ABSTRACT

A seasonal time series with period s is a series which has a pattern that repeats every s time intervals. If a time series contains seasonal· effects with period s we expect observations separated by multiples of s to be similar. For such series, Buys-Ballot (1847) gave a schematic representation of the observations in a two-dimensional table with m rows and s columns, according to the period and season, including the totals and/ or averages. Such two-way tables that display within period pattern, that are similar from period to period are known as Buys-Ballot tables. The emphasis in time series analysis is on model building. Therefore, the ultimate objective of this study is to provide a number of models for analysis of seasonal time series data. Specifically, the study I. reviewed the existing models for time series analysis (including the traditional and ARIMA models) ii. developed a new modeling procedure called Buys-Ballot method for time series decomposition and 111. considered the ARIMA modeling procedure for analysis of purely seasonal series. Some empirical examples are also given to illustrate these methods. The Buys-Ballot estimates are developed for short period series in which the trend and cyclical component are jointly estimated and restricted to a case in which the trend is a ,xiv straight line. In developing the procedure the study gave two alternative models called the Chain Base Estimation (CBE) and Fixed Base Estimation (FBE) methods. Since it takes into consideration all the periodic averages, the FBE method is recommended when it leads to adequate fit in terms of randomness of the residuals. In tenns of forecasts the Buys-Ballot estimates are shown to have outperfonned the traditional methods. The model for the purely seasonal series considered is (]) with one seasonal AR coefficient and one seasonal MA coefficient at the first seasonal lag s. The study considered the stationarity and invertibility conditions for the parameters of the model and obtained expressions for the theoretical autocorrelation function (acf) and for the first few non-zero values of the theoretical partial autocorrelation function · (pacf). The study shows that the process satisfying Equation (D) is stationary if 1$1 < 1 and invertible if 191 < 1. The theoretical acf and pacf of the model are also shown to be nonzero only at multiples of the seasonal lag s. For s = l, the theoretical acf and pacf are those of the nonseasonal ARMA ( 1, 1 ). For s ~ 2, the· coefficients which appear at lags 1, 2, 3.... in ARMA (1,1) now occur at multiples of s (s, 2s, 3s, ...). Hence, for correct identification of the model in Equation (1), one needs to focus attention only at coefficients at multiples of s, with special characteristic d,, = p,. Thirty-six simulated examples are used to illustrate the behaviour of the acf for various combinations of the parameters of the model in Equation (1).