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Hybrid ARIMA Modeling with Stochastic Volatility for Forecasting the Value of Non-Oil and Gas Exports in Indonesia Evatia Suryatin.
Mustika Hadijati.
Zulhan Widya Baskara AbstractAiExport activities consist of oil and gas exports and non-oil and gas exports.
Non-oil and gas exports are one of the sectors that provide the largest foreign exchange contribution to Indonesia, and the movement of non-oil and gas export values has an impact on economic growth.
Therefore, the purpose of this research is to create a model used to predict future non-oil and gas export values.
One mathematical model that can be to predict IndonesiaAos non-oil and gas export values is the combination of the ARIMA model and the stochastic volatility model, also known as Hybrid ARIMA with stochastic volatility.
The Hybrid ARIMA with stochastic volatility modeling has advantages in creating models for data with high volatility and is capable of combining linear patterned data and nonlinear patterned data.
In this study, the best ARIMA .
,1,.
model was obtained with a MAPE value From the residuals of the ARIMA .
,1,.
model, there were signs of heteroscedasticity, so the GARCH model with the best GARCH .
model was used.
In the GARCH .
model, it was found that there was an asymmetric influence, so the EGARCH and GJR-GARCH models were used.
The comparison of EGARCH and GJR-GARCH models was carried out to address the asymmetric residual data pattern.
Based on the research results, the best model used for prediction is the hybrid ARIMA .
,1,.
with EGARCH .
model, with a MAPE value Index TermsAiARIMA.
Hybrid ARIMA.
MAPE.
Non-oil gas exports,Stochastic Volatility I NTRODUCTION Economic growth is defined as a long-term increase in a countryAos capacity to provide various economic goods to its population .
It serves as evidence of a nationAos level of According to Bank Indonesia, 2018 .
, one of the activities that contribute to economic growth is exportation.
The expenditure of goods from a countryAos domestic circulation, which are sent abroad under government regulations in exchange for foreign currency, is referred to as exports .
Indonesia is among the countries impacted by the performance of export activities across various sectors, including non-oil and gas exports.
Non-oil and gas exports contribute significantly to IndonesiaAos foreign exchange earnings.
The cumulative value of IndonesiaAos exports decreased by 6.
94%, and non-oil and gas exports experienced a decline of 4.
82% .
This decline in non-oil and gas exports has consequences for economic performance and growth.
As a result, non-oil and gas export values are a crucial factor in IndonesiaAos economic growth, necessitating the modeling of these values.
Suryatin.
Hadijati.
Baskara are with the Universitas Mataram.
Mataram, 83125 Indonesia e-mail: zulhan.
wb@unram.
Manuscript received October 30, 2023.
accepted January 30, 2024.
Modeling IndonesiaAos non-oil and gas export values can be employed to make forecasts that will inform economic policy decisions in the country.
Forecasting is the process of estimating the magnitude or quantity of something in the future, based on past data, and is analyzed scientifically using statistical methods .
Time series analysis is a statistical method that can be used for forecasting, making it applicable to the modeling of IndonesiaAos non-oil and gas export values.
A time series is a collection of data observations from a fixed source that occur sequentially based on a time index t, with precise time intervals .
Time series analysis uniquely records economic behavior over time.
According to Wei .
, there are two classifications of time series based on their form or function: linear models and nonlinear models.
Autoregressive Integrated Moving Average (ARIMA) is used for linear data modeling.
Autoregressive Integrated Moving Average (ARIMA) is a suitable forecasting method, as it requires only the variable to be forecasted, and provides simple, accurate, and rapid results .
The ARIMA method combines Autoregressive (AR) and Moving Average (MA) models.
However, financial sector data, such as IndonesiaAos non-oil and gas export values, which exhibit random behavior, high volatility, and non-constant variance, indicate data variation .
onditional varianc.
resulting in heteroscedasticity .
As a result, the ARIMA model is insufficient for data exhibiting heteroscedasticity, necessitating the use of a hybrid model.
A hybrid model is an artificial intelligence-based method designed to address difficulties with data containing both linear and nonlinear models .
