Arima model and exponential smoothing method: A comparisonWan Kamarul Ariffin Wan Ahmad and Sabri Ahmad Citation: AIP Conf. Proc. 1522, 1312 (2013); doi: 10.1063/1.4801282 View online: http://dx.doi.org/10.1063/1.4801282 View Table of Contents: http://proceedings.aip.org/dbt/dbt.jsp?KEY=APCPCS&Volume=1522&Issue=1 Published by the AIP Publishing LLC. Additional information on AIP Conf. Proc.Journal Homepage: http://proceedings.aip.org/ Journal Information: http://proceedings.aip.org/about/about_the_proceedings Top downloads: http://proceedings.aip.org/dbt/most_downloaded.jsp?KEY=APCPCS Information for Authors: http://proceedings.aip.org/authors/information_for_authors

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Arima Model and Exponential Smoothing Method : A Comparison

Wan Kamarul Ariffin Wan Ahmad and Sabri Ahmad

Department of Mathematics, Faculty of Science and Technology, Universiti Malaysia Terengganu, 21300 Kuala Terengganu,

Terengganu DI, Malaysia

Abstract.This study shows the comparison between Autoregressive Moving Average (ARIMA) model and Exponential Smoothing Method in making a prediction. The comparison is focused on the ability of both methods in making the forecastswith the different number of data sources and the different length of forecasting period. For this purpose, the data from The Price of Crude Palm Oil (RM/tonne), Exchange Rates of Ringgit Malaysia (RM) in comparison to Great Britain Pound (GBP) and also The Price of SMR 20 Rubber Type (cents/kg) with three different time series are used in the comparison process. Then, forecasting accuracy of each model is measured by examinethe prediction error that producedby using Mean Squared Error (MSE), Mean Absolute Percentage Error (MAPE), and Mean Absolute deviation (MAD). The study shows that the ARIMA model can produce a better prediction for the long-term forecasting with limited data sources, butcannot produce a better prediction for time series with a narrow range of one point to another as in the time series for Exchange Rates. On the contrary, Exponential Smoothing Method can produce a better forecasting for Exchange Rates that has a narrow range of one point to another for its time series, while itcannot produce a better prediction for a longer forecasting period.

Keywords: Mixed Autoregressive Integrated Moving Average (ARIMA) Model, exponential Smoothing Method, forecast accuracy PACS: 02.50.-r

INTRODUCTION

Nowadays, many models or forecasting methods have been introduced by researchers for time series prediction data such as ARIMA model [12], Exponential Smoothing Methods [12], Grey Forecasting model [1], Bayesian Statistical model [13], and others . ARIMA model for example have been applied in many areas include primary energy demand [19], ecological footprint [11], electricity demand [3], and others. While the Exponential Smooting Method also have been used in many areas such that inventory control [14], wind velocity [2], fish production [18] and others. These two methods are among the fast growing methods and have shown a great contribution in the field of forecasting. Nevertheless, each method introduced has its own strengths and weaknesses. This study is focused on the comparison between the ARIMA Model and Exponential Smoothing Method in making a prediction. The comparison is focused on the ability of both methods in creating forecast models with the number of data sources and different long-term forecasting.

The paper [15] presented the study about the comparison between ARIMA model and Autoregressive Neural Network (ANN)to forecast the summer monsoon (June-August) in India and it showed that the ARNN model performed better than ARIMA model. Besides that, the ARIMA model and Vector ARMA modelalso compared with the Fuzzy Time Series Method (i.e. Two-Factor model, Heuristic model and Markov model) to forecast the amount of Taiwan export and this study showed that the ARIMA model generates smaller forecasting errors in longer experiment time period [6]. In this study, Exponential Smoothing Method is selected to compare with ARIMA model because they have performed surprisingly well in forecasting competitiond against more sophisticated approaches [16]. Besides that, although they are relatively simple but robust approaches to forecasting [4].

In order to observe a comparative study between the two models clearly, the time series data used to test the ability of both models in making prediction are The Price of Crude Palm Oil (RM/tonne) [10], Exchange Rates Malaysia Ringgit (MYR) against Great Britain Pound (GBP) [7], and The Price of SMR 20 Rubber Type (cents/kg) [8]. The time series data that were used in this study do not have the effect of seasonal variation where it focused on a component time series that have trend component and also irregular component.

