I am currently working on a MS-ARMA-GARCH model proposed by Dhiman das on this paper, and trying to fit it on simulated data.

So far I understand the model and its construction, but I'm having a hard time while trying to understand his MCMC Algorithm (I'm not a statistician, just a math undergraduate student).

Note that his MS-ARMA(p,q)-GARCH(r,s) model follows the form:

\begin{align} {y_t} &= \gamma {x_t} + {u_t} \\ {u_t} &= \sum\limits_{j = 1}^p {{\phi _j}{u_{t - j}} + {\varepsilon _t} + \sum\limits_{j = 1}^q {{\theta _j}{\varepsilon _{t - j}}} , \hspace{0.5cm}{\varepsilon _t}|{\mathcal{F}_{t - 1}} \sim \mathbf{N}(0,\sigma _t^2)} \\ \sigma _t^2 &= {\mu _0} + {\mu _1}{S_t} + \sum\limits_{j = 1}^r {{\alpha _j}{u_{t - j}}} + \sum\limits_{j = 1}^s {{\beta _j}\sigma _{t - j}^2} \end{align}

Where $\gamma$ is the regression coefficient; $\sigma _{t}^{2}$ is the conditional variance of $\varepsilon$, $S_t$ is an auxiliary variable which follows a Markov Switching model taking integer values in [0,1], ${{\theta }_{j}}$ y ${{\phi }_{j}}$ are ARMA(p,q) model parameters; ${{\alpha }_{j}}$,${{\beta }_{j}}$ are GARCH(r,s) model parameters.

First he follows a Nakatsuma's paper and divides parameters for ARMA ($\delta_1$) and MS-GARCH ($\delta_2$), and treats presample error $\epsilon_0$ as a parameter.

All of these parameters are going to be estimated via MCMC Metropolis-Hastings algorithm.

But, currently I'm stuck trying to understand the proposal distribution for $\gamma$ in Nakatsuma's paper, exactly what are 1,2,3 shown on the picture and how to calculate them.

Nakatsuma proposal distribution for <span class=$\gamma$">

I really appreciate any help you can provide.


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