Predicting volatility of bitcoin returns with ARCH, GARCH and EGARCH models

Abstract

The ınvestment decisions of institutional and individual investors in financial markets are largely influenced by market uncertainty and volatility of the investment instruments. Thus, the prediction of the uncertainty and volatilities of the prices and returns of the investment instruments becomes imperative for successful investment. In this study we seek to identify the best fit model that can predict the volatility of return of Bitcoin, which is in high demand as an investment tool in recent times. Using the opening data of weekly Bitcoin prices for the period of 11.24.2013–03.22.2020, their logarithmic returns were calculated. The stationarity properties of the Bitcoin return series was tested by applying the ADF unit root test and the series were found to be stationary. After reaching the average equation model as ARMA (2.2), it was tested whether there was an ARCH effect in the ARMA (2,2) model. As a result of the applied ARCH-LM test, it is reached that the residuals of the average equation model selected have ARCH effect. Volatility of Bitcoin return series after detection of ARCH effect has been tried to predict with conditional variance models such as ARCH (1), ARCH (2), ARCH (3), GARCH (1,1), GARCH (1,2), GARCH (1,3), GARCH (2,1), GARCH (2,2), EGARCH (1,1) and EGARCH (1,2). While the obtained findings indicate that the best model is in the direction of GARCH (1,1) according to Akaike info criterion, it was found that GARCH (1,1) model does not have ARCH effect as a result of the applied ARCH-LM test. Thus, our empirical findings highlight an ample guide on appropriate modeling of price information in the Bitcoin market.

Introduction

Investment decisions of institutional and individual investors in financial markets are important both in terms of pricing of financial markets and returns of investors. Investment decisions are related to volatility, which is representative of the concept of uncertainty. Volatility is a concept that explains the frequency and magnitude of changes in the prices of financial assets [5, 6, 8, 16, 25, 29]. Generally, financial instruments and their returns are characterized by measures of uncertainty with bubbles and bursts. This is because they mirror firm’s financial status and prospects as highlighted by Balcilar et al. [6]. As a result of the uncertainty brought about by volatility, investors' investment habits may change and this situation may negatively affect both investors and financial markets. This indicates how important the volatility concept is for investors and markets. As a result of being affected by globalization in financial markets and economies, all financial markets act as if there is only one financial market. The biggest example of this is VIX index. The VIX index, known as the fear index worldwide, has become an important indicator of volatility in global markets and plays an important role in investors' decisions. Increases in the level of volatility in commodities, stocks, foreign currency and similar investment instruments may cause uncertainty over prices, which may cause investors to face greater risks.

Although investors behave in different investment decisions, they are more likely to suffer huge loss when the volatility the investment instrument is high.. In this case, it is beneficial for investors to minimize uncertainty and thus decrease in risk level. It is an important issue in terms of finance literature with which models of volatility exposed to different investment instruments in different financial markets can be estimated.

In other words, volatility forecasting models are needed to develop issues such as portfolio optimization, more effective implementation of hedging methods and pricing in derivative instruments [7, 28].

When the literature is examined, the volatility model first appeared in the study of Engle [14] as the autoregressive conditional heteroskedasticity (ARCH) model. In the study of Bollerslev [10], generalized autoregressive conditional heteroscedasticity (GARCH) model was introduced, such that the trajectory of volatility models started to diversify. While the diversity in volatility models causes the financial environment to focus on which model can give better results, changing data and time series prevent a common idea about which model can perform better. This makes a great contribution to the emergence of new models, as well as to try out existing models.

Therefore, this study estimated the volatility of Bitcoin return with various models and evaluated and compared the forecasting performances of the models. Hence, the contribution of this study hinges on the determination of the most suitable model of estimating volatility in Bitcoin returns. Volatility models such as ARCH, GARCH and EGARCH were applied in the study and it was found that the most suitable volatility model according to Akaike info criterion (AIC) was GARCH (1,1).

