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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!

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    $\begingroup$ This is one of the main differences between log normal and normal processes. Plot some moving window sample mean vs the vol over the same sample (using same method if you're using ewma), and see what relationship you get (which is essentially the same as garch, but you don't need to go into al lthe details and can easily just look at it to see)... $\endgroup$ – will May 8 '17 at 8:53
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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.

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