Several models can overcome the limitations of the ARIMA model and are capable of modeling the volatility in IndonesiaAos non-oil and gas export values, including Generalized Autoregressive Conditional Heteroscedasticity (GARCH).
Exponential Generalized Autoregressive Conditional Heteroscedasticity (EGARCH), and the Glosten.
Jaganathan, and Runkle Generalized Autoregressive Conditional Heteroscedasticity (GJR-GARCH) model.
The consideration of these three models for integration with the ARIMA model is based on the patterns generated from the ARIMA residuals when there is an asymmetric effect on the squared ARIMA residuals with GARCH residuals.
Therefore, this study will undertake Hybrid ARIMA Modeling with Stochastic Volatility for Forecasting Non-Oil and Gas Export Values.
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II.
M ETHODOLOGY
The data used in this study are the values of IndonesiaAos nonoil and gas exports, obtained from the official BPS website .
ttps://w.
, which consist of monthly data of IndonesiaAos non-oil and gas export values as variable (Yt ).
This study aims to develop a hybrid ARIMA model with stochastic volatility for forecasting IndonesiaAos non-oil and gas export The method used is time series analysis.
Autoregressive Integrated Moving Average (ARIMA) Autoregressive Integrated Moving Average (ARIMA) is one of the appropriate methods used for forecasting because it only requires the variable to be forecasted in a simple, accurate, and fast manner .
In the 1970s, the ARIMA model gained popularity for its ability to solve problems in various broad situations in the field of econometrics.
Model Autoregressive Integrated Moving Average (ARIMA) is a short-term forecasting model that has good forecasting accuracy.
The difference between the ARIMA method and the ARMA method lies in stationarity, where the ARMA model requires the data to be stationary and does not consider the presence of non-stationary However, the ARIMA method is capable of addressing non-stationary data issues through transformation or differencing processes.
Therefore, the equation for the ARIMA model .
, d, .
is as follows:
GARCH Model High volatility results in a non-constant moving variance.
In time series modeling, the assumption is that the variance should be constant .
To address this issue, .
introduced the Autoregressive Conditional Heteroscedasticity (ARCH) model.
The ARCH method can handle serial correlation and cases of heteroscedasticity.
For the ARCH model, heteroscedasticity occurs because there is high volatility in the time series data.
The ARCH model .
can be expressed in the form of two equations, as Et2 = 0 1 AtOe1 .
p AtOep Et2 = 0 Oc i AtOei Determining the ARCH model requires large values to obtain an appropriate model for time series data.
The use of the Generalized Autoregressive Condition-al Heteroscedasticity (GARCH) model is employed to address the issue of large values in determining the ARCH model .
The GARCH model is an extension of the ARCH model.
The GARCH equation can be expressed in the form of two equations, as Et2 = 0 1 AtOe1 .
p AtOep 1 E12 .
q EtOeq I p (B).
Oe B)d Yt = q (B) Where .
Oe B)d Et2 = 0 Oc i AtOei Oc j EtOei Observation data at time-t Parameter (AR) Parameter (MA) Differencing Operator of orde-d Orde AR Orde MA To validate the best ARIMA model, the Akaike Information Criteria (AIC) is utilized.
Model selection through the AIC method involves choosing the lowest AIC value, indicating the suitable model for a given estimated equation, enabling further measurement of forecasting accuracy using MAPE.
The goodness-of-fit testing of the obtained ARIMA model is based on whether there is evidence of heteroskedasticity.
This is done to ensure that the usage of the ARIMA model is suitable for predictions or whether itAos necessary to employ a combined model to reduce errors from the ARIMA model.
In time series data, the presence of heteroskedasticity can be determined by observing residual patterns and also by conducting the ARCH-LM test.
The ARCH-LM test can detect whether time series data exhibit signs of heteroskedasticity by examining the result values.
If there is evidence of heteroskedasticity, it can be proceeded by constructing a model of the residuals that indicate the presence of heteroskedasticity using the ARCH If the ARCH model is insufficient in addressing the heteroskedasticity issue in the time series data, the process is extended by utilizing the GARCH model.
