where is a sequence of independent and identically distributed (i.i.d.) random variables with mean zero, that is and variance 1, that is , , and (Francq and Zakoian  ). The coefficients , for , must satisfy some regularity conditions to ensure that the unconditional variance of is finite.
2.3.2. Generalized Autoregressive Conditional Heteroscedastic (GARCH) Model
Although the ARCH model is simple, it often requires many parameters to adequately describe the volatility process of a share price return. Some alternative models must be sought. Bollerslev  proposed a useful extension known as the generalized ARCH (GARCH) model. For a return series, , let be the innovation at time t. Then, follows a GARCH (q, p) model if
where again is a sequence of i.i.d. random variance with mean, 0, and variance, 1, and (Tsay  ).
Here, it is understood that , for , and , for . The latter constraint on implies that the unconditional variance of is finite, whereas its conditional variance , evolves over time.
2.3.3. Exponential Generalized Autoregressive Conditional Heteroscedastic (EGARCH)
The EGARCH model represents a major shift from ARCH and GARCH models (Nelson  ). Rather than modeling the variance directly, EGARCH models the natural logarithm of the variance, and so no parameter restrictions are required to ensure that the conditional variance is positive. The EGARCH (q, p) is defined as,
where again, is a sequence of i.i.d. random variance with mean, 0, and variance, 1, and is the asymmetric coefficient.
2.3.4. Glosten, Jagannathan and Runkle (GJR-GARCH) Model
The GJR-GARCH (q, p) model proposed by Glosten, Jagannathan and Runkle  is a variant, represented by
where is an indicator for negative , that is,
and and are nonnegative parameters satisfying conditions similar to those of GARCH models. Also the introduction of indicator parameter of leverage effect, in the model accommodates the leverage effect, since it is supposed that the effect of on the conditional variance is different accordingly to the sign of .
To successfully fit the processes describe by subsections 2.2 and 2.3, the approach is based on Box and Jenkins three iterative procedures which includes model identification, model estimation and diagnostic checking are summarized as follows.
Identification Stage: employs the plots of estimated acf (autocorrelation function) and pacf (partial autocorrelation function) as guides to selecting one or more ARIMA models that appears suitable. At this stage, models whose theoretical acf and pacf most closely resemble the estimated acf and pacf are selected tentatively.
Estimation Stage: The tentatively entertained model is fitted to data and its parameters estimated using maximum likelihood techniques.
Diagnostic checking Stage: Diagnostic checks are carried to help determine if an estimated model is statistically significant targeted at uncovering possible lack-of-fit. If no lack-of-fit is indicated, the model is ready to use. If any inadequacy is found, the iterative cycle of identification, estimation, and diagnostic checking is repeated until a suitable representation is found. (See Box, Jenkins and Reinsel  ; Akpan, Lasisi and Adamu  for more details on the procedures and its application, respectively.)
2.4. Outliers in Time Series
Generally, a time series might contain several, say k outliers of different types and we have the following general outlier model;
where , for an AO, and for an IO at , for a LS, for an TC, and is the size of outlier. For more details on the types of outliers and estimation of the outliers effects (see Moffat and Akpan  ; Sanchez and Pena  ; Box, Jenkins and Reinsel  ; Wei  ; Chen and Liu  ; Chang, Tiao and Chen  ).
Moreover, in financial time series, the residual series, is assumed to be uncorrelated with its own past, so additive, innovative, temporary change and level shift outliers coincide, and where both the mean and variance equations evolves together, we have for example GARCH(1,1) model:
where is the outliers contaminated residuals.
2.5. Methods of Outliers Detection in Heteroscedasticity
One approach for correcting the series for outliers is using standard criteria and then estimates the conditional variance. This approach involves detecting and correcting of outliers before estimating the conditional variance (Carnero, Pena and Ruiz  ). This very method is based on the iterative framework of Chen and Liu  summarized in the following steps;
Step I: Given an ARIMA model fitted to the data, all the potential outliers are detected based on preliminary model parameter estimates.
Step II: Joint estimates of the model parameters and outliers effects are obtained using the accommodated outlier information of step I.
Step III: Outliers are identified and their effects estimated again based on the adjusted estimates of model parameters obtained in Step II.
2.6. Efficiency of Heteroscedastic Models
Efficiency is a measure of quality of an estimator of a model. It is often expressed using variance or mean square error. For the purpose of this study which looks at a unified effect of outliers, unconditional variance and coefficient of kurtosis are considered as the measures of efficiency of estimator of heteroscedastic model. The application of coefficient of kurtosis in this case is to ensure that the existence of heavy-tailed is taken care of.
