Unit root test

In statistics, a unit root test tests whether a time series variable is non-stationary and possesses a unit root. The null hypothesis is generally defined as the presence of a unit root and the alternative hypothesis is either stationarity, trend stationarity or explosive root depending on the test used.

General approach[]

In general, the approach to unit root testing implicitly assumes that the time series to be tested ${\displaystyle [y_{t}]_{t=1}^{T}}$ can be written as,

${\displaystyle y_{t}=D_{t}+z_{t}+\varepsilon _{t}}$

where,

• ${\displaystyle D_{t}}$ is the deterministic component (trend, seasonal component, etc.)
• ${\displaystyle z_{t}}$ is the stochastic component.
• ${\displaystyle \varepsilon _{t}}$ is the stationary error process.

The task of the test is to determine whether the stochastic component contains a unit root or is stationary.^[1]

Main tests[]

Other popular tests include:

• augmented Dickey–Fuller test^[2]

  this is valid in large samples.

• Phillips–Perron test
• KPSS test

  here the null hypothesis is trend stationarity rather than the presence of a unit root.

• ADF-GLS test

Unit root tests are closely linked to serial correlation tests. However, while all processes with a unit root will exhibit serial correlation, not all serially correlated time series will have a unit root. Popular serial correlation tests include:

• Breusch–Godfrey test
• Ljung–Box test
• Durbin–Watson test

Notes[]

References[]

• Bierens, H. J. (2001). "Unit roots". In Baltagi, B. (ed.). A Companion to Econometric Theory. Oxford: Blackwell Publishers. pp. 610–633. "2007 revision"
• Enders, Walter (2004). Applied Econometric Time Series (Second ed.). John Wiley & Sons. pp. 170–175. ISBN 0-471-23065-0.
• Maddala, G. S.; Kim, In-Moo (1998). "Issues in Unit Root Testing". Unit Roots, Cointegration, and Structural Change. Cambridge: Cambridge University Press. pp. 98–154. ISBN 0-521-58782-4.