Cointegration Tests With Structural Breaks

Why Allow a Break in the Long-Run Relation?

A cointegrating relation describes a stable long-run combination of variables, even when each variable separately has a unit root. But an economic relation may change: a tax reform, market liberalization, or change in measurement can shift its intercept or alter the long-run coefficients. If a residual-based test fits one unchanging relation to data from before and after such a change, its residuals can look persistently displaced rather than stationary around zero. The usual no-break residual test can then have poor power and its conventional critical values need not give valid inference for the break-search procedure.

The Gregory–Hansen approach addresses this issue by fitting candidate cointegrating relations with a single break at each of several possible dates. For every date, it tests whether the fitted residuals have a unit root, then uses the most negative statistic across the search. The statistic is not compared with an ordinary Dickey–Fuller or Engle–Granger critical value: the break search and the chosen break specification change its null distribution.

Definition: A series $y_t$ and a vector of series $x_t$ are cointegrated with a break at date $T_b$ in a specified relation if a linear combination of them, after accounting for the permitted break in deterministic terms or long-run coefficients, is stationary. The break date is unknown when it is selected from the data. The Gregory–Hansen null hypothesis is no cointegration; its alternative is cointegration under the particular one-break specification being tested.

Three Break Specifications

Let $y_t$ be the dependent variable and $x_t$ a vector of explanatory variables, typically treated as integrated of order one. For a candidate break date $T_b$, define $D_t(T_b)=0$ when $t\leq T_b$ and $D_t(T_b)=1$ when $t>T_b$. The convention for whether the break observation itself belongs to the first or second regime should be stated and used consistently. The specifications below are long-run regressions; their residuals are then tested for a unit root.

Level shift, C: the intercept can change, while the long-run slopes remain constant. $$ y_t=\mu_1+\mu_2D_t(T_b)+\beta'x_t+u_t. $$ Level shift with a deterministic trend, C/T: the intercept can change and the relation includes a common linear time trend. $$ y_t=\mu_1+\mu_2D_t(T_b)+\gamma t+\beta'x_t+u_t. $$ Regime shift, C/S: the intercept and long-run slopes can change. $$ y_t=\mu_1+\mu_2D_t(T_b)+\beta_1'x_t+\beta_2'\bigl(D_t(T_b)x_t\bigr)+u_t. $$

In C/T, the trend coefficient $\gamma$ is common across regimes: this standard specification includes a deterministic trend as well as an intercept shift. It does not, by itself, allow the slope of the deterministic trend to change. If the substantive hypothesis is a trend-slope break, include a post-break trend term such as $\delta D_t(T_b)(t-T_b)$ in an extended specification and use critical values calibrated for that exact model. Do not interpret an ordinary C/T result as a test for every possible form of trend change.

In C/S, the pre-break long-run coefficient vector is $\beta_1$; after the break it is $\beta_1+\beta_2$. The coefficients in $\beta_2$ are changes, not the post-break slopes themselves. The three alternatives are distinct. A test that allows an intercept shift but not a slope shift can miss cointegration if the long-run slopes changed.

Search and Test Statistic

At each candidate date, estimate the chosen long-run regression and obtain residuals $\hat u_t(T_b)$. Then estimate an augmented Dickey–Fuller regression for those residuals:

\Delta\hat u_t(T_b) =\rho(T_b)\hat u_{t-1}(T_b) +\sum_{j=1}^{k}\psi_j(T_b)\Delta\hat u_{t-j}(T_b) +\varepsilon_t(T_b).

The null is a unit root in the residual process, represented by $H_0:\rho(T_b)=0$; the stationary alternative has $\rho(T_b)<0$. The lagged differences help account for serial correlation. Let $t_{\mathrm{ADF}}(T_b)$ be the t-statistic for $\hat\rho(T_b)$. A Gregory–Hansen ADF statistic is the minimum of these t-statistics over the permitted dates:

\mathrm{ADF}^{*}=\min_{T_b\in\mathcal{T}}t_{\mathrm{ADF}}(T_b).

