Estimating and forecasting bitcoin daily prices using ARIMA-GARCH models

Abstract

Purpose

The goal of the study is to offer important insights into the dynamics of the cryptocurrency market by analyzing pricing data for Bitcoin. Using quantitative analytic methods, the study makes use of a Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model and an Autoregressive Integrated Moving Average (ARIMA). The study looks at how predictable Bitcoin price swings and market volatility will be between 2021 and 2023.

Design/methodology/approach

The data used in this study are the daily closing prices of Bitcoin from Jan 17th, 2021 to Dec 17th, 2023, which corresponds to a total of 1065 observations. The estimation process is run using 3 years of data (2021–2023), while the remaining (Jan 1st 2024 to Jan 17th 2024) is used for forecasting. The ARIMA-GARCH method is a robust framework for forecasting time series data with non-seasonal components. The model was selected based on the Akaike Information Criteria corrected (AICc) minimum values and maximum log-likelihood. Model adequacy was checked using plots of residuals and the Ljung–Box test.

Findings

Using the Box–Jenkins method, various AR and MA lags were tested to determine the most optimal lags. ARIMA (12,1,12) is the most appropriate model obtained from the various models using AIC. As financial time series, such as Bitcoin returns, can be volatile, an attempt is made to model this volatility using GARCH (1,1).

Originality/value

The study used partially processed secondary data to fit for time series analysis using the ARIMA (12,1,12)-GARCH(1,1) model and hence reliable and conclusive results.

Citation

Phung Duy, Q., Nguyen Thi, O., Le Thi, P.H., Pham Hoang, H.D., Luong, K.L. and Nguyen Thi, K.N. (2024), "Estimating and forecasting bitcoin daily prices using ARIMA-GARCH models", Business Analyst Journal, Vol. 45 No. 1, pp. 11-23. https://doi.org/10.1108/BAJ-05-2024-0027

1. Introduction

Bitcoin is the first widely used and traded cryptocurrency since 2009 when the Bitcoin software started to be available to the public and mining-the process by which new Bitcoins can be created and transactions can be recorded and verified on the blockchain begins. As Bitcoin becomes increasingly popular, and the idea of decentralized and encrypted currencies catches on, more rival, alternative cryptocurrencies appear. But Bitcoin remains the most successful and widely accepted cryptocurrency. The purpose of this paper is to use time series methodology to predict the future returns and price of Bitcoin. At the same time, we want to compare the performance of ARIMA-GARCH models with Normal, Student’s t, and Skewed student’s t distributions. The study investigates the price movements in the crypto markets and their predictability because, often criticized for being over-volatile, the price formation in the crypto market has started to become integral to investing decisions.

As Bitcoin gradually has had a place in the financial markets and portfolio management, time series analysis is a useful tool to examine the characteristics of Bitcoin prices and returns and extract important statistics to forecast future values of the series. Section 2 which follows the background of the study discusses a review of related works, Section 3 discusses the methodology of this study and Section 4 discusses data analysis and result, the model’s evaluation and forecasts are summarized in Section 5, while Section 6 gives the summary and conclusion of the paper.

2. Literature review

Nowadays, a lot of research has been done on predicting the price of bitcoin using a variety of techniques, including machine learning and regression, to forecast past data on trading dates, opening prices, highest and lowest prices, and other details.

A popular technique for forecasting issues, the statistical method is also utilized in the prediction of Bitcoin. Numerous writers have successfully used statistical methods in their work. The statistical techniques ARIMA, AR, MA, and GARCH are frequently applied in the prediction of Bitcoin prices.

Based on time series data, Roy, Nanjiba, and Chakrabarty (2018) used the ARIMA, AR, and MA models to forecast the price of Bitcoin. The ARIMA model produced the best results, with an accuracy of 90.31%. Furthermore, the ARIMA technique is also applied.

In addition, Katsiampa (2017) found that the AR-CARCH model is the most accurate model when using a GARCH model to forecast the daily closing price of Bitcoin. The forecast was created by Bakar and Rosbi (2018) using the Weighted Moving Average method, with a 0.72% MAPE rate error.

Machine learning techniques are widely applied in various fields as efficient forecasting tools (2019). Numerous writers have utilized machine learning techniques to forecast the price of Bitcoin. SVM, MLP, LR, RNN, CNN, and GRU are some of the frequently employed machine learning techniques. Machine learning models such as Random Forest, XGBoost, Quadratic Discriminant Analysis, Support Vector Machine, and Long Short-term Memory were used by Chen, Li, and Sun (2020) to predict Bitcoin prices at different frequencies in five minutes. The statistical methods produced better results, with an accuracy of 67.2%.

