Differencestationarity

Differencestationarity, or difference stationarity, is a property of a time series in econometrics and statistics describing that the series becomes stationary after differencing a finite number of times. In the common case, a process is difference-stationary if it is integrated of order one, I(1): its first difference ΔX_t = X_t − X_{t−1} is stationary. This suggests that the nonstationarity is driven by a stochastic trend, such as a random walk with drift, rather than by a deterministic pattern.

A formal representation often used is that an I(1) series can be written as X_t = X_{t−1} + u_t + c, where u_t is a stationary process and c is a constant or drift term. Differencing then yields ΔX_t = u_t, which is stationary. The concept extends to higher orders, with I(d) indicating that differencing d times yields a stationary series. By contrast, trend-stationary processes are nonstationary due to a deterministic trend; removing that trend leaves a stationary residual.

Tests and modeling implications are central in practice. Unit root tests, such as the Dickey–Fuller and Augmented Dickey–Fuller tests, assess the presence of a unit root and thus I(1) behavior. The KPSS test complements this by testing for stationarity around a deterministic trend. Correctly identifying differencestationarity informs model choice, such as ARIMA specifications or detrending strategies, and has implications for cointegration analysis among multiple series.

Examples include macroeconomic levels like GDP, often I(1), whose first differences (growth rates) are typically stationary. See also unit roots, integration order, ARIMA, cointegration, and detrending.