Ljung–Box test

<h2 id="formal-definition">Formal definition</h2>
The Ljung–Box test may be defined as:

H
          
            0
          
        
      
    
    {\displaystyle H_{0}}
  
: The data is not correlated (i.e. the correlations in the population from which the sample is taken are 0, so that any observed correlations in the data result from randomness of the sampling process).

H
 
 a
 
 
 
 
 {\displaystyle H_{a}}
 
: The data exhibit serial correlation.
The test statistic is:<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>

Q
        =
        n
        (
        n
        +
        2
        )
        
          ∑
          
            k
            =
            1
          
          
            h
          
        
        
          
            
              
                
                  
                    ρ
                    ^
                  
                
              
              
                k
              
              
                2
              
            
            
              n
              −
              k
            
          
        
      
    
    {\displaystyle Q=n(n+2)\sum _{k=1}^{h}{\frac {{\hat {\rho }}_{k}^{2}}{n-k}}}

where n is the sample size, 
 
 
 
 
 
 
 
 ρ
 ^
 
 
 
 
 k
 
 
 
 
 {\displaystyle {\hat {\rho }}_{k}}
 
 is the sample autocorrelation at lag k, and h is the number of lags being tested. Under 
 
 
 
 
 H
 
 0
 
 
 
 
 {\displaystyle H_{0}}
 
 the statistic Q asymptotically follows a 
 
 
 
 
 χ
 
 (
 h
 )
 
 
 2
 
 
 
 
 {\displaystyle \chi _{(h)}^{2}}
 
. For <a href="/facts/Significance_level/PMxjdlAu">significance level</a> α, the <a href="/facts/Critical_region/yv9RPF6U">critical region</a> for rejection of the hypothesis of randomness is:

Q
        >
        
          χ
          
            1
            −
            α
            ,
            h
          
          
            2
          
        
      
    
    {\displaystyle Q>\chi _{1-\alpha ,h}^{2}}

where 
 
 
 
 
 χ
 
 1
 −
 α
 ,
 h
 
 
 2
 
 
 
 
 {\displaystyle \chi _{1-\alpha ,h}^{2}}
 
 is the (1 − α)-<a href="/facts/Quantile/oWr1MC3s">quantile</a><a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> of the <a href="/facts/Chi-squared_distribution/wYnWWgUe">chi-squared distribution</a> with h degrees of freedom.
The Ljung–Box test is commonly used in <a href="/facts/Autoregressive_integrated_moving_average/Ep7xCa8W">autoregressive integrated moving average</a> (ARIMA) modeling. Note that it is applied to the <a href="/facts/Errors_and_residuals_in_statistics/Ef0LgAS4">residuals</a> of a fitted ARIMA model, not the original series, and in such applications the hypothesis actually being tested is that the residuals from the ARIMA model have no autocorrelation. When testing the residuals of an estimated ARIMA model, the degrees of freedom need to be adjusted to reflect the parameter estimation. For example, for an ARIMA(p,0,q) model, the degrees of freedom should be set to 
 
 
 
 h
 −
 p
 −
 q
 
 
 {\displaystyle h-p-q}
 
.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>

<h2 id="boxpierce-test">Box–Pierce test</h2>
The Box–Pierce test uses the test statistic, in the notation outlined above, given by<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>

Q
          
            BP
          
        
        =
        n
        
          ∑
          
            k
            =
            1
          
          
            h
          
        
        
          
            
              
                ρ
                ^
              
            
          
          
            k
          
          
            2
          
        
        ,
      
    
    {\displaystyle Q_{\text{BP}}=n\sum _{k=1}^{h}{\hat {\rho }}_{k}^{2},}

and it uses the same critical region as defined above.
Simulation studies have shown that the distribution for the Ljung–Box statistic is closer to a 
 
 
 
 
 χ
 
 (
 h
 )
 
 
 2
 
 
 
 
 {\displaystyle \chi _{(h)}^{2}}
 
 distribution than is the distribution for the Box–Pierce statistic for all sample sizes including small ones.

<h2 id="implementations-in-statistics-packages">Implementations in statistics packages</h2>
<ul><li><a href="/facts/R_(programming_language)/LSrkr8K8">R</a>: the Box.test function in the stats package<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a></li>
<li><a href="/facts/Python_(programming_language)/YbuGqofa">Python</a>: the acorr_ljungbox function in the statsmodels package<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a></li>
<li><a href="/facts/Julia_(programming_language)/AoB0PJ9C">Julia</a>: the Ljung–Box tests and the Box–Pierce tests in the HypothesisTests package<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a></li>
<li><a href="/facts/SPSS/6PGBVUXF">SPSS</a>: the Box-Ljung statistic is included by default in output produced by the IBM SPSS Statistics Forecasting module.</li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Box-Pierce_test/Y8FwSyVW">Q-statistic</a></li>
<li><a href="/facts/Wald%25E2%2580%2593Wolfowitz_runs_test/q42y5dVE">Wald–Wolfowitz runs test</a></li>
<li><a href="/facts/Breusch%25E2%2580%2593Godfrey_test/1epmUIYX">Breusch–Godfrey test</a></li>
<li><a href="/facts/Durbin%25E2%2580%2593Watson_statistic/3m4kYa5d">Durbin–Watson test</a></li></ul>

