Box-Pierce test

The Box-Pierce Q-statistics are given by:

$\text{BP}\left( k \right) = n\sum_{k = 1}^{K}\rho_{a,k}^{2}$, [1]

where:

$\rho_{a,k}^{2}$ is the autocorrelation coefficient at lag $k$ of the residuals ${\widehat{a}}_{t}$.

$n$ is the number of terms in differenced series;

$K$ is the maximum lag being considered, set in JDemetra+ to $24$ (monthly series) or $8$ (quarterly series).

If the residuals are random (which is the case for residuals from a well specified model), they will be distributed as $\chi_{(K - m)}^{2}$ degrees of freedom, where $m$ is the number of parameters in the model which has been fitted to the data.

The Ljung-Box and Box-Pierce tests sometimes fail to reject a poorly fitting model. Therefore, care should be taken not to accept a model on a basis of their results. For the description of autocorrelation concept see section Autocorrelation function and partial autocorrelation function.