Unknown-Break Unit-Root Tests And Break-Date Search

Why Allow a Break When Testing for a Unit Root?

A persistent series can appear to have a unit root when its mean or trend changes once during the sample. For example, an index might follow a stable trend until a policy change, then continue around a different level. A unit-root test that omits this change may have difficulty distinguishing a deterministic break from stochastic persistence. The Zivot–Andrews test addresses this issue by allowing one break under the alternative and estimating its date from the data.

The central feature is the search: the test is run for a set of candidate break dates, and the date giving the strongest evidence against a unit root is selected. Because the date is chosen using the test statistic, the resulting statistic does not have the ordinary Dickey–Fuller distribution. Inference must use critical values for the relevant Zivot–Andrews specification.

Definition: A Zivot–Andrews test is a unit-root test with a null of a unit root without a structural break and an alternative of stationarity around a deterministic path with one unknown break. The break may shift the intercept, change the trend slope, or do both. The test searches over admissible break dates and uses the most negative test statistic for the coefficient on the lagged level.

Three Ways to Specify the Break

Let \(y_t\) be the observed time series, with \(t=1,\ldots,n\). For a candidate break date \(T_B\), define two deterministic indicators:

DU_t(T_B)=\mathbf{1}(t>T_B),\qquad DT_t(T_B)=(t-T_B)\mathbf{1}(t>T_B).

Thus \(DU_t\) is zero through the break date and one afterward. The variable \(DT_t\) is zero through the break date and increases by one in each post-break period. This convention places the first changed period at \(T_B+1\).

Each candidate date is used in an augmented regression. In the equations below, \(\Delta y_t=y_t-y_{t-1}\), \(k\) is the number of lagged differences, and \(\varepsilon_t\) is the regression disturbance. The lagged differences help account for serial correlation in the innovations.

\begin{aligned} \text{Model A:}\quad \Delta y_t &=\mu+\beta t+\theta DU_t(T_B)+\alpha y_{t-1} +\sum_{j=1}^{k}c_j\Delta y_{t-j}+\varepsilon_t,\\ \text{Model B:}\quad \Delta y_t &=\mu+\beta t+\gamma DT_t(T_B)+\alpha y_{t-1} +\sum_{j=1}^{k}c_j\Delta y_{t-j}+\varepsilon_t,\\ \text{Model C:}\quad \Delta y_t &=\mu+\beta t+\theta DU_t(T_B)+\gamma DT_t(T_B)+\alpha y_{t-1} +\sum_{j=1}^{k}c_j\Delta y_{t-j}+\varepsilon_t. \end{aligned}

Model A allows a one-time shift in the intercept of the deterministic path, often called a break in level. Model B allows a change in the trend slope, with no separate level-shift dummy. Model C permits both a level shift and a slope change. In all three models, \(\alpha=0\) corresponds to a unit root in the tested equation; evidence for stationarity is assessed against the left-tailed alternative \(\alpha<0\).

The specifications represent different alternatives, not three interchangeable ways to describe the same break. Choosing among them is a substantive modeling decision. Including a break form unsupported by the process, or omitting a relevant form, can affect the test’s ability to detect stationarity around a break.

Worked Example: Constructing the Break Variables

Suppose a hypothetical annual series has \(n=12\) observations, indexed from \(t=1\) to \(t=12\), and consider the candidate break date \(T_B=8\). Compute the break variables for \(t=7,8,9,10\).

By definition, \(DU_t(8)=\mathbf{1}(t>8)\). Therefore its values at those four dates are \(0,0,1,1\). For the trend-break variable, \(DT_t(8)=(t-8)\mathbf{1}(t>8)\), giving \(0,0,1,2\).

In Model A, the level-shift term \(\theta DU_t(8)\) is zero through year 8 and equals \(\theta\) in years 9 onward. In Models B and C, the slope-break term is zero through year 8, equals \(\gamma\) in year 9, and equals \(2\gamma\) in year 10. These values show how the two indicators encode distinct changes.

Searching Over Candidate Dates

The break date is not fixed in advance. A search procedure considers candidate dates in an interior portion of the sample, excluding a chosen fraction at each end. This trimming avoids dates with too few observations on one side of the break, which can make the regression poorly determined and the test unreliable. The precise trimming range, lag specification, and critical values should follow a consistent implementation.

For each admissible \(T_B\), estimate the chosen regression by least squares and calculate the t-statistic for \(\alpha=0\):

t_\alpha(T_B)=\frac{\widehat{\alpha}(T_B)}{SE\!\left(\widehat{\alpha}(T_B)\right)}.

The Zivot–Andrews statistic is the smallest, or most negative, of these candidate statistics:

ZA=\min_{T_B\in\mathcal{T}} t_\alpha(T_B),

where \(\mathcal{T}\) is the set of admissible break dates. The minimizing date is the estimated break date for this test procedure. The search chooses the candidate most favorable to rejecting the unit-root null; it does not independently establish that a break occurred.

Choose the break form and lag treatment.
Specify Model A, B, or C and select the augmentation lags so the regression errors are adequately controlled for serial correlation.

Set the trimmed candidate range.
Define the admissible dates before examining which candidate gives the smallest statistic.

Estimate at each candidate date.
Construct the date-specific indicators, fit the regression, and record the t-statistic for the lagged-level coefficient.

Compare the minimum with the correct critical value.
Use critical values appropriate to the break form and test design. Reject only if the minimum statistic is more negative than the chosen critical value.

