Unit root test: Difference between revisions

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{{Short description|Time series statistical test}}

In [[statistics]], a '''unit root test''' tests whether a [[time series]] variable is non-stationary using an [[autoregressive]] model. A well-known test that is valid in large samples is the [[augmented Dickey–Fuller test]]. The optimal finite sample tests for a unit root in autoregressive models were developed by [[Denis Sargan]] and [[Alok Bhargava]]. Another test is the [[Phillips–Perron test]]. These tests use the existence of a [[unit root]] as the [[null hypothesis]].

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 [[Stationary process|stationarity]], [[Trend-stationary process|trend stationarity]] or explosive root depending on the test used.

== General approach ==

==See also==

In general, the approach to unit root testing implicitly assumes that the time series to be tested <math>[y_t]_{t=1}^T

* [[Augmented Dickey–Fuller test]]

</math> can be written as,

* [[Dickey–Fuller test]]

:<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]] |year=2014|isbn=978-80-246-2315-3|pages=66}}.</ref>

== Main tests ==

Other popular tests include:

* [[augmented Dickey–Fuller test]]<ref>{{Cite journal | doi = 10.1080/01621459.1979.10482531| title = Distribution of the Estimatorsestimators for Autoregressiveautoregressive Timetime Seriesseries with a Unitunit Rootroot| year = 1979| last1 = Dickey | first1 = D. A. | last2 = Fuller | first2 = W. A. | journal = [[Journal of the American Statistical Association]] | volume = 74| issue = 366a| pages = 427–431}}</ref>

*: this is valid in large samples.

* [[Phillips–Perron test]]

* [[KPSS test]]

*: here the null hypothesis is [[Trend-stationary process|trend stationarity]] rather than the presence of a [[Stationary process|unit root]].

* [[Zivot–Andrews test]]

* [[ADF-GLS test]]

Unit root tests are closely linked to [[Autocorrelation|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]]

* [[Dickey–FullerLjung–Box test]]

* [[Durbin–Watson statistic|Durbin–Watson test]]

==ReferencesNotes==

{{Notelist}}

* {{Cite journal | last1 = Bhargava | first1 = A. | authorlink = Alok Bhargava| title = On the Theory of Testing for Unit Roots in Observed Time Series | journal = [[The Review of Economic Studies]] | volume = 53 | issue = 3 | pages = 369–384 | doi = 10.2307/2297634 | jstor = 2297634| year = 1986 | pmid = | pmc = }}

* Bierens, H.J. (2001). "Unit Roots," Ch. 29 in ''A Companion to Econometric Theory'', editor B. Baltagi, Oxford: [[Blackwell Publishers]], 610–633. [http://econ.la.psu.edu/~hbierens/UNITROOT.PDF "2007 revision"]

* {{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]] | volume = 74| issue = 366a| pages = 427–431}}

*{{cite book |last=Enders |first=Walter |title=Applied Econometric Time Series |publisher=[[John Wiley & Sons]] |year=2004 |edition=Second |pages=170–175 |isbn=0-471-23065-0 }}

*{{citation | first= K. | last= Patterson | title= Unit Root Tests in Time Series | volume= 1 | year= 2011 | publisher= [[Palgrave Macmillan]]}}.

*{{citation | first= K. | last= Patterson | title= Unit Root Tests in Time Series | volume= 2 | year= 2012 | publisher= [[Palgrave Macmillan]]}}.

* {{Cite journal | last1 = Sargan | first1 = J. D. | last2 = Bhargava | first2 = Alok | year = 1983 | title = Testing Residuals from Least Squares Regression for Being Generated by the Gaussian Random Walk | journal = [[Econometrica]] | volume = 51 | issue = 1 | pages = 153–174 | publisher = | jstor = 1912252 | doix = <!-- {{subst:#titleparts:{{subst:PAGENAME}}|0|2}} --> | url = | format = | accessdate = }}

{{Reflist}}

[[Category:Statistical tests]]

[[Category:Time series analysis]]

==References==

*{{cite book |last=Bierens, |first=H. J. (|year=2001). "|chapter=Unit Roots," Ch. 29 inroots ''|title=A Companion to Econometric Theory'', |editor -first=B. |editor-last=Baltagi, |___location=Oxford: |publisher=[[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]] |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:StatisticalTime series statistical tests]]

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