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# 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:

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

## Why is unit root important?

The limiting distributions of estimates and test statistics are considerably different from the stationary situation, so unit-roots are crucial.

Owais Siddiqui