Ljung-Box test

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.

Formal definition

The Ljung-Box test can be defined as follows.:H₀: The data are random.:H_a: The data are not random.

The test statistic is::$Q\; =\; nleft(n+2\; ight)sum\_\{j=1\}^hfrac\{hat\{\; ho\}^2\_j\}\{n-j\}$where "n" is the sample size, $hat\{\; ho\}\_j$ 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.

ee also

* the general term "Portmanteau test"
* the obsolete Box-Pierce test
* the Wald-Wolfowitz runs test

References

*cite article
title = On a Measure of a Lack of Fit in Time Series Models
author = G. M. Ljung
coauthors = G. E. P. Box
journal = Biometrika
year = 1978
volume = 65
pages=297-303
*cite book
title = Introduction to Time Series and Forecasting
edition = 2nd. Ed.
author= Peter Brockwell
coauthors=Richard Davis
publisher=Springer
year=2002
pages= 36