# Initial vector $h$ in Bayesian stochastic volatility models (Jacquier, Polson and Rossi, 1994)

I was going through the paper Jacquier, Polson and Rossi (1994): Bayesian Analysis of Stochastic Volatility Models. While the model seems straight forward to implement. I'm not able to find how the initial vector for $$h$$ is determined.

While the use of regression in order to get $$p(w|\mathbf{h},\mathbf{y})$$ is clear and the iteration component of $$p(h_{t}|h_{t+1},h_{t-1},w,\mathbf{y})$$ is also clear, I'm lost when it comes to finding out the initial vector $$h^{(0)}_t$$.

I don't think the paper says exactly what they do, but I would initialize $\omega$ and $h$ out of the prior. That is, draw $\omega^{(0)}\sim p(\omega)$ and then draw $h^{(0)}\sim p(h\,|\,\omega^{(0)})$.

Drawing $h^{(0)}\sim p(h\,|\,\omega^{(0)})$ means that you fix some initial condition $\ln h_0$ (0 is probably fine), and given your draw $\omega^{(0)}=[\alpha^{(0)},\,\delta^{(0)},\,\sigma^{(0)}_\nu]$, simulate forward the AR(1) law of motion $$\ln h_t = \alpha^{(0)} + \delta^{(0)}\ln h_{t-1}+\sigma^{(0)}_\nu \nu_t,\quad\nu_t\sim N(0,\, 1).$$ This will give you a sequence $\ln h_1$, $\ln h_2$, ..., $\ln h_T$. Exponentiate each term, and you have a draw from the prior $p(h\,|\,\omega)$. Then proceed with the algorithm.

For $m=1,..,M$,

\begin{align} \omega^{(m)}&\sim p(\omega\,|\,h^{(m-1)},\,y) \\ h^{(m)}&\sim p(h\,|\,\omega^{(m)},\,h^{(m-1)},\,y) \end{align}

end

• but in order to get $h$ from $p(h|\omega)$ we need a previous iteration of $h$ itself. you can do $h^{(1)} \sim p(h|h^{(0)}, \omega)$ but you still need an $h^{0}$ Given that the paper talks about priors for $\omega$ it would make sense to me to simulate them first. Aug 1 '18 at 2:32
• You don't need a previous draw of $h$ to draw out of the prior $p(h\,|\,\omega)$, but I see your point. The simulation algorithm for $p(h\,|\,\omega,\,y)$ requires a previous draw of $h$, so my first paragraph below the quote is incomplete. I'll edit the answer.
– jcz
Aug 1 '18 at 2:48
• Thanks, I get it now, It makes sense to me now 1) have a $h^{(0)}_0$ 2) pick $\omega_0$ from priors 3) use the AR(1) process to get the full $h^{(0)}_t$ vector 4) use regression to get $\omega_1$ from $h^{(0)}_t$ 5) use the MCMC algorithm in the paper to get $h^{(1)}_t$ from $h^{(0)}_t$ and $\omega_1$ 6) Repeat. Aug 1 '18 at 2:58
• Yupp, exactly. Of course by "Repeat" you mean "Repeat steps 4 and 5." Steps 1, 2, 3 only have to be done once at the beginning.
– jcz
Aug 1 '18 at 3:02