Where Et2 At Variance at time-t Parameter ARCH Parameter GARCH Error at time-t The GARCH model can be used for data that exhibits the same volatility characteristics, or in other words, equal positive and negative shocks or spikes.
This means that it can capture asymmetry in financial data resulting from changes in volatility .
To test for the presence of asymmetry effects in the data, a Cross-correlation test or cross-correlation is employed by examining the correlation between the squared standard residuals from the ARIMA model and the residuals from the GARCH model.
If the correlation value is not equal to zero, it can be concluded that there is an asymmetry effect in the data .
Exponential GARCH Model Exponential GARCH or EGARCH is a modification of the GARCH model first introduced by Nelson .
The Exponential GARCH model, or EGARCH, is a modification of the GARCH model first introduced by Nelson .
The advantage of the EGARCH model is that it does not require the INTERNATIONAL JOURNAL OF COMPUTING SCIENCE AND APPLIED MATHEMATICS VOL.
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data to be stationary, unlike the ARCH.
GARCH, and GJRGARCH models which have stationarity requirements.
The equation for the EGARCH model is as follows:
AtOep AtOe1 AtOe1 ln Et2 = 0 1 p EtOe1 EtOe1 EtOep AtOeq q 1 ln(EtOe1 ) .
r ln(ErOe1 ) EtOeq AtOe j AtOei ln Et2 = 0 Oc i Oc j Oc k ln Et2 tOei tOei From the equation above, it can be seen that there is a ln conditional variance on the left-hand side, indicating that the asymmetric effect is exponential.
In the EGARCH equation, the use of ln indicates the non-negativity of the variance.
the value of j = 0, it signifies the presence of asymmetry in the data, indicating that j is significant.
The presence of a leverage effect is marked by the hypothesis < 0.
GJR-GARCH Model Glosten, jahathan, and runkle GARCH or another name are GJR-GARCH and TARCH.
That is one of model that was first time known as TARCH (Zakoin.
Glosten.
Jahathan, and Runkle .
make this model more specific that modified by GARCH model which the stationary must be fulfilled.
The similarity of the GJR-GARCH model is:
Forecast Accuracy To select the best model, forecasting accuracy is crucial.
The accuracy of a forecasting method is evaluated based on the forecast errors.
In this study, error measures and relative measures are used, including Mean Absolute Percentage Error (MAPE).
MAPE is used to indicate the magnitude of errors in forecasting by comparing them to the previous values of the series .
The equation for calculating MAPE is as follows: PEt (Percentage Erro.
It is the percentage error, and n represents the number of errors.
MAPE =
Oc |PEt | n t=1 PEt (Percentage Erro.
It is the percentage error, and n represents the number of errors Yt Oe Ft PEt = Observation on period of-t forecast on period of-t For more details, the research was carried out using research steps as follows Figure 1 Oe Et2 = 0 1 AtOe1 .
i AtOei i AtOe1 ItOe1 Oe .
j AtOe j ItOe j 1 EtOe1 .
tOek EtOek Oe Et2 = 0 Oc i AtOei Oc j AtOe j ItOe j Oc k EtOek Where I is a dummy variable with the condition that ItOe1 = 1 if AtOe1 < 0, and ItOe1 = 0 if AtOe1 > 0 .
For the GJR-GARCH model, the asymmetry effect occurs when Yi = 0, indicating that is significantly different from zero.
Hybrid ARIMA with Stochastic Volatility A hybrid model in research combines linear and nonlinear models for a given dataset.
The purpose of using hybrids is to incorporate nonlinear models without discarding linear One application of hybrid models involves combining the ARIMA model with stochastic volatility models.
This leads to the emergence of the Stochastic Volatility Hybrid ARIMA equation as follows:
YCthybrid = LCtARIMA EtV S Where LCtARIMA EtV S I p (B).
Oe B)d Yt = q (B) Et2 , (GARCH.
EGARCH, or GJR-GARCH) The equation above shows that represents the linear component, while represents the nonlinear component.
