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

A commonly used test that is valid in large samples is the augmented Dickey–Fuller test.[2] The optimal finite sample tests for a unit root in autoregressive models were developed by Denis Sargan and Alok Bhargava,[3] by extending the work by John von Neumann, and James Durbin and Geoffrey Watson. In the observed time series cases, for example, Sargan–Bhargava statistics test the unit root null hypothesis in first order autoregressive models against one-sided alternatives, i.e., if the process is stationary or explosive under the alternative hypothesis.

Other popular tests include:

• Phillips–Perron test
• KPSS test (in which the null hypothesis is trend stationarity rather than the presence of a unit root)
• ADF-GLS test
• Zivot–Andrews 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

1. Kočenda, Evžen; Alexandr, Černý (2014), Elements of Time Series Econometrics: An Applied Approach, Karolinum Press, p. 66, ISBN 978-80-246-2315-3.
2. Dickey, D. A.; Fuller, W. A. (1979). "Distribution of the estimators for autoregressive time series with a unit root". Journal of the American Statistical Association. 74 (366a): 427–431. doi:10.1080/01621459.1979.10482531.
3. Sargan, J. D.; Bhargava, Alok (1983). "Testing residuals from least squares regression for being generated by the Gaussian random walk". Econometrica. 51 (1): 153–174. JSTOR 1912252.

References

• Bhargava, A. (1986). "On the theory of testing for unit roots in observed time series". The Review of Economic Studies. 53 (3): 369–384. doi:10.2307/2297634. JSTOR 2297634.
• 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.