1. Introduction
A unit root process is a type of stochastic process that is non-stationary due to the presence of a unit root in its characteristic equation . These processes exhibit long-term dependencies and tend to drift over time rather than reverting to a mean.
In this article, we will cover:
- What a unit root is
- The implications of unit roots in time series
- How to detect unit roots
- How to transform a unit root process into a stationary process
2. What is a Unit Root?
A unit root exists in a time series when the process has a characteristic equation with a root equal to 1. This leads to a situation where shocks to the series have permanent effects.
Mathematical Definition
A simple autoregressive (AR) process of order 1, or AR(1), is given by:
Xt = ϕXt-1 + ϵt
where:
- Xt is the value at time t,
- ϕ is the coefficient of lagged values,
- ϵt is a white noise error term.
Stationary vs. Unit Root Case
- If |ϕ| < 1, the process is stationary and mean-reverting.
- If ϕ =1, the process has a unit root and is non-stationary.
- If |ϕ| > 1, the process is explosive (diverges).
Random Walk: A Classic Unit Root Process
A classic example of a unit root process is a random walk:
Xt = Xt-1 + ϵt
This means each new value is just the previous value plus a random shock. There is no tendency to return to a mean and has increasing variance over time, making it non-stationary.Download python code
Simulating a Random Walk
Implications of Unit Root Processes
- Non-Stationarity: The mean and variance are not constant.
- Persistence of Shocks: Any shock ϵt affects the series permanently.
- Poor Predictability: Traditional models (ARMA, etc.) may not work well.
- Spurious Regressions: If two unrelated unit root processes are regressed, it may falsely suggest a relationship.
How to Detect a Unit Root?
Several statistical tests help determine whether a time series has a unit root.
- Augmented Dickey-Fuller (ADF) Test
- Phillips-Perron (PP) Test
- Kwiatkowski-Phillips-Schmidt-Shin (KPSS) Test
Augmented Dickey-Fuller (ADF) Test
The ADF test checks whether a time series has a unit root.
- Null Hypothesis (H0): The series has a unit root (non-stationary).
- Alternative Hypothesis (H1): The series is stationary.
Phillips-Perron (PP) Test
The PP test is similar to the ADF test but adjusts for heteroscedasticity and autocorrelation.
Kwiatkowski-Phillips-Schmidt-Shin (KPSS) Test
The KPSS test has the opposite null hypothesis:
- Null Hypothesis (H0): The series is stationary.
- Alternative Hypothesis (H1): The series has a unit root (non-stationary).