The candidate-date set $\mathcal{T}$ is trimmed: dates too close to either endpoint are excluded so that each regime has enough observations to estimate the regression. For example, a researcher might require each regime to contain at least 15% of the sample. This is a design choice, not a universal rule. The same set of dates must be used when calibrating critical values. A more negative minimum provides evidence against the no-cointegration null, but its meaning depends on the model specification and the full search.

Why the usual critical values do not apply: even if a unit-root statistic is computed separately at every candidate date, the reported statistic is the minimum across dates. Searching tends to make that minimum more negative than a statistic at a date fixed in advance. In addition, the first-stage residuals are estimated from a cointegrating regression. Therefore, compare $\mathrm{ADF}^{*}$ with Gregory–Hansen critical values for the chosen C, C/T, or C/S model and the appropriate search setup, not with a standard ADF cutoff.

Worked Applications

The following hypothetical examples show how to carry out and report the search. The residual-regression summaries are supplied as sufficient statistics to keep attention on the break procedure. The critical values are explicitly illustrative, model-specific bootstrap cutoffs for the stated setup; they are not universal Gregory–Hansen values.

Worked Example: Testing an Intercept Shift

Suppose annual hypothetical data on a price index $y_t$ and an integrated cost index $x_t$ are available for 1980–2024. The investigator specifies C, requires each regime to contain at least 15% of the observations, and obtains these candidate residual-test statistics: $-2.81$ for 1998, $-3.27$ for 2008, and $-3.04$ for 2015. The 2008 date gives the minimum among the candidates considered.

At 2008, the fitted long-run regression is $y_t=\mu_1+\mu_2D_t+\beta x_t+u_t$. In the residual ADF regression, after accounting for any lagged differences, suppose the effective regressor sum of squares is $S_{xx}=1.000$, the residual variance estimate is $\hat\sigma^2=0.0360$, and the estimated coefficient is $\hat\rho=-0.620$. The standard error and t-statistic are

SE(\hat\rho)=\sqrt{\frac{\hat\sigma^2}{S_{xx}}} =\sqrt{\frac{0.0360}{1.000}}=0.1897, \qquad t_{\mathrm{ADF}}(2008)=\frac{-0.620}{0.1897}=-3.268.

The rounded statistic, $-3.27$, agrees with the candidate-date summary. Suppose the model- and search-specific illustrative 5% bootstrap critical value is $-4.50$. Because $-3.27>-4.50$, the test does not reject no cointegration at the 5% level. The result is not evidence that cointegration is impossible; it means this test does not find sufficiently strong evidence under the specified intercept-shift model and search.

Worked Example: Including a Deterministic Trend

Consider hypothetical quarterly output and capital series. A researcher suspects a change in the long-run intercept and believes a deterministic trend belongs in the relation, so uses C/T. The search yields candidate statistics $-3.58$, $-4.12$, and $-3.76$ at three eligible dates; the most negative is $-4.12$ at the second date.

For that date, suppose the residual ADF regression has $\hat\rho=-0.740$, $S_{xx}=1.600$, and $\hat\sigma^2=0.0625$. Then

SE(\hat\rho)=\sqrt{\frac{0.0625}{1.600}} =\sqrt{0.0390625}=0.1976, \qquad t_{\mathrm{ADF}}^{*}=\frac{-0.740}{0.1976}=-3.745.

This arithmetic conflicts with the stated candidate statistic of $-4.12$, so the summaries cannot all describe the same fitted regression. Checking such consistency is essential. If the supplied coefficient and standard-error summaries are the intended outputs, the statistic is $-3.745$, not $-4.12$. Suppose the other candidate dates have statistics $-3.58$ and $-3.76$; the minimum is therefore $-3.76$. Against an illustrative C/T search critical value of $-5.05$, the minimum does not reject the null.

This example also highlights a modeling point: including $\gamma t$ controls a common deterministic trend; it does not estimate a distinct trend slope in each regime. A hypothesized change in trend slope requires an added post-break trend term and critical values appropriate to that extension.

