I'm reading this paper on using a non-linear Gaussian SSM for measuring regime-switching leverage effect using stock market data. I'm using it as jump-off point for an undergraduate paper. My advisor chose this for me (I major in finance) and it looks like neither he or I know anything about how to implement this. Fortunately I am quite adept at writing R code. My problem is I'm finding it difficult to follow exactly how I could set up the Particle Gibbs sampler since the author is bringing up functions out of nowhere (or I just have inadequate grounding in MCMC methods).
The model is the extended leverage effect model in section 4, where for returns $y_t$,
$y_t = \mu + exp(\frac{x_{t-1}}{2})\varepsilon_t$,
$x_t = \delta_{s_t} + \phi(x_{t-1} - \delta_{s_{t-1}}) + u_t$,
$(\varepsilon_t, u_t) \sim N_2(\mathbf{0}, \mathbf{\Sigma})$ where $\mathbf{\Sigma} = \left[ {\begin{array}{cc} 1 & \rho_{s_t} \sigma_u \\ \rho_{s_t} \sigma_u & \sigma^2_u \\ \end{array} } \right] $
Like I said I'm pretty adept with code, it's just the Bayesian/Gibbs Sampling part that's taking me back. I would greatly appreciate just a few hints on how I could find the $q()$, $g()$, and $f()$ and I think I'll be on track. I'm only targeting to implement the PGAS part, by the way. I tried referring to the original paper on PGAS but it's not much help, to be honest.
