A unit root is a root of the characteristic polynomial of a time-series model that lies on the unit circle. Its presence generally produces
a nonstationary time series. The simplest
example is the random walk
whose autoregressive polynomial has the unit root . Applying the difference
operator gives
which is stationary when the innovations are stationary.
The augmented Dickey-Fuller test is
commonly used to test the null hypothesis that a time series has a unit root.
Dickey, D. A. and Fuller, W. A. "Distribution of the Estimators for Autoregressive Time Series with a Unit Root." J. Amer.
Statist. Assoc.74, 427-431, 1979. https://doi.org/10.1080/01621459.1979.10482531.Hamilton,
J. D. Time
Series Analysis. Princeton, NJ: Princeton University Press, 1994.
has the unit root 
. Applying the difference
operator gives
