Nonstationary

Nonstationary is a term used in statistics and time series analysis to describe processes whose statistical properties change over time. In a stationary process, the joint distribution of observations is invariant to time shifts; in the weaker sense, a process is weakly stationary if its mean, variance, and autocovariance depend only on the lag between observations, not on the time at which they are observed. Nonstationary means that at least one of these moments changes with time.

Common forms of nonstationarity include deterministic trends, where the mean grows or declines over time; stochastic or unit-root trends, such as random walks where shocks have a permanent effect on the level; structural breaks, where the governing behavior abruptly changes at certain times; and evolving variance or volatility, where dispersion changes over time. Nonstationarity can also arise from seasonality with changing amplitude or from evolving relationships among variables.

The presence of nonstationarity has important implications for modeling and inference. Many standard statistical methods assume stationarity, and applying them to nonstationary data can lead to biased estimates, spurious relationships, or unreliable forecasts. Detecting nonstationarity often involves unit-root tests like the Augmented Dickey-Fuller or Phillips-Perron tests, and tests for level or trend stationarity such as the KPSS test. Structural-break tests may be used to identify regime changes.

Common remedies include differencing the series to remove a stochastic trend, detrending deterministic components, applying transformations, or modeling within a framework that accommodates nonstationarity, such as cointegration and error-correction models for integrated series, or adopting locally stationary models for slowly evolving processes.