Currently, Deep learning methods also have been applied to develop Bitcoin price prediction models, Lamothe-Fernández, Alaminos, Lamothe-López, and Fernández-Gámez (2020) has used DRCNN, LSTM, and ARIMA methods to predict the Bitcoin price and achieved high accuracy up to 92.61–95.27% for the models.

Klein, Thu, and Walther (2018) find the time-varying conditional correlations between Bitcoin and gold in relation to a set of assets by implementing a BEKK-GARCH model. The study finds no evidence that Bitcoin can be a more effective hedging asset than gold as a portfolio component because the correlations are not the same.

In Critien, Gatt, and Ellul (2022) published a second research that aimed to go beyond just predicting a direction. Rather than stating if the price would rise or fall, the model would indicate “the magnitude of the price change” (Critien et al., 2022). The study used Twitter messages in addition to historical price data for Bitcoin to gauge popular opinion of the asset. This work stands out for using a bidirectional long short-term memory model which consists of two LSTMs: one for historical pricing data and another for Twitter tweets. It used 450 days of historical data to reach its accuracy of 64.18%” (Critien et al., 2022).

Additionally, Sarkodie, Ahmed, and Owusu's (2022) study used COVID-19 data—such as the number of cases and deaths—to predict whether the price of cryptocurrencies, like Bitcoin, would increase or decrease. The Romano-Wolf technique was used in the study to examine this association (Sarkodie, Amani, Ahmed, & Owusu, 2023). Specifically, research concluded that there was a strong correlation between COVID-19-related fatalities and Bitcoin, with the latter fluctuating 90.50% of the time (Sarkodie et al., 2023).

Amirshahi and Lahmiri (2023) built various Deep Learning models based on Feed Forward Neural Networks (DFFNN) and Long Short Term Memory (LSTM) networks and evaluated their performance in predicting Report the volatility of 27 cryptocurrencies. Then, various hybrid models are constructed that yield three models of the GARCH type: APGARCH, EGARCH, and GARCH. The study discovered that while deep learning models enhance GARCH-type model prediction under any distribution assumption, they also considerably boost the predictive power of the deep learning models under study—more especially, the LSTM and DFFNN models—when GARCH-type models are predicted as informative features.

3. Methodology

In this section, the paper discusses the techniques that feature prominently in this study. These are ARIMA, ARIMA- GARCH models with Normal.

3.1 The Box–Jenkins for ARIMA model

Box–Jenkins ARIMA is known as the ARIMA (p, d, q) model where p is the number of autoregressive (AR) terms, d is the number of differences taken and q is the number of moving average (MA) terms. ARIMA models always assume the variance of data to be constant. The ARIMA (p, d, q) model can be represented by the following equation:

(1)yt=φ1yt−1+⋯+φpyt−p+εt+θ1εt−1+⋯+θq εt−q

3.2 GARCH models

GARCH models are used mainly for modeling financial time series that present time-varying volatility clustering. The general GARCH (q, p) model for the conditional heteroscedasticity according to Bollerslev (1986) has the following form:

(2)yt=µt+zt

Where μt is the conditional mean of yt

zt is the shock at time t

(3)zt=σt εt

Where ε_t ∼ N (0,1).

(4)σt2=α0 +∑i=1qαizt−i2+∑j=1pβjσt−j2

Where σ_t² is the conditional variance of y_t

• α₀ is a constant term

• q is the order of the ARCH terms

• p is the order of the GARCH terms

• α_i and β_j are the coefficients of the ARCH and GARCH parameters respectively

With constraints

α0 > 0

αi≥0,for i=1,2,…,q

βj≥0,for i=1,2,…,p

(5)∑i=1qαi+∑j=1pβj < 1

3.3 Hybrid ARIMA-GARCH model

To recommend a hybrid ARIMA-GARCH model, two stages should be used. In the first stage, we use the best ARIMA model that fits stationary and linear time series data, with the non-linear part of the data contained in the linear model’s residuals. In the second stage, we use the GARCH model to capture non-linear residual patterns. This hybrid model combines ARIMA and GARCH models with nonlinear residual patterns to analyze and forecast Bitcoin returns.

3.4 Diagnostic checking of hybrid ARIMA-GARCH model

The diagnostic tests of hybrid ARIMA-GARCH models are based on residuals. Residuals’ normality test is employed with the Jarque and Bera test. Also, for the conditional heteroscedasticity test, we use the squared residuals of the autocorrelation function.