After performing predictions and obtaining results using the Stochastic Volatility Hybrid ARIMA, an evaluation of forecasting accuracy will be conducted.
Fig.
1: flowchart INTERNATIONAL JOURNAL OF COMPUTING SCIENCE AND APPLIED MATHEMATICS VOL.
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F INDING AND D ISCUSSION
ARIMA Modelling By using the in-sample data, a time series plot of IndonesiaAos non-oil and gas export values (Yt ) is created, which is the first step to be taken before developing an ARIMA model, to understand the patterns formed in the data, resulting in the following time series plot:
Fig.
3: ACF plot after first differencing Fig.
2: The Non-oil and Gas Export Values From Figure 2, it can be observed that the non-oil and gas export values data show an increasing trend or non-constant Additionally, the fluctuations in IndonesiaAos nonoil and gas export values occur due to unavoidable volatility in the financial market, which is sensitive to changes.
From the time series plot above, it can be concluded that the data is not stationary concerning the mean and variance.
Through using Box-Cox as result that the rounded value ( ) is obtained at 00 using a 95% confidence level with an upper limit value of 41 and a lower limit of 0.
34, so that the data is not stationary for the variance, so a natural logarithmic transformation is The natural logarithmic transformation of the data looks again at the Box-Cox using ln Yt .
According to Wei .
if a data obtains a rounded value ( ) of 0.
Oo it is necessary to carry out a root transformation process ( lnYt ).
For data cases like this, it is used lnYt as initial data so that only one transformation is used in making a linear model from the ARIMA method.
Seeing the changes Oo obtained after the transformation of the resulting root ( lnYt ) data is stationary concerning the variance, then proceed with analyzing whether the data is stationary concerning the mean.
For stationary concerning the mean ADF test carried out.
From ADF test the stationary data of average is done by a process of ARIMA model can be made through an analysis ACF and PACF plots:
From Figure 3.
plot A Oo2n the red line is the critical value for knowing autocorrelation and the blue line is interval time .
it can be seen that there is a cut off at first lag, because it is calculated from the initial five lags only one lag is significant.
In the ACF plot,it can be seen that after the initial five lags there are several significant lags but do not form a seasonal pattern so it is sufficient to use the ARIMA process.
After viewing the ACF plot it continues by analyzing the pattern of PACF plot.
From Figures 3 and 4, the ACF and PACF plots, it can be seen that the generated data pattern for ACF experiences a cut- Fig.
4: PACF plot after first differencing off at the first lag, and for PACF experiences a cut-off at the first lag as well.
Therefore, it can be inferred that the pattern contains ARIMA .
,1,.
ARIMA .
,1,.
, and ARIMA .
,1,.
The selection of the best ARIMA model is performed using the smallest AIC value.
TABLE I: ARIMA Model Selection Model ARIMA .
,1,.
ARIMA .
,1,.
ARIMA .
,1,.
Estimation P-Value 0,4982 0,0000 -0,4925 0,0000 -0,294 0,266 0,0022 0,041 Description AIC Significant 0,0075 Significant 0,0075 Significant 0,0074 In Table I above, it can be seen that ARIMA .
,1,.
is a significant model and has the smallest AIC value among the ARIMA .
,1,.
and ARIMA .
,1,.
models, even though all three models tested are significant.
After obtaining the best model, the prediction of the ARIMA model is then carried Prediction of ARIMA The ARIMA prediction is carried out after obtaining the best ARIMA model.
For instance, predicting the value of IndonesiaAos non-oil and gas exports from August 2021 (YC212 ) with the number of time origins used .
=211, by substituting the value of t with t = 212 as follows:
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TABLE i: AIC Value of Garch lnY212 = lnY211 I1 lnY211 Oe I1 lnY210 e e211 ln Yt = lnYtOe1 Oe 0, 294 lnYtOe1 0, 294 lnYtOe2 et 0, 266etOe1 By using the same steps for the period from September 2021 to July 2022, a comparison between the actual data and prediction results yields a MAPE value of 13.
Using the residuals from the ARIMA .
,1,.
model, we can test whether there are signs of heteroskedasticity.
