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CPS1145 Adeniji Oyebimpe Emmanuel et al.
2. Methodology
We define rt as a (t x1) vector of assests log- returns up to time t that is:
rt = t = tzt (1)

where zt follows a particular probability distribution, andt is the square root
of the conditional variance. The mean equation of the model ie the
deterministic aspect of the series follows Autoregressive Model, AR(p),
yt= 1 yt-1+ t (2)

the Standard GARCH (p,q) model by Bollerslev in (1986) is given as :

(3)

where 0 > 0, i ≥0 (for i=1, ---, q),  j ≥0 (for j=1, …, p) is sufficient for
conditional variance to be positive and stationarity.
To capture asymmetry observed in series, a new class of ARCH model was
introduced: The asymmetric power ARCH (APARCH) model by Ding et’al
(1993), the exponential GARCH (EGARCH) model by Nelson (1991), the GJR by
Glosten et al. (1993). This model can generate many model when the
parameters are relaxed and is expressed as:

(4)

where 0 > 0,  ≥0, i ≥0, i =1,...q , -1 <i <1, i=1,...q,j 0, j =1,… p
The i parameter permit us to catch the asymmetric effects. The conditional
standard deviation process and the asymmetric absolute residuals in the
model were imposed in term of a Box-transformation. The well-known
Leverage effect is the asymmetric response of volatility to negative and
positive shocks. Harvey (2013) developed three sets of volatility models that
take into account robust capturing occasional changes in financial series
known as jumps, he considered the EGARCH and AEGARCH types of the Beta-
GARCH models each with two distributions assumptions applied.

The BetaEGARCH model specified without the leverage effect is given:
(5)

Introducing the leverage effect we have the Beta-AEGARCH model (ie EGAS):
(6)

where lt-1 =sgn (-zt )( µt+1) when student-t distribution is considered and lt-1
=sgn (-zt )( µt+1) for skewed student-t distribution. Again, combing the same
student-t with EGAS model leads to Beta-t AEGARCH ie AEGAS model.

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