 Open Access
 Authors : Zul Amry
 Paper ID : IJERTV10IS030100
 Volume & Issue : Volume 10, Issue 03 (March 2021)
 Published (First Online): 19032021
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Time Series Modeling and Forecasting for Indonesian Coffee Export
Zul Amry
Department of Mathematics, State University of Medan, Indonesia
=1
=1
AbstractThis paper to build a time series forecasting model for data of Indonesian coffee export from 1976 until 2019. The method used in this research is the Box Jenkins method. The autocorre
tor as the value for the parameter that maximizes the likelihood function. If = (1, 2, , ) represents a random sample from (; ),then the likelihood function is:
lation function (ACF) and the partial autocorrelation function (PACF) are used for stationary test and model idenfication. Ljunc Box Q statistics are used for diagnostic test, whereas to show the
() =
(; )
(2)
accuracy model are used Root Mean Squared Error (RMSE), Mean Absolute Error (MAE) and Mean Absolute Percentage Error (MAPE).
KeywordsARIMA model, coffee export, forecasting, Box Jenkins method.
The maximum likelihood estimator, that is , is a value of
that satisfies:
(1, 2, , ; ) = (1, 2, , ; ) (3)
k
k

INTRODUCTION The autocorrelation between Yt and Yt+k is defined as
Indonesia is one of the largest coffee exporting countries
Cov(Yt ,Yt k )
(4)
in the world and the demand for coffee exports to increase every year. In order for this demand to be fulfilled, it is
Var(Yt ) Var(Yt k )
necessary to forecast the demand coffee in the future. One of the tools for forecasting is the time series model forecasting.
that can be estimated from sample data by
nk
Time series model forecasting is a type of forecasting that uses past observational data, investigates its behavior and is
k
k
k
Yt Y Yt k Y
n
n
t 1
(5)
extrapolated into the future. The Autoregressive Integrated Moving Average (ARIMA) model developed by Box and Jenkins (1976) and has been widely used in various fields as a
0
Y
t
t
t 1
Y 2
statistical model, especially related to forecasting problems. In connection with the brief description above, this paper focuses on constructing a time series forecasting model with the ARIMA model to be applied to Indonesian coffee export

MATERIALS AND METHODS
and the set k , k 0,1, 2, is called the ACF.
as
as
The partial autocorrelation between Yt and Yt+k is defined
Cov Yt Y , Yt k Y k
t
t
t
t
The material used in this paper consists of coffee export
kk
Var Y Y Var Y Y
(6)
data and the theories of statistical related to forecasting time series models. Coffee export data is annual data of Indonesia
t t t k
t k
coffee export from 1975 to 2019. The method used is the Box Jenkins method. Some statistical theories in time series
where
Y k =
1Ytk1 2Ytk2 k 1Yt1
andkk
can be
t
t
analysis are ARIMA model, ACF, PACF, maximum
estimated from sample data by
likelihood method, LjungBox Q statistics, RMSE, MAE and MAPE.
1 1
2
k 2
1
The ARIMA (p, d, q) model of the time series
{1, 2, } is defined as
() = () (1)
1
k 1
kk 1
1
k 2
1
k 3
k 3
1
2
k
where is the backward shift operator,
= 1
, = 1
2
1
1
2
k 2
k 1
is the backward difference, = 1 1 2
, = 1 1 22
1
k 1
k 2
1
k 3
k 3
1
k 2
1
Principle of maximum likelihood yields a choice of the estima
(7)
kk
kk
and the set , k 0,1, 2, is called the PACF.
The ACF and PACF use to identify the models. The following Table 1 summarizes how to identify the model of the stationary data by using of characteristics for the ACF and PACF.
Table 1: Characteristics for the ACF and PACF
Model
ACF, k
PACF, kk
AR(p)
Damped exponential and / or sine functions
kk =0 for k>p
MA(q)
k = 0 for k >q
Dominated by damped exponential and/or sine function
ARMA(p,q)
Damped exponential and/or sine functions after lag (qp)
Dominated by damped exponential and/or sine function after lag (pq)
Diagnostics checking aims to conclude whether the forecasting model which obtained is adequate. the way is to test the assumption of residual independence between lags. If the residual is whit noise, then the model is adequate. The hypothesis is 0: 1 = = = 0 vs 1: , 0 and tested with LjungBox Q Statistic
Figure 1: Plot of x
Figure 2: Plot ACF of x
To overcome this condition, it is transformed to ; is the first differences of , that is = 1. The graph of the
data in Figure 3 below and the ACF plot in Figure 4 with a
= ( 2)
2
~2( ) (8)
muffled sine wave shape, indicates that the time series data is
=1 ()
where n is the sample size, 2 is the autocorrelation of residuals at lag k and K is the number of lags being tested, and
stationary.
1
1
reject 0 at the level , if > 2
( ).
The measures to determine the accuracy of a forecasting model in this research is RMSE, MAE and MAPE defined respectively as follows:
RMSE
n
ESS
n
(9)
Figure 3: Plot of y
t
t
Yt Y
MAE t 1
n
(10)
n
t 1
Yt Yt
Yt
n
t 1
Yt Yt
Yt
MAPE (11)
n
where =The actual value at time ; = The forecast value at time ; n =The number of observations and ESS=the error sum of square.