ARCH-LM Test The test used to determine whether there is an ARCH effect or not is the Lagrange Multiplier test.
With the statistic test:
LM = NR2
No.
Model
AIC
GARCH.
GARCH.
-2,6874
-2,6872
From Table 3, it can be seen that a model GARCH .
has the smallest AIC value, so it can be used for further testing.
In the follow-up tests, it will identify if there are asymmetric There is or not an asymmetric effect characterized by a correlation that is not equal to zero.
Asymmetric Testing From the LM test result, the value obtained is 24,991 with a N.
value is 3,841, where=0,05.
This means that the LM value > N.
, which implies that there are signs of heteroskedasticity in the residuals of the ARIMA .
,1,.
In conclusion, the residuals from the ARIMA .
,1,.
model can be modeled using the ARCH or GARCH method.
GARCH Modelling In the squared residuals of the ARIMA model from the ARCH-LM test, there are signs of heteroscedasticity.
Therefore, the ARIMA residuals are used to create a GARCH model with the following possible models:
In financial data, an asymmetric effect occurs because there is a positive or negative correlation between the present time value and future volatility values, which can result in the GARCH model needing to be more precise in modeling.
To test whether there is an asymmetrical effect on the data, the Correlation test is used with the correlation between the standard squared residuals from ARIMA and the residuals from the GARCH model, where if the correlation value is not equal to zero, then it can be concluded that there is an asymmetric effect on the data.
In asymmetric testing, a crosscorrelation test is used, where the squared residuals of ARIMA .
,1,.
are correlated with the residuals of GARCH .
TABLE II: GARCH Model Selection Model GARCH .
GARCH .
GARCH .
GARCH .
GARCH .
GARCH .
GARCH .
GARCH .
Parameter 0 Estimation 0,0005759 0,210620 0,000010 0,999000 0,000010 0,000000 0,999000 0,005408 0,189039 0,068394 0,000013 0,154832 0,844168 0,000013 0,000000 0,157811 0,841189 0,000011 0,000000 0,000000 0,999000 0,000013 0,000000 0,000000 0,143370 0,855630 P-Value 0,00000 0,007217 0,000736 0,000000 0,000736 1,000000 1,000000 0,0000 0,10289 0,28035 0,00000 0,00000 0,00000 0,34368 1,00000 0,00000 0,00000 0,00000 1,00000 1,00000 0,00000 0,39201 1,00000 1,00000 0,00000 0,00000 Description Significant Not Significant Significant Significant Significant Not Significant Significant Significant Not Significant Not Significant Significant Significant Significant Not Significant Not Significant Significant Significant Significant Not Significant Not Significant Significant Not Significant Not Significant Not Significant Significant Significant Fig.
5: CCF Plot From Figure 5 above, it is known that the squared ARIMA .
,1,.
residual and GARCH .
residual models contain asymmetric effects, where the cross-correlation plot produces cross-correlation results.
Therefore, it is necessary to use modified methods of GARCH, such as EGARCH and GJRGARCH, to create models for data experiencing asymmetric EGARCH Model Through the significance test of the parameters in the table 2, it is found that the GARCH .
and GARCH .
models are significant.
Therefore, the best model needs to be selected using the smallest AIC value as follows:
The use of the EGARCH model is one method that can model the asymmetric effects that occur in time series data, so several possible EGARCH models can be used as follows:
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TABLE IV: EGARCH Model Selection TABLE VI: AIC Value of EGARCH Model Parameter Estimation P-Value Description EGARCH .
-0,1229
-0,0514
0,9753
-0,1836
0,000736
0,000000
0,000000
0,000000
Significant Significant Significant Significant EGARCH .
-0,209637
-0,50654
0,601031
0,356302
-0,206116
0,000000
0,000000
0,000000
0,000000
0,000000
Significant Significant Significant Significant Significant EGARCH .
