Testing Cointegration When Hedge Ratios Change
Summary
The document explains why a rolling regression can make a pairs spread appear stationary even when the apparent mean reversion comes from repeatedly updating the hedge ratio. A short lookback window can adjust the ratio to recent price movements, pulling the constructed spread toward zero and making stationarity tests misleading as evidence of a stable relationship.
It identifies cointegration as the relevant property for pairs trading: two instruments are cointegrated when a linear combination of their price series is stationary for some hedge ratio. The Engle–Granger two-step method is offered as one statistical test. The answer does not give implementation details, test results, or guidance on selecting a rolling window. Its main caution is that a visually or statistically stable spread under frequent ratio updates does not by itself establish genuine mean reversion.
Key ideas
- A rolling hedge ratio can make a spread appear stationary by adapting to recent prices.
- Cointegration describes price series with a stationary linear combination.
- The hedge ratio is the coefficient used to form that combination.
- The Engle–Granger two-step method is one way to test for cointegration.
- Apparent spread stationarity alone does not show that the relationship is stable.
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Full text
# Answer by Sebastian (score 1) # Pairs trading using dynamic hedge ratio - how to tell if stationarity of spread is due to genuine cointegration or shifting of hedge ratio? 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? ## Answer by Sebastian (score 1) https://quant.stackexchange.com/a/70251 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.
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