*2.1. Model Specification*

Time-series models with non-Gaussian error were previously considered, in some detail, by Li and McLeod (1988), and earlier in Lawrance and Lewis (1980) and Ledolter (1979). In this article, specifically, the AR and TAR models are further explored. The AR(p) model is defined as follows:

$$RV\_t = \sum\_{i=1}^{p} \varphi\_i \ast RV\_{t-i} + \varepsilon\_{t \text{ \textquotedblleft}} \tag{2}$$

where *εt* is the random error, assumed to follow a Gamma distribution; thus, *εt* ∼ <sup>Γ</sup>(*<sup>α</sup>*, *β*), where the density function is defined as

$$f(\mathbf{x}) = \frac{1}{\Gamma(\alpha) \* \beta^a} \* \mathbf{x}^{a-1} \* e^{-\frac{\mathbf{x}}{\beta}}.\tag{3}$$

It should be noted that it is assumed that there is no drift term in the AR model; yet, the drift term could be easily incorporated into the model. The TAR(p) model, similar to that introduced in Tong (1978) but with a modification in the random error term, is defined as follows:

$$\begin{aligned} RV\_t &= \sum\_{i=1}^p q\_{1,i} \ast RV\_{t-i} + \varepsilon\_{1,t} \qquad \text{if } RV\_{t-d} \le T \text{ .} \\ RV\_t &= \sum\_{i=1}^p q\_{2,i} \ast RV\_{t-i} + \varepsilon\_{2,t} \qquad \text{if } RV\_{t-d} > T \text{ .} \end{aligned} \tag{4}$$

where *d* ≥ 1 is the lag of the model and *T* is the threshold, such that the model is divided into two regimes, according to the observations at *d* time periods earlier. The pivot element *RVt*−*<sup>d</sup>* determines which regime *RVt* falls into, with *RVt* falling into the first regime if *RVt*−*<sup>d</sup>* is less than or equal to the threshold and into the second regime, otherwise. Each regime follows an AR(p) model, as defined above, with different AR and Gamma parameters.
