Unit Root

What is Unit Root?

A unit root (also known as a difference stationary process or a unit root process) is a stochastic trend in a time series that is frequently referred to as a “random walk with drift.” If a time series has a unit root, it exhibits a predictable, systematic pattern.

When we try to model a time series with a unit root directly, we run into three major issues:

A series with a unit root does not revert to the mean, unlike stationary time series.
Time series with unit roots frequently exhibit erroneous relationships.
When we apply an ARMA model, the predicted parameters have an unequal distribution dependent on the model. The number of observations in the sample and the presence of a time trend (a Dickey-Fuller distribution). This limits our ability to model series well.
These problems can be addressed by modelling the first differences of a unit root time series, which are their changes from one period to the next. Modelling first differences also can address time trends and seasonality.

Example of Unit Root:

A random walk is a special case of a wider class of time series known as unit root processes. They are called this because when expressed using lag polynomials, one of their roots is=1, as in:

$ \left ( 1-L \right )\left ( 1-0.65L \right )y_{t}= \epsilon _{t} $