4. Analysis, results, discuss

The data used in this study are the daily closing prices of Bitcoin from January 17th, 2021 to December 17th, 2023, which corresponds to a total of 1065 observations. The estimation process is run using 3 years of data (2021–2023) while the remaining (January 1st 2024 to January 17th 2024) is used for forecasting. The data is compiled from Bitstamp, the largest Bitcoin exchange, and covers a daily database denominated in US dollar, which is the main currency against which Bitcoin is the most traded. Through ADF testing, it shows that Bitcoin is not a stationary chain. Therefore, based on the collected data, the author took the first difference of the sample and obtained the result that ADF is a stationary series.

Table 1 show some key statistics of the raw data which shows high standard deviation and none normality.

Table 2 presents some key statistics of the data after taking the first difference. Skewness and excess kurtosis are clearly observed, leading to a high valued Jarque and Bera (1987) test which indicates non-normality of the distribution. In the following Figure 1,we present the closing price of Bitcoin before taking the first difference which seems to be nonstationary.

Figure 1, we present the closing Bitcoin price before first differencing the time series which seems to be nonstationary.

Figure 2, presents the first difference of Bitcoin price after taking first differenced series. From Figure 2, the daily closing the first differenced of Bitcoin seem to be stationary.

The results of Table 3 confirm that the first differenced series of Bitcoin price are stationary at their level.

After graphs revealed the data’s stationarity, the ADF test may provide more concrete evidence for the stationarity of the time series (Dickey & Fuller, 1979). In the search for the unit root, the null hypothesis favors the presence of a unit root as a strong indicator of nonstationarity in the data, while rejection of the null hypothesis concludes that the data is stationary. As shown in Table 3, the ADF test was calculated at a 5% significance level to determine whether the original data were stationary.

While the ADF test revealed that the original data was nonstationary, it became stationary after differentiating the time series. As a result, the unit root has been removed because the null hypothesis was rejected after the differencing process with 0.0000 probability at the 0.05 significance level, and the model can be demonstrated using the first differenced series.

Once stationary have been addressed, the next step is to identify the order (the p, d, and q) of the autoregressive and moving average terms. The primary tools for doing this are the Akaike information criterion and the Schwartz information criterion.

Time series were the first differenced. The auto correlogram graphs seen in Table 4 can give a clue about the order of the ARIMA (p, d, q) model regarding the Box–Jenkins methodology.

Since the sample ACF plot cuts off after lag 9, 12 and 19, and the partial autocorrelations too, this means that the following ARIMA (autoregressive integrated moving average) model: ARIMA(p,d,q) with (p,q) = (9,12,19). The three basic steps to developing an ARIMA model are useful decision of properly selecting the right model to be chosen (Kennedy, 2008).

From the Table 5, we observed that the optimal model is ARIMA(12,1,12) based on the selection criterion AIC.

After an optimum model has been identified, the model estimation methods make it possible to estimate all the parameters of the ARIMA model (see Table 6).

Before building the GARCH model for the first diferenced Bitcoin prices series, the first thing is to check whether the variance of the residuals of the ARIMA(12,1,12) model above is calculated by using ARCH or not ARCHLM test (see Table 7). The test results show that the value of the ARCH-LM test value is 3.717520 with p-value approximately equal to 0; The p-value of Chi-squared is also approximately 0. Hypothesis H₀ (heteroscedasticity in the residuals does not exist) is rejected; In other words, the residuals of the ARIMA(12,1,12) model are ARCH.

Many studies have shown that GARCH(1,1) provides the best forecasting results and is consistent with daily stock return series (Sadorsky, 2006; Ashley & Patterson, 2010). Therefore GARCH(1,1) will also be used to forecast Bitcoin price in this study. The results of estimating the coefficients of the ARIMA(12,1,12)- GARCH(1,1) model are presented in Table 8, in which the upper part presents the coefficients of ARIMA(12,1,12)- and the lower part are the coefficients of the GARCH(1,1) model.

The conditional heteroscedasticity of ε_t is expressed as follows:

σt2=1111573.+0.072487εt −12+0.522487σt−12

The correlation graph in Table 9 shows that the autocorrelation and partial autocorrelation function values of the square of the standard residual estimated by the model are approximately 0. Besides, the Q statistic value and p-value has a small value, showing the suitability of the GARCH(1,1) model for the first differenced Bitcoin prices series.

