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The '''Ljung–Box test''' (named for [[Greta M. Ljung]] and [[George E. P. Box]]) is a type of [[statistical test]] of whether any of a group of [[autocorrelation]]s of a [[time series]] are different from zero. Instead of testing [[randomness]] at each distinct lag, it tests the "overall" randomness based on a number of lags, and is therefore a [[portmanteau test]].
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This test is sometimes known as the '''Ljung–Box Q test''', and it is closely connected to the '''Box–Pierce test''' (which is named after [[George E. P. Box]] and David A. Pierce). In fact, the Ljung–Box test statistic was described explicitly in the paper that led to the use of the Box-Pierce statistic,<ref name=BP/><ref name=LB/> and from which that statistic takes its name. The Box-Pierce test statistic is a simplified version of the Ljung–Box statistic for which subsequent simulation studies have shown poor performance.
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The Ljung–Box test is widely applied in [[econometrics]] and other applications of [[time series analysis]].
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==Formal definition==
The Ljung–Box test can be defined as follows.
: '''H<sub>0</sub>:''' The data are independently distributed (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<sub>a</sub>:''' The data are not independently distributed.
 
The test statistic is:<ref name=LB/>
:<math>
Q = n\left(n+2\right)\sum_{k=1}^h\frac{\hat{\rho}^2_k}{n-k}
</math>
where ''n'' is the sample size, <math>\hat{\rho}_k</math> is the sample autocorrelation at lag ''k'', and ''h'' is the number of lags being tested. For [[significance level]] α, the [[critical region]] for rejection of the hypothesis of randomness is
 
: <math>
Q > \chi_{1-\alpha,h}^2
</math>
 
where <math>\chi_{1-\alpha,h}^2</math> is the α-[[quantile]] of the [[chi-squared distribution]] with ''h'' degrees of freedom.
 
The Ljung–Box test is commonly used in [[autoregressive integrated moving average]] (ARIMA) modeling. Note that it is applied to the [[errors and residuals in statistics|residuals]] 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 <math>m - p - q</math>.<ref>{{cite book|last=Davidson|first=James|title=Econometric Theory|year=2000|publisher=Blackwell Publishing|isbn=0631215840|page=162}}</ref>
 
==Box-Pierce test==
The Box-Pierce test uses the test statistic, in the notation outlined above, given by<ref name=BP/>
:<math>
Q_\text{BP} = n \sum_{k=1}^h \hat{\rho}^2_k,
</math>
and it uses the same critical region as defined above.
 
Simulation studies have shown that the Ljung–Box statistic is better for all sample sizes including small ones.{{Citation needed|date=June 2011}}
 
==See also==
* [[Q-statistic]]
* [[Wald–Wolfowitz runs test]]
 
{{refimprove|date=June 2011}}
 
==References==
{{reflist |refs=
.<ref name=BP>Box, G. E. P. and Pierce, D. A. (1970) "Distribution of Residual Autocorrelations in Autoregressive-Integrated Moving Average Time Series Models", ''[[Journal of the American Statistical Association]]'', 65: 1509–1526. {{jstor|2284333}}</ref>
 
<ref name=LB>{{cite journal
|doi = 10.1093/biomet/65.2.297
|title = On a Measure of a Lack of Fit in Time Series Models
|author = G. M. Ljung
|coauthors = G. E. P. Box
|journal = [[Biometrika]]
|year = 1978
|volume = 65
|pages=297–303
|issue = 2}}</ref>
}}
 
==Further reading==
*{{cite book
|title = Introduction to Time Series and Forecasting
|edition = 2nd.
|author= Peter Brockwell
|coauthors=Richard Davis
|publisher=Springer
|year=2002
|page= 36
|isbn = 0-387-94719-1}}
 
==External links==
*[http://www.answers.com/topic/box-pierce-statistic Box–Pierce test] on [[answers.com]]
 
{{NIST-PD}}
{{Statistics|analysis}}
 
{{DEFAULTSORT:Ljung-Box test}}
[[Category:Statistical tests]]
[[Category:Time series analysis]]
[[Category:Time domain analysis]]
 
[[de:Box-Ljung-Test]]

Latest revision as of 02:16, 23 July 2014

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