Unit Root Testing With a Structural Break

Time Series · Hard · Free problem

Suppose you have a time series $X_t$ for $t = 1, \ldots, T$ that may contain a level shift at some unknown time $\tau$. You want to test whether $X_t$ has a unit root or is trend-stationary, but the usual Augmented Dickey-Fuller (ADF) test has low power when a structural break is present.

  1. Write down an augmented Dickey-Fuller regression that includes a break dummy $D_t(\tau) = \mathbf{1}(t > \tau)$ to allow for a level shift at time $\tau$. Clearly state the null and alternative hypotheses.
  1. Since $\tau$ is unknown, outline a grid-search procedure over candidate break dates to find the most favorable evidence against the unit root null. What test statistic do you compute at each candidate $\tau$, and how do you select the final statistic?
  1. Explain why the critical values for this procedure are nonstandard -- they differ from both the usual ADF tables and the standard normal. How would you obtain appropriate critical values in practice?

Hints

  1. Start by writing down the standard ADF regression, then think about what additional regressor you need to capture a one-time level shift at time $\tau$.
  2. Since $\tau$ is unknown, you cannot condition on it -- consider computing the ADF $t$-statistic at every candidate break date and taking the most extreme value (Zivot-Andrews approach).
  3. The critical values are more negative than standard ADF tables because you are taking the infimum of many correlated test statistics. Simulate random walks under $H_0$, apply the same grid-search, and tabulate the resulting distribution.

Worked Solution

How to Think About It: The classic ADF test assumes the data-generating process has no structural break. If there actually is a level shift and you ignore it, the ADF test confuses the break for evidence of a unit root -- it loses power dramatically. Perron (1989) showed that ignoring a single mean shift can cause the ADF test to fail to reject a unit root even when the series is clearly stationary around a broken trend. The fix is to embed the break directly into the ADF regression and search over possible break dates. The cost of searching is that you are running many tests, which inflates the chance of finding a spurious rejection, so you need special critical values.

Key Insight: The core tension is between power (allowing the break so the test can detect stationarity) and size control (searching over break dates introduces a supremum-type statistic whose distribution is shifted to the left relative to a single ADF $t$-statistic).

The Method:

  1. Set up the break-augmented ADF regression. For each candidate break date $\tau$, define the dummy $D_t(\tau) = \mathbf{1}(t > \tau)$ and estimate:

$\Delta X_t = \alpha + \beta t + \gamma D_t(\tau) + \rho X_{t-1} + \sum_{j=1}^{p} \phi_j \Delta X_{t-j} + \varepsilon_t$

The null hypothesis is $H_0: \rho = 0$ (unit root with a possible level shift), and the alternative is $H_1: \rho < 0$ (trend-stationary with a level shift). The lag length $p$ is chosen by AIC or BIC, and may be re-selected at each $\tau$.

2. Grid search over candidate break dates. Restrict the search to an interior subset of the sample, typically $\tau \in [0.15T, \, 0.85T]$, so you have enough observations on both sides. For each candidate $\tau_k$ on the grid: - Estimate the regression above. - Record the $t$-statistic for $\hat{\rho}$, call it $t_{\text{ADF}}(\tau_k)$.

  1. Select the final test statistic. The Zivot-Andrews (1992) approach takes the infimum (most negative) $t$-statistic across all break dates:

$t_{\text{ZA}} = \inf_{\tau \in [0.15T, \, 0.85T]} t_{\text{ADF}}(\tau)$

This selects the break date that gives the strongest evidence against the unit root. You reject $H_0$ if $t_{\text{ZA}}$ is more negative than the appropriate critical value.

4. Obtain nonstandard critical values. The critical values differ from ordinary ADF tables for two reasons: - The break dummy adds a nuisance parameter under both $H_0$ and $H_1$. - The infimum over $\tau$ shifts the entire distribution to the left -- you are taking the minimum of many correlated $t$-statistics, so the distribution has a heavier left tail than a single ADF statistic.

In practice, critical values are obtained by Monte Carlo simulation: - Simulate $B$ (say 10,000) samples of length $T$ from a random walk (the null DGP): $X_t = X_{t-1} + \varepsilon_t$, $\varepsilon_t \sim N(0,1)$. - For each simulated sample, run the full grid-search procedure and record $t_{\text{ZA}}$. - The 1%, 5%, and 10% critical values are the corresponding quantiles of the simulated $t_{\text{ZA}}$ distribution.

For the Zivot-Andrews Model C (level and trend break), typical 5% critical values are around $-5.08$, compared to roughly $-3.45$ for the standard ADF. Published tables from Zivot and Andrews (1992) and subsequent updates are widely available, but for nonstandard sample sizes or modified specifications you should simulate your own.

Practical Considerations:

  • Model variants: Zivot-Andrews defined three models -- Model A (level break only), Model B (trend slope break only), Model C (both). Choose based on economic context. Model A is most common in macro/finance applications.
  • Multiple breaks: The single-break framework can be extended (Lumsdaine-Papell, 1997), but each additional break date searched inflates the critical values further.
  • Trimming fraction: The 15% trimming is standard but somewhat arbitrary. Narrower trimming (e.g., 5%) lets you detect breaks near endpoints but makes critical values more extreme.
  • Lag selection: Re-selecting $p$ at each $\tau$ is more robust but computationally heavier. A common shortcut is to fix $p$ from the full-sample ADF and reuse it.
  • Endogenous vs. exogenous break: The Zivot-Andrews test treats $\tau$ as endogenous (estimated from data). If you know the break date from external information (e.g., a policy change on a known date), just run a single ADF regression at that $\tau$ using Perron (1989) critical values, which are less extreme.

Answer: Augment the ADF regression with a level-shift dummy $D_t(\tau)$, search over candidate break dates in $[0.15T, 0.85T]$, and take the infimum $t$-statistic across the grid. Reject the unit root null if this infimum exceeds the Zivot-Andrews critical value (roughly $-5.08$ at 5% for Model C), which is more negative than standard ADF critical values because the search over break dates shifts the null distribution leftward. Obtain critical values by Monte Carlo simulation under a pure random walk DGP with the same grid-search procedure.

Intuition

The fundamental issue here is that structural breaks and unit roots can masquerade as each other. A stationary series with a level shift looks like it has a unit root if you ignore the break -- the shift creates apparent persistence. Conversely, a genuine unit root can produce what looks like a break simply through its random wandering. This is why Perron's critique of the original Nelson-Plosser results was so influential: once you allowed for a break at the Great Depression, many macro series that appeared to have unit roots turned out to be trend-stationary.

The grid-search approach (Zivot-Andrews) is the practical workhorse because in most applications you do not know the break date in advance. The price you pay is statistical: searching over many break dates inflates the test's critical values, making it harder to reject the null. This is a classic multiple-testing tradeoff. In practice, the Zivot-Andrews critical values are about 1.5 units more negative than standard ADF values, which means you need substantially stronger evidence to reject. When you encounter this in real work -- say, testing whether a rates spread has a unit root around a regime change -- always simulate your own critical values if the sample size or model specification deviates from the published tables.

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