With the help of the knowledge about
the asymptotic distribution of the autocorrelations we can derive
a statistic to test the hypothesis of white noise. One can either
test the original series or the residuals of an ARMA(
)
process. The number of estimated parameters is in the first case
and in the second case
.
Under the null hypothesis it holds for every
Studies show that in small samples poorly approximates the
asymptotic distribution. This results from the fact that
is a biased estimator for
. The
bias is stronger for small
, and thus an asymptotically
equivalent statistic can be defined as