Cointegration and pairs

Two walks that stay together

Two price series $x_t$ and $y_t$ that are each integrated of order one can still share a long-run link: if some combination $z_t=y_t-\alpha-\beta x_t$ is stationary, the series are cointegrated. Each wanders, but they wander together, and the gap $z_t$ keeps returning to its mean. This is the statistical basis of pairs trading: when the spread is unusually wide, you short the expensive leg and buy the cheap one, expecting it to close. Cointegration is about price levels over the long run. It differs from correlation, which measures whether daily returns move together and says nothing about whether the levels stay close.

z_t=y_t-\alpha-\beta x_t,\qquad Z_t=\frac{z_t-\bar z}{s_z},\qquad t_{1/2}=\frac{\ln(1/2)}{\ln\varphi},\qquad \Delta y_t=-\lambda\,z_{t-1}+\dots

(1)

The Engle-Granger test has two steps. First regress $y_t$ on $x_t$ by least squares to estimate $\alpha$ and the hedge ratio $\beta$ . Then run an ADF test on the residuals $z_t$ . Because the residuals come from a fitted regression, the ordinary ADF critical values are too lenient and the Engle-Granger values apply: about $-3.34$ at 5% for two series with a constant. If the unit root is rejected, the pair is cointegrated and the spread is mean-reverting with half-life $\ln(1/2)/\ln\varphi$ , where $\varphi$ is the spread's AR(1) coefficient. The error-correction view says that when yesterday's spread was positive, $y$ falls or $x$ rises today. To trade, standardise the spread into a z-score, enter when $|Z|$ exceeds about 2, exit near 0, and size the legs by $\beta$ . The risks are real: the relationship can break, $\beta$ drifts, costs eat small spreads, and testing thousands of pairs will find many that pass by chance, so choose pairs for an economic reason and confirm out of sample.

A spread and its signalSuppose regressing

y

x

gave

\alpha=10

and

\beta=1.5

. When

x=40

and

y=73

the spread is

z=73-10-1.5\cdot40=3

. If the spread has mean

0

and standard deviation

1.5

, the z-score is

Z=2

y

is expensive relative to

x

. The trade shorts 100 units of

y

and buys

1.5\cdot100=150

units of

x

. If the spread's AR(1) coefficient is

0.95

per day, the half-life is

\ln(0.5)/\ln(0.95)\approx13.5

days, so you should expect to wait weeks, not hours.

Cointegrated: each series wanders, but a linear combination of them is stationary.
Engle-Granger: regress, then test the residuals with ADF using the stricter Engle-Granger critical values.
Pairs trade: trade the z-score of the spread, size legs by $\beta$ , and expect the spread to close over about one half-life.

Lab · what's graded

The spread15 pts
The z-score10 pts
The half-life of the spread15 pts
Sizing the legs10 pts
Reading the test15 pts
Match the vocabulary15 pts
Building a pairs trade10 pts
Correlation is not cointegration10 pts

Two walks that stay together

z_t=y_t-\alpha-\beta x_t,\qquad Z_t=\frac{z_t-\bar z}{s_z},\qquad t_{1/2}=\frac{\ln(1/2)}{\ln\varphi},\qquad \Delta y_t=-\lambda\,z_{t-1}+\dots

(1)

A spread and its signalSuppose regressing

y

x

gave

\alpha=10

and

\beta=1.5

. When

x=40

and

y=73

the spread is

z=73-10-1.5\cdot40=3

. If the spread has mean

0

and standard deviation

1.5

, the z-score is

Z=2

y

is expensive relative to

x

. The trade shorts 100 units of

y

and buys

1.5\cdot100=150

units of

x

. If the spread's AR(1) coefficient is

0.95

per day, the half-life is

\ln(0.5)/\ln(0.95)\approx13.5

days, so you should expect to wait weeks, not hours.

Cointegrated: each series wanders, but a linear combination of them is stationary.
Engle-Granger: regress, then test the residuals with ADF using the stricter Engle-Granger critical values.
Pairs trade: trade the z-score of the spread, size legs by $\beta$ , and expect the spread to close over about one half-life.

Lab · what's graded

The spread15 pts
The z-score10 pts
The half-life of the spread15 pts
Sizing the legs10 pts
Reading the test15 pts
Match the vocabulary15 pts
Building a pairs trade10 pts
Correlation is not cointegration10 pts

Two walks that stay together

Lab · what's graded

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Cointegration and pairs

Two walks that stay together

Lab · what's graded

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