Page 423 - Contributed Paper Session (CPS) - Volume 4

CPS2509 D.L Sepato et al.
This study, despite the different types of outliers focused on four basic and
commonly used types of outliers in time series namely, the Additive Outlier
(AO), Innovational Outlier (IO), Temporary Change (TC) and Level Shift (LS)
outlier discussed in Fox (1972) proved that the power of the test performed
better than that of the iid assumption. Studies by (Andersen et al., (2007);
Boudt et al. (2013); Laurent et al. (2016), and Verhoeven & McAleer (2004)
caution that the consequence of neglecting jumps (outliers) in GARCH models
usually overestimate the volatility during several days, if not weeks, after the
occurrence of these jumps. Though these models have enjoyed the success in
studying the evolution of financial time series modelling, they are unable to
explain the high frequency and size of extreme jumps commonly occurring in
practice which may wrongly suggest conditional heteroscedasticity (Carnero
et al., 2007).

2. Methodology
Data used in this study consist of the daily closing prices of the primary
South African indices the JSE index obtained from the Johannesburg Stock
Exchange (JSE) ranging from January 3, 2011, to April 29, 2016. The percentage
log returns are calculated as: = 100 ∗ ( − 1), where Xt is the daily

closing price of JSE index stock market at time t, and Xt-1 denotes lagged
stock price on day t-1, this is used to forecast ARCH/GARCH (p, q) models.

2.1. General outlier detection in ARCH and GARCH models
An outlier detection procedure used by Fox (1972) follows the following
steps to detect outliers in a time series data:
1. Model a GARCH (1, 1) using the original data under the assumption
that there are no outliers
= + ,|−1~(0, ℎ) (3)
′
−1 is the division up to time t. In practice, consists of a constant
term
2. The outlier effect () and the residual variance, are computed from
the residuals.
3. Using (2), the test statistics ()is calculated for all possible = 1,… ,
where i denotes outlier type. When () = ⁄() this indicates an IO,
1
() = 1 implies an AO, () =
1 ⁄1 − ()which implies a LS outlier. Then = () ⇒ = + ,
where the IO
( )
( )
= and = . AO = and = () , LS = and =

()(1 − ) . This will test the null hypothesis for an outlier
−1 ( )

0: = = = 0 against : ≠ 0, : ≠ 0 ,: ≠ 0 (4)
This tests the likelihood ratio test statistics for testing H0 vs. HA, HI,
and HL respectively, = / for i = I, A, and L where is the standard
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