Sankhya: The Indian Journal of Statistics1997, Volume 59, Series B, Pt. 2, 142-155
ON MULTIPLICATIVE VERSUS NON-MULTIPLICATIVE SEASONAL MODELS
BRAJENDRA C. SUTRADHAR, Memorial University of Newfoundland
SUMMARY. In time series analysis, stationary periodic data are usually parametrized in terms of seasonal multiplicative models. There are, however, situations where a non-multiplicative model may be more appropriate for the periodic data. For example, an employment data collected over a long period of time involving several recessions and regular economic periods will generally follow a seasonal pattern but the seasonality will not be multiplicative by nature. Similar to this socio-economic time series, if it happens that a data set really follows a non-multiplicative seasonal model, then the search for a possible multiplicative model for the data will be fruitless. This raises a fundamental question for testing for the adequacy of a specific multiplicative seasonal model of suitable order against any non-multiplicative seasonal model with the same order, before searching for an alternative multiplicative model. This paper is mainly concerned about the tests for the multiplicative versus non-multiplicative seasonal models when it is known that these models are generated based on Gaussian noises. Two tests are developed. The construction of the first test is quite similar to the test for independence in a classical contingency table, and the test statistic has asymptotically chi-square distribution under the null hypothesis. The computation of the test statistic is quite simple, which makes it practically appealing. The second test is developed based on suitable score functions under the null hypothesis.This test, unlike the likelihood ratio test, does not require the specification of an alternative non-multiplicative seasonal model. The test is asymptotically locally optimal, and the test statistic has asymptotically $\chi^2$ distribution under the null hypothesis, with $m$ degrees of freedom, where $m$ is the number of independent restrictions over the parameters, specified under the null hypothesis.
AMS (1991) subject classification. 62G10, 62M10
Key words and phrases. Hypothesis testing, regular and seasonal autocovariance functions, Pearson type chi-square test, asymptotically locally optimal test, consistent estimates.
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