Common mistake 1 with Structural breaks & stability

Allowing one break when there were obviously three.

Common mistake 2 with Structural breaks & stability

Finding a break and then reporting the full-sample regression anyway.

Common mistake 3 with Structural breaks & stability

Searching for a break, finding one, and then testing hypotheses on the split samples at conventional critical values. The split was chosen by the data, so those tests are no longer valid at their nominal size.

Common mistake 4 with Structural breaks & stability

Reporting the break date without its uncertainty. The date is an estimate with a sampling distribution, often several periods wide, and quoting it exactly implies knowledge the test does not have.

EconStatsCraft › Structural breaks & stability

Structural breaks & stability

Zivot–Andrews finds a unit root when an unknown break is allowed; CUSUM and recursive coefficients show whether a regression relationship holds steady across the sample.

What it needs

Series to test
Dependent variable
Independent variables

The intuition

A series that jumps once and then trends steadily looks, to a Dickey–Fuller test, exactly like a random walk - the test cannot tell a permanent level shift from an accumulation of shocks. Perron's point in 1989 was that much of the "evidence" for unit roots in macro series was really evidence of unmodelled breaks.

When to use it

Any long time series that spans a crisis, a policy change, a reunification, a pandemic. Standard unit-root tests are biased towards "non-stationary" when a break is present.

Where the estimator comes from

Zivot–Andrews runs the ADF regression at every candidate break date \lambda, adding a break dummy DU_t(\lambda), and takes the minimum t-statistic on the lagged level: \inf_\lambda t_{\hat{\alpha}}(\lambda). Because the date was chosen to be most favourable to rejection, the critical values are far more negative than standard ADF values. CUSUM instead accumulates recursive residuals, W_t=\sum_{j=k+1}^{t} w_j/\hat{\sigma}; under stability the path is a random walk with known bounds, and a systematic drift out of the band signals the coefficients moved.

The model

\Delta y_t=\mu+\theta\,DU_t(\lambda)+\beta t+\alpha y_{t-1}+\sum_{i=1}^{k}c_i\,\Delta y_{t-i}+e_t

What it assumes

Zivot–Andrews allows exactly ONE break, chosen by the data.
CUSUM assumes the model is otherwise correctly specified.
The break is in the parameters, not in the variance alone. A shift in volatility with a stable mean relationship will trip some of these tests while leaving the coefficients entirely valid.
There is enough data either side of the break to estimate both regimes. The test trims the ends of the sample for exactly this reason, so a break in the first or last fifteen per cent cannot be found at all.

Statistical properties

Zivot–Andrews allows exactly one break, and only under the alternative - a limitation Lee & Strazicich later addressed.
The estimated break date is consistent but its confidence interval is usually wide.
CUSUM detects gradual drift well and sudden jumps poorly; CUSUM-of-squares is the complement, detecting variance shifts.
Recursive estimates are noisy at the start of the sample by construction, since they use very few observations.

How to read the output

If ADF said "unit root" but Zivot–Andrews rejects, your series is stationary around a broken trend - a very different economic story.
The break date should correspond to something real. If it does not, be suspicious.
A CUSUM path that leaves the band marks the point where the relationship changed.
Recursive coefficients that drift steadily are the same warning drawn differently.
The break date is estimated, so it has sampling uncertainty of its own - often several periods - and quoting it to the exact quarter overstates what the test knows.
Failing to reject is the common outcome and is not evidence of stability. These tests search over every candidate date, which costs power, so they find only fairly large breaks.
Compare the break date against what you know happened. A statistically identified break with no corresponding event is more likely a sampling artefact than a discovery.

A worked reading

On log GDP the test returns t = -4.2706 for a level break (5% critical value -4.811, p = 0.205), -4.2712 for a trend break (5% critical -4.406, p = 0.072) and -4.3856 allowing both (5% critical -5.073, p = 0.269). The three specifications put the break at observations 169, 131 and 118 of 200 respectively - they disagree, which is itself informative, because a well-identified break shows up in the same place however it is modelled. None rejects at 5%, and only the trend-break version comes close. So the honest reading is that allowing one break does NOT rescue stationarity here: the series behaves like a unit root whether or not a break is permitted, and it should be differenced. That is the more common outcome and the more useful lesson - the interesting case, where a break flips the verdict, is worth looking for precisely because it is not the default.

Symptom → diagnosis → what to do

If you see this	It means	Do this
ADF fails to reject but Zivot–Andrews does	Stationary around a broken trend, not a unit root	Model with a break dummy; do not difference
Break date matches no real event	Probably an artefact	Be sceptical; check robustness to the window
CUSUM leaves the band	Coefficients are not constant	Split the sample or add interactions with a regime indicator
Recursive coefficients drift steadily	Slow structural change	Consider a rolling window or time-varying parameters

Common mistakes

Allowing one break when there were obviously three.
Finding a break and then reporting the full-sample regression anyway.
Searching for a break, finding one, and then testing hypotheses on the split samples at conventional critical values. The split was chosen by the data, so those tests are no longer valid at their nominal size.
Reporting the break date without its uncertainty. The date is an estimate with a sampling distribution, often several periods wide, and quoting it exactly implies knowledge the test does not have.

Where to read this properly

Perron (1989) - the great crash, the oil shock, and the unit-root hypothesis
Zivot & Andrews (1992)
Brown, Durbin & Evans (1975) - CUSUM
Bai & Perron (1998) - multiple unknown breaks

Related Tests & breaks

Monte Carlo experiment · F-test, Wald & Chow

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Structural breaks & stability

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

Related Tests & breaks

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