I'm very new to pairs trading, and am trying it out on a few dozen pairs.

It seems very natural to me to use a dynamic hedge ratio, as it seems likely that the ratio will move over time.

To accomplish this, I am using rolling linear regression (so I choose a lookback period of, say, 100 hours and I keep shifting this 100-hour window forward, run linear regression on that window to determine the "current" hedge ratio).

I have noticed, though, that by doing this, it seems like I can make a "stationary" spread out of just about any pair. I realize this is likely because part of the "stationarity" is due to the self-correcting nature of a rolling window regression, which over time will make the spread return to 0 by changing the hedge ratio, not because the spread actually reverted to the mean.

How can I address this? How can I tell if my spread is stationary due to real mean reversion, or just the shifting hedge ratio? Is there a better way of finding a hedge ratio?

I realize there's a lot loaded in this question, and I'll be happy to give a bounty to anyone who takes the time to respond deeply. Thank you!

Related question and discussion here: Pairs Trading - isn't any spread stationary if your rolling lin-reg window is small enough?


1 Answer 1


You are right. If the window is small enough every spread looks stationary. What you need is that the spread is steady enough.

The mathematical property which implies that pairs-trading works is called co-integration. You can test this statistically, for example, by the Engle–Granger two-step method.

For your context here the definition: Two instruments $x$ and $y$ are co-integrated if $\omega$ exists such that $x+\omega y$ is stationary. Where $\omega$ is the mentioned hedge ratio.

  • $\begingroup$ That's exactly what I arrived at. However - testing the spread for stationarity is the same as testing the series for cointegration, right? That is the two-step method by Engle and Granger, am I correct? I greatly appreciate you confirming my suspicion, but it seems that the problem remains of deciding an appropriate window size to, for example, run a cointegration test (because these things can be fluid and change over time, no? Just as the hedge ratio can change over time?) $\endgroup$ Mar 22, 2022 at 2:13
  • $\begingroup$ Vladimir: Just to re-iterate, if you don't estimate the hedge ratio but rather, use a constant, such as 1.0, and then test for stationarity of residuals, this might improve the lack of stability issue. Also, check out Paul Teetor's PLS approach to pairs. He introduces the idea of PLS in order to deal with the issue that which stock is $y$ and which is $x$ ( in the E&G cointegration framework ) is an arbitrary choice. It's on the net somewhere. If you can't find it, let me know. It might help with stability because there is no choice made as to which stock is y and which is x. $\endgroup$
    – mark leeds
    Mar 22, 2022 at 3:54

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.