J.Q. Smith and Antonio Santos
with a density given by
1 y12 (α1 - φm0)2
f (α11Dι) ∖ —α1Λ exp -.i,2—. exp - ,—2 ʌ (38)
exp (2) ∖ 2β exP(αɪ)/ y 2 φ2 Cо + σ^ J
The constant c can be obtained using numerical integration. In this way the mean
and standard deviation associated to the distribution of α1 |D1 can also be obtained.
By varying the value of y1 , it can be appreciated how the results obtained using the
APF with a first or second order approximation deviate from the exact results. In
Figure 4 shows the mean and standard deviation evolution associated with the pos-
terior distribution of α1 |D1 using the three techniques described above for different
values of y1 , which vary between 0 and 9: values compatible with most financial
time series. The parameters used were β = 1.0, φ = 0.97, ση = 0.15, m0 = 0 and
C0 = 0.3. It can easily be seen in Figure 4 that the APF based on a first order
approximation of the log-likelihood is not robust to outliers when compared with an
higher order approximation. We note that, with the first order approximation, for
values of y1 greater than 8 the estimated standard deviation is zero, indicating that
the continuous density f (α 1 ∣D 1) is approximated by a single point. This represents
the extreme case of sample impoverishment.
7 Conclusion
We have demonstrated that it is possible to develop APFs based on a second order
Taylor series approximation, which unlike their first order analogues perform well for
series with extreme observations, which are fairly common in financial time series.
We are now developing this procedure for time series whose likelihood is not log-
concave. Preliminary results are encouraging and will be given in a future paper.
G.E.M.F - F.E.U.C.
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