Proceedings of the 20th National Symposium on Mathematical SciencesAIP Conf. Proc. 1522, 1312-1321 (2013); doi: 10.1063/1.4801282

© 2013 AIP Publishing LLC 978-0-7354-1150-0/$30.00

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The effectiveness of both forecasting models used are measured by comparing the forecast error generated by the two models. The smaller the prediction error, the better the forecast value produced by the model. Prediction error measure used to test the effectiveness of the ARIMA model and Exponential Smoothing Method in this study are Mean Squared Error (MSE) [12], Mean Absolute Percentage Error (MAPE) [12] and Mean Absolute Deviation (MAD) [9].

BOX-JENKINS METHODOLOGY

The Box-Jenkins approach is synonymous with the general ARIMA modelling. This approach was introduced by G. E. P. Box and G. M. Jenkins in 1976. ARIMA modelling is commonly applied to time-serie analysis, forecasting and control [12]. The term ARIMA is in short stands for the combination that comprises of Autoregresive/ Integrated/Moving Average Model [12]. This model can be used when the time series data is in stationary state and there is no missing data within the time series data. In the ARIMA analysis, anidentified underlying process is performed based on observations to a time series data in generating good model that shows the process-generating mechanism precisely [19]. According to [12], the basic model of the Box-Jenkins methodology are :

The Autoregressive (AR) Model

In the AR model, the current value of the variable is defined as a function of its previous values plus an error term. Mathematically, it is written as :

21 2(1 ... )p

p t tB B B y� � � � �� � � � � � (1)

The Moving Average (MA) Model

The Moving Average (MA) model links the current values of the time series to random error that have occured in the previous periods rather than the values of the actual series themselves. The Moving Average (MA) can be written as :

2

1 2(1 ... )qt q ty B B B� � � � �� � � � � �

(2)

The Mixed Autoregressive Moving Average (ARMA) Model

From the above, we can see that both the AR and the MA models. However, there are series that fit better by combining both models. Such model is known as the mixed ARMA model. Under the assumption of stationarity, the mixed Autoregressive (AR) and Moving Average (MA) model of the Box-Jenkins methodology is known as ARMA model. In other words the series ty is assumed stationary (no need for differencing) and the ARMA model is written as :

2 21 2 1 2(1 ... ) (1 ... )p q

p t q tB B B y B B B� � � � � � � �� � � � � � � � � �

(3)

The Mixed Autoregressive Integrated Moving Average (ARIMA) Model

When the stationarity assumption of the variable is not met, then the ARIMA model is formulated. In this formulation, it is necessary that the data series needs, firstly, to be differenced in order to achieve stationarity. The model, thus, obtained is represented in general term as ARIMA ( , , )p d q , where as stated earlier the symbol ‘ d ’ denotes the number of time the variable ty needs to be differenced in order to achieved stationarity.

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THE STAGES IN THE ARIMA MODEL DEVELOPMENT

The basis of the Box-Jenkins modelling approach consists of three main stages. There are model identification, model estimation and validation, and model application [12].

Model Identification

Firstly, we need to identify the class of model most suitable to be applied to the given data set. This is done by computing, analysing, and plotting various statisics based on statistical data. Common statistics used to identify the model type is autocorrelation (ACF) and the partial aurocorrelation (PACF) coefficients. However, model identification based on observing the ACF and PACF method can be the trickiest part of the application of the Box-Jenkins methodology caused by it is very difficult to exactly pinpoint the exact model type based on the ACF and PACF alone. So, in order to select the best fitted model, one needs to run several models and by applying certain statistical test procedures one should be able to determine the best fitted model [12]. In this study, SPSS 11.5software package is used to select the best fitted model.

Then, the simulated data should be ensured that they are stationary. A stationary time series has the property that is statistical characteristics such as the mean and the autocorrelation structure are constant over time. If the series is non-stationary, then it needs to be differenced to achieve stationary. Once a time series is identified as stationary, AR and MA parameters should be estimated according to its partial autocorrelation (PACF) and autocorrelation (ACF) function respectively. If the ACF function of a stationary series died off smoothly at a geometric rate and the PACF declined geometrically after one lag, then a first-order AR model is appropriate. Similarly, if the ACF function of a stationary series died off smoothly at a geometric rate after one lag and the PACF declined geometrically, a first-order MA process would seem appropriate [11].In Box and Jenkins [paper 1976], it also stated that the ARIMA model can be used when the time series is stationary and there is no missing data in the within the time series [19].