For ARCH(q) model which is equivalent to GARCH(q, 0) model, the unconditional variance is given as follows:
For GARCH(q, p) model, the unconditional variance is expressed thus:
For EGARCH(q, p) model, the unconditional variance is expressed as follows:
where exp represent natural exponential function.
For GJR-GARCH(q, p) model
Kurtosis coefficient for a centered (that is, zero-mean) distribution is defined as the ratio of the fourth-order moment, which is assumed to exist, to the squared second-order moment (Francq and Zakoian  ). This coefficient is equal to 3 for a normal distribution (mesokurtosis). For heavy-tailed distribution, the coefficient is greater than 3 (leptokurtosis). The excess kurtosis of GARCH(1,1) model can be obtained as follows:
Note that, , , and , where is the excess kurtosis of the innovation, .
provided that exists.
provided that and . the excess kurtosis of , if it exists, is then
This excess kurtosis can be written in an informative expression. Considering the case where follows a normal distribution, ,
where the superscript, g, is used to denote the Gaussian distribution. The same idea applies to other GARCH-type models (Tsay  ).
3. Results and Discussion
Data collection is based on secondary source as documented in the records of Nigerian Stock Exchange. The documented data on the daily closing share prices of the sampled banks (Skye bank, Sterling bank, Unity bank and Zenith bank) from January 3, 2006 to November 24, 2016 were purchased from the Nigerian Stock Exchange and delivered through email@example.com. Since the data were obtained from a credible and secured source therefore reliable.
3.2. Interpretation of Time Plot
The share prices of the four prominent Nigerian banks considered are found to be nonstationary given the random fluctuations away from the common mean (see Figures 1-4).
To achieve stationarity, Equation (1) was applied to the share price series and these transformed series were found to cluster round the common mean and thus indicated the presence of heteroscedasticity (see Figures 5-8).
3.3. Modeling Joint ARIMA-GARCH-Type Processes of Return Series of Nigerian Banks
Based on Box and Jenkins procedures, out of the several models identified tentatively, the following joint ARIMA-GARCH-type models with respect to both normal (norm) and student-t (std) distributions in (Table 1) were considered and selected on the grounds of smallest information criteria and model adequacy (see Table 2).
3.4. Identification of Outliers in the Residual Series of ARIMA Models Fitted to the Return Series of Nigerian Bank
Here, we examined the residuals series of the fitted ARIMA models for detection
Figure 1. Share price series of Skye Bank.
Figure 2. Share price series of Sterling Bank.
Figure 3. Share price series of Unity Bank.
Figure 4. Share price series of Zenith Bank.
Figure 5. Return series of Skye Bank.
Figure 6. Return series of Sterling Bank.
Figure 7. Return series of Unity Bank.
Figure 8. Return series of Zenith Bank.
Table 1. Output of ARIMA-GARCH-type models of returns series of Nigerian Banks.
of possible potential outliers in the returns series of the banks under study. The iterative procedure of Chen and Liu  was applied and those statistics that are in absolute value higher than a threshold (critical value, C) identify the time point of a potential outlier. In this study, C = 4 is chosen on the condition that the number of observations, T ≥ 450 and where C = 4 is not sufficient, C = 5 is used.
Table 2. Diagnostic Checking for ARIMA-GARCH-type models of returns series of Nigerian Banks
LB = Ljung-Box, LM = Lagrange Multiplier.
3.4.1. Identification of Outliers in the Residual Series of ARIMA (1, 1, 0) Model Fitted to the Return Series of Skye Bank
About twenty six (26) different outliers were identified to have contaminated the residuals series of ARIMA(1,1,0) model using the critical value, C = 4; six (6) innovation outliers (IO), six (6) additive outliers (AO) and fourteen (14) temporary change (TC) as indicated in (Table 3).
3.4.2. Identification of Outliers in the Residual Series of ARIMA (2, 1, 0) Model Fitted to the Return Series of Sterling Bank
About seven (7) different outliers were identified to have contaminated the residual series of ARIMA(2,1,0) model using the critical value, C = 5 one (1) innovation outlier (IO), four (4) additive outliers (AO) and two (2) temporary change (TC) as shown in (Table 4).
3.4.3. Identification of Outliers in the Residual Series of ARIMA (1, 1,0) Model Fitted to the Return Series of Unity Bank
About thirty three (33) different outliers were identified to have contaminated the residual series of ARIMA(1,1,0) model using the critical value, C = 5; two (2) innovation outliers (IO), six (6) additive outliers, fifteen (15) temporary change (TC) and ten (10) level shift (LS) as indicated in (Table 5).
