Trendstationary

Trend-stationary, or trend-stationary process, refers to a time series that is non-stationary because of a deterministic trend, rather than a stochastic trend. If the deterministic trend is removed, the remaining component is stationary. In contrast, a difference-stationary process becomes stationary only after differencing.

A typical trend-stationary form is Y_t = mu + beta t + u_t, where mu and beta define a deterministic trend and u_t is a stationary process (for example, white noise or an ARMA process). The key property is that the observed non-stationarity is explained by a predictable trend, and once detrended, the series behaves like a stationary series.

Modeling and forecasting with trend-stationarity usually involve two steps. First, remove the deterministic trend by regressing Y_t on time (and possibly on higher-order terms or structural components). Second, analyze or model the residuals u_t with a stationary model such as ARMA. Forecasts then combine the trend component with forecasts of the residuals. This approach contrasts with difference-stationary models, where differencing is required to achieve stationarity.

Testing for trend-stationarity often relies on tests that involve a deterministic trend. The KPSS test assesses the null hypothesis of stationarity around a deterministic trend, while augmenting the series with a trend term in unit-root tests (e.g., ADF with trend) to compare conclusions. Caution is advised in the presence of structural breaks or regime changes, which can mimic or mask trend-stationarity.

Applications include macroeconomic series commonly modeled with a deterministic growth trend, such as certain GDP series, where the focus is on the behavior of fluctuations around a trend.