Cointegration

Cointegration is a property of two or more time series that each wander without a fixed level but are tied together in the long run. Individually, each series is non-stationary. Taken together in the right proportions, their combination is stationary: it fluctuates around a stable mean and tends to return to it. The series may drift far from where they started, but they do not drift far from each other.

The concept was developed by Clive Granger and Robert Engle, whose joint paper appeared in 1987. Granger shared the 2003 Nobel Memorial Prize in Economic Sciences for the work. In economics it is used to study relationships such as those between consumption and income or between interest rates of different maturities. In trading it is the statistical basis of pairs trading and much of statistical arbitrage.

A series is said to be integrated of order one if it is non-stationary but its changes are stationary. Most asset prices behave this way, in the manner of a random walk. Two such series $x_t$ and $y_t$ are cointegrated if there is a coefficient $\beta$ for which the following combination is stationary:

• $y_t$ and $x_t$ are the two series, for example the prices of two shares.
• $\beta$ is the cointegrating coefficient, which in trading serves as the hedge ratio.
• $\alpha$ is a constant.
• $u_t$ is the spread, or the deviation from the long-run relationship.

Cointegration requires that $u_t$ satisfy stationarity. If it does, any gap that opens between the two series is temporary.

Suppose that for two shares the estimated relationship has $\alpha = 5$ and $\beta = 2.2$. Today share Y trades at $50 and share X at $20. The spread is 50 − 5 − (2.2 × 20) = 50 − 5 − 44 = 1.0. Historically the spread has averaged zero with a standard deviation of 0.5, so today's value is two standard deviations above its mean, a z-score of 2.

A trader who relies on the relationship would sell one share of Y and buy 2.2 shares of X, expecting the spread to fall back toward zero. If it does, the position gains about $1 per unit regardless of whether both shares rose or fell in the meantime. If the relationship has broken, the spread may keep widening.

The two ideas are often confused. Correlation measures whether the returns of two assets tend to move in the same direction over short intervals. Cointegration concerns the levels of the prices over the long run. Two shares can have highly correlated daily returns and still drift apart without limit if one grows slightly faster. Two cointegrated series can have low short-term correlation and still be bound together.

A related danger is spurious regression. Regressing one random walk on another, unrelated one often produces a high R-squared and apparently significant coefficients. The result is meaningless unless the residuals are stationary, which is exactly what a cointegration test checks.

• Engle-Granger two-step method. One series is regressed on the other by linear regression, and the residuals are tested for a unit root with the augmented Dickey-Fuller test. Special critical values are used because the residuals come from an estimated equation. The outcome can depend on which series is placed on the left-hand side.
• Johansen test. The Johansen test handles several series at once and estimates how many independent cointegrating relationships exist among them. It does not require choosing a dependent variable.

When series are cointegrated, their short-run behavior can be written as an error correction model, in which each period's change depends partly on the previous period's spread. The coefficient on the spread shows how fast the gap closes.

Cointegration is most plausible where there is an economic reason for prices to stay together. Examples include a futures contract and its underlying asset, two share classes of one company, funds tracking the same index, and producers of the same commodity. A trading rule built on a cointegrated pair buys the spread when it is unusually low and sells it when unusually high, a form of mean reversion strategy. The speed of reversion sets the expected holding period and decides whether the expected gain exceeds trading costs.

With more than two assets, a basket can be formed whose combined value is stationary. Index arbitrage and many equity market-neutral strategies rest on relationships of this type.

A test result describes the past sample. Relationships end when a company changes its business, is taken over, or faces a shock that does not affect its partner. A widening spread is then a permanent shift, and a strategy that keeps adding to the position suffers large losses. The estimated coefficient $\beta$ also moves over time, and many traders re-estimate it on a rolling basis.

Searching across thousands of possible pairs guarantees that some will pass a test by chance. Pairs found this way often fail afterward, so results should be checked on data not used in the search. The tests have low power when reversion is slow, and they can be misled by structural breaks. Short-selling costs, borrowing limits and the capital needed to hold a diverging position all reduce what can be earned in practice.