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

the following equations,

^f (^αt^,k^lDt)   α  ^f (yt∖at) ^f (^αt∖^αt-1,k)                               (¹3)

≤  g (yt↑^αt^,μt,_k)^f (^at∖^at-1,k)                         (¹4)

=  g ^yyt^μ^k,k) g ^a*t∖a_t-1,k^, yt^, μt,k)                  (¹5)

a g (at,k∖Dt)                                   (16)

To implement rejection sampling, k is first sampled from a density proportional
to g (y_t∣μ_t,_k}, then a_t is drawn from g (at∖at_-1 ,_k,y_t,μ_t,_k), which is equivalent of

sampling from g (αt, k|Dt).

As this is an approximating density, the pair (k, αt) is

accepted with probability

f (at,k∣Dt)
g (at, k∖Dt)

(17)

which can be rewritten as

^f (yt∖^at)
g ^yyt∖a_t,μk_k)

(18)

Pitt and Shephard (1999, 2001) developed these results and applied them to the

SV model in (1)-(2). They used μ_t,_k = φα_t-1 ,_k and the first order approximation

log g (yt∖at,^μt,_k) = const

at

2

^yt²

2^β2 exp ^μk_ttk)

μt,k         (19)

(20)

and

g yt ∖μ_tkk a exp

^μμt,k ^μt,k∖ f ^yt (^{1 + μ}t,k)
tʒf^e exp   2β² exp (μtkk)

where

„.._,,.+ ⅛(____y2__1

^μt^{k   μ}t^{k +}2 ' ² exp (μt,k)

(21)

(22)