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Two gaps in the optimality theory supporting current ARIMA model-based seasonal adjustment are addressed. The main one concerns the requirement that decomposition components be uncorrelated with one another after they are minimally differenced to stationarity. Uncorrelatedness has been assumed but not formally verified. We verify it by introducing a model compatibility criterion fitting current practice that specifies how the ARIMA models of the seasonal decomposition components are to be compatible with the ARIMA model of the observed seasonal series. This criterion always supports the assumption of uncorrelated components for a stationary decomposition involving the differenced observed series and the similarly differenced and therefore overdifferenced stationary component series. We verify the requirement by proving that overdifferencing can be corrected for and that doing this preserves uncorrelatedness. Then we investigate whether correlated components are also allowed by the compatibility criterion and give a complete description of the allowed correlation structures for two-component decompositions. We also discuss their impracticality.
David F. Findley. (2012). Uncorrelatedness and Other Correlation Options for Differenced Seasonal Decomposition Components of ARIMA Model Decompositions. Center for Statistical Research & Methodology Research Report Series (Statistics #2012-06). U.S. Census Bureau. Available online at <http://www.census.gov/srd/papers/pdf/rrs2012-06.pdf>.
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