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The seasonal autoregressive moving average  SARMA  models have been widely adopted for modeling many time series encountered in economic, hydrology, meteorological, and environmental studies which exhibited strong seasonal behavior with a period  s . If the model is adequate, the autocorrelations in the errors at the seasonal and the nonseasonal lags will be zero. Despite the popularity uses of the portmanteau tests for the  SARMA  models, the diagnostic checking at the seasonal lags    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$$1s,2s,3s,\ldots ,ms$$\end{document}    1  s  ,  2  s  ,  3  s  ,  …  ,  m  s, where  m  is the largest lag considered for autocorrelation and  s  is the seasonal period, has not yet received as much attention as it deserves. In this paper, we devise seasonal portmanteau test statistics to test whether the seasonal autocorrelations at multiple lags  s  of time series are different from zero. Simulation studies are performed to assess the performance of the asymptotic distribution results of the proposed statistics in finite samples. Results suggest to use the proposed tests as complementary to those classical tests found in literature. An illustrative application is given to demonstrate the usefulness of this test.

Portmanteau test statistics for seasonal serial correlation in time series models