# Particle Gibbs Sampler For Regime-Switching Nonlinear Gaussian SSM

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.

• Are you familiar with standard particle filtering? standard particle Gibbs? standard Gibbs sampling? in my very humble opinion, this is a relatively complicated research topic for a finance undergrad, even if you are only focusing on the programming. This particle model also doesn't just plug in naturally to standard particle filtering due to the time subscripts. If it helps, I had a post about the regular particle Gibbs sampler a while back: stats.stackexchange.com/questions/351249/… Commented Nov 21, 2020 at 22:47