The remainder of this study follows with a stylized review of related literature is presented in "Literature review" section. Subsequently, "Data and methodology" section focuses on data and econometrics procedures. "Empirical findings and discussions" section provides empirical results discussion. Finally, concluding remarks are rendered in "Conclusion" section with study based implications.

Literature review

When the literature is analyzed, all of the existing volatility models have been tried on different financial instruments and new opinions are raised which one gives better results. For this reason, instead of the volatility models tested only on Bitcoin prices, the studies on volatility models tested on different investment instruments in financial markets will also be mentioned. The motivation of the previous studies varies but centred on the modelling of the volatility characteristics and predictability of several investment instruments and prices.

The first volatility modeling related to exchange rates was introduced in Bollerslev [11]. In the study of Bollerslev [11], it was investigated which model works better in order to estimate the volatility in Dollar/Mark and Dollar/Pound exchange rates. As a result of the applied ARCH and GARCH models, it was found that the GARCH (1,1) model predicted the daily exchange rate changes most effectively.

In the study of West and Cho [33], the short- and long-term prediction of dollar rate volatility was tested. The findings are that the GARCH model gives better results in the short run in the dollar rate volatility, while it is not sufficient in the long-term forecasts. Fong [15] attempted to estimate Japanese securities return volatility using ARCH and SWARCH methods. As a result of a comparison of the applied methods, it has been reached that the SWARCH model explains the data better.

Murari's (2015) study, the volatility of the Indian Rubi against the Euro, the US Dollar, the Yen and the Pound was modeled. As a result of the GARCH method applied for the period 2000–2013, it has been reached that the GARCH (2,1) model is a suitable model for finding symmetrical effects. In the study of Kumar [20], which is a similar study, the model with which the volatility of the Euro exchange rate in the spot market can be analyzed most effectively was tested. In the study in which ARCH group models were used, it was found that GARCH (1,1) model was the most suitable model in forecasting estimation.

In Birau et al. [9], the most appropriate model was tried to be determined in the estimation of the volatility structure for the Indian Bombay exchange bank index. As a result of the obtained findings, the most effective model in the Indian Bombay exchange bank index volatility prediction was determined as the GARCH (1,1) model.

Yildirim [34] study, in which the volatility estimation in crude oil prices is tested, which model is best tested, ARCH and GARCH models were applied and the findings obtained were found to be able to test the volatility in crude oil prices with the most effective GARCH (1,1) model.

In a recent study of Katsiampa [19], it was investigated which model could best be applied in estimating the volatility in Bitcoin prices. The findings obtained in the study where models such as GARCH, EGARCH, TGARCH, AP GARCH, C-GARCH and asymmetric component GARCH are applied are the CGARCH model that is the most successful model that measures the volatility in Bitcoin prices. In a similar study, Şahin and Özkan [31] investigated which model is best for estimating volatility for Bitcoin prices. The findings obtained as a result of the applied ARCH, GARCH, ARCHM, EGARCH and TARCH models have been reached to be the TARCH model of the best volatility forecast model in Bitcoin prices.

In the study of Amjad and Sahah [3], it was investigated which model would be more accurate to use in terms of estimating past prices and future prices of Bitcoin. As a result of the obtained results, it has been reached that the ARIMA method will give more accurate results.

In the study of Jang and Lee [18], the Bitcoin price of Bayesian neural network technique is modeling and prediction has been found to give better results than other linear and nonlinear models.

Sutiksno et al. [30], the model with which the Bitcoin prices can be estimated has been investigated and as a result of the methods such as ARIMA, NNAR models and the α-Sutte indicator method, it has been reached that the Bitcoin prices can be estimated more effectively with the α-Sutte indicator method. In McNally [23] study, it is tried to be guessed in which trend the Bitcoin price will be priced. As a result of comparing the predictive findings of ARIMA and deep learning methods, it was found that deep learning method gave better results. Furthermore, Lahmiri and Bekiros [21], the most effective model for predicting the most traded crypto money prices findings have been reached that it is a deep learning technique.

