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CPS1909 Retius C. et al.
we fit the GPD, Pearson type-IV and stable distributions to the standardized
residuals from the APARCH model.
Combining APARCH (1,1) with heavy-tailed distributions
APARCH (1,1)-GPD model
To fit the GPD model, we check whether the tail of the standardized
residuals follow a Pareto distribution. It can be shown using the Pareto
quantile plot that the tail of the data is almost a straight line confirming that
the standardized residuals may follow a generalized Pareto distribution. The
mean excess and the parameter stability plots were used to come up with a
reasonable high threshold . The suitable threshold must lie where there is a
positive change in the mean excess. We use the parameter stability plot to
check the threshold where the parameters are most stable. The estimated
parameters are more stable when ≥ 1.7. There are 635 observations above
the threshold. Since the exceedances above the threshold cannot be assumed
to be independent and identically distributed, declustering was performed.
We fit GPD (model 1) to the declustered exceedances. The maximum
likelihood (ML) parameter estimates with standard errors in brackets are
̂
obtained as = -0.0647 (0.1254) and ̂ = 0.3793 (0.0701) with 635
observations in the tail. The probability plot and the quantile plot suggest that
the exceedances seems to follow the GPD model. Thus, all the diagnostic plots
suggest that the exceedances follows the GPD model.
APARCH (1,1)-PIVD model
The PIVD is fitted to the standardized residuals extracted from the APARCH
(1,1) model with normal innovations. The parameters are estimated using the
method of maximum likelihood. The maximum likelihood procedure is carried
out using R package PearsonDS. The ML estimates of the Pearson type-IV
distribution fitted to the standardized residuals of the APARCH(1,1) model
̂
with normal innovations are ̂ = 12.66, ̂ = 11.7361, = −2.2198 and ̂ =
4.2221. The calculated AD statistic value is 0.2868 with a corresponding -
value of 0.9477. The the value of ̂ = 12.6666 > 0.5 , thus satisfying the
condition for a PIVD. The AD statistic is significant, thus the PIVD is a good fit
of the standardized residuals extracted from the APARCH(1,1) model.
APARCH (1,1)- SD model
The stable distribution (SD) is also fitted to the extracted standardized
residuals of the APARCH (1,1) model. The model is referred to as APARCH
(1,1)-SD model. The ML parameter estimates of a stable distribution fitted to
̂
the standardized residuals of APARCH (1,1) model are ̂ = 1.9163, =
̂
−1.0000, ̂ = 0.6784 and = 0.0778 . The calculated AD statistic value is
1.0652 with a corresponding -value of 0.3248. The value of the index of
stability (̂) is 1.9163 which is less than 2. This suggested that the tail of the
standardized residuals follows a Pareto law indicating the distribution is
̂
heavy-tailed and also has infinite variance. The stable skewedness () is -1,
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