-0,223122
0,019252
-0,067247
0,955972
0,073507
-0,275930
0,000000
0,000000
0,000000
0,000000
0,000000
0,000000
Significant Significant Significant Significant Significant Significant 0
-0,5130599
0,049999
-0,152500
0,184208
0,712357
0,200551
-0,117625
0,110073
0,453136
0,034917
0,000033
0,000000
0,211283
0,190278
Not Significant Not Significant Significant Significant Significant Not Significant Not Significant EGARCH .
In Table 4, it can be seen that EGARCH .
EGARCH .
, and EGARCH .
have significant estimation values.
Before concluding the best EGARCH model, observations on the GJR-GARCH model were carried out.
No.
Model
AIC
EGARCH.
EGARCH.
EGARCH.
-0,07277
1,28440
1,27730
From Table 6, it can be seen that the EGARCH .
model has the smallest AIC value of -0.
07277, which means that the EGARCH .
model indicates the appropriate model to be used for making predictions.
EGARCH Prediction The prediction results for the EGARCH .
model, using the t-value replaced with t in the following equation:
AtOe1 AtOe1 Oe 0, 1836 ln Et2 = Oe 0, 1229 Oe 0, 0514 EtOe1 EtOe1 0, 9753 ln EtOe1 They have an impacted sign effect of -0,1836.
A sign effect means a negative impact on volatility, resulting in a decline of 0,1836.
However, side effects are affected by utility impacts, which means that volatility affects the t Oe 1.
Therefore, the prediction results are obtained for the EGARCH model for the next 12 periods.
Then, a hybrid process is performed to achieve the optimal prediction results.
GJR-GARCH Model GJR-GARCH model is one of the same model as EGARCH that is used for data that have asymmetric effect, but the GJRGARCH method must qualified to be stationary in the process, which is different from EGARCH.
For this data case.
GJRGARCH can use the:
TABLE V: GJR-GARCH Model Model Parameter Estimation P-Value Description GJR-EGARCH .
0,000385
0,000000
0,925486
0,036367
0,009239
1,0000
0,000000
0,273499
Significant Not Significant Significant Not Significant GJR-EGARCH .
0,000480
0,000000
0,590296
0,312080
0,053621
0,010653
1,0000
0,213128
0,5168199
0,262351
Significant Not Significant Not Significant Not Significant Not Significant GJR-EGARCH .
0,0001399
0,000000
0,000000
0,964584
0,213853
-0,184678
0,080724
1,00000
1,00000
0,000000
0,063915
0,107568
Not Significant Not Significant Not Significant Significant Not Significant Not Significant 0
0,000139
0,00000
0,00000
0,964584
0,0000
0,213853
-0,184678
0,092877
1,00000
1,00000
0,00000
0,999991
0,064062
0,107676
Not Significant Not Significant Not Significant Significant Not Significant Not Significant Not Significant GJR-EGARCH .
In Table 5.
for the estimation of the GJR-GARCH model it can be seen that there is no significant model, so the model that can be used to make predictions is the EGARCH model.
Next, using the Akaike Information Criteria (AIC), the AIC results for EGARCH .
are obtained as shown below:
Hybrid ARIMA-EGARCH After obtaining the prediction results using the ARIMA method and the EGARCH method, the models are combined as follows:
YCthybrid = ln YtOe1 Oe 0, 294 ln YtOe1 0, 294 ln YtOe2 et 0, 266etOe1 Oe 0, 1229 AtOe1 Oe 0, 0514 EtOe1 AtOe1 Oe 0, 1836 0, 9753 ln EtOe1 EtOe1 the forecast results for 12 periods from August 2021 to July 2022 are as follows:
TABLE VII: Hybrid ARIMA-EGARCH Prediction Period
Actual Data
Prediction Residual August 2021
September 2021
October 2021
November 2021
December 2021
January 2022
February 2022
March 2022
April 2022
May 2022
June 2022
July 2022
1185,34
128,12
1373,53
1712,94
1321,89
-1825,1
-780,67
4688,69
5330,27
-700,87
3729,22
3168,62
Table 7 above, obtained prediction results with the hybrid ARIMA-EGARCH method for export grade data Indonesian INTERNATIONAL JOURNAL OF COMPUTING SCIENCE AND APPLIED MATHEMATICS VOL.
