**5. Conclusions**

In this article, the model fitting of a non-Gaussian model on the realized volatility is explored. As the definition of realized volatility requires it to be positive, previous works established a Wishart model (a multi-variate analog of the chi-square distribution) that belongs to the Gamma family; considering this selection, a univariate Gamma random error is proposed and the AR and TAR models are explored. MLE estimation, based on the AIC and BIC, and with some adjustment, is proposed. A profile likelihood method, which replaces the Gamma parameters with their MLE counterparts, is used to reduce the dimension of the estimation and a non-gradient numerical optimization method is employed, as the calculation of gradient may not be feasible. A penalty method is introduced into the likelihood function, to enforce the non-negative constraint imposed by Gamma random error. The proposed process manages to find the true model with a rather insignificant bias and MSE, when the true model is AR or TAR. Finally, the model is tested on the empirical realized volatility data of 30 stocks and managed to fit one third of the cases quite well, suggesting that the model may have the potential to be further generalized, in order to act as a good supplement for current Gaussian random error models. The lack of fit may be improved by considering higher AR orders or a denser initial point selection for the Nelder–Mead method, which requires more computational time. Other possible directions of improvement include using a better method (instead of AIC or BIC) to reduce the ambiguity in choosing the model and possibly using other AR structures, such as the Heterogeneous Auto-Regressive (HAR) model. Other time-series models with non-Gaussian error may also be considered and the model fitting methodology proposed in this article could possibly be extended to these models without difficulty.

**Author Contributions:** Conceptualization, Z.Z. and W.K.L.; methodology, Z.Z. and W.K.L.; software, Z.Z.; validation, Z.Z. and W.K.L.; formal analysis, Z.Z. and W.K.L.; investigation, Z.Z. and W.K.L.; resources, W.K.L.; data curation, W.K.L.; writing—original draft preparation, Z.Z.; writing—review and editing, W.K.L.; visualization, Z.Z.; supervision, W.K.L.; project administration, W.K.L.; funding acquisition, W.K.L.

**Funding:** This research was funded by HK SAR GRF gran<sup>t</sup> number 17304417 and 17303315.

**Acknowledgments:** Special thanks to the referees of the journal for the invaluable advice they provide to better this report.

**Conflicts of Interest:** The authors declare no conflict of interest.
