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.
This study was prompted by the fact that the presence of outliers in discrete-time stochastic series may result in model misspecification, biases in parameter estimation and in addition, it is difficult to identify some outliers due to masking effects. However, the iterative approach which involves joint estimation of outliers effects and model parameters appears to be a panacea for masking effects. Considering the dataset on credit to private sector in Nigeria from 1981 to 2014, we found that ARIMA (1, 1, 1) model fitted well to the series without considering the presence of outliers. Using the iterative procedure method to reduce masking effects, the following outliers, IO (t = 24), AO (t = 33) and TC (t = 22) were identified. Adjusting the series for outliers and iterating further, ARIMA (2, 0, 1) model alongside AO (t = 33) and TC (t = 22) outliers was found to fit the series better than ARIMA (1, 1, 1) model. The implication is that in the presence of outliers, ARIMA (1, 1, 1) model was misspecified, the order of integration was wrong and by extension, autocorrelation and partial autocorrelation functions were misleading, and the estimated parameters were biased.
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
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