A goodness-of-fit process for ARMA(p,q)(p,q) models based on a modified residual autocorrelation sequence

The asymptotic distribution of the usual goodness-of-fit process for ARMA(p,q)(p,q) models based on the residual autocorrelations is obtained. This distribution depends on unknown parameters. This fact motivates the introduction of a modified goodness-of-fit process based on a transformation of the residual autocorrelations. This new process is shown to converge weakly to the Brownian bridge. Thus, functionals based on the modified process are adequate for goodness-of-fit purposes. The behavior of these functionals is analyzed and compared to that of the standard Ljung–Box and cumulative periodogram statistics. The new method is found to perform better in some simulation experiments at the cost of some increased numerical complexity.