We study the integral of the Frobenius norm as a measure of the discrepancy between two multivariate spectral densities. Such a measure can be used to fit time series models, and ensures proximity between model and process at all frequencies of the spectral density - this is more demanding than Kullback-Leibler discrepancy, which is instead related to one-step ahead forecasting performance. We develop new asymptotic results for linear and quadratic functionals of the periodogram, and make two applications of the total Frobenius norm: (i) fitting time series models, and (ii) testing whether model residuals are white noise. Model fitting results are further specialized to the case of atomic structural time series models, wherein co-integration rank testing is formally developed. Both applications are studied through simulation studies, as well as illustrations on in ation and construction data.