International Journal of Statistics and Applications 2019, 9(5): 153-159 DOI: 10.5923/j.statistics.20190905.04 Choice between Mixed and Multiplicative Models in Time Series Decomposition Eleazar C. Nwogu 1 , Iheanyi S. Iwueze 1 , Kelechukwu C. N. Dozie 2,* , Hope I. Mbachu 2 1 Department of Statistics Federal University of Technology, Owerri, Imo State, Nigeria 2 Department of Statistics Imo State University, Owerri, Nigeria Abstract This study discusses the condition(s) under which the mixed model best describes the pattern in an observed time series data, while comparing it with those of the additive and multiplicative models. Existing studies have focused on how to choose between additive and multiplicative models, with little or no emphasis on the mixed model. The ultimate objective of this study is therefore, to propose a statistical test for choosing between mixed and multiplicative models when the trending curve is linear. in descriptive time series analysis. The method adopted in this study is the Buys-Ballot procedure developed for choice of model by [1]. Results show that the column/seasonal variance of the Buys-Ballot table is, for the mixed model, a constant multiple of the square of seasonal effect and for the multiplicative model, a quadratic (in j) function of the square of the seasonal effects. Therefore, test for the choice between mixed and multiplicative models has been proposed based on the column/seasonal variances of the Buys-Ballot table. have been used to illustrate the applicability of the proposed test, Using empirical examples, the proposed test statistic identified the mixed model correctly in 98 out of the 100 simulations. Keywords Choice of Model, Time Series Decomposition, Mixed Model, Multiplicative Model, Buys-Ballot Table 1. Introduction One of the greatest challenges identified in the use of descriptive method of time series analysis is choice of appropriate model for decomposition of any study data. That is, when to use any of the three models for analysis is uncertain. And it is clear that; use of wrong model will certainly lead to erroneous estimates of the components. The three models most commonly used for time series decomposition are the Additive Model: t t t t t X T S C e     (1) Multiplicative Model: t t t t t X T S C e     (2) and Mixed Model t t t t t X T S C e     (3) where for time t,   t X ,t 1, 2, ..., n  , is the observed time * Corresponding author: kcndozie@yahoo.com (Kelechukwu C. N. Dozie) Published online at http://journal.sapub.org/statistics Copyright © 2019 The Author(s). Published by Scientific & Academic Publishing This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/ series, t T is the trend, t S is the seasonal effect, t C is the cyclical and t e is the irregular component [2,3]. For short period time series data the cyclical component is superimposed into the trend and the observed time series   t X ,t 1, 2, ..., n  can be decomposed into the trend-cycle component   t M , seasonal component   t S and the irregular/residual component   t e , [3]. Therefore, the decomposition models are Additive Model: t t t t X M S e    (4) Multiplicative Model: t t t t X M S e    (5) and Mixed Model t t t t X M S e    . (6) It is always assumed that the seasonal effect, when it exists, has period s, that is, it repeats after s time periods. t s t S S , for all t   (7) For Equation (4), it is convenient to make the further assumption that the sum of the seasonal components over a