possibility of using a second order
approximation without developing it. The main problems are that it can be more
algebraically intensive and a perfect envelope cannot be defined. So SIR must be
used instead. The main idea here is to perform a Gaussian approximation to the
log-likelihood and combine it with a Gaussian transition density.

Within an APF approach the way forward is to develop an approximation to
the likelihood function, g yy_t∖a_t,μ_t^β,, and an approximation of the target density,
g (α_t,k∖D_t), which is factorized in g yyt∖∣t>t,k) and g (α_t∣α_t- 1 ,_k,y_t,μ_t,_k}, from which
first and second stage weights are defined. The approximation g yy_t∖at,μ_tk) is de-

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

<<   8   9   10   11   12   13   14   >>

More intriguing information