The asymptotic distribution of the residual autocovariance matrices in the class of periodic vector autoregressive time series models with structured parameterization is derived. Diagnostic checking with portmanteau test statistics represents a useful application of the result. Under the assumption that the periodic white noise process of the periodic vector autoregressive time series model is composed of independent random variables, we demonstrate that the finite sample distributions of the Hosking-Li-McLeod portmanteau test statistics can be approximated by those of weighted sums of independent chi-square random variables. The quantiles of the asymptotic distribution can be computed using the Imhof algorithm or other exact methods. Thus, using the (single) chi-square distribution for these test statistics appears inadequate in general, although it is often recommended in practice for diagnostic methods of that kind. A simulation study provides empirical evidence.