1.0 Introduction
1.1 Background
One of the aims of time series is description of a series. Description includes the examination of trend, seasonality, cycles, turning point, changes in levels and so on that may influence the series. This is an important preliminary to modeling, when it has to be determined whether and how to de-trend seasonally adjust, to transform, to deal with outliers, and whether to fit a model to the entire history or only part of it.
In the examination of trend, seasonality and cycles, a time series is often described as having trend, seasonal effect, cyclic patterns and the irregular or random components. Since emphasis in time series analysis is on model building, the additive model and multiplicative model are usually considered. Symbolically, the models are respectively written as
Additive Xt= Tt+St+Ct+et, t=1,2,…,n…………………….(1.1)
Multiplicative Xt= TtStCtet, t=1,2,…,n…………………….(1.2)
For time t, Xt denote the observed value of the series, Tt is the trend, St seasonal variation, Ct is the cyclical variation, and et is the irregular component of the series.
Trend or secular trend may be loosely defined as long-term changes in the mean or the general direction in which the graph of a time series appears to be going over a long interval of time. Trend may be upward (growth) or downward (decline) but its major characteristic is that it maintains a regular pattern for long period.
Seasonal variation refers to the regular periodic movements in time series associated whit the time of the year. Such movements are due to recurring event which take place annually.
Cyclical variation refers to long term oscillations or swings about the trend. Only long period set of data will show cyclical fluctuation of any appreciable magnitude.
Irregular or random variation refers to variation due to some sporadic movement that occurs at some time. It is usually what is left of a set of data after the systematic components (trend, seasonal and cyclical components) have been moved.
If short period of time is involved, the cyclical components are superimposed into the trend [Chatfield (2004), Kendal and Ord (1990)] and we obtain a trend-cycle component denoted by Mt. in this case, equations (1.1) and (1.2) may be written as:
Xt = Mt+St+et, t=1,2,…,n………………………………………(1.4)
Xt = MtStet , t=1,2,…n……………………………………………(1.5)
Using (1.4) or (1.5), we can decompose the series into its components parts. A summary of the traditional methods of time series decomposition will be given in section 1.2.
1.2 Traditional Method of Time Series Decomposition.
The methods available for analysis of seasonal time series data in the time domain include the descriptive method and fitting of probability models [see Box et-al (1994), Chatfield (2004)]. in the descriptive method, the traditional practice is to estimate and isolate the components existing in a study series. Usually, the curve fitting by least squares, which is adjudged the most objective method, is used to estimate the trend.
The de-trended series is then used to estimate the seasonal effects. Methods for estimating the seasonal effect include monthly or quarterly average, ratio-to-trend, ratio-to-moving average method and link-relative method. Using the de-trended and de-seasonalized series, the estimates of the cyclical component are obtained by calculating a moving average of appropriate order among other methods.
The whole process of (a) fitting a trend curve by some method and de-trending the series (b) using the de-trended series to estimate the seasonal indices involved in the traditional method are laborious. In order to address these problems, which are associated with traditional method (i.e. de-trending a series before computing the estimates of the seasonal effects), lwueze and Nwogu (2004) proposed the Buys-Ballot procedure for time series decomposition .The development of the Buys-Ballot estimation procedure was based on the table proposed by Buys-Ballot in(1847).
1.3 Statement of Problem
Observations have been made that people find it difficult to analyze time series data. The question is why?
The researcher is therefore motivated by this observed problem and decided to find out easiest procedure or method of analyzing time series data.
1.4 Aim and Objective
The main objective of this project work is to compare the Buys-Ballot procedure and least square estimation meyhod specifically the project will:
- Determine the slope for two methods.
- Determine the seasonal indices using the two methods.
- Compare the results in (i) and (ii) above.
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