Decomposition with Mixed Model when Trend-Cycle Component is Exponential

This paper discusses the decomposition of an observed time series data which admits the mixed model when trend-cycle component is exponential. Once the estimates of the trend-cycle and seasonal components, are obtained, estimates of the residuals or irregular component can be obtained from the observed series either by successive subtraction (for the Additive model) or by successive division (for the Multiplicative model) of the estimates of trend-cycle and seasonal components. However, for the mixed model, estimates of the residuals obtained either by successive subtraction from or by successive division of the observed series are contaminated by estimates of the estimates of trend-cycle and seasonal components. The purpose of this study is to show how estimates of residuals uncontaminated by the estimates of trend-cycle and seasonal components can be obtained. The Buys-Ballot procedure for time series decomposition was adopted in this study. The procedure is based on row, column and overall means and variances of the Buys-Ballot table. When trend-cycle component is exponential estimates of the trend-cycle component and seasonal indices are derived from these row, column and overall means of the Buys-Ballot table. Simulated series were used to illustrate results. Evaluation of the estimates of the residuals shows that the fitted models adequately describe the patterns in the simulated series. The proposed procedure has been recommended for decomposition of any observed series that admits the mixed model and exponential trend-cycle component.