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CPS2509 D.L Sepato et al.
deviation of the estimate. Under the null hypothesis of no outliers,
these statistics are asymptotically distributed as N (0, 1) (Jesús Sánchez
& Pena, 2003).
4. The maximum of the absolute value of these test statistics,
max==1,…,|̂ ()| is computed. If the value of the test statistic
exceeds the pre-specified critical value, C (significant) then an outlier is
detected. Thus, the point t where occurs is the point detected as
having the outlier. The procedure is to compute the effect of outliers
on residuals following (Tsay, 1988) where the actual parameters and
2
are known in the modelling procedure estimated by any consistent
estimator by employing: AO: = {:1≤≤}|,| , IO:
={:1≤≤}|,|, LS: = {:1≤≤}|,|, (4,5,6) where
these are used as testing criterions for outlier detection. Comparing
the test statistics with critical value C the existence of outliers can be
detected, where the time points at which the above maxima occur are
timings of the corresponding outliers.
5. The final step is to determine whether the observations are outliers and
by removing each outlier from the series by deducting the value of the
effect of , then apply the GARCH modelling procedure to obtain the
most adequate model and use it for forecasting future values of the
series.

2.2. Information criterion for model selection between the candidate
models
The model with the highest subsequent probability is the one that
minimises BIC, a desirable model is one that minimises the AIC or the BIC
(Ngailo, 2011).

2.3. Model diagnostics
This section discusses the model diagnostic tests for the selected model
such as runs test for independence, normality tests, Test for autocorrelations
and heteroscedasticity in that respect.

2.4. Forecasting performance evaluation
This study uses three evaluation measures to evaluate the forecast
accuracy of JSE top 40 index for the proposed model ARMA-GARCH model
applied, namely the Root Mean Square Error (RMSE), Mean Absolute Error
(MAE), and Mean Absolute Percentage Error (MAPE) statistics.

3. Result
This section provides and discuss the results that quantify, summarise and
check the distributional properties of the financial time series.

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