Investigating a question: “Does commodity price volatility scale with price level?”

Question

I'm trying to answer a simply posed question using a GARCH model: can we expect larger price shocks in a commodity when it's price is higher? (i.e., may we expect larger price shocks at \$100 per barrel oil versus \$20 per barrel?) My initial exploration of this question have involved using a multivariate GARCH model relating the return series and the price series, but the more I dive into the GARCH modeling method, the less I think it is appropriate. I'm essentially trying to see if the series variance is dependent on the series mean.

I already have a way of tackling this question via price regimes and change point analysis, ideally I would figure out a way to use GARCH volatility models as well. Many thanks!

Answer 1

Besides the useful commnent by @will on how to do this without a GARCH model, you could proceed as follows: include the price level (or some transformation of it, e.g. power or log) in the conditional variance equation. Something like this:

\begin{aligned} r_t &= \mu_t+u_t, \\ u_t &= \sigma_t\varepsilon_t, \\ \sigma_t^2 &= \omega + \alpha_1 u_{t-1}^2 + \beta_1 \sigma_{t-1}^2 + \gamma_1 g(P_{t-1},P_{t-2},\dots), \\ \varepsilon_t &\sim i.i.d.(0,1), \\ \end{aligned}

where $g(P_{t-1},P_{t-2},\dots) = P_{t-1}$ or $g(P_{t-1},P_{t-2},\dots) = \sqrt{P_{t-1}}$, or $g(P_{t-1},P_{t-2},\dots) = 0.6 P_{t-1} + 0.4 P_{t-2}$, or something similar. The only difference from the vanilla GARCH model would be the $g(P_{t-1},P_{t-2},\dots)$ term.