2
$\begingroup$

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 $h^{(0)}_t$.

$\endgroup$
1
$\begingroup$

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

$\endgroup$
4
  • $\begingroup$ 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. $\endgroup$
    – Sahil Puri
    Aug 1 '18 at 2:32
  • $\begingroup$ 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. $\endgroup$
    – jcz
    Aug 1 '18 at 2:48
  • 1
    $\begingroup$ 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. $\endgroup$
    – Sahil Puri
    Aug 1 '18 at 2:58
  • $\begingroup$ 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. $\endgroup$
    – jcz
    Aug 1 '18 at 3:02

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.