Unit root

In probability theory and statistics, a unit root is a feature of some that can cause problems in statistical inference involving time series models. A linear has a unit root if 1 is a root of the process's characteristic equation. Such a process is non-stationary but does not always have a trend.

If the other roots of the characteristic equation lie inside the unit circle—that is, have a modulus (absolute value) less than one—then the first difference of the process will be stationary; otherwise, the process will need to be differenced multiple times to become stationary. Due to this characteristic, unit root processes are also called difference stationary.

Unit root processes may sometimes be confused with trend-stationary processes; while they share many properties, they are different in many aspects. It is possible for a time series to be non-stationary, have no unit-root yet be trend-stationary. In both unit root and trend-stationary processes, the mean can be growing or decreasing over time; however, in the presence of a shock, trend-stationary processes are mean-reverting (i.e. transitory, the time series will converge again towards the growing mean, which was not affected by the shock) while unit-root processes have a permanent impact on the mean (i.e. no convergence over time).

If a root of the process's characteristic equation is larger than 1, then it is called an explosive process, even though such processes are often inaccurately called unit roots processes.

The presence of a unit root can be tested using a unit root test.

Consider a discrete-time ${\displaystyle \{y_{t},t=1,\ldots ,\infty \}}$, and suppose that it can be written as an autoregressive process of order p:

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