Cointegration and pairs

Two prices that stay together

Prices wander: they behave like random walks, so each is non-stationary. Two wandering series are cointegrated when some combination of them does not wander, the spread $s_t=y_t-\beta x_t$ keeps returning to a mean. That is the idea behind pairs trading. The Engle–Granger test has two steps. First regress $y$ on $x$ by least squares to get the hedge ratio $\hat\beta$ and the spread. Then run an augmented Dickey–Fuller test on the spread: $\Delta s_t=\gamma s_{t-1}+\varepsilon_t$ , and ask whether $\gamma<0$ . The t-statistic must be below special critical values, about $-3.34$ at 5% for two series, which are stricter than the usual ones because the spread was estimated. In statsmodels this is coint(y, x).

s_t=y_t-\hat\beta x_t,\qquad \Delta s_t=\gamma s_{t-1}+\varepsilon_t,\qquad \phi=1+\gamma,\qquad t_{1/2}=\frac{\ln 0.5}{\ln\phi}

(1)

A five-day spread by handTake

x=1,2,3,4,5

and

y=3,5,6,9,10

. Regressing gives

\hat\beta=18/10=1.8

, so the spread

y-1.8x

1.2,1.4,0.6,1.8,1.0

with mean

1.2

and standard deviation

0.447

. Today's spread

1.0

has z-score

(1.0-1.2)/0.447=-0.45

: it is a bit below normal. If the spread follows

s_t=0.9\,s_{t-1}+\varepsilon

, a gap closes by half in

\ln0.5/\ln0.9=6.58

days. A test regression

\Delta s=\gamma s_{t-1}

with

\hat\gamma=-0.1

and standard error

0.04

has

t=-2.5

, above the critical

-3.34

, so you cannot call the pair cointegrated. On real daily data since 2000 (

6{,}696

days), regressing one log price on another gives

R^2

0.85

0.93

for pairs such as KO and PEP, XOM and CVX, or HD and LOW, yet the Engle–Granger statistics are

-2.69

-1.80

and

-3.97

, so only HD and LOW pass at 5% (

p=0.008

). Even that fragile: each half of the sample alone fails (

t=-1.58

and

-2.63

) and the hedge ratio moves from

0.13

0.93

. The full-sample spread has

\phi=0.998

, a half-life of about

332

days, too slow to trade. For comparison, in

1000

simulated pairs of independent random walks,

85\%

show a slope with

|t|>1.96

High R-squared is not evidenceTwo wandering prices always look related when you regress one on the other. Only the stationarity of the spread, tested with the Engle–Granger critical values, says the link is real. Check stability across sub-periods before trading it.

Two steps: regress for the hedge ratio, then test the spread for a unit root with Engle–Granger values.
The half-life tells you whether a real mean reversion is fast enough to trade.

Lab · what's graded

The hedge ratio10 pts
The mean spread10 pts
A z-score15 pts
A half-life15 pts
A Dickey–Fuller statistic15 pts
Reading the test10 pts
Passing is not enough25 pts

Two prices that stay together

s_t=y_t-\hat\beta x_t,\qquad \Delta s_t=\gamma s_{t-1}+\varepsilon_t,\qquad \phi=1+\gamma,\qquad t_{1/2}=\frac{\ln 0.5}{\ln\phi}

(1)

A five-day spread by handTake

x=1,2,3,4,5

and

y=3,5,6,9,10

. Regressing gives

\hat\beta=18/10=1.8

, so the spread

y-1.8x

1.2,1.4,0.6,1.8,1.0

with mean

1.2

and standard deviation

0.447

. Today's spread

1.0

has z-score

(1.0-1.2)/0.447=-0.45

: it is a bit below normal. If the spread follows

s_t=0.9\,s_{t-1}+\varepsilon

, a gap closes by half in

\ln0.5/\ln0.9=6.58

days. A test regression

\Delta s=\gamma s_{t-1}

with

\hat\gamma=-0.1

and standard error

0.04

has

t=-2.5

, above the critical

-3.34

, so you cannot call the pair cointegrated. On real daily data since 2000 (

6{,}696

days), regressing one log price on another gives

R^2

0.85

0.93

for pairs such as KO and PEP, XOM and CVX, or HD and LOW, yet the Engle–Granger statistics are

-2.69

-1.80

and

-3.97

, so only HD and LOW pass at 5% (

p=0.008

). Even that fragile: each half of the sample alone fails (

t=-1.58

and

-2.63

) and the hedge ratio moves from

0.13

0.93

. The full-sample spread has

\phi=0.998

, a half-life of about

332

days, too slow to trade. For comparison, in

1000

simulated pairs of independent random walks,

85\%

show a slope with

|t|>1.96

Two steps: regress for the hedge ratio, then test the spread for a unit root with Engle–Granger values.
The half-life tells you whether a real mean reversion is fast enough to trade.

Lab · what's graded

The hedge ratio10 pts
The mean spread10 pts
A z-score15 pts
A half-life15 pts
A Dickey–Fuller statistic15 pts
Reading the test10 pts
Passing is not enough25 pts

Two prices that stay together

Lab · what's graded

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Cointegration and pairs

Two prices that stay together

Lab · what's graded

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