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 fit 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) fitting time series models, and (ii) testing whether model residuals are white noise. Model fitting 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.

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