Correlation Compared With Cointegration

Two non-stationary series can share a stationary linear combination even when correlation on their price levels shows nothing reliable.

Correlation measures whether two series move together over the sample used to compute it. Cointegration is a stronger and different claim: two series can each be individually non-stationary, containing what statisticians call a unit root, and still have a specific linear combination of the two that is stationary, meaning it reverts around a fixed level instead of wandering without bound. Correlation computed on price levels cannot detect this relationship, and mistaking one for the other is one of the more expensive statistical errors in pairs trading.

A non-stationary series, one that behaves like a random walk, has a variance that grows with time and no fixed level to revert toward. Regress one such series on another completely unrelated one and the regression can still produce an R-squared above 0.9 with an apparently significant t-statistic, purely because both series are trending over the sample window. This is the spurious regression result described by Granger and Newbold in 1974, and it applies whether the trend is a genuine drift or nothing more than the accumulated randomness of a random walk.

Correlation computed on price levels is exposed to exactly the same failure. Two prices can show a correlation above 0.9 across a multi-year sample for no reason connected to either business, simply because both happened to drift upward or downward over that period. The number looks like evidence of a relationship. It is closer to an artifact of the fact that neither series has a fixed mean to return to.

Two series are cointegrated when each one individually is integrated of order one, meaning it becomes stationary after being differenced once, but some specific linear combination of the two is stationary on its own, integrated of order zero. That stationary combination is the spread a pairs trade holds, and its defining property is that deviations from its own mean tend to shrink back rather than drift away indefinitely. Correlation says nothing about whether that stationary combination exists. Two series can be highly correlated in their day to day moves and still have no cointegrating relationship at all, drifting apart from each other without limit over a longer horizon.

The reverse is also true. Two series can show weak correlation in short-window returns and still be cointegrated, if their short-term noise is large relative to the slow-moving equilibrium pulling them together. Correlation and cointegration answer different questions about the same pair of series, one about co-movement over the sample, the other about whether a long-run anchor exists at all.