Portmanteau test

In statistics, a portmanteau test tests whether any of a group of autocorrelations of a time series are different from zero. The term portmanteau test refers both to the Ljung-Box test and to the (now obsolete) Box-Pierce test. The portmanteau test is useful in working with ARIMA models.

The Ljung-Box test statistic is calculated as $$Q=T(T+2)\sum_{k=1}^s r_k^2/(T-k)$$

If the sample value of Q exceeds the critical value of a chi-square distribution with s degrees of freedom, then at least one value of r is statistically different from zero at the specified significance level. (The Null Hypothesis is that none of the autocorrelation coefficients up to lag s are different from zero.)

The Ljung-Box (1978) test is an improvement over the Box-Pierce (1970) test, whose statistic was

$$ Q = T \sum^s_{k=1} r^2_k $$

The problem with the Box-Pierce statistic was bad performance in small samples. The Ljung-Box statistic is better for all sample sizes including small ones.

T=number of observations

s=length of coefficients to test autocorrelation

$$r_k$$=autocorrelation coefficient (for lag k)

Q=Portmanteau Test statistic