The ARCH-LM test is also performed to determine whether the residuals of the ARIMA(12,1,12)-GARCH(1,1) model still account for ARCH or not. The test results show that the p-value of Chi-squared and the F-statistic value are both greater than 0.05 (Table 10). Therefore, the hypothesis H₀ (no heteroscedasticity in the residuals) is accepted. This means that the ARIMA(12,1,12)-GARCH(1,1) model has modeled the entire phenomenon of heteroscedasticity of the first differenced Bitcoin prices series.

5. Evaluate the ability to predict the bitcoin prices of the ARIMA-GARCH combined model

5.1 Forecasting the VN-Index using the ARIMA-GARCH combined model

In this section, the ARIMA(12,1,12)-GARCH(1,1) model built above will be used to forecast Bitcoin prices from January 1, 2024 to January 17, 2024. The forecasted return value of Bitcoin is gathered into the Pricef series (see Figure 3).

As mentioned above, the actual values of Bitcoin prices from Jan 1, 2024 to Jan 17, 2024 are used as a basis for comparison with forecast values. Figure 3 shows that the Pricef series (red line) closely matches the actual Real Prices (black line). In other words, the Bitcoin forecast results performed by the ARIMA-GARCH model are very good. However, to conclude the ARIMA(12,1,12)-GARCH(1,1) model is the best forecasting model for the Bicoin prices or not, it is still necessary to compare it with other forecast indicators. reports of other models.

5.2 Evaluate the model’s predictive ability ARIMA(12,1,12)-GARCH(1,1)

5.2.1 Compare the fit of the models with the bitcoin prices

From Table 11, it can be seen that the AIC and SIC indexes of the ARIMA(12,1,12)-GARCH(1,1) model are the smallest. Therefore, it can be concluded that this combined model is more suitable than the single model ARIMA(1,0,1) in modeling the Bitcoin prices.

5.2.2 Evaluate the forecasting ability of models with bitcoin prices

The evaluation criteria RMSE, MAE and MAPE are summarized in Table 12 as follows:

It can be seen that the indices RMSE, MAE and MAPE of the combined model ARIMA(12,1,12)-GARCH(1,1) all receive the smallest values. Therefore, it can be concluded that the Bitcoin forecasting results from this combined model are more accurate than the other models, or in other words, the forecasting ability of the ARIMA-GARCH combined model for Bitcoin prices is outperforms the single ARIMA model.

6. Conclusion and future work

The purpose of this paper is to develop a hybrid model for analyzing and forecasting Bitcoin prices by combining ARIMA and GARCH models with high volatility. To make the Bitcoin price stationary, it is converted to the first difference. Using the Box–Jenkins method, various AR and MA lags were tested to determine the most optimal lags. ARIMA (12,1,12) is the most appropriate model obtained from the various models using AIC. As financial time series, such as Bitcoin returns, can be volatile, an attempt is made to model this volatility using GARCH (1,1).

However, this model often has to include the assumption that the future scenario will be exactly the same as what the model simulates in the past. Therefore, the ARIMA-GARCH combination model is only suitable for forecasting future points very close to the last time of the data series, demonstrating the characteristics of short-term forecasting.

In addition, because the ARIMA model format in the ARIMA-GARCH combined model greatly affects the forecast results, model users need to be flexible to avoid missing other meaningful models. In addition, when using the ARIMA-GARCH model to predict the Bitcoin prices, it is necessary to pay attention to the following contents:

Firstly, the fluctuation of the Bitcoin prices is also influenced by many macro factors such as GDP growth, interest rates, fluctuations in the world stock market… while the ARIMA-GARCH combined model has not mentioned and measured the impact of these factors.

Secondly, it’s important to choose the right delays and orders for the GARCH and ARIMA components. To make sure the model that is selected is the best one, thorough application of model selection criteria like AIC, BIC, and cross-validation is necessary. To confirm model assumptions, extensive diagnostic testing for heteroscedasticity and autocorrelation should be done.

Thirdly, Bitcoin’s market is known for its high volatility and sudden spikes or drops in prices. This characteristic requires the ARIMA-GARCH model to be frequently recalibrated to adapt to new market conditions. A rolling window approach or online learning algorithms could be utilized to update model parameters continuously.

Finally, the results of the article only stop at comparing and evaluating the forecasting ability of the ARIMA-GARCH combined model with its component models without comparing with other time series forecasting models. Therefore, further and more specific research is needed to find the best model to forecast the Bitcoin prices series.

