Unit root test

<h2 id="general-approach">General approach</h2>
<p>In general, the approach to unit root testing implicitly assumes that the time series to be tested  
  
    
      
        [
        
          y
          
            t
          
        
        
          ]
          
            t
            =
            1
          
          
            T
          
        
      
    
    {\displaystyle [y_{t}]_{t=1}^{T}}
  
 can be written as,
</p>

y
          
            t
          
        
        =
        
          D
          
            t
          
        
        +
        
          z
          
            t
          
        
        +
        
          ε
          
            t
          
        
      
    
    {\displaystyle y_{t}=D_{t}+z_{t}+\varepsilon _{t}}

<p>where, 
</p>
<ul><li>
  
    
      
        
          D
          
            t
          
        
      
    
    {\displaystyle D_{t}}
  
  is the deterministic component (trend, seasonal component, etc.)</li>
<li>
  
    
      
        
          z
          
            t
          
        
      
    
    {\displaystyle z_{t}}
  
 is the stochastic component.</li>
<li>
  
    
      
        
          ε
          
            t
          
        
      
    
    {\displaystyle \varepsilon _{t}}
  
 is the stationary error process.</li></ul>
<p>The task of the test is to determine whether the stochastic component contains a unit root or is stationary.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>
</p>
<h2 id="main-tests">Main tests</h2>
<p>Other popular tests include:
</p>
<ul><li><a href="/facts/Augmented_Dickey%25E2%2580%2593Fuller_test/Du639qXu">augmented Dickey–Fuller test</a><a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>
this is valid in large samples.</li>
<li><a href="/facts/Phillips%25E2%2580%2593Perron_test/dQa2gGxr">Phillips–Perron test</a></li>
<li><a href="/facts/KPSS_test/gTxsI7uF">KPSS test</a>
here the null hypothesis is <a href="/facts/Trend-stationary_process/FaX9GYFN">trend stationarity</a> rather than the presence of a <a href="/facts/Stationary_process/vozdgfEX">unit root</a>.</li>
<li><a href="/facts/ADF-GLS_test/0aLXWnsp">ADF-GLS test</a></li></ul>
<p>Unit root tests are closely linked to <a href="/facts/Autocorrelation/dm4MqK0A">serial correlation</a> 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:
</p>
<ul><li><a href="/facts/Breusch%25E2%2580%2593Godfrey_test/1epmUIYX">Breusch–Godfrey test</a></li>
<li><a href="/facts/Ljung%25E2%2580%2593Box_test/Y8FwSyVW">Ljung–Box test</a></li>
<li><a href="/facts/Durbin%25E2%2580%2593Watson_statistic/3m4kYa5d">Durbin–Watson test</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li>Bierens, H. J. (2001). "Unit roots". In Baltagi, B. (ed.). <i>A Companion to Econometric Theory</i>. Oxford: <a href="/facts/Blackwell_Publishers/XUkBW8kE">Blackwell Publishers</a>. pp. 610–633. <a href="http://econ.la.psu.edu/~hbierens/UNITROOT.PDF">"2007 revision"</a></li>
<li>Enders, Walter (2004). <a href="https://archive.org/details/appliedeconometr00ende_0/page/170"><i>Applied Econometric Time Series</i></a> (Second ed.). <a href="/facts/John_Wiley_%2526_Sons/53g3LqJc">John Wiley & Sons</a>. pp. <a href="https://archive.org/details/appliedeconometr00ende_0/page/170">170–175</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-471-23065-0.</li>
<li><a href="/facts/G._S._Maddala/x3BUEmbJ">Maddala, G. S.</a>; Kim, In-Moo (1998). "Issues in Unit Root Testing". <a href="https://archive.org/details/unitrootscointeg00madd"><i>Unit Roots, Cointegration, and Structural Change</i></a>. Cambridge: Cambridge University Press. pp. <a href="https://archive.org/details/unitrootscointeg00madd/page/n116">98</a>–154. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-58782-4.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Kočenda, Evžen; Alexandr, Černý (2014), Elements of Time Series Econometrics: An Applied Approach, Karolinum Press, p. 66, ISBN 978-80-246-2315-3. <a href="978-80-246-2315-3" target="_blank">978-80-246-2315-3</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>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. <a href="/wiki/Journal_of_the_American_Statistical_Association" target="_blank">/wiki/Journal_of_the_American_Statistical_Association</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
</ol>

Unit root test open-in-new

Unit root test