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The LjungâBox test (named for Greta M. Ljung and George E. P. Box) is a type of statistical test of whether any of a group of autocorrelations 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.
This test is sometimes known as the LjungâBox Q test, and it is closely connected to the BoxâPierce test (which is named after George E. P. Box and David A. Pierce). In fact, the LjungâBox test statistic was described explicitly in the paper that led to the use of the BoxâPierce statistic, [1] [2] and from which that statistic takes its name. The BoxâPierce test statistic is a simplified version of the LjungâBox statistic for which subsequent simulation studies have shown poor performance. [3]
The LjungâBox test is widely applied in econometrics and other applications of time series analysis. A similar assessment can be also carried out with the BreuschâGodfrey test and the DurbinâWatson test.
The LjungâBox test may be defined as:
The test statistic is: [2]
where n is the sample size, is the sample autocorrelation at lag k, and h is the number of lags being tested. Under the statistic Q asymptotically follows a . For significance level α, the critical region for rejection of the hypothesis of randomness is:
where is the (1 â α)- quantile [4] of the chi-squared distribution with h degrees of freedom.
The LjungâBox test is commonly used in autoregressive integrated moving average (ARIMA) modeling. Note that it is applied to the 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 . [5]
The BoxâPierce test uses the test statistic, in the notation outlined above, given by [1]
and it uses the same critical region as defined above.
Simulation studies have shown that the distribution for the LjungâBox statistic is closer to a distribution than is the distribution for the BoxâPierce statistic for all sample sizes including small ones.[ citation needed]
Box.test
function in the stats package
[6]acorr_ljungbox
function in the statsmodels
package
[7]HypothesisTests
package
[8]This article incorporates public domain material from the National Institute of Standards and Technology