Using Variance Ratios to Assess Mean Reversion
Summary
The document explains what variance ratio tests can and cannot establish about a price series. Rejecting the null of uncorrelated increments shows that the series departs from a random-walk-type benchmark, but it does not by itself prove mean reversion. Different covariance patterns can produce deviations from the null, so a specific interpretation requires assumptions about the alternative model.
It also gives a heuristic reading of variance ratios across horizons: a roughly constant ratio is associated with random-walk behavior, rising ratios with trending behavior, and falling ratios with mean reversion. The discussion is conceptual and supplies no empirical data or formal decision thresholds. These patterns should therefore be treated as clues rather than a definitive classification, especially when the series has changing variance and does not meet stationarity assumptions.
Key ideas
- Rejecting uncorrelated increments does not alone establish mean reversion.
- Different covariance structures can produce similar variance ratio deviations.
- A falling variance ratio across horizons is associated with mean-reverting behavior.
- Interpreting variance ratios requires assumptions about the alternative process.
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Full text
# Using variance ratios to test for mean reversion
# Using variance ratios to test for mean reversion
Can you use the variance ratio test to determine whether or not a time series is mean reverting? I'm using the `Lo.Mac` function in the `vrtest` library in `R`.
I've used the test to reject geometric Brownian motion as a price process. Does that indicate that I have a mean reverting process or does it only indicate that the assumptions of geometric Brownian motion are not satisfied?
My plot of variance ratios looks like this:
I don't quite understand the interpretation of this plot!
Note: I don't want to use a unit root test for stationarity because the process has nonconstant variance. It is not second order stationary although I believe that it is mean reverting.
## Answer by Ryogi (score 10, accepted)
https://quant.stackexchange.com/a/7667
It only indicates that the null hypothesis of uncorrelated increments is violated.
For the sake of simplicity, assume a time series is stationary. Then a sufficient statistic for arbitrary variance ratios is its covariance function. In general, a given deviation from the null can originate from different covariance functions, which in turn, entails that making any specific claim about mean reversion is not trivial. I find that this point is often overlooked in the financial literature. I expect that when phrased properly a similar statement can be made about non-stationary series.
That being said, mapping abnormal variance ratios to mean reversion/trend following is not impossible but it requires making specific assumptions about the alternative model.
## Answer by 4pie0 (score 10)
https://quant.stackexchange.com/a/7671
of course you can use this test to elaborate on this matter. Basically this test measures the ratio of variance of series in period `tn` to `n*variance` of `t` preriod
$\frac{Var(tn)}{nVar(t)}$
in short:
- constant ratio for random walk
- increasing for series with trend
- decreasing for mean reverting process, more decreasing (faster) - better mean reversionShown 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.