Decomposition with a Mixed Model When the Trend-Cycle Component Is Quadratic

Abstract

This study develops a rigorous procedure for the decomposition of a time series whose underlying structure follows a mixed model with a quadratic trend-cycle component. Conventional approaches, namely subtracting the sum of the estimated trend and seasonal components from the observed series or dividing the observed series by their product, often fail to produce an adequate decomposition under mixed model assumptions. To address this limitation, a method based on the statistical properties of the Buys--Ballot table is proposed. When the appropriate trend-cycle component is quadratic, estimates of the trend parameters and seasonal indices are derived from the row and column means and variances of the Buys--Ballot table. The results show that the proposed approach provides an adequate fit to the observed data and yields stable and interpretable seasonal indices. The method is therefore recommended for the decomposition of time series exhibiting quadratic trend-cycle behaviour under mixed models.