Standard diagnostic procedures for assessing the goodness of fit of linear models include tests of the null hypothesis of no residual autocorrelation against the alternative of linear dependence. The literature proposes several portmanteau tests for residual autocorrelation in SARMA models, mainly focusing on two cases: short-term and single-seasonal autocorrelation. Since many time series exhibit multiple seasonal patterns, whose periodic components may interact with one another, diagnostic tests for residual autocorrelation in a multi-seasonal framework are needed. However, the literature still lacks portmanteau tests specifically designed for multiple seasonal autocorrelation. This paper addresses this gap by extending classical portmanteau tests to settings with multiple seasonalities. In addition, the proposed approach jointly tests for both short-term and multiple seasonal residual autocorrelation. Test statistics and their asymptotic distributions are defined. Then, Monte Carlo simulations are used to evaluate the performance of the tests and the consistency with the expected results is assessed using statistical tests. The results suggest that the proposed extension can serve as a useful goodness-of-fit diagnostic tool for the class of mSARIMA models. An application to road traffic data is also provided.

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