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⏳Intro to Time Series

Key Concepts of Unit Root Tests

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Why This Matters

Unit root tests are the gatekeepers of time series analysis—they determine whether your data is stationary or non-stationary, which fundamentally shapes every modeling decision that follows. Get this wrong, and you'll end up with spurious regressions, invalid hypothesis tests, and forecasts that look impressive but mean nothing. You're being tested on your ability to choose the right test for the situation, interpret results correctly, and understand why stationarity matters for inference.

These tests embody core principles of hypothesis testing, asymptotic theory, and model specification. The key insight? Different tests have different null hypotheses, different assumptions about error structure, and different strengths in various contexts. Don't just memorize which test does what—know when each test is your best tool and what it means when tests disagree with each other.

Tests with Unit Root as Null Hypothesis

These tests assume non-stationarity until proven otherwise. Rejecting the null means you have evidence of stationarity—a crucial distinction that trips up many students.

Augmented Dickey-Fuller (ADF) Test

  • Null hypothesis: unit root present—rejection indicates the series is stationary, which is often what you're hoping for
  • Lagged differences included as regressors to absorb autocorrelation in the error term, with lag length typically selected via information criteria
  • Flexible specification allows you to test with no deterministic terms, a constant only, or a constant plus linear trend

Phillips-Perron (PP) Test

  • Non-parametric correction for serial correlation—adjusts test statistics directly rather than adding lagged terms like ADF
  • Robust to heteroskedasticity in the error term, making it preferable when variance changes over time
  • Same null hypothesis as ADF (unit root present), so interpretation of rejection is identical

Dickey-Fuller GLS (DF-GLS) Test

  • GLS detrending removes deterministic components before testing, substantially improving power over standard ADF
  • Superior small-sample performance—when you have limited observations, this is often your best unit root test
  • Different critical values than ADF, so don't accidentally use the wrong 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 your go-to choice.

Tests with Stationarity as Null Hypothesis

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

Kwiatkowski-Phillips-Schmidt-Shin (KPSS) Test

  • Null hypothesis: stationarity—rejection suggests the series has a unit root, reversing the logic of ADF/PP interpretation
  • Complementary role in confirmatory testing: use alongside ADF to cross-check conclusions when stakes are high
  • Level vs. trend versions available depending on whether you're testing stationarity around a constant or around a deterministic trend

Compare: ADF vs. KPSS—these tests 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 indicates your series is near the boundary or you need more data. FRQs love asking about this scenario.

Tests for Structural Breaks

Standard unit root tests can be fooled by structural breaks—a one-time shift in level or trend that mimics non-stationarity. These tests account for that possibility.

Zivot-Andrews Test

  • Endogenous break detection—identifies a single structural break point from the data 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, with the alternative being stationarity around a broken trend

Compare: ADF vs. Zivot-Andrews—if your series experienced a major shock (policy change, financial crisis, regime shift), standard ADF may falsely indicate a unit root when the series is actually stationary with a break. Zivot-Andrews handles this by estimating the break point endogenously. Essential 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 and contrast 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?