Distinguishing Mean Reversion in GARCH Volatility from the Mean
Summary
The discussion clarifies that mean reversion can refer to different parts of a time-series model. In a setup with a conditional mean and time-varying volatility, the modeled level may revert toward a long-run mean, while conditional variance may separately revert toward its own long-run level. The cited claim about the sum of GARCH coefficients concerns the volatility process, rather than establishing mean reversion in the asset price or return level.
The response interprets the paper's focus on volatility shocks and their half-life as evidence that it is discussing variance reversion. It cautions that the paper may not be a reliable source and recommends consulting stronger financial econometrics references. The answer does not derive the GARCH stationarity or persistence conditions, assess the paper in detail, or explain how to test mean reversion in the conditional mean; those questions require separate analysis.
Key ideas
- Mean reversion can describe the conditional mean or the conditional variance, and these are distinct properties.
- The cited GARCH coefficient condition concerns persistence and reversion in conditional variance.
- A volatility result does not by itself show that an asset's price or return level mean-reverts.
- The response advises checking the paper's claims against stronger financial econometrics sources.
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Full text
# Does it makes sense to use GARCH to measure mean reversion? # Does it makes sense to use GARCH to measure mean reversion? I am doing my final paper at my bachelor. For this, I am testing mean reversion in an asset. I found this paper (Mean reversion in international markets: evidence from G.A.R.C.H. and half-life volatility model) where the author uses GARCH to test for mean reversion and he wrote this: "The generalised A.R.C.H. model is denoted as the G.A.R.C.H. process, and in G.A.R.C.H. model we sum up both the A.R.C.H. (α) and G.A.R.C.H. (β) coefficients. In the G.A.R.C.H. model, if the sum of coefficients is less than 1 (α+β<1) then the indices of the time seriesdemonstrate the mean reversion process." Does it make sense? I asked this for GPT and he disagreed. As I am still not a specialist, I have doubts. What do you think? ## Answer by Richard Hardy (score 1, accepted) https://quant.stackexchange.com/a/80417 Suppose you are modelling a variable $x$ where $$ x_t=\mu_t+\sigma_t z_t $$ with $\mu_t$ being the conditional mean of $x$, $\sigma_t$ the conditional variance of $x$ and $z_t\sim i.i.d.(0,1)$ (zero mean and unit variance). When you talk about mean reversion, you probably have in mind $x_t$ converging to the average value $\mu$ of $\mu_t$ (if such a $\mu$ exists), if not for the never ending shocks $z_t$. Mean reversion could also be mentioned in the context of conditional variance, namely, $\sigma^2_t$ reverting to its long-term mean $\sigma^2$ (if such $\sigma^2$ exists), again, if not for the shocks $z_t$. I have not perused the paper, but it is probably the latter thing they are talking about in the quote. E.g. in the abstract, they say > An important aim is to measure and compare the speed of mean reversion and half-life of volatility shocks of emerging and developed markets. (Emphasis is mine.) I would also say that this does not look like the highest quality work by experts of the field. (I could be wrong, but I do not think I am.) I would not be surprised if they got some things wrong. If you want to learn about mean reversion or GARCH models, there should be better sources to study from. I would look for some high-ranking journals in finance or financial econometrics and see what they have to offer.
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