Common mistake 1 with Panel cointegration tests

Skipping the unit-root step. Cointegration is a statement about I(1) variables; on stationary ones the test is not interpretable.

Common mistake 2 with Panel cointegration tests

Reading a non-rejection as proof of no relationship. These tests have low power in short panels - at thirty periods and fifteen entities they catch a slow-adjusting relationship well under half the time.

Common mistake 3 with Panel cointegration tests

Using printed critical values on a macro panel. Countries share global shocks; the tables assume they do not, and the size distortion from ignoring that is severe.

Common mistake 4 with Panel cointegration tests

Running it on more than a handful of regressors. The critical values and the power both deteriorate as the cointegrating space grows.

Common mistake 5 with Panel cointegration tests

Treating rejection as proof of causation. Cointegration says the series share a destination; it says nothing about which one moves first.

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Panel cointegration tests

Before you estimate a long-run relationship in a panel, test that there is one. Six statistics from the two families the field uses - Westerlund's error-correction tests and Pedroni's residual tests - with p-values bootstrapped from your own panel, so a common global shock does not invalidate them the way a printed table would.

What it needs

Entity identifier
Time identifier
Dependent variable - In levels, not differences - the whole question is whether the levels are tied.
Long-run partners

The intuition

Two drunks walking home are each unpredictable, but if they are walking home TOGETHER the distance between them stays bounded - and that bounded distance is the whole of cointegration. Granger showed that regressing one wanderer on another that it is not tied to produces high R-squared and enormous t-statistics out of nothing at all, which is why the test has to come before the regression. In a panel you have many pairs of drunks, and testing each one separately wastes information: a relationship too weak to detect in any single country over thirty years can be visible across twenty countries at once. The difficulty is that the countries are not independent - a global recession moves all of them - so the classic tabulated critical values, which assume they are, do not apply.

When to use it

Before any long-run panel estimate. If the variables are I(1) and NOT cointegrated, a regression in levels is spurious and the t-statistics are meaningless however good they look. If they are cointegrated, differencing throws the long-run information away. This test is what decides which.

Where the estimator comes from

Westerlund's tests come from the conditional error-correction model \Delta y_{it}=\delta_i'd_t+\alpha_i y_{i,t-1}+\gamma_i'x_{i,t-1}+\sum_{j=1}^{p}\lambda_{ij}\Delta y_{i,t-j}+\sum_{j=0}^{p}\psi_{ij}'\Delta x_{i,t-j}+e_{it}, in which the long run is \beta_i=-\gamma_i/\alpha_i and cointegration is exactly \alpha_i<0. The group-mean statistics average the entity evidence, G_t=N^{-1}\sum_i\hat\alpha_i/\mathrm{se}(\hat\alpha_i) and G_a=N^{-1}\sum_i T\hat\alpha_i/(1-\sum_j\hat\lambda_{ij}), and reject when SOME entity cointegrates. The panel statistics impose \alpha_i=\alpha: partial the entity-specific nuisance terms out of both \Delta y_{it} and y_{i,t-1}, stack, and regress, giving P_t=\hat\alpha/\mathrm{se}(\hat\alpha) and P_a=T\hat\alpha. Pedroni's route is residual-based instead: run y_{it}=a_i+b_i'x_{it}+\hat e_{it} in levels and test \hat e_{it} for a unit root, averaging the ADF t-statistics (group) or pooling to one common root (panel).

The model

\Delta y_{it}=\delta_i'd_t+\alpha_i\left(y_{i,t-1}-\beta_i'x_{i,t-1}\right)+\sum_j\lambda_{ij}\Delta y_{i,t-j}+\sum_j\psi_{ij}'\Delta x_{i,t-j}+e_{it}\qquad H_0:\alpha_i=0\ \forall i

What it assumes

Every series is I(1). Test that first with a panel unit-root test - a cointegration test on stationary variables answers a question nobody asked.
The panel is balanced enough to average over. Entities with gaps are dropped rather than filled, and the count is reported.
Enough periods for a unit-root regression per entity. These are large-T tests; below about twenty-five periods a non-rejection means very little.
Entities may be cross-sectionally dependent - the bootstrap is there precisely so they can be. Published critical-value tables assume independence and are wrong when a common shock hits everybody.
The short-run lag order soaks up serial correlation. Too few and the test over-rejects; too many and it runs out of periods.

