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).
ELEAZAR, C (2022). Some Contributions To The Study Of Seasonal Time Series Models. Repository.mouau.edu.ng: Retrieved Nov 01, 2024, from https://repository.mouau.edu.ng/work/view/some-contributions-to-the-study-of-seasonal-time-series-models-7-2
CHUKWUNENYE, ELEAZAR. "Some Contributions To The Study Of Seasonal Time Series Models" Repository.mouau.edu.ng. Repository.mouau.edu.ng, 05 Dec. 2022, https://repository.mouau.edu.ng/work/view/some-contributions-to-the-study-of-seasonal-time-series-models-7-2. Accessed 01 Nov. 2024.
CHUKWUNENYE, ELEAZAR. "Some Contributions To The Study Of Seasonal Time Series Models". Repository.mouau.edu.ng, Repository.mouau.edu.ng, 05 Dec. 2022. Web. 01 Nov. 2024. < https://repository.mouau.edu.ng/work/view/some-contributions-to-the-study-of-seasonal-time-series-models-7-2 >.
CHUKWUNENYE, ELEAZAR. "Some Contributions To The Study Of Seasonal Time Series Models" Repository.mouau.edu.ng (2022). Accessed 01 Nov. 2024. https://repository.mouau.edu.ng/work/view/some-contributions-to-the-study-of-seasonal-time-series-models-7-2