nonstacionary

Nonstationary is a property of a time series in which its statistical characteristics—such as the mean, variance, or autocovariance—change over time. In time series analysis, many methods assume stationarity, meaning that these properties are constant over time. A process is strictly stationary if its joint distributions are invariant to time shifts; weak (or second-order) stationarity requires a constant mean, finite variance, and an autocovariance that depends only on the lag.

Nonstationarity can arise in several forms. A deterministic trend produces a time-varying mean, often modeled with a regression on time or removed by detrending. A stochastic trend, or unit root process, has a random walk component and typically requires differencing to achieve stationarity; such series are described as integrated of order one, I(1), or higher. Structural breaks cause abrupt changes in mean or variance at unknown or known times. Seasonal nonstationarity involves evolving seasonal patterns. Variance nonstationarity (heteroskedasticity) means the variability changes over time, as in ARCH effects; this is often treated separately from mean nonstationarity.

Detection and handling: tests such as the augmented Dickey-Fuller (ADF) and Phillips-Perron test assess unit roots, while the KPSS test assesses stationarity. If nonstationary due to a trend or a unit root, differencing or detrending is commonly used to obtain a stationary series, enabling standard modeling with ARIMA or regression. In multivariate settings, nonstationarity can be addressed via cointegration, where certain linear combinations of nonstationary series are stationary.

Applications and caveats: nonstationarity is common in economics, finance, and climate data. Failing to account for nonstationarity can produce spurious relationships, so appropriate transformation and testing are essential.