##### Modeling Heteroscedasticity in the Presence of Serial Correlations in Discrete-time Stochastic Series: A GARCH-in-Mean Approach

*Asian Journal of Probability and Statistics*,
Page 1-16

**DOI:**
10.9734/ajpas/2019/v4i230109

**Background: **In modeling heteroscedasticity of returns, it is often assumed that the series are uncorrelated. In practice, such series with small time periods between observations can be observed to contain significant serial correlations, hence the motivation for this research.

**Aim:** The aim of this research is to assess the existence of serial correlations in the return series of Zenith Bank Plc, which is targeted at identifying their effects on the parameter estimates of heteroscedastic models.

**Materials and Methods:** The data were obtained from the Nigerian Stock Exchange spanning from January 3, 2006, to November 24, 2016, having 2690 observations. The hybridized Autoregressive Integrated Moving Average-Generalized Autoregressive Conditional Heteroscedasticity (ARIMA-GARCH-type) models such as Autoregressive Integrated Moving Average-Generalized Autoregressive Conditional Heteroscedasticity (ARIMA-GARCH), Autoregressive Integrated Moving Average-Exponential Generalized Autoregressive Conditional Heteroscedasticity (ARIMA-EGARCH) and the Autoregressive Integrated Moving Average-Glosten, Jagannathan and Runkle Generalized Autoregressive Conditional Heteroscedastic (ARIMA-GJRGARCH) under normal and student-t distributions were employed to model the conditional variance while the GARCH-in-Mean-GARCH-type model corresponding to the selected ARIMA-GARCH-type model was applied to appraise the possible existence of serial correlations.

**Results:** The findings of this study showed that heteroscedasticity exists and appeared to be adequately captured by ARIMA(2,1,1)-EGARCH(1,1) model under student-t distribution but failed to account for the presence of serial correlations in the series. Meanwhile, its counterpart, GARCH-in-Mean-EGARCH(1,1) model under student-t distribution sufficiently appraised the existence of serial correlations.

**Conclusion:** One remarkable implication is that the estimates of the parameters of ARIMA-GARCH-type model are likely to be biased when the presence of serial correlations is ignored. Also, the application of GARCH-in-Mean-GARCH-type model possibly provides the feedback mechanism or interaction between the variance and mean equations.