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⏳Intro to Time Series

Key Concepts of Unit Root Tests

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Why This Matters

Unit root tests determine whether your time series data is stationary or non-stationary, and that distinction shapes every modeling decision that follows. If you get this wrong, you risk spurious regressions, invalid hypothesis tests, and meaningless forecasts.

These tests draw on core principles of hypothesis testing, asymptotic theory, and model specification. The thing to internalize is that different tests have different null hypotheses, different assumptions about error structure, and different strengths depending on context. Don't just memorize which test does what. Know when each test is your best tool and what it means when two tests disagree.

Tests with Unit Root as Null Hypothesis

These tests assume non-stationarity until proven otherwise. Rejecting the null means you have evidence of stationarity. That distinction trips up a lot of students, so keep it front and center.

Augmented Dickey-Fuller (ADF) Test

  • Null hypothesis: unit root present. Rejection indicates the series is stationary.
  • Lagged differences of the dependent variable are included as regressors to absorb autocorrelation in the error term. You choose the number of lags, typically using an information criterion like AIC or BIC.
  • Flexible specification: you can test with no deterministic terms, a constant only, or a constant plus a linear trend. The right choice depends on what you believe about the data-generating process.

Phillips-Perron (PP) Test

  • Uses a non-parametric correction for serial correlation. Instead of adding lagged terms like ADF does, it adjusts the test statistic directly.
  • Robust to heteroskedasticity in the error term, which makes it a better choice when variance changes over time.
  • Shares the same null hypothesis as ADF (unit root present), so you interpret rejection the same way.

Dickey-Fuller GLS (DF-GLS) Test

  • Applies GLS detrending to remove deterministic components (mean or trend) before running the test. This substantially improves statistical power over standard ADF.
  • Superior small-sample performance. When you have limited observations, this is often your strongest option for detecting stationarity.
  • Uses different critical values than ADF. Double-check that you're reading the correct table when interpreting results.

Compare: ADF vs. DF-GLS both test the null of a unit root, but DF-GLS applies GLS detrending first for greater power. If you're working with a short series and need maximum sensitivity to detect stationarity, DF-GLS is the better choice.

Tests with Stationarity as Null Hypothesis

The KPSS test flips the logic. Here, rejecting the null means evidence of a unit root, which is the opposite interpretation from ADF and PP.

Kwiatkowski-Phillips-Schmidt-Shin (KPSS) Test

  • Null hypothesis: stationarity. Rejection suggests the series has a unit root.
  • Plays a complementary role in confirmatory testing. Pairing KPSS with ADF lets you cross-check your conclusions, which is especially useful when the stakes are high or results are ambiguous.
  • Comes in level and trend versions. The level version tests stationarity around a constant; the trend version tests stationarity around a deterministic trend.

Compare: ADF and KPSS have opposite null hypotheses. When ADF fails to reject (suggesting unit root) and KPSS also fails to reject (suggesting stationarity), you have conflicting evidence. This often means the series is near the stationarity/non-stationarity boundary, or you simply need more data. Exam questions love this scenario.

Tests for Structural Breaks

Standard unit root tests can be fooled by structural breaks. A one-time shift in level or trend can mimic non-stationarity even when the series is actually stationary around that break. These tests account for that possibility.

Zivot-Andrews Test

  • Endogenous break detection: the test identifies a single structural break point from the data itself, rather than requiring you to specify it in advance.
  • Three model specifications: break in intercept only, break in trend only, or break in both intercept and trend.
  • Null hypothesis: unit root (even allowing for a break). The alternative is stationarity around a broken trend or intercept.

Compare: If your series experienced a major shock (a policy change, financial crisis, or regime shift), standard ADF may falsely indicate a unit root when the series is actually stationary with a structural break. Zivot-Andrews handles this by estimating the break point endogenously. This matters a lot for economic and financial data.

Quick Reference Table

ConceptBest Examples
Unit root as null hypothesisADF, PP, DF-GLS, Zivot-Andrews
Stationarity as null hypothesisKPSS
Handles autocorrelation parametricallyADF, DF-GLS
Handles autocorrelation non-parametricallyPP
Robust to heteroskedasticityPP
Best power in small samplesDF-GLS
Accounts for structural breaksZivot-Andrews
Confirmatory/complementary testingKPSS (paired with ADF or PP)

Self-Check Questions

  1. You run an ADF test and fail to reject the null, then run a KPSS test and reject the null. What do both results suggest about your series, and are they consistent?

  2. Which two tests share the same null hypothesis but differ in how they handle serial correlation, one parametrically and one non-parametrically?

  3. Your dataset contains only 50 observations and you suspect the series may be stationary. Which unit root test would give you the best chance of detecting stationarity, and why?

  4. Compare the ADF and Zivot-Andrews tests. In what scenario would ADF give misleading results that Zivot-Andrews would correctly identify?

  5. A colleague claims that rejecting the null in a KPSS test proves the series is stationary. What's wrong with this interpretation, and what does KPSS rejection actually indicate?