The paper provides an understanding of the theoretical and empirical illustration of the working of various classes of ARCH family models used in the study. It equally exploits the potential benefits derivable from using this family type model. It dwells heavily into the best time series model among autoregressive moving average (ARMA), Autoregressive conditional heteroscedasticity (ARCH), Generalized Autoregressive conditional heteroscedasticity (GARCH), Integrated Generalized Autoregressive conditional heteroscedasticity (IGARCH), Threshold Generalized Autoregressive conditional heteroscedasticity (TGARCH) and Exponential Generalized Autoregressive conditional heteroscedasticity (EGARCH) models, and determined the models which actually give the best forecast performance. The mathematical background of all these models was set up and promptly illustrated using monthly data of the number of patients admitted for malaria at Ladoke Akintola teaching hospital, Osogbo. It covers the period of five years (2012 January to December 2016), obtained from the hospital record of LAUTECH, Osogbo. Stationarity tests (graph, unit root, and correlogram) were conducted before proceeding to parameter estimations. The grid search using Akaike information criteria (AIC) and Performance measures indices were used to determine the best model. So also, performance measure indices were cross-tabulated with the models. In all, out of seven performance measure indices used, the EGARCH (1,1) Model is best in the six of the indices. From these results, EGARCH (1,1) is recommended for would-be investors, forecasters, and other categories of users.

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