This is the case of an AR time series model where the predicted value is expected to equal the previous period plus a random error:
xt = b0 + xt-1 + εt
When b0 is not equal to zero, the model is a random walk with a drift, but the key characteristic is a b1 = 1.
The expected value of the error is still zero.
The mean reverting level for a random walk is not covariance stationary and the technique of first differencing is frequently used to transform an AR model with one time lag variable (AR1) into a model that is covariance stationary.
If an AR time series is covariance stationary, then the serial correlations for the lag variables are insignificant or they rapidly drop to zero as the number of time period lags rises.
When the lag coefficient is not statistically different from 1, a unit root exists.
Dickey-Fuller test = applied to AR1 model to test for a unit root.
If a unit root is present, then the model is not covariance stationary; if this is the case, the independent variable must be transformed, so you can re-model.
