HARCH and EMA-HARCH:

Although the HARCH [30] and EMA-HARCH [29] processes were constructed in order to capture the information from high frequency data, we have also studied these models (in addition to the 5 presented in detail) in the present context with daily data. The results we obtained with the 4 parameter processes HARCH(3) and EMA-HARCH(3) and the 5 parameter processes HARCH(4) and EMA-HARCH(4) (all implemented with tail emphasized optimization) showed marginal improvement when compared with tail emphasized GARCH(1,1) (model 5). We have however not emphasized these models in the present study because we did not see the kind of improvement per degree of freedom to warrant their use with daily data. The comparative improvement of these models over tail emphasized GARCH(1,1) when run with high frequency data - which they were designed for - is currently under investigation to be presented in a separate paper.