I'm trying to model GARCH volatility on electricity prices. Typically the first step is to use prices to obtain log returns to make them stationary. I have encountered a small problem however: electricity prices can go negative. So returns defined as

\begin{array}{cc} r_t:=\log(P_t / P_{t-1}) \end{array}

will produce some undefined values. I have gotten around it by using differences

\begin{array}{cc} r_t:=P_t - P_{t-1}, \end{array}

but I'm wondering if there is a better method out there.

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    $\begingroup$ This is not about GARCH but about defining returns in presence of negative prices. I suggest you remove the garch tag. $\endgroup$ Commented Jan 23, 2017 at 18:03
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    $\begingroup$ As Richard previously noted, the solution is probably to approach your assumptions on the distribution of changes in electricity prices. I have a hard time believing that the distribution of continuous changes in electricity prices is normal. I'd hazard a guess as a layperson that electricity prices themselves are much more normally distributed than their continuous rate of change. Rather than modeling the volatility of change in price, you can just directly model the volatility of prices. $\endgroup$
    – milkmotel
    Commented Jan 23, 2017 at 19:42
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    $\begingroup$ You don't. The concept of 'returns' does not make sense for products that can have negative prices. You should work with price differences instead. $\endgroup$ Commented Apr 28, 2020 at 9:18

1 Answer 1


are hyperbolic sine transformation, could also not to take log of p(t)/p(t-1),



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