# 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.

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.

• I tend to agree with you. However, I found the following quote: "The mean values of the individual firm variance ratios are shown in Table 5. They suggest some long-horizon mean reversion for individual stock prices." Page 19 of albany.edu/faculty/faugere/PhDcourse/meanreversion1.pdf I'm having trouble understanding why certain variance ratios would imply mean reversion. Apr 5, 2013 at 18:02
• In the abstract of the same paper: "It demonstrates that variance ratios are among the most powerful tests for detecting mean reversion in stock prices." Apr 5, 2013 at 18:05
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
$\frac{Var(tn)}{nVar(t)}$