Besides that, the purpose of identification of an ARIMA ( , , )p d q model is to determine the value for p , d , and q where the p denotes the order of autoregressive (AR), d demotes the degree of differencing involved to achieve stationary in the series and q denotes the order of moving average (MA) [12].

Model Estimation and Validation

According to [12], the Box-Jenkins models are estimated based on sample statistic that must be tested to ensure their validity as estimates of the true population parameter values. In this case there are three components of the validation process. (1) Statistical validation or residual diagnostics. (2) Parameter validation. (3) Model validation. While three common statistical measure used when validating the ARIMA model are the AIC (Akaike’s Information Criterion), BIC (Bayesian Information Criterion) and the Box-Pierce Q Statistic.

Then, the estimated values need to be validated. The Box-Jenkins framework assumes that the residuals (error terms) are not correlated with each other, i.e. it is assumed that there is no systematic pattern in the residuals. In short, the residuals are independent to one another. When there is systematic pattern in the behaviour of the residuals, then the model is said to be mis-specified. Mis-specification is a symptom that indicates that important parameters may have been omitted or alternatively unimportant parameter(s) is included in the model. A procedure to check mis-specification is to check for the presence of correlation among the residuals, carried out by calculating the chi-squared value of the terms. One such test is the Ljung-Box statistic, given as:

2

1( 2)

hk

k

rQ T T

n k�

� ��

(5)

which is approximately distributed as a chi-square distribution with ( )h p q� � degrees of freedom. In this equation, T is the number of observations in the time series, h is the maximum lags being tested, p is the number of AR terms, q is the number of MA terms, n is the sample size, and kr is the sample autocorrelation of the residual terms [12].

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The statistics indicate that, if each kr is close to zero, then Q will be relatively small, and likewise for large kr ’s then Q will also be large. If the residuals are white noise the Q statistics is distributed as 2 with ( )h p q� � degrees of freedom. In other words if the calculated chi-squared value is larger than the tabulated 2 for ( )h p q� � degrees of freedom then reject 0H (which states that the residuals are white noise) and accept 1H (which states that the residuals are not white noise). By accepting the 1H the model is therefore, considered mis-specified or inadequate. Likewise, if the Q statistic is less than the tabulated 2 with ( )h p q� � degrees of freedom, accept the

0H (the model is adequate). When the 0H is rejected (the model is mis-specified), then the best course of action is to try other model variations [12].

Model Application

If all test criteria are met and that the model’s fitness has been confirmed, it is the ready to be used to generate the forecasts value. The forecast values may be in terms of single-valued items or in terms of confidence intervals. The confidence interval estimates provide the probabilistic measures of certainty and uncertainty associated with the forecast value [12]. In this study, the forecast values was taken and further calculations are done based on these data to be compared with the forecast values generated by Exponential Smoothing Method.

EXPONENTIAL SMOOTHING METHOD

The Exponential Smoothing Methods are relatively simple but robust approaches of forecasting [4]. Conceptually, the application of the exponential smoothing technique involves the building of and re-estimating new models as new information/data point is obtained. This means that, as new data point is obtained, it is incorporated into the existing data series. Now by using the expanded series, a new model is then re-estimated and a new forecast value is subsequently generated [12].

According to [12], exponential smoothing method has five types. There are :

Single Exponential Smoothing

It is the simplest form of model within the family of the exponential smoothing techniques. The model requires only one parameter, that is the smoothing constant, � , to generate the fitted values and hence forecast. The advantage of this procedure over moving average is that it takes into account the most recent forecast. In single exponential smoothing models, the forecast for the next and all subsequent periods are determined by adjusting the current period forecast by portion of the difference between the current forecast and the current actual value. This is described in terms of minimum errors/ residuals. Hence, if the recent forecast proves to be accurate, then it seems reasonable to base the subsequent forecasts on these estimates. Likewise, if recent predictions have been subjected to large eorrors, then new forecasts will also take into consideration.

