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

J.Q. Smith and Antonio Santos

must be more evenly distributed than those from the first order approximation
because the second order approximation allows a better approximation of the target
distribution.

5 Approximations based on maximum likelihood

estimates

The rather cumbersome second order methods described above all use the approxi-
mation based on a Taylor series expansion around the point μ_t,_k = φα_t- 1 ,_k suggested
by Pitt and Shephard (1999, 2001). For likelihoods associated with extreme obser-
vations, this is not where we expect the posterior density to centre its weight (Dawid
1973). For the class of SV models the weight should be more closely centred around
the maximum, α⅛ = log (¾2), of the likelihood function. We therefore propose using
the Taylor series approximation above in (24), but around o⅛.

There are two main advantages in using this approximation. Firstly, the algebra
needed to implement the procedure is greatly simplified. Secondly, these procedures
can be extended to include the cases where the likelihood is no longer log-concave.
We will focus here on the first advantage. Using α*_t = log (y2/β²) in (24), as
l⁰ (o⅛) = 0, we have

log g⁽ytαt^,α↑) = ^l(α) + 2^l'' ^(α)^(αt^{- α})^{2                (31)}

The algebra is simpler because we are able to combine the logarithm of the kernel
of two Gaussian densities, one given by the transition density and the other given
by 21⁰⁰ (α↑) (α_t — tit.)², which is the log-kernel of a Gaussian density with mean α↑
and variance — 1 /l" (α_t) = 2. Furthermore, the definition of the first weights is also

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

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