This study is different from the previous literature. The current study applied return volatility prediction models instead of the volatility prediction model in price movements which were commonly used in the literature. Also, while the ARCH and GARCH models, which are common in the selection of the suitable volatility model in the literature, are preferred for the Bitcoin returns, the EGARCH model was applied in addition to the ARCH and GARCH models.

Data and methodology

In this section, the econometric structure of the data set and models used in the research will be discussed. In this study, it is aimed to obtain the most suitable model in order to predict the variability in Bitcoin prices. A series of analyses were applied to weekly data from 11.24.2013 to 03.22.2020 from the investing.com website. The applied analyses were carried out using the EViews 9 program.

Before testing volatility prediction models, the logarithmic return of weekly Bitcoin prices was calculated and volatility prediction models were tested in the light of the series obtained. While calculating the logarithmic return of weekly Bitcoin prices, the following formula was used [32]:

$$r = {\text{log}}\left( {\frac{{x_{t} }}{{x_{t - 1} }}} \right)$$

(1)

Volatility models such as ARCH, GARCH and EGARCH were applied to the weekly logarithmic data obtained and the best model was chosen as a result of the findings obtained. Explanation of the econometric infrastructure of the three mentioned models is important for the effectiveness of the study.

In the modeling of financial time series with variable volatility, variance ARCH model, which is one of the first models, accepting that it is not fixed, is one of the most common models in the literature.

In the study of Engle [14], the error terms in the period t of $u_{t}$ it has been suggested that the variance is consecutively dependent on the variance of $u_{t}$ in the previous periods and the ARCH model has been developed [27: 341].

Under conditions of $\omega > 0;\alpha_{i} \ge 0;\mathop \sum \nolimits_{i = 1}^{q} \alpha_{i} < 1$, the general ARCH (q) process is shown in Eq. (2) format:

$$h_{t} = Var\left( {u_{tq} } \right) = \sigma_{t}^{2} = \omega + \alpha_{1} u_{t - 1}^{2} + \alpha_{2} u_{t - 2}^{2} + \alpha_{q} u_{t - q}^{2} = \omega + \sum\nolimits_{i = 1}^{q} {\alpha_{i} u_{t - i}^{2} }$$

(2)

The ARCH model developed in the Engle [14] study could contain parameter estimates with negative variance. In order to prevent this limitation, Bollerslev [10] study generalized autoregressive conditional heteroscedasticity (GARCH) model was developed. The GARCH model allows both autoregressive and moving average terms to be used in conditional variance modeling [4: 53].

Under $\omega > 0;\alpha_{i} \ge 0;\beta_{j} \ge { }0; \mathop \sum \nolimits_{i = 1}^{q} \alpha_{i} + \mathop \sum \nolimits_{j = 1}^{P} \beta_{j} < 1$ conditions, the GARCH (p, q) model process is expressed in Eq. (3):

$$h_{t} = \omega + \sum\nolimits_{j = 1}^{P} {\beta_{j} h_{t - j} } + \sum\nolimits_{i = 1}^{q} {\alpha_{i} u_{t - i}^{2} }$$

(3)

GARCH model against positive and negative shocks assumes volatility reacts symmetrically and leverage may be inadequate in modeling the effect. In order to eliminate these deficiencies, the EGARCH model was developed in the study of Nelson [26].

In the EGARCH model, the natural logarithm of conditional variance has its own delay values are $\frac{{\left| {u_{t - i} } \right|}}{{\sqrt {h_{t - i} } }}$ conditional to the standardized error term rather than the interpolated error term frame[12, 27: 344]. The developed EGARCH model is expressed in Eq. (4) [26]:

$$\log (h_{t} ) = \omega + \sum\nolimits_{j = 1}^{P} {\beta_{j} \log (h_{t - j} )} + \sum\nolimits_{i = 1}^{q} {\alpha_{i} \frac{{\left| {u_{t - i} } \right|}}{{\sqrt {h_{t - i} } }}} + \sum\nolimits_{i = 1}^{q} {\gamma_{i} \frac{{\left| {u_{t - i} } \right|}}{{\sqrt {h_{t - i} } }}}$$

(4)

These models are applied because they are identified to be the most commonly and best performing volatility models [1, 2].

