A new diagnostic method for VARMA(p,q) time series models is introduced. The procedure is based on a statistic that generalizes to a multivariate setting the properties of the usual univariate ARMA(p,q) residual correlations. A multiple version of the cumulative periodogram statistic is also suggested. Simulation studies and one real data application are presented.

A goodness-of-fit approach for multivariate VARMA(p, q) models is presented. The idea is to consider a stochastic process based on a modified residual correlation matrix sequence, that is shown to converge to the Brownian bridge. Standard criteria based on this new random function, as for instance the Kolmogorov–Smirnov and Cramér–von Mises statistics, will have then a pivotal null asymptotic distribution. The properties of these two methods are investigated by simulation. As compared with the traditional methods in this area, their size does not depend critically on the choice of any lag parameter value, and they have better power properties.

We study genuine savings as an indicator of long-term welfare for Sweden for the period 1850 to 2000. Sweden has developed long series of comprehensive ‘green’ national accounts for this entire period and is, therefore, interesting as a testing ground for the hypotheses linking green accounting and sustainability. We find support for the weakest of the hypotheses in the theoretical literature on weak sustainability and genuine savings, namely that genuine savings are correlated with future economic well-being. However, the stronger hypotheses in this literature are not supported: there is no one-to-one relationship between genuine savings and prosperity, there is no indication that the relationship becomes stronger for longer time horizons, or with more comprehensive savings measures. The findings suggest that genuine savings, at least as currently measured in national accounts and satellite accounts, may not be a good forward-looking indicator of future prosperity.