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

J.Q. Smith and Antonio Santos

statistics that characterize the distribution. Secondly, we must be able to imple-
ment these procedures sequentially as states evolve over time and new information
becomes available. This implementation needs to be efficient and the approxima-
tions need to remain good as we move through the sequence of states.

There are several ways of sampling from f (α_t∣D_t). Typically we simulate from
an approximating density g(α_t∣∙). After a draw is obtained from the approximating
density, it is modified to make it a draw from f(α_t∣D_t). The two most popular
techniques for performing this modification are sampling importance resampling
(SIR) and rejection sampling/Markov chain Monte Carlo (Gilks, Richardson, and
Spiegelhalter 1996; Gamerman 1997; Robert and Casella 1999; Doucet, de Freitas,
and Gordon 2001; Liu 2001).

There are always errors associated with the approximation of distributions with
continuous support from a discrete mass function. However, ignoring this aspect
of approximating error, the effective implementation of the particle filter depends
on how well we approximate f (α_t∣D_t) by g (α_t∣∙). If we could sample directly from
f (α_t∣D_t), then the sample would be independent and identically distributed and the
numerical approximation would depend only on the number of draws. On the other
hand, when it is not possible to sample directly from f (αt∣Dt), it becomes crucial
to define good approximations g (α_t∣∙). Here we suggest how this might be done for
a stock market return series.

We essentially use the SIR method to sample from the distribution of interest.
Taking into account the structure of the model, equation (3) can be used to define
the approximating density g (α_t∣∙), and subsequently its associated modifications.
When a set of particles is used to approximate f (αt-1 ∣Dt-1), {αt-1,1, . . . , αt-1,m}
with respective weights {πt-1,1 , . . . , πt-1,m}, each particle is used to define the den-
sity f ( αt∣Dt ) α f ( yt∣αt ) f ( αt∣αt- ɪ ,j ), j = 1 ,...,m. The approximating density

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

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