Diagnostic Checks in Multiple Time Series Modelling

The multivariate relation between sample covariance matrices of errors and their residuals is an important tool in goodness-of-fit methods. This paper generalizes a widely used relation between sample covariance matrices of errors and their residuals proposed by Hosking (J Am Stat Assoc 75(371): 602-608, 1980 [6]). Consequently, the asymptotic distribution of the residual correlation matrices is introduced. As an extension of Box and Pierce (J Am Stat Assoc 65(332): 1509-1526, 1970 [11]), the asymptotic distribution recommends a graphical diagnostic method to select a proper VARMA(p, q) model. Several examples and simulations illustrate the findings.