Assume there are \(T\) daily returns and realized measures in the period \(t\). Let \(r_i\) and \(RM_i\) be the \(i^{th}\) daily return and daily realized measure respectively (with \(i=1, \ldots,T\)).
The most basic heavy model is the one with lag matrices p of \(\left( \begin{array}{ccc} 0 & 1 \\ 0 & 1 \end{array} \right)\) and q of \(\left( \begin{array}{ccc} 1 & 0 \\ 0 & 1 \end{array} \right)\). This can be reprensented by the following equations:
$$
\mbox{var}{\left(r_t \right)} = h_t = w + \alpha RM_{t-1} + \beta h_{t-1}; w,\alpha \geq 0, \beta \in [0,1]
$$
$$
\mbox{E}{\left(RM_t \right)} = \mu_t = w_R + \alpha_R RM_{t-1} + \beta_R \mu_{t-1}; w_R,\alpha_R, \beta_R \geq 0, \alpha_R+\beta_R \in [0,1]
$$
Equivalently, they can be presented in terms of matrix notation as below:
$$
\left( \begin{array}{ccc} h_t \\ \mu_t \end{array} \right) = \left( \begin{array}{ccc} w \\ w_R \end{array} \right) + \left( \begin{array}{ccc} 0 & \alpha \\ 0 & \alpha_R \end{array} \right) \left( \begin{array}{ccc} r^2_{t-1} \\ RM_{t-1} \end{array} \right) + \left( \begin{array}{ccc} \beta & 0 \\ 0 & \beta_R \end{array} \right) \left( \begin{array}{ccc} h_{t-1} \\ \mu_{t-1} \end{array} \right)
$$
In this version, the parameters vector to be estimated is \(\left( w, w_R,\alpha, \alpha_R, \beta, \beta_R \right) \).
In terms of startingvalues, Shephard and Sheppard recommend for this version of the Heavy model to set \(\beta\) be around 0.6 and sum of \(\alpha\)+\(\beta\) to be close to but slightly less than one.
In general, the lag length for the model innovation and the conditional covariance can be greater than 1. Consider, for example, matrix p is \(\left( \begin{array}{ccc} 0 & 2 \\ 0 & 1 \end{array} \right)\) and matrix q is the same as above. Matrix notation will be as below:
$$
\left( \begin{array}{ccc} h_t \\ \mu_t \end{array} \right) = \left( \begin{array}{ccc} w \\ w_R \end{array} \right) + \left( \begin{array}{ccc} 0 & \alpha_1 \\ 0 & \alpha_R \end{array} \right) \left( \begin{array}{ccc} r^2_{t-1} \\ RM_{t-1} \end{array} \right) +\left( \begin{array}{ccc} 0 & \alpha_2 \\ 0 & 0 \end{array} \right) \left( \begin{array}{ccc} r^2_{t-2} \\ RM_{t-2} \end{array} \right) + \left( \begin{array}{ccc} \beta & 0 \\ 0 & \beta_R \end{array} \right) \left( \begin{array}{ccc} h_{t-1} \\ \mu_{t-1} \end{array} \right)$$
In this version, the parameters vector to be estimated is \(\left( w, w_R,\alpha_1, \alpha_R, \alpha_2, \beta, \beta_R \right) \).