**1. Introduction**

As the financial market and investment instruments grow more sophisticated, the need for the proper risk managemen<sup>t</sup> of financial activities and the modeling of financial volatility has become more crucial. As there is no unique and unambiguous definition for volatility, observable quantities (such as daily high-lows or intra-day price changes) are used to approximate the quantity, thus dividing volatility modeling techniques into two sub-groups: Parametric and non-parametric (Anderson et al. 2002; Zheng et al. 2014). The first group are the traditional parametric latent volatility models, such as the Generalized Autoregressive Conditional Heteroscedastic (GARCH) model or the Stochastic Variance (SV) model. However, these parametric models have become increasingly restrictive in use, due to growing complexity. As mentioned in McAleer and Medeiros (2008), as the traditional standard latent volatility models cannot adequately describe the slowly decreasing auto-correlations of squared returns and as the usage of Gaussian standardized error has been criticized by many, the Realized Volatility (RV) model, as an alternative non-parametric method, has received increasing attention. In its simplest form, the RV model can be simply defined as

$$RV\_t = \sum\_{i=0}^{n\_t} r\_{t,i}^2 \tag{1}$$

where *RVt* denotes the realized volatility at day *t*, *rt*,*<sup>i</sup>* denotes the *i*th intra-period return at day *t*, and *nt* is the number of high frequency data observed. It has been shown that the RV model more accurately measures the 'true volatility' than daily squared returns (Anderson et al. 1999; Kambouroudis et al. 2016) and it is among the best for modeling the volatilities of the U.S. and E.U. stock indices (Kambouroudis et al. 2016). It is also a good measure for market risk, due to its ability to show clustering and fat-tail behavior for price fluctuations (Zheng et al. 2014).

A lot of work has been done towards constructing the realized volatility; see McAleer and Medeiros (2008) for a review. The focus of this article is to consider possible probabilistic models, given *RVt*; particularly if it is possible to model it with a non-Gaussian random error structure. As models based on the Wishart distribution have been proposed for multi-variate realized volatility (Golosnoy et al. 2012) and multi-variate stochastic volatility (Gouriéroux et al. 2009), and as the Wishart distribution is the multi-variate analog of the chi-square distribution (which is a member of the Gamma distribution family), a Gamma random error structure in the univariate case has become of interest. Thus, traditional Autoregressive (AR) and Threshold-type non-linear AR (TAR) models with Gamma random error are explored. This article can be regarded as an extension of Li and McLeod (1988).

#### **2. Materials and Methods**

This section aims to provide the specification for the proposed model and the fitting methodology. It will also briefly touch on the materials and methods for conducting the empirical data analysis.
