# Unit root test

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{{Short description|Time series statistical test}}
In [statistics](/source/statistics), a '''unit root test''' tests whether a [time series](/source/time_series) variable is non-stationary and possesses a [unit root](/source/unit_root). The null hypothesis is generally defined as the presence of a unit root and the alternative hypothesis is either [stationarity](/source/Stationary_process), [trend stationarity](/source/Trend-stationary_process) 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  <math>[y_t]_{t=1}^T
</math> can be written as,

:<math>y_t = D_t + z_t + \varepsilon_t </math>

where, 
* <math>D_t

</math>  is the deterministic component (trend, seasonal component, etc.) 
* <math>z_t
</math> is the stochastic component.  
* <math>\varepsilon_t
</math> is the stationary error process. 
The task of the test is to determine whether the stochastic component contains a unit root or is stationary.<ref>{{Citation |title=Elements of Time Series Econometrics: An Applied Approach|last1=Kočenda|first1=Evžen |last2= Alexandr| first2= Černý |publisher= [Karolinum Press](/source/Karolinum_Press) |year=2014|isbn=978-80-246-2315-3|pages=66}}.</ref>

== Main tests ==

Other popular tests include:
* [augmented Dickey–Fuller test](/source/augmented_Dickey%E2%80%93Fuller_test)<ref>{{Cite journal | doi = 10.1080/01621459.1979.10482531| title = Distribution of the estimators for autoregressive time series with a unit root| year = 1979| last1 = Dickey | first1 = D. A. | last2 = Fuller | first2 = W. A. | journal = [Journal of the American Statistical Association](/source/Journal_of_the_American_Statistical_Association) | volume = 74| issue = 366a| pages = 427–431}}</ref>
*: this is valid in large samples.
* [Phillips–Perron test](/source/Phillips%E2%80%93Perron_test)
* [KPSS test](/source/KPSS_test)
*: here the null hypothesis is [trend stationarity](/source/Trend-stationary_process) rather than the presence of a [unit root](/source/Stationary_process).
* [ADF-GLS test](/source/ADF-GLS_test)
Unit root tests are closely linked to [serial correlation](/source/Autocorrelation) 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](/source/Breusch%E2%80%93Godfrey_test)
* [Ljung–Box test](/source/Ljung%E2%80%93Box_test)
* [Durbin–Watson test](/source/Durbin%E2%80%93Watson_statistic)

==Notes==
{{Notelist}}

{{Reflist}}

==References==
*{{cite book |last=Bierens |first=H. J. |year=2001 |chapter=Unit roots |title=A Companion to Econometric Theory |editor-first=B. |editor-last=Baltagi |location=Oxford |publisher=[Blackwell Publishers](/source/Blackwell_Publishers) |pages=610–633 }} [http://econ.la.psu.edu/~hbierens/UNITROOT.PDF "2007 revision"] {{Webarchive|url=https://web.archive.org/web/20140617113943/http://econ.la.psu.edu/~hbierens/UNITROOT.PDF |date=2014-06-17 }}
*{{cite book |last=Enders |first=Walter |title=Applied Econometric Time Series |publisher=[John Wiley & Sons](/source/John_Wiley_%26_Sons) |year=2004 |edition=Second |pages=[https://archive.org/details/appliedeconometr00ende_0/page/170 170–175] |isbn=0-471-23065-0 |url-access=registration |url=https://archive.org/details/appliedeconometr00ende_0/page/170 }}
*{{cite book |last=Maddala |first=G. S. |authorlink=G. S. Maddala |last2=Kim |first2=In-Moo |chapter=Issues in Unit Root Testing |title=Unit Roots, Cointegration, and Structural Change |url=https://archive.org/details/unitrootscointeg00madd |url-access=limited |location=Cambridge |publisher=Cambridge University Press |year=1998 |isbn=0-521-58782-4 |pages=[https://archive.org/details/unitrootscointeg00madd/page/n116 98]–154 }}

Category:Time series statistical tests

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