The equation for single exponential smoothed statistics is given as,

(1 )t m t tF y F� �� � � � (6)

where t mF � is the single exponentially smoothed value in period t m� for 1,2,3,4,...m � , ty is the actual value in time period t , � is the unknown smoothing constant to be determined with value lying between 0 and 1 (0 1)�� � , selected by forecaster or alternatively determined by the data, and tF is the forecast for period t .

Double Exponential Smoothing

This technique is also known as Brown’s method. It is useful for series that exhibits a linear trend characteristics. The main advantage of the double exponential smoothing method over single exponential is its ability to generate multiple-ahead-forecasts. In other words, the equation are able to generate the one, two, three, and so forth ahead-forecast values. To demonstrate the method the following notations will be used:

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Let, tS be the exponentially smoothed value of ty at time t

'tS be the double exponentially smoothed value of ty at time t

Generally, there are four main equations involved. Equation (7) computes the single exponentially smoothed value,

1(1 )t t tS y S� � �� � � (7)

Equation (8) computes the double exponentially smoothed value,

' '1(1 )t t tS y S� � �� � � (8)

Equation (9) computes the difference between the exponentially smoothed value,

'2t t ta S S� � (9)

Equation (10) computes the adjustment factor,

'( )1t t tb S S�

�� �

� (10)

The main difficulty encountered when using this method is the determination of the size of � . The crtiterion to

choose the best alpha (� ) value is based on the lowest MSE value produced. Forecast for m-step-ahead are computed using the equation,

t m t tF a b m� � � (11)

where t mF � is the forecast at period m made in period t , for 1,2,3,4,...m �

Holt’s Method

This method is to overcome the problem of Brown’s Method because it has only one smoothing constant used and as such the estimated linear trend values obtained are sensitive to random influences. The Holt’s method has two-parametersto handle data with linear trend. This technique not only smoothes the trend and the slope directly by using different smoothing constants but also provides more flexibility in selecting the rates at which trend and slopes are tracked.

The application of the Holt’s method requires three equations: Equation (12) is the equation of exponentially smoothed series,

1 1(1 )( )t t t tS y S T� � � �� � � � (12) Equation (13) is the equation of trend estimate,

1 1( ) (1 )t t t tT S S T � �� � � � (13) Equation (14) is the eqution of m period into the future,

t m t tF S T m� � � � (14)

The � and are the parameters to be determined with values range from 0 to 1.

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Adaptive Response Rate Exponential Smoothing (ARRES)

This method is an attempt to overcome the problem of Single Exponential Smoothing and Double Exponential Smoothing techniques where the value of alpha (� ) is assumed constant for all time period without considering thattime events may take place which will affect the subsequent data behaviour. This technique incorporate the effect of the changing pattern of the data series into the model.

The ARRES technique comprises of the following basic equations,

1 (1 )t t t t tF y F� �� � � � (15)

Basically, this equation has similar meaning to the single exponential smoothing equation, except for the parameter value alpha (� ) which is identified by the subcript t . This indicates that the value of the alpha is only appropriate at a particular period t , and may be different at different value of t . The values of t� are estimated using the following equations,

tt

t

EAE

� �

(16)

where,

1(1 ) ; 0 1t t tE e E �� � � � � (17)

1(1 )t t tAE e AE �� � �

(18) and that,

t t te y F� � The value of tE is defined as the smoothed average error, and tAE is the smoothed absolute error. In the

application of this method, one does not require to find the best alpha, � . This is because there is no single best �value which happens to vary over time. For this reason the appropriate symbol used is t� .

Holt-Winter’s Trend and Seasonality

Holt-Winter’s Trend and Seasonality is one such technique that takes into account the trend and seasonality factors. This method consists of three basic equations that define the level component, the trend component, and the seasonality component. Two assumptions can be made with regard to the relationship of these components are multiplicative effect and additive effect.

