Nonstationarity

Nonstationarity refers to a property of a stochastic process in which its statistical characteristics change over time. In a strictly stationary process, the joint distribution is invariant to shifts in time; in a weaker, second-order sense, the mean, variance, and autocovariances do not depend on the time at which they are computed. Nonstationary processes violate these conditions, making standard time series methods that assume constant moments less reliable.

Nonstationarity can arise from several sources. Deterministic trends produce a changing mean over time, such as a steadily increasing series. Stochastic trends, or unit-root processes, exhibit persistence where shocks have lasting effects, akin to a random walk. Structural breaks or regime shifts cause abrupt changes in the underlying dynamics. Seasonal patterns introduce time-varying means in a periodic form, and changes in variance over time (heteroskedasticity) span nonstationarity in a broader sense. Some processes may exhibit multiple forms simultaneously.

Detection and consequences: Nonstationarity can render conventional inference unreliable and can lead to spurious relationships if unrelated nonstationary series are regressed together. Unit root tests (such as the augmented Dickey-Fuller test) and structural-break tests help diagnose nonstationarity, while visual inspection of plots and autocorrelation functions is also informative. Handling strategies include differencing or detrending the data, transforming the series, or using models that accommodate nonstationarity, such as cointegration frameworks and error-correction models.

Remedies and modeling: If a nonstationary series is cointegrated with others, a stationary combination may exist, enabling meaningful long-run inferences. Otherwise, modeling often relies on differenced data or specialized nonstationary time series methods. Nonstationarity is a common consideration across economics, climatology, finance, and engineering, influencing both modeling choices and interpretation.