Unit root tests are crucial for understanding time series stationarity. They help determine if shocks have permanent effects on a series, impacting its mean and variance over time. Non-stationarity can lead to misleading statistical results, making these tests essential for valid analysis.

The Augmented Dickey-Fuller (ADF) and Kwiatkowski-Phillips-Schmidt-Shin (KPSS) tests are key tools for assessing stationarity. ADF tests for unit roots, while KPSS tests for trend-stationarity. Using both tests provides a more robust assessment of a time series' properties.

Unit Root Tests

Unit roots and time series stationarity

• Unit root
  • Characteristic of a time series that becomes non-stationary
  • Presence implies shocks have a permanent effect on the series (random walk)
• Implications for stationarity
  • Stationary series has constant mean, variance, and autocovariance over time
  • Non-stationary series with unit roots exhibit trending behavior (upward or downward drift)
  • Non-stationarity can lead to spurious regression results and invalid statistical inference (misleading relationships)

Application of ADF test

• Augmented Dickey-Fuller (ADF) test
  • Tests null hypothesis that a unit root is present in a time series
  • Extends Dickey-Fuller test by including lagged differences to account for serial correlation (removes autocorrelation)
• Conducting the ADF test
  • Estimate the ADF regression equation: $\Delta y_t = \alpha + \beta t + \gamma y_{t-1} + \sum_{i=1}^{p} \delta_i \Delta y_{t-i} + \varepsilon_t$
    • $\Delta y_t$: First difference of the series (removes trend)
    • $\alpha$: Constant term (intercept)
    • $\beta t$: Time trend (captures deterministic trend)
    • $\gamma y_{t-1}$: Lagged level of the series (tests for unit root)
    • $\sum_{i=1}^{p} \delta_i \Delta y_{t-i}$: Lagged differences, p is lag order (captures short-run dynamics)
  • Test significance of coefficient $\gamma$ using ADF test statistic (t-statistic)

Interpretation of ADF test results

• Interpreting ADF test results
  • Compare ADF test statistic with critical values at different significance levels (1%, 5%, 10%)
  • If test statistic is more negative than critical value, reject null hypothesis of a unit root (series is stationary)
  • Failure to reject null hypothesis suggests presence of a unit root and non-stationarity (needs differencing)
• Conclusions about stationarity
  • Rejecting null hypothesis implies series is stationary (no unit root)
  • Failing to reject null hypothesis suggests series is non-stationary and may require differencing (remove unit root)

KPSS test for stationarity

• Kwiatkowski-Phillips-Schmidt-Shin (KPSS) test
  • Tests null hypothesis that a time series is trend-stationary
  • Complements ADF test by reversing null and alternative hypotheses (stationarity vs unit root)
• Conducting the KPSS test
  • Estimate KPSS test statistic based on residuals from regression of series on constant and time trend
  • Compare test statistic with critical values to determine whether to reject null hypothesis of stationarity (large statistic)

ADF vs KPSS test comparison

• Comparison of ADF and KPSS tests
  • Null hypotheses
    • ADF: Null hypothesis of a unit root, non-stationarity (common)
    • KPSS: Null hypothesis of trend-stationarity (less common)
  • Power of the tests
    • ADF test has low power against near unit root processes (close to 1)
    • KPSS test has higher power in detecting departures from stationarity (more sensitive)
• Strengths and limitations
  • ADF test
    • Widely used and easy to interpret (popular choice)
    • May have low power in small samples or near unit root processes (weak performance)
  • KPSS test
    • Useful for confirming stationarity when ADF test fails to reject null (complementary)
    • Sensitive to choice of lag truncation parameter (bandwidth)
• Combining ADF and KPSS tests
  • Use both tests to gather more evidence about stationarity of a series (robustness check)
  • If both tests agree (ADF rejects null and KPSS fails to reject), stronger conclusions can be drawn about stationarity (confirmatory results)