Multiplicative Effect Assumption

The multiplicative in nature assumption pertaining to the components of the models can be made. The equations are represented as follows,

Level component :

1 1(1 )( )tt t t

t s

yL L b

S� � � �

�

� � � �

(18)

Trend component :

1 1( ) (1 )t t t tb L L b � �� � � � (19)

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Seasonality component :

(1 )tt t s

t

yS S

L� � �� � �

(20)

The m-step-ahead forecast is calculated as,

( )t m t t t s mF L b m S� � �� � � (21) where ty is the actual values which include seasonality, tL is the level component of the series which comprising

of the smoothed values but does not include the seasonality component, tb is the estimate of the trend component,

tS is the estimate of the seasonality component, S is the length of seasonality, � is the smoothing constant for level (0 1)�� � , is the smoothing constant for the trend estimate (0 1) � � , � is the smoothing constant for seasonality estimate (0 1)�� � , m is the number of step-ahead to be forecast, and t mF � is forecast for m-step-ahead.

Additive Effect Assumption

The additive effect assumption pertaining to the components of the models can also be made. The basic equation are,

Level Component :

1 1( ) (1 )( )t t t s t tL y S L b� �� � �� � � � � (22) Trend Component :

� �1 1(1 )t t t tb L L b � �� � � �

(23) Seasonality component :

� � � �1t t t t sS y L S� � �� � � �

(24) The m-step-ahead forecast is calculated as,

t m t t t s mF L b m S� � �� � � (25) As can be seen, the trend component is the same to that of the multiplicative effect assumption. For the rest of

the equations, absolute change is calculated rather than the relative change. The initial values for the level and trend components are similarly calculated except for the seasonal effects which are calculated as

1 1 1 2 2 2, ,..., s s sS y L S y L S y L� � � � � � .

THE STEPS OF EXPONENTIAL SMOOTHING METHOD

In this study, the SPSS 11.5 software package is used to apply the Exponential Smoothing Method. There are three steps in order to use this method :

The Identification of Component in the Time Series

In this step, the type of components which exist in the time series are identified. There are four types of component which are trend component, cyclical component, seasonal component and irregular component [12]. This is because in order to select the type of Exponential Smoothing Method, it depends on the type of component that exist in the time series. The time series data used in the study have two type of components which are trend component and irregular component.

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The Determination of Parameter Values

After that, the parameter values are determined. We can choose the values of parameter by grid search or specify particular values for the parameters. In this study, grid search is used to find the values of parameter and these values are searched by the software package. Then, the software package will display only 10 best models for grid search. When this is selected, the parameter value (s) and sum of squared erorrs (SSE) are displayed only for the 10 parameter combinations with the lowest SSE., regardless of the number of parameter combinations tested [17].

The Application of the Model

Once the parameter value(s) is determined, the process of forecasting can be run. It will be display the forecast values depend the period as it set.

EVALUATING THE MODELS

In order to measure the accuracy of both models – ARIMA model and Exponential Smoothing Method in making the forecasts, they can be determined by using the test accuracy of forecasting models. The accuracy of forecasting model is determined from size of forecasting error. The best model produces the lowest size of forecasting error. The criteria chosen to select the best model in making the forecasts are mean square error (MSE), mean absolute percentage error (MAPE) and mean absolute deviation (MAD).

2

1( )

MSE =

T

t tt

y y

T�

� 2)

(26)

0

( )100MAPE = T

t t

t t

y yT y�

� )

(27)

1

1MAD = T

t tt

y yT �

� ty

(28)

where t is time period, T is total number of observations, ty is the actual value and tyty is forecasted value at time t .

RESULT AND DISCUSSION

In general, the result of the forecasting accuracy which determine the best model is simplified in the following table.

TABLE (1). The results of Crude Palm Oil's Price (RM) forecasting accuracy No The Forecasting of Price of Crude Palm Oil (RM) The Best Model 1 Upcoming 12 months forecasting by 60 sets of data ARIMA Model 2 Upcoming 18 months forecasting by 60 sets of data Exponential Smoothing Method 3 Upcoming 24 months forecasting by 60 sets of data ARIMA Model 4 Upcoming 12 months forecasting by 48 sets of data (4 years) ARIMA Model 5 Upcoming 12 months forecasting by 60 sets of data (5 years) ARIMA Model6 Upcoming 12 months forecasting by 72 sets of data (6 years) ARIMA Model

The forecasting of Crude Palm Oil’s price (RM) in Table (1), it was found that the ARIMA model produces

more accurate forecast than the Exponential Smoothing Method for all except the 18-month-ahead forecasts with 60 observed data. This shows that the ARIMA model is more accurate in generating forecast value for this time series data.

