Box–Pierce test

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 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 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, 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.

The Ljung–Box test is widely applied in econometrics and other applications of time series analysis. A similar assessment can be also carried out with the Breusch–Godfrey test and the Durbin–Watson test.

The Ljung–Box test may be defined as:

The test statistic is:

where n is the sample size, ${\displaystyle {\hat {\rho }}_{k}}$ is the sample autocorrelation at lag k, and h is the number of lags being tested. Under ${\displaystyle H_{0}}$ the statistic Q follows a ${\displaystyle \chi _{(h)}^{2}}$. For significance level α, the critical region for rejection of the hypothesis of randomness is.

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