# Stationary distribution of AR(1) process with AR(1) shocks

I am trying to find the stationary distribution of an AR(1) process, where the shock terms themselves are an AR(1) process. That is, the process moves subject to the following 2 equations: $$\begin{equation} z_{t+1} = \rho z_t + \sigma_t \eta_{z,t+1} \end{equation}$$ $$\begin{equation} \sigma^2_{t+1} = \max\left\{\nu \sigma_t^2 + \sigma \eta_{\sigma,t+1},\: 0 \right\} \end{equation}$$ where $$\eta_{\sigma, t+1}, \eta_{z,t+1} \sim N(0,1)$$ are IID.

I believe that the stationary distribution of $$\sigma_t \eta_{z,t+1}$$ is a linear combination of $$\chi$$-square distributions. (Based on answers here)

Does this mean that the stationary distribution of $$z_{t+1}$$ is also the same linear combination of $$\chi$$-square distributions?

• This looks more like some kind of stochastic volatility shocks rather than AR(1) shocks. – Richard Hardy Sep 18 at 6:17
• Yeah I'm trying to prove an existence result for the Bansal and Yaron stochastic volatility model. Does that change the nature of the problem? – Flintro Sep 18 at 6:44
• I do not think altering the name changes the nature of the problem, but it changes the relevance of your title (currently a little misleading). – Richard Hardy Sep 18 at 7:23