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Showing content from https://github.com/ilyaZar/SVmodelExamples below:

GitHub - ilyaZar/SVmodelExamples

This is an example of an SV model to be estimated via PMMH using RcppSMC in a as a package. Also, RcppSMC is used to obtain for log-likelihood estimates for a standard toy SV model applied to real data or simulated data.

A pure R implementation of Particle Gibbs with and without ancestor sampling for the same model is provided too.

The SV model equations (measurements and latent state transition) are given as:

1. $X_t|(X_{t-1} = x_{t-1})\sim\mathcal{N}\left(x_t|\phi x_{t-1}, \sigma^2\right);,$
2. $Y_t|(X_t = x_t) \sim \mathcal{N}\left(y_t|0, \beta^2\exp(x_t)\right);.$

Assuming that the variance parameters $\sigma^2$ and $\beta^2$ are unknown, but $\phi$ is fixed, we consider a Bayesian setting with standard inverse Gamma priors on the parameters:

1. $\sigma^2\sim\mathcal{IG}(a=0.01, b=0.01);,$
2. $\beta^2\sim\mathcal{IG}(a=0.01, b=0.01);.$

This prior setup is conjugate to the model from 1. and 2. such that full conditional Gibbs blocks are obtained with distribution for the parameter in closed form according to:

1. $p\left(\sigma^2|x_{0:T}, y_{1:T}\right)=p(\sigma^2|x_{0:T})=\mathcal{IG}\left(\sigma^2|a+\frac{T}{2}, b+\frac{1}{2}\sum_{t=1}^T(x_t-\phi x_{t-1})^2\right);,$
2. $p\left(\beta^2|x_{0:T}, y_{1:T}\right)=\mathcal{IG}\left(\beta^2|a+\frac{1}{2}, b+\frac{1}{2}\sum_{t=1}^T\exp\left(-x_t\right)y_t^2\right);.$

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