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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?

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    $\begingroup$ No expert on this lit, but it may be important to make sure you are comparing mean reversion in the same object (as Richard Hardy says below). ARCH and GARCH do, I believe, typically show mean reversion in volatility, while mean reversion in the level (e.g., stock price, as opposed to the vol of the stock price) is a different question. (It looks like the \alfa and \beta criterion you mention may be hinting at stationarity of the volatility, which makes sense for thinking about mean reversion in vol, I guess. As an aside, I can't imagine what a process with non-stationary vol looks like!) $\endgroup$ Commented Aug 27, 2024 at 13:20

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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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