We started the estimation procedure with unit root test to establish stationarity characteristics of the series. This is important because the basic assumption of the ARCH and GARCH models is that the series are stationary. This is also the reason for the use of the return series. After the stationarity test, we proceed to the ARCH test, using ARCH-LM test to examine the volatility of the series. Then, we applied the ARCH, GARCH, and the EGARCH models to estimate and forecast the volitility of the series.

Empirical findings and discussions

In this section, various tests were conducted for the model selection of weekly Bitcoin return volatility for the period of 11.24.2013–03.22.2020, the findings of the applied tests. ARCH, GARCH and EGARCH results were included in order to select the most suitable model and the findings were interpreted.

Firstly, when the time series related to logarithmic returns is examined as reported in Table 1, the median return is 0.004408, the minimum return is -0.536040 and the maximum return is 0.533432. On the other hand, the height of the kurtosis 5.913069 indicates that the series is pointed, while the skewness is 0.026048 indicates that the series is symmetrical. The fact that the series mean value is very close to 0 indicates that the series may be stationary.

Table 1 Descriptive statistics.

Stationarity of the logarithmic return series was tested as presented in Table 2 after obtaining and interpreting descriptive statistical values. The periodic presence of volatility on the chart was tested before applying the augmented Dickey–Fuller [13] test, which tests whether the series contains unit roots.

Table 2 Unit root test at I (0) level of logarithmic returns of bitcoin price movements

Figure 1 clearly shows the volatility cluster in the Bitcoin logarithmic return series. Large changes in logarithmic returns follow large movements and small changes follow small movements. This situation clearly reveals the clustering of volatility and volatility of Bitcoin prices.

Fig. 1

Logarithmic returns of bitcoin price movements

Since the time series is used in the study, the stationarity of the Bitcoin return series should be tested before volatility models are tested. In order to test the stationarity, augmented Dickey and Fuller [13] test was applied and it was tested whether the series contained unit root. If the series contains unit root as a result of the unit root test, this means that the series is not stationary. Since non-stationary time series give inaccurate results, series should be made stationary when they are not stationary. Unit root results for the Bitcoin return series are given in Table 2.

When the results of the Bitcoin return series are analyzed, it is found that the ADF test statistics are higher than the critical values of Mac-Kinnon (1996) as absolute values such as none, constant, constant and trending. On the other hand, the results obtained were found to be significant at the level of 1%, 5% and 10% significance. This indicates that there is no unit root in the Bitcoin return series and the series is stationary. Thus, the ARCH effect for the Bitcoin return series was tested after it was reached that the series were stationary at their own levels. While testing the ARCH effect, ARCH-LM (ARCH–Lagrange multiple) test was applied.

In order to determine the presence of the ARCH effect in the return series, the appropriate conditional equation needs to be established using the Least Squares (OLS) method and whether autocorrelation should be tested. As a result of the least squares (OLS) method applied, it was found that the most suitable model for the conditional average equation is ARMA (2,2). As a result of the Least Squares (OLS) method applied, the conditional average equation is given below.

$$\begin{aligned} {\text{RBTC}} = & 0.00{7768} - {1}.0{6}0{\text{144AR}}\left( {1} \right) - 0.{\text{999224AR}}\left( {2} \right) + {1}.0{\text{51866MA}}\left( {1} \right) \\ & \; + 0.{\text{999869MA}}\left( {2} \right) \\ \end{aligned}$$

As a result of the applied model, AR (1) + AR (2) < 1 and MA (1) + MA (2) < 1 condition has been reached. On the other hand, the parameters AR (1), AR (2), MA (1) and MA (2) < 0.05 have been reached. This indicates that the parameters are statistically significant.

ARCH-LM test was applied to the conditional average equation and the results are shown in Table 3.

Table 3 ARCH-LM test

As a result of the applied ARCH-LM test, probability values were found to be significant at 1% level. This indicates that H0: β1 = β2 = … = βn = 0 null hypothesis is rejected. This indicates that the $H_{0}$ hypothesis, which indicates equal variance, will be rejected and the existence of the ARCH effect will be accepted [14].

