Bootstrapped Multivariate Spectral Test for Goodness‐of‐Fit of Weak Vector Autoregressive Models

We propose a Cramér‐von Mises type test for diagnostic checking of weak vector autoregressive models, in which the errors are assumed to be uncorrelated but not necessarily independent. The test statistic is constructed from the integrated squared distance between the sample periodogram of the residuals and a constant spectrum. Unlike time domain portmanteau tests, which rely on the first residual autocorrelations, the proposed spectral test is sensitive to correlations at all lags. We study the asymptotic behavior of the test statistic and show that, due to the dependent structure of the errors and the effect of parameter estimation, the limiting distribution is not asymptotically pivotal. To address this issue, we employ a blockwise random weighting bootstrap to approximate critical values and establish its asymptotic validity. Finite sample performance is assessed through extensive Monte Carlo simulations and illustrated with a real data application.