Ljung–Box test

Formal definition

The Ljung–Box test may be defined as: : 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: The data exhibit serial correlation.

The test statistic is: : Q = n(n+2)\sum_{k=1}^h\frac{\hat{\rho}^2_k}{n-k} where n is the sample size, \hat{\rho}k is the sample autocorrelation at lag k, and h is the number of lags being tested. Under H_0 the statistic Q asymptotically follows a \chi^2{(h)}. For significance level α, the critical region for rejection of the hypothesis of randomness is:

: Q \chi_{1-\alpha,h}^2

where \chi_{1-\alpha,h}^2 is the (1 − 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 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 h - p - q.

Box–Pierce test

The Box–Pierce test uses the test statistic, in the notation outlined above, given by : Q_\text{BP} = n \sum_{k=1}^h \hat{\rho}^2_k, 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 \chi^2_{(h)} distribution than is the distribution for the Box–Pierce statistic for all sample sizes including small ones.

Implementations in statistics packages

• R: the Box.test function in the stats package
• Python: the acorr_ljungbox function in the statsmodels package
• Julia: the Ljung–Box tests and the Box–Pierce tests in the HypothesisTests package
• SPSS: the Box-Ljung statistic is included by default in output produced by the IBM SPSS Statistics Forecasting module.

References

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References

1. (1979). "Some power studies of a portmanteau test of time series model specification.". Biometrika.
2. (2002-03-08). "Introduction to Time Series and Forecasting". Taylor & Francis.
3. Davidson, James. (2000). "Econometric Theory". Blackwell.
4. "R: Box–Pierce and Ljung–Box Tests".
5. "Python: Ljung–Box Tests".
6. "Time series tests".
7. (1970). "Distribution of Residual Autocorrelations in Autoregressive-Integrated Moving Average Time Series Models". [[Journal of the American Statistical Association]].