<h2 id="further-reading">Further reading</h2>
<ul><li>Brockwell, Peter; Davis, Richard (2002). <a href="https://books.google.com/books?id=VHB4OSAmwcUC&pg=PA35">Introduction to Time Series and Forecasting</a> (2nd ed.). Springer. pp. 35–38. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-94719-8.</li>
<li>Enders, Walter (2010). <a href="https://books.google.com/books?id=Tc4RPwAACAAJ&pg=PA69">Applied Econometric Time Series</a> (Third ed.). New York: Wiley. pp. 69–70. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0470-50539-7.</li>
<li><a href="/facts/Fumio_Hayashi/dhfhfa7D">Hayashi, Fumio</a> (2000). <a href="https://books.google.com/books?id=QyIW8WUIyzcC&pg=PA142">Econometrics</a>. Princeton University Press. pp. 142–144. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-691-01018-2.</li></ul>
<h2 id="external-links">External links</h2>
 This article incorporates <a href="/facts/Copyright_status_of_works_by_the_federal_government_of_the_United_States/Q1jvr96g">public domain material</a> from the <a href="https://www.nist.gov">National Institute of Standards and Technology</a>

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

<ol>
<li id="fn:1">Box, G. E. P.; Pierce, D. A. (1970). "Distribution of Residual Autocorrelations in Autoregressive-Integrated Moving Average Time Series Models". Journal of the American Statistical Association. 65 (332): 1509–1526. doi:10.1080/01621459.1970.10481180. JSTOR 2284333. <a href="/wiki/Journal_of_the_American_Statistical_Association" target="_blank">/wiki/Journal_of_the_American_Statistical_Association</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">G. M. Ljung; G. E. P. Box (1978). "On a Measure of a Lack of Fit in Time Series Models". Biometrika. 65 (2): 297–303. doi:10.1093/biomet/65.2.297. <a href="/wiki/Biometrika" target="_blank">/wiki/Biometrika</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Davies, Neville; Newbold, Paul (1979). "Some power studies of a portmanteau test of time series model specification". Biometrika. 66 (1): 153–155. doi:10.1093/biomet/66.1.153. <a href="https://academic.oup.com/biomet/article-abstract/66/1/153/223702" target="_blank">https://academic.oup.com/biomet/article-abstract/66/1/153/223702</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">G. M. Ljung; G. E. P. Box (1978). "On a Measure of a Lack of Fit in Time Series Models". Biometrika. 65 (2): 297–303. doi:10.1093/biomet/65.2.297. <a href="/wiki/Biometrika" target="_blank">/wiki/Biometrika</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Brockwell, Peter J.; Davis, Richard A.; Davis, R. J. (2002-03-08). Introduction to Time Series and Forecasting. Taylor & Francis. p. 36. ISBN 978-0-387-95351-9. <a href="978-0-387-95351-9" target="_blank">978-0-387-95351-9</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Davidson, James (2000). Econometric Theory. Blackwell. p. 162. ISBN 978-0-631-21584-4. <a href="978-0-631-21584-4" target="_blank">978-0-631-21584-4</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Box, G. E. P.; Pierce, D. A. (1970). "Distribution of Residual Autocorrelations in Autoregressive-Integrated Moving Average Time Series Models". Journal of the American Statistical Association. 65 (332): 1509–1526. doi:10.1080/01621459.1970.10481180. JSTOR 2284333. <a href="/wiki/Journal_of_the_American_Statistical_Association" target="_blank">/wiki/Journal_of_the_American_Statistical_Association</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">"R: Box–Pierce and Ljung–Box Tests". stat.ethz.ch. Retrieved 2016-06-05. <a href="https://stat.ethz.ch/R-manual/R-devel/library/stats/html/box.test.html" target="_blank">https://stat.ethz.ch/R-manual/R-devel/library/stats/html/box.test.html</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">"Python: Ljung–Box Tests". statsmodels.org. Retrieved 2018-07-23. <a href="https://www.statsmodels.org/dev/generated/statsmodels.stats.diagnostic.acorr_ljungbox.html" target="_blank">https://www.statsmodels.org/dev/generated/statsmodels.stats.diagnostic.acorr_ljungbox.html</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">"Time series tests". juliastats.org. Retrieved 2020-02-04. <a href="https://juliastats.org/HypothesisTests.jl/latest/time_series/" target="_blank">https://juliastats.org/HypothesisTests.jl/latest/time_series/</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
</ol>

Ljung–Box test open-in-new

Ljung–Box test