Ljung–Box test

The Ljung–Box test may be defined as:

H₀: 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_a: The data are not independently distributed; they exhibit serial correlation.

The test statistic is:[2]

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

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 asymptotically follows a ${\displaystyle \chi _{(h)}^{2}}$. For significance level α, the critical region for rejection of the hypothesis of randomness is:

${\displaystyle Q>\chi _{1-\alpha ,h}^{2}}$

where ${\displaystyle \chi _{1-\alpha ,h}^{2}}$ is the (1 − α)-quantile[4] 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 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 ${\displaystyle h-p-q}$.[5]

The Box–Pierce test uses the test statistic, in the notation outlined above, given by[1]

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

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 ${\displaystyle \chi _{(h)}^{2}}$ distribution than is the distribution for the Box–Pierce statistic for all sample sizes including small ones.

• R: the Box.test function in the stats package[6]
• Python: the acorr_ljungbox function in the statsmodels package[7]
• Julia: the Ljung–Box tests and the Box–Pierce tests in the HypothesisTests package[8]

• Q-statistic
• Wald–Wolfowitz runs test
• Breusch–Godfrey test
• Durbin–Watson test

• Brockwell, Peter; Davis, Richard (2002). Introduction to Time Series and Forecasting (2nd ed.). Springer. pp. 35–38. ISBN 978-0-387-94719-8.
• Enders, Walter (2010). Applied Econometric Time Series (Third ed.). New York: Wiley. pp. 69–70. ISBN 978-0470-50539-7.
• Hayashi, Fumio (2000). Econometrics. Princeton University Press. pp. 142–144. ISBN 978-0-691-01018-2.

 This article incorporates public domain material from the National Institute of Standards and Technology.