How do you test for stationarity in a time series?

To test for stationarity in a time series, you primarily use statistical tests and visual analysis to determine whether the series’ statistical properties (mean, variance, autocorrelation) remain constant over time. The most common methods are the Augmented Dickey-Fuller (ADF) test, the Kwiatkowski-Phillips-Schmidt-Shin (KPSS) test, and visual inspection of plots like rolling statistics or autocorrelation functions (ACF). These approaches help identify trends, seasonality, or structural breaks that violate stationarity assumptions, which is critical because many time series models (e.g., ARIMA) require stationary data to work reliably.

The ADF test is a hypothesis test where the null hypothesis assumes the presence of a unit root (non-stationary). If the test statistic is more negative than a critical value (or the p-value is below a threshold like 0.05), you reject the null and conclude the series is stationary. For example, in Python, you can use statsmodels.tsa.stattools.adfuller() to run this test. The KPSS test flips the hypothesis: the null assumes stationarity, and a test statistic exceeding critical values indicates non-stationarity. Combining ADF and KPSS helps avoid false conclusions—e.g., if ADF rejects non-stationarity but KPSS also rejects stationarity, the series might have a deterministic trend. Tools like statsmodels.tsa.stattools.kpss() implement this.

Visual methods complement statistical tests. Plotting the time series can reveal obvious trends or seasonality. For instance, monthly sales data with a consistent upward trend or quarterly spikes would fail stationarity. Rolling statistics (e.g., a 12-month rolling mean and variance) can show whether these metrics change over time. The ACF plot of a stationary series decays rapidly, while a non-stationary one shows slow decay. If tests or visuals indicate non-stationarity, transformations like differencing (computing period-over-period changes) or log transformations are applied. For example, differencing stock price data once often removes trends. Always retest after transformations to confirm stationarity.