Ljung-Box test

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In statistics, there are a large number of tests of randomness. The Ljung-Box test 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.

[edit] Formal definition

The Ljung-Box test can be defined as follows.

H0: The data are random.
Ha: The data are not random.

The test statistic is:

Q = n\left(n+2\right)\sum_{j=1}^h\frac{\hat{\rho}^2_j}{n-j}

where n is the sample size, \hat{\rho}_j is the sample autocorrelation at lag j, and h is the number of lags being tested. For significance level α, the critical region for rejection of the hypothesis of randomness is rejected if

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

where \chi_{1-\alpha,h}^2 is the α-quantile of the chi-square 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.

[edit] See also

[edit] References

  • G. M. Ljung; G. E. P. Box. "On a Measure of a Lack of Fit in Time Series Models", pp. 297-303. 
  • Peter Brockwell; Richard Davis (2002). Introduction to Time Series and Forecasting, 2nd. Ed., Springer, 36. 

This article incorporates text from a public domain publication of the National Institute of Standards and Technology, a U.S. government agency.Portmanteau test