After the acceptance of the presence of ARCH effect on the return series, ARCH and GARCH models were tested for the selection of the suitable ARCH type model, and EGARCH model, which is an effective model in determining the asymmetric effect of the shocks on volatility, was also applied.

The coefficients of the applied volatility models and Akaike info criterion, Schwarz criterion and Hannan–Quinn criterion values are given in Table 4.

Table 4 ARCH, GARCH and EGARCH models forecast results

The coefficients of the applied ARCH, GARCH and EGARCH models yielded statistically significant results at the level of 1%, 5% and 10%. On the other hand, as a result of applied ARCH, GARCH and EGARCH models, it has been reached that there is no autocorrelation at 5% error level. The most suitable model for Bitcoin returns is selected according to Akaike Information Criterion (AIC) values. According to Akaike ınformation criterion (AIC), the lowest value was observed to be − 1.724.458. In this case, GARCH (1,1) has been found to be the best model that can predict Bitcoin returns volatility.

For the GARCH (1,1) model, which can best predict the volatility of the Bitcoin return series, the existence of the ARCH effect has been tested and the results are shown in Table 5.

Table 5 ARCH-LM test

As a result of the applied ARCH-LM test, it was reached that both probability values were greater than 5% and in this case, it was found that there was no ARCH effect between the error terms of the GARCH (1,1) model.

Conclusion

Corporate and financial investors desire to earn income from their investments while some of the investors try to avoid the risk of volatility, volatility is perceived as an opportunity for others. Therefore, although investors' perspectives are different, the concept of volatility is the same for all types of investors. Investment decisions, which have a significant impact especially on the direction of the markets, are made taking into account volatility.

For this reason, the importance of volatility models is increasing in the financial literature. Besides researching which volatility model is more suitable for which investment tool and time series, it is important for markets and investors to participate in the literature in their new models.

The fact that ARCH and GARCH models are insufficient to predict the volatility of financial assets that give asymmetrical reactions caused the emergence of models such as EGARCH and TGARCH. It is known that EGARCH and TGARCH models are effective in measuring the effect of negative and positive shocks on volatility. On the other hand, there is a difference between GARCH and GARCH models. While the lever effect is exponential in the EGARCH model, the lever effect is quadratic in the TGARCH model [22, 27].

Therefore, in addition to ARCH and GARCH models, EGARCH model was also applied for Bitcoin returns in the study. The choice of the most suitable model among the volatility models has been determined according to Akaike info criterion and it has been reached that the most common model for Bitcoin returns is the GARCH (1,1) model. GARCH (1,1) is followed by GARCH (1,3) and EGARCH (1,1), respectively.

The γ parameter in the EGARCH (1,1) model is statistically significant at the level of 5%, which indicates that the shocks formed have an asymmetrical effect on return volatility. In cases where GARCH (1,1) and GARCH (1,3) models are insufficient, EGARCH (1,1) volatility model can be suggested. This finding supports the submissions of Alhassan and Kilishi [2] and Alao et al. [1] that the asymmetric GARCH models outperform the symmetric models. Thus, this study contributes to the volatility forecasting models debate which is need in times of proliferation in the extant literature to methodological advancement to amerloirte issues such as portfolio optimization, more effective implementation of hedging methods and pricing in derivative instruments.

The findings are similar to some of the studies in the literature and support the findings of studies such as West and Cho [33], Kumar [20], Birau et al. [9], Yıldırım [34]. Using only Bitcoin returns as crypto money in the study can be considered as the most important constraint of the study. Thus, our empirical findings highlights an ample guide on appropriate modelling of price information in the Bitcoin market.

The limited of this study is that it considered only Bitcoint among several other cryptocurrencies. Hence, further study to examine the volatility modelling of other cryptocurrencies such as Ethereum, Riple, Tether, Litecoin, Eos and Tezos, which are not included in the analysis, will make a great contribution to the literature.