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Choosing the Appropriate Model in Time Series Data when Trend-Cycle Component is Linear




This Study discusses the choice of appropriate model in time series decomposition when trend-cycle component is linear. The method adopted in this study is Buys-Ballot procedure developed for choice of model, among other uses based on row, column and overall means and variances of the Buys-Ballot table. Empirical example is taken from monthly records of number of birth rate at Salvation hospital Owerri, Imo State, Nigeria over the period of ten years  The ultimate objective is this study is therefore, to choose  the appropriate model of the monthly number of birth rate over the period under investigation. This study is limited to the time series when trend-cycle component is linear. Result indicates that, the appropriate model is additive.


1.1 Background of Study

Decomposition method involves the separation of an observed time series into components consisting of trend (long term direction), the seasonal (systematic, calendar related movements), cyclical (long term oscillations or swings about the trend) and irregular (unsystematic, short term fluctuations) components. Time series analysis includes the examination of trend, seasonality, cycles, turning points, changes in level, trend and scale and so on that may influence the series. This is an important preliminary to modelling, when it has to be decided whether and how to seasonally adjust, to transform, and to deal with outliers and whether to fit a model. In the examination of trend, seasonality and cycles, a time series is often described as having trends, seasonal effects, cyclic pattern and irregular or random component. The classical decomposition procedure is sometime called procedure of decomposing time series. Its applications is usually predicated on time series models. As stated in the literature, classical decomposition procedure has attracted so much research attention. The objectives of the classical decomposition method have been mentioned in several studies. The advantages of classical decomposition method are; it is used to investigate the presence of trend, seasonal, cyclical and error components in time series analysis. The time series models are;

Additive Model:                                                                               (1)

Multiplicative Model:                                                                        (2)

Mixed Model:                                                                                    (3)

For short term period in which cyclical and trend components are jointly combined Chatfield (2004) and the observed time series  can be decomposed into the trend-cycle component , seasonal component  and the irregular component .  Therefore, the decomposition models are

Additive Model:


Multiplicative Model:


and Mixed Model

.                                                                                             (6)

An important aspect of descriptive time series analysis is the choice of model in time series decomposition. As the literature indicates, choice of model in descriptive time series has attracted so much research attention. Different approaches to determine choice of model like the use of sequence plot (time plot) as well as other techniques have continue to evolve. Among them are the use of the coefficient of variation of seasonal differences (CV) and seasonal quotient by Puerto and Rivera (2001). The test for constant variances by Iwueze and Nwogu (2014). Proposed Chi-Square test by Nwogu, et al, (2019) and Dozie, et al, (2020) etc. proposed a test for choice of model based on Chi-Square distribution. Although time series data does not satisfy all the assumptions of most common statistical test, the Chi-Square test appears to be the most efficient among them. The proposed test is able to distinguish between the mixed and multiplicative models with a high degree of confidence.

For a series arranged in Buys-Ballot table for time series decomposition, Dozie and Nwanya (2020) compared the periodic, seasonal and overall means for mixed and multiplicative models in time series. In their summary, they showed that the expected values of linear trend cycle and seasonal components are the same for both mixed and multiplicative models.

To choose between additive and multiplicative models. Chatfield (2004) observed that, if the seasonal variation is independent of the absolute level of the time series, but it takes approximately the same magnitude each year then the appropriate model is additive and for the multiplicative model, the seasonal changes increase with  the overall trend. Linde (2005) stated the difference between additive and multiplicative models. In his opinion, if the seasonal variation is independent of the absolute level of the time series, but it takes appropriately the same magnitude each year then the appropriate model is additive. For multiplicative model, the seasonal variation takes the same relative magnitude each year.  

 1.2 Statement of Problem

Almost all the literatures on choice of models for time series decomposition emphasized choice between additive and multiplicative models. Little or no literature is available on when to use the mixed model. There is tendency to wrongly apply the multiplicative model, when the actual model should be mixed model.

1.3   Rationale for Study

The rationale of the study is to choose the appropriate model in time series decomposition for a study series which may be used to reduce error rising from use of wrong model.

 1.4 Aim and Objectives

The aim of this study is to choose the appropriate model in time series data.

The specific objectives are:

  1. To estimate trend parameters
  2. To estimate seasonal indices

Limitations of Study

This work is limited to series when trend-cycle component is linear.

  Definition of Concepts

  1. Decomposition: It is an observed time series denoted by . A time series is decomposed when the objective is to remove the seasonal variations. When a time series is decomposed, it is then divided into four non-observed components. A certain relation is assumed between these four components. The decomposition is either of the additive model or multiplicative model or mixed model.
  2. Buys Ballot: A seasonal time series data, that is conventionally arrange into periods and seasons. By arranging a seasonal series of length n into m periods and s – seasons. With m representing number of periods/years while 5 are seasons/columns. This two-dimensional arrangement of a series is referred to as the buys Ballot table.
  3. Additive Model: The additive model assumes that the components are independent of one another. In the additive model the series is expressed as the sum of the component such as


The additive model is useful when the seasonal variation is relatively constant over time.

  1. Multiplicative Model: The multiplicative model assumes that the four components of the time series are due to different causes but they not necessarily independent and they can affect each other. It is expressed as


This model is useful when the seasonal variation increase over time.

  1. Mixed Model: The mixed model is the combinations of additive and multiplicative models. It is expressed as

The model is selected, if the series contains values close or equal to zero.

  1. Trend: This refers to the average long run direction of the time series. This phenomenon is normally observed in most of the series relating to economic and Business example an upward tendency is normally noticed in term series relating to population, production and sales of products, prices, incomes etc. While a downward tendency is observed in time series relating to death, epidemic etc.
  2. Seasonal Component: It is a fixed and known period, season component is the fluctuation in a time series whose pattern is repeated every year. It is similar to cyclical component whose complete cycle is one year such movements are due to recurring events, which take place yearly. Good example of a seasonal time series is monthly rainfall; similar pattern is repeated annually.
  3. Cyclical Component: This pattern is the long-term oscillation or savings about the trend. The cycle may or may not follow similar pattern after equal interval of time.
  4. Irregular component: This refers to the sporadic changes in time series due to choice events, such as strikes, flood etc.
  5. Constant Variance: Random error is typically assumed to be normally distribution with zero mean and a constant variance. There are various tests that may be performed on the residuals for testing if they have constant variance. It is usually sufficient to visually interpret a residual versus fitted values plot.

Outline of Study

The study is divided into five chapters, Chapter One treats Introduction including the Background of the study, Statement of problem, Rationale for study, Aim and Objectives, Limitation of study, Definition of concepts, Outline of study. Chapter Two presents a review of existing Literatures Chapter Three discusses the methodology including Buys-Ballot procedure for choice of models, when trend – cycle component is linear Chapter Four contains Results. Chapter Five contains Summary, Conclusion and Recommendation.

Pages:  51

Category: Project

Format:  Word & PDF               

Chapters: 1-5                                          

Source: Imsuinfo

Material contains Table of Content, Abstract and References.


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