Page 251 - Contributed Paper Session (CPS) - Volume 6

CPS1909 Retius C. et al.
combining dynamic volatility models with heavy-tailed distributions in
modelling South African financial data. In this paper, we extend the work of
Paolella (2016) by proposing an APARCH (1,1)-GPD model, APARCH (1,1)-
PIVD model and compare it with the APARCH-SD model to the daily
FTSE/JSE All Share Price returns. We estimate VaR and then select the more
robust model using the Kupiec likelihood ratio test.
The rest of the paper is organised as follows. In section 2, we provide some
background theory on APARCH(1,1), generalized Pareto distribution, Pearson
type-IV distribution, stable distribution, VaR and backtesting. The data used in
this study is described in Section 3. Section 4 presents the empirical results
and discussions. Finally, Section 5 concludes this work.

2. Methodology
In this section, we present some background theory on the APARCH(1,1)
model combined with generalized Pareto, Pearson type-IV and stable
distributions. We also discuss VaR and backtesting procedures.
APARCH (1,1) model
Ding et al. (1993), introduced the APARCH model as an extension of the
GARCH model. The APARCH generalized both the ARCH and GARCH models.
The structure of the volatility equation is given by

= + ∑ (| − | + ) + ∑ (1)
−
−

=1 =1
where > 0, ≥ 0, ≥ 0, and 0 ≤ ∑ + ∑ ≤ 1. and are the

=1
=1
ARCH and GARCH coefficients respectively and is the leverage coefficient.

When is positive, it implies that the negative shocks has stronger impact on

price volatility than the positive shocks. is a positive real number which
functions as the symmetric power transformation of . Considering the case,

where = 1 for = = 1, then the volatility equation becomes:
= + (| −1 | + ) +
1 −1
1
1 −1

(2)
Generalized Pareto distribution
The two‐parameter generalized Pareto distribution (GPD), with scale
parameter β and shape parameter ξ, has the following distribution function
1 − (1 + ) −1/ if ≠ 0
() = {
1 − −( ) if = 0

(3)
Where = − are the exceedances above the threshold and > 0 when
≥ 0, 0 ≤ ≤ −/ξ when < 0, and the scale parameter > 0 (Tsay, 2013).
Threshold selection

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