Unitroot

A unit root is a property of a time series indicating that shocks to the series have a permanent, rather than temporary, effect on its level. If a series has a unit root, it is non-stationary and typically exhibits a stochastic trend. In an autoregressive representation, a unit root occurs when a root of the characteristic equation equals 1. For example, an AR(1) model y_t = ρ y_{t-1} + ε_t has a unit root when ρ = 1, which yields a random walk (with or without drift).

More generally, many time series are integrated of order one, I(1): differencing once yields a stationary series. This framework helps distinguish permanent from transitory changes and guides modeling and inference. Non-stationary behavior due to a unit root can complicate statistical procedures that assume stationarity, including biased standard errors and spurious regressions.

Unit root tests are used to assess the presence of a unit root. Common tests include the Dickey-Fuller test and its augmented form (ADF), the Phillips-Perron test, the KPSS test for stationarity, and the DF-GLS test. Near-unit-root or highly persistent processes, where the autoregressive root is close to 1 but not equal to it, can pose additional challenges for inference.

Modeling implications include differencing the series to achieve stationarity, or employing cointegration if multiple I(1) series share a long-run relationship. Understanding whether a series has a unit root informs the choice between short-run dynamics and long-run equilibrium modeling.