In ilyaZar/SVmodelRcppSMC: RcppSMC package example for the Stochastic volatitly model

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SVmodelRcppSMC

Model description:

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

1. $X_t = f(X_{t-1}) = \phi X_{t-1} + \varepsilon_t, ~\varepsilon\sim\mathcal{N}(0, \sigma^2)$
2. $Y_t = \beta\exp(X_t/2)\epsilon_t,\;~\epsilon_t\sim\mathcal{N}(0, 1)\;.$

It follows that

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 $\phi < 1$, the latent state process is stationary, i.e. $\mathbb{E}(X_t)=\mathbb{E}(X_{t-1})\equiv \mu_x$ and $\mathbb{V}(X_t)=\mathbb{V}(X_{t-1})=\sigma_x^2$, we have

\begin{align} \mathbb{E}(X_t)&=\phi\mathbb{E}(X_{t-1}) + \mathbb{E}(\varepsilon_t) \Rightarrow \mu_x(1-\phi_x)=0 \Rightarrow \mu_x=\frac{0}{1-\phi_x}=0\;.\ \mathbb{V}(X_t)&=\phi^2\mathbb{V}(X_{t-1}) + \mathbb{V}(\varepsilon_t) \Rightarrow \mathbb{V}(X_t)(1-\phi^2)=\sigma^2\;.\ \Rightarrow \mathbb{V}(X_t)&=\frac{\sigma^2}{1-\phi^2}\;. \end{align}

Thus, the initial (stationary) distribution of $X_0$ is $$ X_0\sim\mathcal{N}\left(0, \frac{\sigma^2}{1-\phi^2}\right)\;. $$

Prior distributions and full condtional Gibbs blocks

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

ilyaZar/SVmodelRcppSMC documentation built on Dec. 20, 2021, 6:57 p.m.