Testing Whether Commodity Volatility Depends on Price Level
Summary
The document asks whether commodity price shocks tend to be larger when the price level is higher, using oil prices as an illustration. It notes that the question concerns dependence between a series’ variance and its mean, and that price regimes or change point analysis are alternative ways to investigate it.
The proposed GARCH extension adds a function of lagged prices to the conditional variance equation alongside the usual shock and prior variance terms. The price input could be a lagged level, a transformation such as its square root, or a weighted combination of recent levels. This provides a direct way to test whether price level helps explain conditional volatility. The note offers a model specification rather than empirical results, and does not discuss estimation, constraints, diagnostics, or how to establish that the relationship is stable or causal.
Key ideas
- A GARCH variance equation can include lagged price level as an explanatory term.
- The price input can be transformed or formed from multiple lagged observations.
- Price regimes and change point analysis are presented as alternative approaches.
- The document proposes a specification but supplies no empirical findings or model diagnostics.
Tags
Full text
# Investigating a question: "Does commodity price volatility scale with price level?"
# Investigating a question: "Does commodity price volatility scale with price level?"
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 by Richard Hardy (score 1)
https://quant.stackexchange.com/a/34118
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.