CONTRUCTION OF FORECASTING MODEL The graph of the original data in Figure 1 shows an
increasing trend and the ACF plot in Figure 2 shows a slow decline, this indicates that the time series data is not stationary.
Figure 4: Plot ACF o y
Figure 5: Plot PACF of y
Furthermore, the construction of forecasting model consists of identification of model, estimation of parameter, diagnostic test, and accuracy of model. Based the ACF plot in Figure 4 and the PACF plot in Figure 5, they are interrupted after lag 1, then the possible models for data are ARMA (1, 0), ARMA (0, 1) or ARMA (1,1) or ARIMA (1,1,0), ARMA
(0,1,1) or ARMA (1, 1, 1) for data. Results of estimation of parameters use likelihood maximum method presented in the Table 2 below:
Tabel 2: Estimation of parameter
Model
estimate value of parameters
1
ARIMA (1,1,0)
0.3433
–
ARMA (0,1,1)
–
0.4693
ARIMA (1,1,1)
0.0588
0.5131
Results of diagnostic test use LjuncBox Q statistics to ARIMA (1,1,0), ARMA (0,1,1) and ARMA (1, 1, 1) at the
level = 0.05 and degrees of freedom=15 with the value
0.95
0.95
2 (15) = 24.996 presented in the Figure 6, Figure 7,
Figure 8, and Table 3 below:
Figure 6: Diagnostic test of ARIMA (1,1,0) model
Figure 7: Diagnostic test of ARIMA (0,1,1) model
Figure 8: Diagnostic test of ARIMA (1,1,1) model
Tabel 3: Diagnostic test
Model
Qvalue
Decision for 0
ARIMA (1,1,0)
20.7242
No reject
ARMA (0,1,1)
15.5253
No reject
ARIMA (1,1,1)
15.7478
No reject
the results of diagnostic test in the Table 3 above concluded that all models were adequate.
Results the accuracy of the model use RMSE, MAE and MAPE presented in the Table 4 below:
Tabel 4: Accuracy; RMSE, MAE and MAPE
Model
Accuracy
RMSE
MAE
MAPE
ARIMA (1,1,0)
7.7129
6.2174
16.9425
ARMA (0,1,1)
7.3466
6.0200
15.8802
ARIMA (1,1,1)
7.3236
6.0147
15.7893
Next, based on the smallest values of RMSE, MAE and MAPE in the Table 4 above, it is concluded that the most suitable model is the ARIMA (1,1,1) model, that is
1() 1 = 1()
(1 1) (1 ) = (1 1)
(1 1) ( 1) = 1
1 1 + 11 = 1
1 11 + 12 = 11
1.0588 1 + 0.0588 2 = + 0.53111
= 1.0588 1 0.0588 2 + 0.53111 +

CONCLUSION
Time series data of Indonesian coffee export from 1976 until 2019 is not stationary, but stationary for one level difference data, so that the data analyzed is the difference of one level and the results are returned to the original data. Based on calculations and analysis of data it is concluded that the most suitable model is the ARIMA (1,1,1).
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