Journal of Current Researches on Business and Economics

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doi: 10.26579/jocrebe.76

Journal of Current Researches

on Business and Economics

(JoCReBE)

ISSN: 2547-9628

http://www.jocrebe.com

Investigation of Seasonal Movements in Time Series by a Variance

Analysis Method

Saadettin AYDIN1 Keywords Time series, seasonal movements, Variance analysis. Abstract

In this study, a test is examined through a variance analysis of seasonal movements, which is one of the elements of a statistical time series. Assumptions are given for this test and then the procedure is explained. In this assumption, the period of the series to be examined is considered to be certain. For example, if the series is based on a month, the period of the series is determined as 12. The effect of the month factor is determined by proportioning the variance consisting of the month factor to the variance consisting of the irregular movement, and the effect of the year factor is determined by proportioning the variance consisting of the year factor to the variance consisting of the irregular movement.

Article History Received 12 May, 2020 Accepted 27 Jun, 2020 1. Introduction

As it is known, a statistical time series consists of four components. These components are trend (general tendency), cyclical fluctuations, seasonal and irregular (accidental) movements. It is possible to examine each of these components separately. In our study, before examining the seasonal movements, we will examine the variance analysis test, which is one of the tests that determine whether there is any seasonal effect in these series.

Besides various climatic and social events, some tests can be done if we want to make sure that a statistical time series is definitely under the influence of the season. These tests can be applied to monthly series that are clear of seasonal effects. In a series that will be examined with the variance analysis method below, a test that reveals whether there is an effect of a month and year will be examined.

2. Assumption

This test is based on a variance analysis (Laloire, 1972; 113-115). We will accept that the period of the series to be examined is known. Essentially, the economic series are those that are issued annually. The period difference is uncommon. Because we will use the periods as m = 12 or m = 4 depending on whether they are of 12 months or 3 months.

1_{Corresponding Author. ORCID: 0000-0002-9559-0730. Dr., University of Health Sciences. Faculty}

Member of Gülhane Medical Faculty, saadettin.aydin@sbu.edu.tr

Year: 2020 Volume: 10 Issue: 2

Research Article/Araştırma Makalesi

For cited: Aydın, S. (2020). Investigation of Seasonal Movements in Time Series by a Variance Analysis

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126 Aydın, S. (2020). Investigation of Seasonal Movements in Time Series by a Variance Analysis Method

In addition, it will be considered that the economic time series to be examined has not a trend, or the trend cannot be revealed.

This description is important as it may also be possible to examine a non-seasonal series. By showing the effects of various months of the trend in the same year, this test will reveal the loss as if it is seasonal if there is a loss between these effects. In practice, it will also be taken into account that the series does not change when performing operations on the adapted trend deviations in the case that the serial trend is linear, or on the secondary differences in the case that a parabolic trend is low.

Considering that there is a two-factor analysis of variance, all Yt observations

arranged in the Buys-Ballot (Guitton, 1971; 296-297) Table will be separated from itself and the corresponding month and year effects.

Yt = tji

where j is an index indicating the year, and j = 1, 2, 3, …, n i is an index indicating the month, and i = 1, 2, 3, …, 12

To avoid complexity, operations are carried out over full years and in this case, observation of mn (m = 12) is determined.

The value of month i of any year j is expressed as follows. Yji = Sji + Rji (1)

Sji = ai + bj (2)

In the equation (1) and (2)

ai indicates the effect of the month,

bj indicates the effect of the year,

Rji indicates the irregular variables.

Rji's are independent from each other and exhibit a normal distribution

characteristic, the averages of which are zero. Rji ~N(0,5)

Two significant results are tested here.

 If the month results have any significance, the series is seasonal.

 If the results of the year have significance and if the operation is performed

on the real values series, the series shows the annual axis. If there is an operation on the difference of Zt = Yt - Yt - 1, the serial trend shows the

changes. On the other hand,

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Journal of Current Researches on Business and Economics, 2020, 10 (2), 125-130. 127

Wherein, the value indicates the general average on the observations mn

in the following formula:

is the average of the same months of various years. in the following equation:

is the annual average of the sum average of the monthly values of any year.

3. Procedure

Yij values are influenced by the factors j and i, and the total variance of these values

(with Wt indicating the total variance) can be found as follows:

In the equation (10);

indicates the variance consisting of the month factors,

indicates the variance consisting of the year factors,

indicates the variance consisting of the irregular factors,

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Let the variance estimates be VA, VY and VR, respectively, the sum of squares of

which are WA, WY and WR. In that case, the following ratios are tested.

If these rates are zero, it follows a Fisher-Snedecor distribution.

The month results or significant results with a fixed significance α will be expressed as follows.

The effect of the month:

The ratio of is compared to .

Wherein

If the resulting ratio is smaller than the value , then it means that the

seasonal factor has not an effect in the series, otherwise the seasonal factor has an effect.

The effect of the year:

The ratio of is compared to .

Wherein .

If the resulting ratio is smaller than the value , then it means that the

seasonal factor has not an effect in the series, otherwise the seasonal factor has an effect.

Table 1. Description and Expression of the variance estimates

Sum of squares Degree of

freedom Description of the variance estimates Expression of the variance estimates Variance consisting of the month factor Variance consisting of the year factor Residual variance

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Journal of Current Researches on Business and Economics, 2020, 10 (2), 125-130. 129

Table 2. Operation table for the test using the variance analysis

4. Result

As can be seen from the descriptions in the study, it is understood by the test that we criticize whether there is any seasonal effect in a seasonal movement examined. The variances consisting of the month and year factors are proportioned to the residual variance, and the effect of the month and year factor is investigated. Thus, it is determined whether there is a seasonal effect in the series to be obtained.

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References

Guitton, H. (1971). Statistique. Dalloz, Paris

Kaufman, H. & Graboillot, J. L. (1975). Les technique de la prévision a court terme, Dunod, Paris

Laloire, J. C. (1972). Methodes du Traitement des Chroniques. Dunod, Paris

Zhang, G.P. (2007). A neural network ensemble method with jittered training data for time series forecasting. Information Sciences, 177 pages: 5329–5346. Zhang, G.P. (2003). Time series forecasting using a hybrid ARIMA and neural

network model. Neurocomputing, 50 pages: 159–175.

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