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oil and gas from August 2021 until July 2022.
the prediction result of the hybrid ARIMA-EGARCH method, we can calculate of MAPE grade to know the accuracy grade of the prediction result.
Hybrid ARIMA-EGARCH Forecasting Accuracy Next, the forecasting accuracy calculation is performed using the MAPE (Mean Absolute Percentage Erro.
as follows:
MAPE = 9, 35158%
Because the forecasting accuracy obtained is very accurate, being less than 10%, it shows that it can reduce the initial ARIMA error, which was 13.
20824%, so the model can be used for predicting non-oil export values.
IV.
CONCLUSION
Based on the results of research for data on the fluctuating value of Indonesian non-oil and gas exports, a combined Arima model was used.
the use of a hybrid model can reduce the errors that exist in the ARIMA model by considering asymmetric effects to choose the best model that can be combined with the ARIMA model.
In this study, the best model was obtained.
Hybrid ARIMA-EGARCH with a MAPE value of 9,35158% which shows that the prediction results obtained are very accurate and can later be used to predict the value of non-oil and gas exports in the future.
R EFERENCES
Jhingan.
Ekonomi Pembangunan dan Perencanaan.
Rajawali Pers.
Jakarta, 2013.
Indonesia.
AuLaporan Perekonomian Indonesia 2018,Ay https://w.
id/id/publikasi/laporan/Documents/14 LPI2018, 2018, last accessed 22/07/2022.
Amir.
AuInovasi Pendidikan Melalui Problem Based Learning,Ay Kencana Prenada Media Group, 2009.
Statistik.
AuStatistika
Perdagangan Luar Negeri Indonesia
Ekspor
Menurut Kode
ISIC
,2019-2020,Ay https://w.
id/publication/2021/07/06/9f3d9054c2f29bc478d56cd1, 2019, last accessed 25/03/2022.
Sudjana.
Metode Statistika.
Tarsito.
Bandung, .
Cryer.
Time Series Analysis.
PWS-Kent Publishing Co.
Boston, 1986.
Wei.
Time Series Analysis: Univariate and Multivariate Methods, 2nd Edition.
Addison Wesley.
Boston, 2006.
Hartati.
AuPenggunaan metode ARIMA dalam meramal pergerakan Inflasi,Ay Jurnal Matematika Sains Dan Teknologi, vol.
18, no.
1, pp.
1Ae10, 2017.
Eliyawati.
Hidayat, and D.
Azizah.
AuPenerapan Model GARCH (Generalized Autoregressive Conditional Heteroscedasticit.
Untuk Menguji Pasar Modal Efisien di Indonesia,Ay Administrasi Bisnis (JAB), 2014.
Zhang.
AuTime series forecasting using a hybrid arima and neural network model,Ay Neurocomputing, vol.
50, pp.
159Ae175, 2003.
Engle.
AuAutoregressive conditional heteroscedasticity with estimates of the variance of united kingdom inflation,Ay Econometrica:
Journal of the econometric society, pp.
987Ae1007, 1982.
Bollerslev.
AuGeneralized autoregressive conditional heteroskedasticity,Ay Journal of econometrics, vol.
31, no.
3, pp.
307Ae327, 1986.
Mubarokah.
Fitrianto, and F.
Affendi.
AuPerbandingan Model GARCH Simetris dan Asimetris pada Data Kurs Harian,Ay Indonesian Journal of Statistics and Its Applications, vol.
4, no.
4, pp.
627Ae637, .
Abiyani.
AuPeramalan data saham s&p 500 index menggunakan model threshold autoregressive conditional heteroscedastic .
,Ay Ph.
Universitas Negeri Malang, 2013.
Ekananda.
Analisis Data Time Series.
Mitra Wacana Media.
Jakarta, .
Maricar.
AuAnalisa perbandingan nilai akurasi moving average dan exponential smoothing untuk sistem peramalan pendapatan pada perusahaan xyz,Ay Jurnal Sistem dan Informatika (JSI), vol.
13, no.
36Ae45, 2019.