Worked Example: Allowing Long-Run Slopes to Change

Suppose hypothetical data relate a commodity-price index $y_t$ to two integrated exchange-rate measures, collected in $x_t$. The researcher uses C/S because the slopes as well as the intercept may have changed. Candidate dates give residual ADF statistics $-4.88$, $-5.61$, and $-5.14$; the middle date is selected by the minimum rule.

At that date, suppose the residual ADF regression gives $\hat\rho=-1.080$, $S_{xx}=2.250$, and $\hat\sigma^2=0.0900$, with $S_{xx}$ and the variance calculated after accounting for included lagged differences. Then

SE(\hat\rho)=\sqrt{\frac{0.0900}{2.250}} =\sqrt{0.0400}=0.2000, \qquad t_{\mathrm{ADF}}^{*}=\frac{-1.080}{0.2000}=-5.400.

The recomputed minimum is $-5.40$, not $-5.61$; the latter cannot be retained alongside these regression summaries. Using the consistent statistics, the minimum across the three dates is $-5.40$. If the illustrative C/S critical value for this search is $-5.20$, then $-5.40<-5.20$, so the null is rejected at the corresponding level. The evidence supports a stationary long-run relation under a regime shift in this specification. It does not establish that the particular break date is known without uncertainty, nor that a causal effect has been identified.

Why the Break Search Changes the Conclusion

The issue can be seen directly from a simple omitted-break relation. Suppose the true cointegrating residual $u_t$ is stationary, but the intercept changes by $\mu_2$ after $T_b$. A regression that omits the change can leave a residual approximately equal to $u_t+\mu_2D_t(T_b)$. Before the break, its mean is approximately zero; afterward, its mean is approximately $\mu_2$. Unless $\mu_2=0$, this residual does not fluctuate around one stable mean over the full sample. Thus, a no-break residual-based test may fail to detect the stationary deviations around the changing relation. The break-adjusted regression removes the modeled shift before testing the residual dynamics.

The break date is selected using the same data as the unit-root test. Consequently, reporting only the most favorable date and applying a fixed-date test cutoff overstates the evidence. The minimum statistic and its break-search critical values address this selection step. The same principle applies to choosing among C, C/T, and C/S after looking at the results: if the specification is selected based on which test rejects, inference should account for that additional search or the model choice should be justified in advance.

Choose the economic break hypothesis.
Decide whether the plausible change concerns the intercept, a relation with a deterministic trend, or the long-run slopes as well.

Set the search design.
Specify the trimming rule, candidate dates, residual ADF lag treatment, and the exact regression form before interpreting the result.

Estimate and test at every eligible date.
Fit the long-run regression, calculate its residuals, and obtain the residual ADF statistic for each candidate break.

Use matching critical values.
Take the minimum statistic and compare it with values calibrated for that break specification and search, rather than a conventional no-break residual-test cutoff.

State a conditional conclusion.
Report the specification, search range, selected date, statistic, critical value, and whether the null is rejected. Describe the result as evidence under the modeled break, not proof of a unique long-run mechanism.

Common Pitfall

A common mistake is to find a visually plausible break date, fit the relation only at that date, and compare the resulting residual ADF statistic with an ordinary Engle–Granger cutoff. That treats the date as if it had been fixed independently of the data. The correction is to search across the prespecified eligible dates, use the minimum statistic, and compare it with a critical value for the same model and search procedure. A second mistake is to call a non-rejection proof that no long-run relation exists. Non-rejection can reflect limited power, an unsuitable break specification, or a break pattern outside the model.

Key takeaway: Gregory–Hansen tests assess cointegration when one unknown break may alter the long-run relation. The break form, trimming rule, and search are part of the test: the minimum residual ADF statistic must be judged using critical values calibrated to those choices.

Check Your Understanding

Answer each question by identifying the model and the inference step involved.

In the C/S specification, what does $\beta_2$ represent, and what is the post-break long-run slope vector?
Why can a conventional residual-based test have difficulty when the cointegrating intercept changes once?
Why is the minimum ADF statistic across candidate break dates not compared with an ordinary fixed-date ADF critical value?
Does the standard C/T specification allow the deterministic trend slope to change? What additional term would represent such a change?
A break-adjusted statistic does not cross its model-specific critical value. What can and cannot be concluded about cointegration?