3.4.4. Identification of Outliers in the Residual Series of ARIMA (2, 1, 1) Model Fitted to the Return Series of Zenith Bank
About forty two (42) different outliers were identified to have contaminated the residual series of ARIMA(2,1,1) model using the critical value, C = 5; thirteen (13) innovation outliers (IO), nine (9) additive outliers and twenty (20) temporary change (TC) as shown in (Table 6).
Table 3. Outliers identified in the residual series of ARIMA(1, 1, 0) model fitted to return series of Skye Bank.
Table 4. Outliers identified in the residual series of ARIMA(2, 1, 0) model fitted to return series of Sterling Bank.
Table 5. Outliers identified in the residual series of ARIMA(1, 1, 0) model fitted to return series of Unity Bank.
Table 6. Outliers identified in the residual series of ARIMA(2, 1, 1) model fitted to return series of Zenith Bank.
3.5. Modeling Joint ARIMA-GARCH-Type Processes of Outlier Adjusted Return Series of Nigerian Banks
However, with the identified outliers being adjusted for, we obtained a new series (outlier adjusted return series). Again, Box and Jenkins iterative procedures were applied and those models in (Table 7) were selected based on smallest information criteria and model adequacy (Table 8).
3.6. Determination of Efficiency of ARIMA-GARCH-Type Models of Returns Series of Nigerian Banks
The ARIMA(1, 1, 0)-GARCH(1, 1)-norm model fitted to both outlier adjusted return series and outlier contaminated series with the same unconditional variance, 0.0016. Again, the value of kurtosis, 2.9465 captured by GARCH(1, 1)-norm model on outlier adjusted return series is nearly the value accommodated by the normal distribution while GARCH(1, 1)-norm model on outlier contaminated series seems inferior with excess kurtosis of 132.8707 (see Table 9).
From (Table 10), evidence shows that ARIMA(2, 1, 2)-EGARCH(1, 1)-std model fitted to outlier adjusted return series appeared to be more efficient given that the unconditional variance and kurtosis coefficient are smaller than that of the ARIMA(2, 1, 0)-EGARCH(1, 1)-norm model fitted to the outlier contaminated series.
From (Table 11), evidence shows that ARIMA(1, 1, 0)-GJR-GARCH(1, 0)-norm model fitted to outlier adjusted return series appeared to be more efficient given that the unconditional variance and kurtosis value are smaller than that of the ARIMA(0,1,1)-GARCH(1,1)-norm model fitted to the outlier contaminated series.
The ARIMA(2, 1, 1)-EGARCH(1, 1)-std model fitted to both outlier adjusted return series and outlier contaminated series converges to respective unconditional variances of 5.118125e−5 and 5.887684e−36 with corresponding kurtosis values of 3.5746 and 26.3794. Though, the unconditional variance of ARIMA(2, 1, 1)-EGARCH(1, 1)-std model fitted to the outlier contaminated series is smaller than that of the outlier adjusted return series, the model for the outlier adjusted return series is more efficient given that the kurtosis value is near three, the value occupied by normal distribution (Table 12).
Table 7. Output of ARIMA-GARCH-type models of outlier adjusted returns series of Nigerian Banks.
Our study has shown that the use of minimum unconditional variance as a measure of efficiency of heteroscedastic models in the presence of outliers is not heuristic as outliers are capable of inflating or reducing the unconditional variance. To this end, the use of kurtosis coefficient as a measure of heteroscedastic
Table 8. Diagnostic checking for ARIMA-GARCH-type models of outlier adjusted returns series of Nigerian Banks.
LB = Ljung-Box, LM = Lagrange Multiplier.
Table 9. Efficiency of ARIMA-GARCH-type model of Skye Bank.
Table 10. Efficiency of ARIMA-GARCH-type model of Sterling Bank.
Table 11. Efficiency of ARIMA-GARCH-type model of Unity Bank.
Table 12. Efficiency of ARIMA-GARCH-type model of Zenith Bank.
models in the presence of outliers becomes more tractable irrespective of the choice of distribution of the innovations. In addition, this study highlights that the applicability of iterative methods in time series modeling and in gauging model efficiency yet failed to consider the application of iterative methods in forecasting. It is recommended that further studies be extended to focus mainly on the application of iterative methods in maximum likelihood estimation of GARCH parameters.
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