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TABLE (2). The results of RM against the GBP’s exchange rates forecasting accuracy No The Forecasting of RM against the GBP’s Exchange Rates The Best Model 1 Upcoming 12 months forecasting by 60 sets of data Exponential Smoothing Method 2 Upcoming 18 months forecasting by 60 sets of data ARIMA Model 3 Upcoming 24 months forecasting by 60 sets of data Exponential Smoothing Method 4 Upcoming 12 months forecasting by 84 sets of data (7 years) ARIMA Model 5 Upcoming 12 months forecasting by 120 sets of data (10 years) Exponential Smoothing Method 6 Upcoming 12 months forecasting by 156 sets of data (13 years) Exponential Smoothing Method

While the forecasting of Exchange Rates Ringgit Malaysia (RM) against the currency of Great Britain Pound

(GBP) as in Table (2), it was found that the Exponential Smoothing Method produces more accurate for all forecasting values except for the 18-month-ahead forecasts with 60 observed data and the 12-month-ahead forecasts with 84 observed data (7 years). This shows that the Exponential Smoothing Method is more accurate for this time series data.

TABLE (3). The results of Latex SMR 20’s Price forecasting accuracy No The Forecasting Latex SMR 20’s Price The Best Model 1 Upcoming 12 months forecasting by 60 sets of data Exponential Smoothing Method 2 Upcoming 18 months forecasting by 60 sets of data ARIMA Model 3 Upcoming 24 months forecasting by 60 sets of data ARIMA Model 4 Upcoming 12 months forecasting by 84 sets of data (7 years) Exponential Smoothing Method 5 Upcoming 12 months forecasting by 108 sets of data (9 years) Exponential Smoothing Method 6 Upcoming 12 months forecasting by 132 sets of data (11 years) ARIMA Model

For the forecasting of Latex SMR 20’s price as in Table (3), it can be seen that ARIMA model produces more

accurate for the 18-month and 24-month-ahead forecasts with 60 observed data. Then, ARIMA model also produces more accurate for the 12-month-ahead forecasts into the future with 132 observed data (11 years). While the Exponential Smoothing Method produces more accurate prediction for the 12-month-ahead forecasts with 60 observed data. Exponential Smoothing Method also yield more accurate forecasting results for the 12-month-ahead forecasts into the future with 84 and 108 observed data. This shows that the ARIMA model can produce more accurate forecasting for the longer period ahead. In addition, ARIMA model can produce more accurate for the 12-month-ahead forecasts with more number of observed data for this time series.

From the results of this study, it can be observed that the ARIMA model produces more accurate prediction than Exponential Smoothing Method for forecasting a longer period with a limited number of data observations. It is shown that the ARIMA model is suitable for forecasting for longer forecasting period. While the Exponential Smoothing Method cannot produce an accurate prediction for the long term forecasting when the time series data sources are limited.

However, Exponential Smoothing Method is more accurate in the forecasting of Exchange Rates Ringgit Malaysia (RM) compared to Great Britain Pound (GBP) except the 18-month-ahead with 60 observed data and 12-month-ahead with 84 observed data. If we observe the pattern of this time series, its range is small between one point to another. This shows that Exponential Smoothing Method is suitable for forecasting time series data that has a narrow range.

It can be found that the Exponential Smoothing Method produces the same results for the forecasting of Crude Palm Oil’s price (RM) for 12-month-ahead with 60 and 72 observed data. This shows that the Exponential Smoothing Method can not produce a different prediction results for data observations that have small differences. This implies weakness of this method in making a prediction.

CONCLUSIONS

This study have shown that Autoregressive Moving Average (ARIMA) model can produce a better forecast for the long period. However, this model cannot produce a good prediction for time series that has a narrow range between one point to another.

In contrast, the Exponential Smoothing Method can produce better forecasting time series with a narrow range between one point to another. However, this method cannot produce a better prediction for a long period of time data series.

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In conclusion, both Autoregressive Moving Average (ARIMA) model and Exponential Smoothing Method have its own advantages and disadvantages. However, it can be used according to the type of time series data. It is hoped that this study will give you some meaningful contribution in forecasting disciplines and serve as a reference to researchers who conducted the study of forecasting.

ACKNOWLEDGEMENTS

The author would like to thank anonymous referees for their helpful comments.

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