Stationarity Tests

Stationarity tests are crucial in time series analysis, helping to determine if a series is stable over time. Key tests like ADF, KPSS, and PP assess unit roots and stationarity, guiding model selection and ensuring valid results in data analysis.

1. Augmented Dickey-Fuller (ADF) test

   • Tests for the presence of a unit root in a univariate time series.
   • Null hypothesis states that the time series has a unit root (non-stationary).
   • Incorporates lagged terms to account for autocorrelation in the residuals.
   • Provides critical values to determine the significance of the test statistic.
   • Commonly used in econometrics and finance for time series analysis.

2. Kwiatkowski-Phillips-Schmidt-Shin (KPSS) test

   • Tests for stationarity around a deterministic trend.
   • Null hypothesis states that the time series is stationary.
   • Unlike ADF, it focuses on the level of stationarity rather than the presence of a unit root.
   • Can be used in conjunction with ADF for a comprehensive analysis.
   • Critical values are also provided to assess the test statistic.

3. Phillips-Perron (PP) test

   • Another test for unit roots, similar to the ADF test.
   • Adjusts for serial correlation and heteroskedasticity in the error terms.
   • Null hypothesis indicates the presence of a unit root (non-stationary).
   • Provides a robust alternative to ADF, especially in the presence of autocorrelation.
   • Critical values are used to evaluate the significance of the test statistic.

4. Ljung-Box test

   • Tests for the presence of autocorrelation in a time series.
   • Null hypothesis states that there is no autocorrelation at any lag.
   • Useful for checking the adequacy of a fitted time series model.
   • Can be applied to residuals from a model to assess model fit.
   • Provides a Q-statistic that follows a chi-squared distribution.

5. Autocorrelation Function (ACF) plot

   • Visual representation of the correlation between a time series and its lagged values.
   • Helps identify the presence of autocorrelation in the data.
   • Useful for determining the appropriate lag length for time series models.
   • ACF values close to 1 indicate strong correlation, while values near 0 suggest weak correlation.
   • Can reveal seasonal patterns and trends in the data.

6. Partial Autocorrelation Function (PACF) plot

   • Measures the correlation between a time series and its lagged values, controlling for intermediate lags.
   • Helps identify the direct relationship between a variable and its lags.
   • Useful for determining the order of autoregressive models (AR).
   • PACF values that drop to zero after a certain lag indicate the appropriate number of lags to include.
   • Complements the ACF plot in model identification.

7. Unit root tests

   • Statistical tests used to determine if a time series is non-stationary due to a unit root.
   • Common tests include ADF, KPSS, and PP tests.
   • Essential for ensuring the validity of time series models, as non-stationary data can lead to spurious results.
   • Results guide the need for differencing or transformation of the data.
   • Helps in understanding the underlying properties of the time series.

8. Variance ratio test

   • Tests for the random walk hypothesis in time series data.
   • Compares the variance of the time series over different time intervals.
   • Null hypothesis states that the series follows a random walk (non-stationary).
   • Useful in financial markets to assess the efficiency of stock prices.
   • Provides a test statistic that can be compared to critical values for significance.

9. Zivot-Andrews test

   • Tests for unit roots in the presence of structural breaks in the time series.
   • Allows for the identification of changes in the data generating process.
   • Null hypothesis indicates the presence of a unit root with no structural break.
   • Useful for analyzing economic time series that may experience sudden shifts.
   • Provides critical values adjusted for the presence of breaks.

10. Seasonal unit root tests

    • Tests specifically designed to detect unit roots in seasonal time series data.
    • Important for series that exhibit periodic fluctuations, such as monthly or quarterly data.
    • Common tests include the HEGY test and the Canova-Hansen test.
    • Helps in identifying the need for seasonal differencing to achieve stationarity.
    • Critical for accurate modeling of seasonal patterns in time series analysis.