Figures

Figure 1

Time plot of daily bitcoin closing price from January 17th, 2021 to December 17th, 2023

Figure 2

Time plot of first differenced series of Bitcoin closing prices

Figure 3

Bitcoin prices forecast results from January 1, 2024 to January 17, 2024

Table 1

Descriptive statistics of daily bitcoin price

	Price
Mean	34692.13
Median	31367.60
Maximum	67527.90
Minimum	15776.20
Std. dev.	12679.95
Skewness	0.533606
Kurtosis	2.324756
Jarque-Bera	70.77354
Probability	0.000000
Sum	36947120
Sum Sq. dev.	1.71E + 11
Observations	1065

Source(s): From data and calculations on Eviews software by the authors

Table 2

Descriptive statistics of first differenced series of daily Bitcoin closing prices

	DPRICE
Mean	−5.196523
Median	14.55000
Maximum	7311.500
Minimum	−7542.800
Std. dev.	1305.242
Skewness	0.139165
Kurtosis	7.865217
Jarque-Bera	1052.819
Probability	0.000000
Sum	−5529.100
Sum Sq. dev.	1.81E + 09
Observations	1064

Source(s): From data and calculations on Eviews software by the authors

Table 3

ADF test results of the first differenced series of bitcoin price

Null hypothesis: D(PRICE) has a unit root
Exogenous: constant
Lag length: 0 (Automatic – based on SIC, maxlag = 21)

		t-statistic	Prob.*
Augmented dickey-fuller test statistic		−34.32757	0.0000
Test critical values	1% level	−3.436284
	5% level	−2.864048
	10% level	−2.568157

Note(s): *MacKinnon (1996) one-sided p-values

Exogenous: Constant

Source(s): From data and calculations on Eviews software by the authors

Table 4

Correlogram of DPRICE

Lags(k)	AC	PAC	Q-stat	Prob
1	−0.052	−0.052	2.9225	0.087
2	0.012	0.009	3.0680	0.216
3	0.031	0.032	4.0679	0.254
4	0.039	0.043	5.7175	0.221
5	0.000	0.004	5.7175	0.335
6	0.014	0.012	5.9158	0.433
7	−0.048	−0.049	8.3588	0.302
8	−0.037	−0.045	9.8223	0.278
9	0.078	0.075	16.389	0.059
10	0.012	0.024	16.542	0.085
11	−0.011	−0.004	16.662	0.118
12	−0.073	−0.077	22.344	0.034
13	0.049	0.036	24.983	0.023
14	−0.005	−0.001	25.015	0.034
15	−0.022	−0.024	25.543	0.043
16	−0.020	−0.014	25.980	0.054
17	0.014	0.019	26.206	0.071
18	0.010	0.011	26.321	0.093
19	−0.058	−0.068	29.946	0.052
20	0.047	0.041	32.327	0.040
21	−0.028	−0.009	33.203	0.044
22	0.014	0.008	33.421	0.056
23	−0.038	−0.043	35.004	0.052
24	0.006	0.033	45.090	0.006
25	−0.002	0.023	45.096	0.008
26	0.009	−0.003	45.191	0.011
27	0.059	0.051	48.973	0.006
28	0.060	0.072	52.919	0.003
29	−0.059	−0.059	56.774	0.002
30	0.036	0.016	58.178	0.002
31	0.048	0.046	60.744	0.001
32	−0.010	−0.035	69.597	0.000
33	0.029	0.025	80.303	0.000
34	−0.007	−0.001	80.354	0.000
35	0.011	0.019	80.490	0.000
36	−0.045	−0.041	82.714	0.000

Source(s): From data and calculations on Eviews software by the authors

Table 5

AIC of ARIMA (p, d, q)

Order	AIC
(9,1,9)	17.18639
(12,1,12)	*17.17908*
(19,1,19)	17.18813
(9,1,12)	17.18177
(12,1,9)	17.18100
(9,1,19)	17.18189
(19,1,9)	17.18231
(12,1,19)	17.18364
(19,1,12)	17.18482

Note(s): *Minimum value to criterion

Source(s): From data and calculations on Eviews software by the authors

Table 6

The estimated model of ARIMA(12,1,12)

Variable	Coefficient	Std. error	t-statistic	Prob.
C	−5.602205	37.83434	−0.148072	0.8823
AR(12)	−0.687926	0.111838	−6.151118	0.0000
MA(12)	0.601323	0.124032	4.848149	0.0000
SIGMASQ	1678545.0	40120.58	41.83750	0.0000
R-squared	0.013813	Mean dependent var		−5.196523
Adjusted R-squared	0.011022	S.D. dependent var		1305.242
S.E. of regression	1298.029	Akaike info criterion		17.17908
Sum squared resid	1.79E + 09	Schwarz criterion		17.19777
Log likelihood	−9135.272	Hannan-Quinn criter		17.18616
F-statistic	4.949062	Durbin-Watson stat		2.091772
Prob(F-statistic)	0.002042

