**An Experiment on Autoregressive and Threshold Autoregressive Models with Non-Gaussian Error with Application to Realized Volatility**

#### **Ziyi Zhang †,‡ and Wai Keung Li \*,‡**

Department of Statistics and Actuarial Science, University of Hong Kong, Hong Kong, China; u3514050@connect.hku.hk or saas@hku.hk


Received: 27 February 2019; Accepted: 11 June 2019; Published: 17 June 2019

**Abstract:** This article explores the fitting of Autoregressive (AR) and Threshold AR (TAR) models with a non-Gaussian error structure. This is motivated by the problem of finding a possible probabilistic model for the realized volatility. A Gamma random error is proposed to cater for the non-negativity of the realized volatility. With many good properties, such as consistency even for non-Gaussian errors, the maximum likelihood estimate is applied. Furthermore, a non-gradient numerical Nelder–Mead method for optimization and a penalty method, introduced for the non-negative constraint imposed by the Gamma distribution, are used. In the simulation experiments, the proposed fitting method found the true model with a rather insignificant bias and mean square error (MSE), given the true AR or TAR model. The AR and TAR models with Gamma random error are then tested on empirical realized volatility data of 30 stocks, where one third of the cases are fitted quite well, suggesting that the model may have potential as a supplement for current Gaussian random error models with proper adaptation.

**Keywords:** Autoregressive Model; non-Gaussian error; realized volatility; Threshold Autoregressive Model
