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

J.Q. Smith and Antonio Santos

g (α_t∣α_t- 1 ,j) = f (α_t∣α_t- 1 ,j) is used to define a tentative draw from f (α_t∖D_t). The
“plain” SIR algorithm can be expressed using the following steps:

1. Generate a new set of particles using the transition equation, αt,j = gt(αt∣αt-1,j),
j = 1, . . . , m. A new set of weights is calculated for each of the m particles
using the formula

wt,j = f(yt∣αt,j)

^wt,j

∏t,j = ÿ™-----^, j = ^{1 ,}...^,m                    (6)

i=1 ^wt,i

2. Resample from {α_t,1 , . . . , α_t,m} using the weights {π_t,1, . . . , π_t,m}, and thus
obtain a new set of particles with equal weights. These are then used in the
next iteration.

Although this method was regarded as a considerable breakthrough, it is now
widely recognized that this algorithm suffers from several weaknesses:

1. sample impoverishment (so the quality of the approximation thus deteriorates
as time passes);

2. a lack of robustness regarding outliers, and

3. typically poor approximation of the tails of the posterior distribution.

Improvements to the basic SIR algorithm focusing on robust filters to outliers
were recently proposed by Pitt and Shephard (1999, 2001). This algorithm - called
the auxiliary particle filter (APF) - is widely recognized as an important improve-
ment on the basic algorithm when implemented in time series such as those in
finance, where the weaknesses referred to above become critical.

From a sequential perspective the main objective is to update the particles at t -
1, and the respective weights, {α_t-1,1, . . . , α_t-1,m} and {π_t-1,1, . . . , π_t-1,m}. Using the

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

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