Common mistake 1 with Cointegration

Testing for cointegration between stationary series. Meaningless - cointegration is about I(1) series.

Common mistake 2 with Cointegration

Reporting Engle–Granger only. Swapping which variable is on the left can flip the conclusion.

Common mistake 3 with Cointegration

Finding cointegration between unrelated trending series. With enough series you will find it by chance.

Common mistake 4 with Cointegration

Concluding from a failure to reject that the series are definitely not cointegrated. These tests have low power, particularly when adjustment is slow, so a non-rejection is weak evidence rather than a finding.

EconStatsCraft › Cointegration

Cointegration

Engle–Granger, Johansen trace and maximum-eigenvalue tests, plus an error-correction model with the speed of adjustment.

What it needs

Series - Two or more, all I(1) - check with the unit-root module first.

The intuition

Two drunks staggering randomly will end up arbitrarily far apart. Two drunks tied by a rope also stagger, but never further than the rope - the DIFFERENCE between them is well-behaved even though neither of them is. Cointegration is that rope, and finding it is what makes a regression in levels legitimate rather than spurious.

When to use it

When your unit-root tests said two or more series are I(1) and economic theory says they should be tied together - consumption and income, spot and futures prices, prices in two markets.

Where the estimator comes from

If y_t and x_t are both I(1) but z_t=y_t-\beta x_t is I(0), they are cointegrated. Engle–Granger estimates \beta by OLS in levels (superconsistent even though inference on it is invalid) and then tests whether \hat{z}_t has a unit root - using special critical values, because the residual is estimated rather than observed. Johansen instead works from the VAR, testing the rank of \Pi=\alpha\beta' by maximum likelihood, which handles several relationships and does not privilege a left-hand-side variable.

The model

y_t-\beta x_t=u_t\ \text{with } u_t\sim I(0)\qquad \Delta y_t=\gamma\,(y_{t-1}-\beta x_{t-1})+\lambda\,\Delta x_t+\varepsilon_t

What it assumes

Every series is I(1). Test that first - cointegration is meaningless otherwise.
Engle–Granger assumes exactly one cointegrating relationship and depends on which variable you put on the left.
Johansen handles several relationships and is symmetric, but is sensitive to lag length.
The sample is long enough for the asymptotics. Cointegration tests are large-sample procedures and their critical values are badly wrong in short series - fifty observations is few, and under thirty the test is close to uninformative.

Statistical properties

Cointegration is only meaningful between I(1) series. Testing it on I(0) series is a category error.
With n variables there can be up to n−1 cointegrating relationships.
Engle–Granger can give different answers depending on which variable is normalised - a real limitation.
Johansen is more powerful but sensitive to lag length and deterministic terms.
Both suffer from low power in short samples: a run of 50 quarters is rarely enough.

How to read the output

Engle–Granger: a small p-value means cointegration.
Johansen trace: work down the table until you fail to reject. That row gives the rank r.
r = 0 means no long-run relationship - model in differences.
γ in the ECM is the speed of adjustment. Negative and significant confirms error correction.
Half-life: how long it takes for half of a deviation from equilibrium to be corrected.

A worked reading

Both series are I(1), so the test is well posed - and then it does not reject. Engle-Granger gives τ = -2.8674 with p = 0.1452, against a 5% critical value of -3.367. Johansen agrees from the other direction: the trace statistic for r ≤ 0 is 7.7374 against a 5% critical value of 15.494, so the rank is 0. The error-correction term is -0.0158 with p = 0.2643 - the right sign, and not distinguishable from zero. All three point the same way, which is the reassuring case even though the answer is negative: these two series are not cointegrated, and a regression of one level on the other would be spurious. The right model is in first differences. It is worth seeing a clean non-rejection, because the temptation with cointegration is to read a negative ECM coefficient as confirmation regardless of its standard error, and here that would have produced a long-run money-output relationship out of nothing.

Symptom → diagnosis → what to do

If you see this	It means	Do this
Series are not all I(1)	Test invalid	Check integration orders; use ARDL bounds for a mix
Engle–Granger and Johansen disagree	Normalisation or lag sensitivity	Report both; prefer Johansen
Cointegration found between unrelated series	Chance, or a common trend	Demand an economic story before believing it
ECM coefficient positive	Not error-correcting	Re-check the sign convention
\|γ\| > 1 in the ECM	Overshooting	Possible but unusual - inspect the residual plot

Common mistakes

Testing for cointegration between stationary series. Meaningless - cointegration is about I(1) series.
Reporting Engle–Granger only. Swapping which variable is on the left can flip the conclusion.
Finding cointegration between unrelated trending series. With enough series you will find it by chance.
Concluding from a failure to reject that the series are definitely not cointegrated. These tests have low power, particularly when adjustment is slow, so a non-rejection is weak evidence rather than a finding.

Where to read this properly

Granger (1981); Engle & Granger (1987) - the Nobel-winning papers
Johansen (1988, 1991)
Enders (2014), Applied Econometric Time Series, 4th ed., Wiley, chapter 6

Related Time series - several series

Yield curve (Nelson–Siegel / Svensson) · Dynamic factor model · Dynamic regression (ARIMAX) · Structural VAR · VAR · Granger causality · VECM · ARDL & bounds test

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Cointegration

What it needs

The intuition

When to use it

Where the estimator comes from

The model

What it assumes

Statistical properties

How to read the output

A worked reading

Symptom → diagnosis → what to do

Common mistakes

Where to read this properly

See also

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