This study traced the patterns of discrete time series over time with respect to GARCH effect and asymmetric GARCH effect. Particularly, we paid attention to the weakness of the GARCH model in modeling the asymmetry of GARCH effect. In order to handle this weakness, we applied the sign and size bias test which comprises sign bias test, negative size bias test, positive size bias test, and Lagrange Multiplier test in order to identify the asymmetric effect in the residual series of the GARCH model. Where the asymmetric effect is present and significant, we fit the asymmetric GARCH models. Exploring the share price returns of Zenith bank plc obtained from the Nigerian Stock Exchange from January 4, 2006 to May 26, 2015, our findings indicated the presence of GARCH effect and was adequately captured by GARCH(0,1) model. Also, the sign and size bias test for asymmetric GARCH effect on the residual series of GARCH(0,1) model showed a joint significance as indicated by the Lagrange Multiplier test. Moreover, the asymmetric GARCH effect was adequately captured by EGARCH(0,1) and TGARCH(0,1) models. In addition, the significance of the size bias test indicated that the size of negative and positive returns has an impact on the predicted heteroscedasticity. Hence, we concluded that GARCH(0,1) model adequately predicted the GARCH effect but failed to capture the asymmetric effect in the share price returns of the discrete series. However, this was complemented by both EGARCH(0,1) and TGARCH(0,1) models with the size of both the negative and positive effects taken into consideration.

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This study looks at a possible combination of both the ARMA and ARCH-types models to form a single model such as ARMA-ARCH that will completely model the linear and non-linear features of financial data. The data used for this study are daily closing share prices of First Bank of Nigeria plc from January 4, 2000 to December 31, 2013 and were obtained from the Nigerian Stock Exchange. The share price series was found to be nonstationary while the returns series which is the first difference of log of the share price series was found to be stationary. This study provides evidence to show that ARMA(2,2) model is found to be adequate in the modeling the linear dependence in the returns of First Bank of Nigeria while the ARCH(1) model is adequate in modeling the changing conditional variance in the returns of First Bank of Nigeria. Therefore, combining the two models results in a single ARMA(2,2)-ARCH(1) model that completely models the returns series of First Bank of Nigeria. Keywords: ARMA model; ARCH model; linear dependence; conditional variance; First Bank of Nigeria IntroductionLinear time series models are not good models for describing certain characteristics of a volatility series in that in ARMA models, it is assumed that linear dependence is present in the observations. Also, assumption of homoscedasticity is not appropriate when using financial data. For instance, returns typically exhibit linear dependence as such ARMA models are natural candidates for modeling the linear dependence in financial data. However, financial data frequently exhibit volatility clustering leading to the violation of the assumption of constant variance thus making a way for the

The purpose of this study is to apply white noise process in measuring model adequacy targeted at confirming the assumption of independence. This ensures that no autocorrelation exists in any time series under consideration, and that the autoregressive integrated moving average (ARIMA) model entertained is able to capture the linear structure in such series. The study explored the share price series of Union bank of Nigeria, Unity bank, and Wema bank obtained from Nigerian Stock Exchange from January 3, 2006 to November 24, 2016 comprising 2690 observations. ARIMA models were used to model the linear dependence in the data while autocorrelation function (ACF), partial autocorrelation function (PACF), and Ljung-Box test were applied in checking the adequacy of the selected models. The findings revealed that ARIMA(1,1,0) model adequately captured the linear dependence in the return series of both Union and Unity banks while ARIMA(2,1,0) model was sufficient for that of Wema bank. Also, evidence from ACF, PACF and Ljung-Box test revealed that the residual series of the fitted models were white noise, thus satisfying the conditions for stationarity.

This study investigated a scenario where both the deterministic and stochastic trends coexist in a single realization. On exploring the monthly internally generated revenue of Akwa Ibom State in Nigeria from January, 2010 to December, 2014, we found that the deterministic trend with ARMA (1, 0) model adequately described the coexistence of both the deterministic and stochastic trends.

The study explored both Box and Jenkins, and iterative outlier detection procedures in determining the efficiency of ARIMA-GARCH-type models in the presence of outliers using the daily closing share price returns series of four prominent banks in Nigeria (Skye (Polaris) bank, Sterling bank, Unity bank and Zenith bank) from January 3, 2006 to November 24, 2016. The series consists of 2690 observations for each bank. The data were obtained from the Nigerian Stock Exchange. Unconditional variance and kurtosis coefficient were used as criteria for measuring the efficiency of ARIMA-GARCH-type models and our findings revealed that kurtosis is a better criterion (as it is a true measure of outliers) than the unconditional variance (as it can be depleted or amplified by outliers). Specifically, the strength of this study is in showing the applicability and relevance of iterative methods in time series modeling.