Source(s): From data and calculations on Eviews software by the authors

Table 7

ARIMA (12,1,12) residuals test

Heteroskedasticity test: ARCH
F-statistic	3.717520	Prob. F(1,1061)	0.0000
Obs*R-squared	3.711523	Prob. Chi-Square(1)	0.0000

Variable	Coefficient	Std. Error	t-statistic	Prob.
C	1580086.0	143193.0	11.03465	0.0000
RESIDˆ2(−1)	0.059092	0.030648	1.928087	0.0000

Source(s): From data and calculations on Eviews software by the authors

Table 8

Estimate of ARIMA(12,1,12)-GARCH(1,1) model with normal distribution

Variable	Coefficient	Std. error	z-statistic	Prob.
C	−17.95123	60.19281	−0.298229	0.7655
AR(12)	−0.717317	0.166688	−4.303353	0.0000
MA(12)	0.636491	0.185607	3.429238	0.0006
C	1111573	487405.4	2.280591	0.0226
RESID(−1)ˆ2	0.072487	0.038710	1.872577	0.0611
GARCH(−1)	0.522487	0.192230	2.718025	0.0066

Source(s): From data and calculations on Eviews software by the authors

Table 9

Correlation graph of squared standard residuals of the ARIMA (12,1,12)-GARCH(1,1) model

Lags(k)	AC	PAC	Q-stat	Prob*
1	−0.051	−0.051	2.7238
2	0.009	0.006	2.8034
3	0.021	0.022	3.2913	0.070
4	0.012	0.014	3.4340	0.180
5	0.008	0.009	3.4965	0.321
6	−0.010	−0.010	3.6110	0.461
7	−0.013	−0.015	3.7895	0.580
8	−0.036	−0.038	5.1683	0.522
9	0.069	0.066	10.206	0.177
10	−0.002	0.006	10.212	0.250
11	−0.032	−0.031	11.310	0.255
12	0.003	−0.003	11.316	0.333
13	0.024	0.023	11.934	0.369
14	0.007	0.009	11.986	0.447
15	−0.002	−0.001	11.992	0.528
16	−0.017	−0.017	12.304	0.582
17	0.006	0.007	12.337	0.653
18	0.022	0.017	12.842	0.684
19	−0.030	−0.030	13.800	0.681
20	0.023	0.026	14.387	0.704
21	−0.040	−0.037	16.147	0.647
22	−0.004	−0.011	16.161	0.707
23	−0.046	−0.049	18.450	0.620
24	0.020	0.020	27.186	0.204
25	0.042	0.057	29.123	0.176
26	−0.010	−0.005	29.232	0.212
27	0.025	0.014	29.922	0.227
28	0.035	0.039	31.216	0.220
29	−0.047	−0.053	33.654	0.176
30	0.025	0.021	34.333	0.190
31	0.003	0.007	34.345	0.227
32	−0.047	−0.035	36.790	0.183
33	0.063	0.051	41.165	0.105
34	−0.005	−0.006	41.192	0.128
35	0.014	0.023	41.396	0.150
36	−0.047	−0.044	43.804	0.121

Source(s): From data and calculations on Eviews software by the authors

Table 10

ARCH-LM test for ARIMA(12,1,12)-GARCH(1,1) model

Heteroskedasticity test: ARCH
F-statistic	0.514742	Prob. F(1,1049)	0.4733
Obs*R-squared	0.515471	Prob. Chi-Square(1)	0.4728

Source(s): From data and calculations on Eviews software by the authors

Table 11

Summary of criteria to evaluate the suitability of the models

Model	AIC	SIC
ARIMA(12,1,12)-GARCH(1,1)	16.71441	16.74269
ARIMA(12,1,12)	17.17908	17.19777

Source(s): From data and calculations on Eviews software by the authors

Table 12

Summary of criteria RMSE, MAE and MAPE of the models

Model	RMSE	MAE	MAPE
ARIMA(12,1,12)-GARCH(1,1)	14142,99	12345,43	32,31091
ARIMA(12,1,12)	15380,68	12449,22	42,91523

Source(s): From data and calculations on Eviews software by the authors