Statistical properties

The group-mean and panel statistics test DIFFERENT alternatives and it matters which one a rejection came from. Group: cointegration for some entities. Panel: cointegration for the panel with a common adjustment speed. A rejection of the first without the second is evidence of heterogeneity.
Every one of these has an asymptotic normal limit after standardising by tabulated moments, and those tables assume entities are cross-sectionally independent. They are not, in any macro panel, so this app bootstraps instead: the null distribution is built from the panel itself by resampling differences in blocks across the whole cross-section at once, which preserves both serial and cross-sectional dependence.
Measured on 200 simulated panels with no cointegration and a common factor linking every entity, the bootstrap p-values gave rejection rates of 4.0% (Gt), 5.5% (Pt), 1.0% (Ga), 8.5% (Pa), 2.5% (panel ADF) and 0.0% (group ADF) against a nominal 5% - so Gt and Pt are correctly sized, Pa is mildly liberal, and the residual-based statistics are conservative.
Power is the reason Westerlund wrote these. Against a strong alternative all six reject essentially always, but against a WEAK one - adjustment speeds between -0.12 and -0.04, fifteen entities, thirty periods - Pt caught it 21% of the time, Gt 16%, and Pedroni's group ADF only 3.5%. Residual-based tests impose a common factor restriction that the error-correction tests do not, and that restriction costs power.
The same simulation at N = 15, T = 30 put the size of Gt at 7.5% and Pt at 9.5%. Small panels make these tests mildly liberal, which is one more reason a marginal p-value near 0.05 in a short panel should not settle an argument.
A failure to reject is not evidence of no relationship. With low power it is the expected outcome whether or not the relationship exists, which is why the method warns rather than concluding when T is short.

How to read the output

H₀ is NO cointegration in every row. A small p-value is evidence FOR a long-run relationship.
Gt, Ga and the Pedroni group statistic reject if the relationship holds for SOME entities. Pt and Pa reject if it holds for the panel as a whole with one common adjustment speed.
A split between the two families is information, not a broken test: it says the long-run tie exists in part of the panel, so pooling the adjustment speed is what fails.
All six statistics are negative under the alternative. More negative is stronger evidence.
The bar chart shows each entity on its own. A panel test borrows strength across entities, so it can reject when most individual entities cannot - that is the entire reason for running a panel test.

A worked reading

On twenty-four countries of import demand over forty-five years - built to be I(1) in levels and cointegrated in all twenty-four - every statistic rejects at the 0.5% level: Gt = -3.62, Pt = -18.69, Pedroni group ADF = -3.78. The entity bar chart is the instructive part: far fewer than twenty-four countries clear the dashed significance line on their own forty-five observations, even though all twenty-four genuinely cointegrate. That gap between what one country can prove and what twenty-four can prove together is the whole argument for panel tests.

Symptom → diagnosis → what to do

If you see this	It means	Do this
All six reject	Cointegration	Estimate the long run - Mean Group or PMG - rather than differencing
Group tests reject, panel tests do not	The tie exists for some entities only	Mean Group. A common adjustment speed is what the panel tests could not find
Panel tests reject, group tests do not	A weak but common relationship	Pooling bought the power. PMG is defensible, but check the entity bar chart
Nothing rejects and T is small	Low power, not necessarily no relationship	Report the non-rejection as inconclusive and say so
Pedroni rejects and Westerlund does not	The common-factor restriction happens to hold	Unusual. Look at whether the levels regression is driven by one entity
Results swing with the lag order	Serial correlation is not being absorbed	Try one more lag; if it still swings, the panel is too short for this test

Common mistakes

Skipping the unit-root step. Cointegration is a statement about I(1) variables; on stationary ones the test is not interpretable.
Reading a non-rejection as proof of no relationship. These tests have low power in short panels - at thirty periods and fifteen entities they catch a slow-adjusting relationship well under half the time.
Using printed critical values on a macro panel. Countries share global shocks; the tables assume they do not, and the size distortion from ignoring that is severe.
Running it on more than a handful of regressors. The critical values and the power both deteriorate as the cointegrating space grows.
Treating rejection as proof of causation. Cointegration says the series share a destination; it says nothing about which one moves first.

Where to read this properly

Westerlund (2007), "Testing for Error Correction in Panel Data", Oxford Bulletin of Economics and Statistics 69(6).
Pedroni (1999, 2004), "Critical Values for Cointegration Tests in Heterogeneous Panels" and "Panel Cointegration", Econometric Theory 20(3).
Kao (1999), "Spurious Regression and Residual-Based Tests for Cointegration in Panel Data", Journal of Econometrics 90(1).
Persyn and Westerlund (2008), "Error-Correction-Based Cointegration Tests for Panel Data", Stata Journal 8(2) - the xtwest command.
Granger and Newbold (1974), "Spurious Regressions in Econometrics", Journal of Econometrics 2(2).

Related Panel data

Panel regression · Heterogeneous panel: Mean Group & PMG · Gravity model of trade · Dynamic panel GMM · Panel diagnostics · Panel unit-root tests

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Panel cointegration tests

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

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