Why the Critical Values Are Nonstandard

If the break date were fixed in advance, the test would still require critical values appropriate to its unit-root regression. In the Zivot–Andrews procedure, however, the date is selected by minimizing the statistic across candidates. Under the unit-root null, the break date is not identified as a true date in a stationary process. Searching across dates changes the sampling distribution of the minimum.

The key relationship is direct: the reported statistic is a minimum over a collection of correlated statistics, not one statistic evaluated at a prespecified date. Consequently, its null distribution is not the ordinary Dickey–Fuller distribution. Using ordinary Dickey–Fuller critical values ignores the search and can produce misleading rejection decisions.

Zivot–Andrews critical values depend on the break specification, as well as the conventions used for trimming and augmentation. As an illustration, commonly reported asymptotic 5% critical values are approximately \(-4.80\) for Model A, \(-4.42\) for Model B, and \(-5.08\) for Model C. These are reference values, not universal constants: use values or p-values matched to the implementation being reported.

Worked Example: Choosing the Date and Making the Test Decision

Consider a hypothetical series of 80 annual observations. For Model A, suppose dates 20 through 61 are admissible, and the fitted regressions at three candidate dates give the following t-statistics for \(\alpha\): \(-4.02\) at date 30, \(-4.76\) at date 45, and \(-4.40\) at date 55. Suppose all other candidate dates give statistics greater than \(-4.76\).

The search statistic is the minimum:

ZA=\min\{-4.02,-4.76,-4.40,\ldots\}=-4.76.

The selected date is \(T_B=45\). Using the illustrative Model A 5% critical value of \(-4.80\), the decision is not to reject the unit-root null at the 5% level, because \(-4.76>-4.80\). The result is close to that threshold, but closeness does not change the decision rule. It also does not prove that the series has a unit root; it means this test does not provide sufficient evidence against the null at that level.

Worked Example: Comparing Alternative Break Forms

Suppose a second hypothetical series has 100 quarterly observations. After using the same stated trimming rule and a suitable augmentation specification, the minimum statistics are \(-4.70\) for Model A, \(-4.35\) for Model B, and \(-5.21\) for Model C. Use the illustrative 5% critical values above to assess each model’s test decision.

For Model A, \(-4.70>-4.80\), so the unit-root null is not rejected at 5%. For Model B, \(-4.35>-4.42\), so the null is not rejected. For Model C, \(-5.21<-5.08\), so the null is rejected at 5% under the specification allowing both a level shift and a slope change.

This comparison does not by itself prove that the true process has both kinds of break. Each test evaluates a different alternative, and the reported critical values must match the corresponding model. A rejection under Model C is evidence against the unit-root null in that specification; deciding whether a level shift, slope change, or particular historical event explains the result requires additional analysis.

Assumptions, Interpretation, and Limits

The test is designed for a single break in the deterministic component and a unit-root null. Its usual regression-based implementation relies on a suitable lag augmentation: the remaining disturbance should not retain problematic serial correlation, and the innovations must meet the regularity conditions behind the critical values used. The deterministic trend and break forms should also be credible descriptions of the series. The test is not a general test for any number of breaks, nonlinear trends, or arbitrary changes in the stochastic process.

Lag choice matters because too few lagged differences can leave serial correlation in the regression disturbance, while unnecessary lags can reduce precision. The break-date search and lag procedure should therefore be reported, rather than only the final statistic. The estimated date is the date selected by this test under its chosen specification; it should not be treated as an exact measurement of when an economic event began.

A rejection says the evidence is inconsistent with a unit root under the test’s null and provides evidence for stationarity around a one-break deterministic path. It does not establish causation, identify the event responsible for a break, or guarantee that the series will be stable outside the sample. Failure to reject likewise does not establish a unit root.

Common pitfall: Treating the most negative candidate t-statistic as an ordinary Dickey–Fuller statistic and comparing it with an ordinary Dickey–Fuller critical value. The search over dates changes the null distribution. Use the nonstandard critical value for the selected Zivot–Andrews break form and search design.

Practical Reporting

A clear report identifies the tested series and its frequency, the break form, lag selection, trimming range, minimizing statistic, selected date, and the critical value or p-value used. State the significance level and the conclusion in context. For example: “For the quarterly index, the Model C statistic was \(-5.21\), with the minimum at 2008 Q2; using the matched 5% critical value of \(-5.08\), we reject the unit-root null in favor of stationarity around a single level and slope break.”

This wording makes the inferential claim explicit without turning the selected date into a known fact. If more than one break is plausible, or if the substantive question concerns breaks under the unit-root null, a different procedure may be needed. The Zivot–Andrews test addresses its specific one-break alternative, not every form of structural change.

Check Your Understanding

Use the definitions and decision rule in this tutorial to answer the following.

For a candidate date \(T_B=20\), what are \(DU_{22}(T_B)\) and \(DT_{22}(T_B)\)?
Which model allows a change in trend slope but does not include a separate level-shift indicator?
If the candidate statistics are \(-3.9\), \(-4.6\), and \(-4.2\), what is the search statistic and which candidate date is selected?
Why should an ordinary Dickey–Fuller critical value not be used for the selected Zivot–Andrews statistic?
At a 5% critical value of \(-4.80\), does a Model A statistic of \(-4.76\) reject the unit-root null?
What does rejection establish, and what does it not establish, about the selected break date and its cause?