Cointegration, Not Correlation

Question

You want to build a pairs trade.

Explain why the correlation of two price series is the wrong tool and cointegration is the right one, including the spurious-regression problem. Describe the Engle-Granger two-step test, how you form the spread and the hedge ratio, and how you would trade and size the position.

Accepted Answer

Why correlation failsCorrelation measures the co-movement of returns over a window and is unstable from period to period. Worse, run on price levels it is actively misleading. Two independent random walks will, over any finite sample, often display a high correlation and a regression of one on the other will show a large $R^2$ and a hugely "significant" t-statistic — even though there is no relationship at all. This is the spurious regression problem (Granger-Newbold): standard regression inference is invalid when the series are non-stationary (I(1)), because the residuals are themselves non-stationary and the usual standard errors are meaningless.

What cointegration says insteadTwo I(1) series $P_A$ and $P_B$ are cointegrated if there is a coefficient $\beta$ such that the combination \begin{equation} s_t = P_{A,t} - \beta\,P_{B,t} \end{equation} is stationary (I(0)). Economically, the two prices share a common stochastic trend and the spread $s_t$ wanders around a stable mean — it mean-reverts. That mean reversion, not co-movement, is what a pairs trade actually monetises.

The Engle-Granger two-step test

 	Step 1Regress one price on the other by OLS, $P_{A,t} = \alpha + \beta P_{B,t} + s_t$. The slope $\beta$ is the hedge ratio and the residual series is the spread.
 	Step 2Run an ADF test on the residual $s_t$. If you can reject a unit root, the spread is stationary and the pair is cointegrated. (Because $\beta$ was estimated, the ADF critical values must be the Engle-Granger ones, not the standard ADF table.)


Trading and sizingStandardise the spread into a z-score using a trailing mean and standard deviation, $z_t = (s_t - \mu)/\sigma$. Enter when the spread is stretched — short the spread (short $A$, long $\beta$ units of $B$) when $z$ is high, long the spread when $z$ is low — and exit as it reverts toward zero, with a stop for the case where the relationship breaks. The hedge ratio $\beta$ sets the relative position sizes so the combined book is roughly insensitive to the common trend. The natural horizon comes from the spread's half-life of mean reversion: fit $\Delta s_t = \lambda s_{t-1} + \epsilon_t$ (or an AR(1) with coefficient $\phi$) and use \begin{equation} t_{1/2} = -\frac{\ln 2}{\ln \phi} \end{equation} A short half-life means fast reversion and a higher-turnover trade.

CaveatsCointegration is not permanent — structural breaks, mergers or regime shifts can destroy it, so the relationship must be re-tested out-of-sample and protected with stops. For more than two assets, the Johansen test estimates the cointegrating relationships jointly.

Title	Category	Subcategory	Difficulty	Status
A One-Step GARCH(1,1) Forecast	Machine Learning	Time Series	Medium	Unlock
Identifying an ARIMA Model	Machine Learning	Time Series	Medium	Unlock
Is This Series Stationary?	Machine Learning	Time Series	Easy	Unlock
The Unit Root	Machine Learning	Time Series	Hard	Unlock

The Interview Guide for Quants and Traders

Cointegration, Not Correlation

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