MODELING CRYPTOCURRENCIES VOLATILITY USING GARCH MODELS: A COMPARISON BASED ON NORMAL AND STUDENT’S T-ERROR DISTRIBUTION

This study measures the volatility of cryptocurrency by utilizing the symmetric (GARCH 1,1) and asymmetric (EGARCH, TGARCH, PGARCH) model of GARCH family using a daily database designated in different digital monetary standards. The results for an explicit set of currencies for entire period provide evidence of volatile nature of cryptocurrency and in most of the cases, the PGARCH is a better-fitted model with student’s t distribution. The findings show positive shocks heavily affected conditional volatility as a contrast with negative stuns. Those additional analyses can be provided further support their findings and worthwhile information for economic thespians who are engrossed in adding cryptocurrency to their equity portfolios or are snooping about the capabilities of cryptocurrency as a financial asset.


Introduction
Cryptocurrency is a controversial issue for researchers in the recent era, this is just because of its volatile and digital nature is considered as an important concept for many economic and financial applications, such as portfolio optimization and risk management (Bhosale and Mavale 2018). Cryptocurrency is a secure virtual medium of exchange in the form of digital currency by using cryptography to secure and verify the specific transaction (Mukhopadhyay, Skjellum et al. 2016, Chu, Chan et al. 2017, Chuen, Lee et al. 2017. The digital currency eliminates the centralized system or third parties and their high fee of the transaction (Canetti, Dodis et al. 2007). This advanced digital type of currency looks for after to enhance regular money related structure to back trades without incorporation of revealed in unattainable, while making

Methodology
When modeling volatility deal GARCH family Models, the adequacy of the mean equation is significantly important. Mean equation is given below: This study has to use GARCH type models, i.e. GARCH, EGARCH, PGARCH & TGARCH, each model which has a divergent purpose, with normal error distribution technique to measure the volatility of cryptocurrencies. Specifically, by using GARCH, EGARCH, PGARCH & TGARCH, we modeled the variance for the above Mean equations. The compassion of the result's estimation the distribution assumptions are checked the estimation results of the model. In 1982 Engle examined modeling volatility using conditionally heteroscedastic regression with the Autoregressive Conditional Heteroskedasticity (ARCH) model, but the large lag length is the major problem with such modeling which means large numbers of parameters are required to predict volatility. While using the Generalized Autoregressive Conditional Model (GARCH), conditional variance allows depending upon its lag, typically lessen the number of obligatory ARCH lags when measuring the volatility.

Symmetric Models GARCH (1, 1) Model
The Generalized Autoregressive conditional Model by (Bollerslev 1986) denotes by GARCH (p,q) has: In this equation denotes the logarithmic return of the financial time series in respect t time, μ is mean value of return's representative, is shows the mean equation's error term as well as it can riven into two stochastic pieces i.e. and , depicts independent and identical distributional zero mean, is assumed to have normal distribution with limitation and is dependent standard deviation. GARCH (1, 1) model has presented by the following equation: A positive variance guaranteed in almost all the cases, under the following limitations ω>0 and , but a new GARCH extension models which deals with the weakness of GARCH (1, 1) model and capture diverse features of the financial time series, α + β<1 show the persistency of data. (Nelson 1990, Nelson 1991 introduced the Exponential GARCH model which measured the leverage effects (the asymmetry in return volatility). The following equation gives the general form of EGARCH (p,q):

Asymmetric Models Exponential GARCH (1, 1) Model (EGARCH)
In this equation, γ is representative of a leverage effect or the asymmetric response parameter that can appear with negative or positive sign to depict the future uncertainty. EGARCH (1, 1) shows in the following equation:

The Threshold GARCH (1, 1) Model (TGARCH)
The following equation represents the Threshold GARCH model (Zakoian 1994): Only "I" is a new term in this equation which represents the dummy variable. The threshold GARCH model and the GJR-GARCH model (Glosten, Jagannathan et al. 1993) are almost the same. In TGARCH (1, 1) model, (positive shocks) and (negative shocks) produce a differential effect on volatility.

The Power GARCH Model (PGARCH)
The variance equation of Asymmetric Power ARCH (APARCH (p,q)) Model introduced by (Ding, Granger et al. 1993, Ling and McAleer 2002, Tully and Lucey 2007 in the following equation: Where ω>0, δ>0, ≥0, -1< and ≥0 are shows constant, power parameter, ARCH term, Leverage effect as well as GARCH term respectively. With the change of δ's power the results become different at the power 1 the conditional standard deviation will be drawn, and at the power 2 leverage effects will be estimated and it's become (above equation) classic GARCH model as (Kovačić 2007).

Distribution model and selection criteria
In the analysis of this study Normal Gaussian Distribution which introduced by Carl Friedrich Gauss in 1809 is used (Alspach andSorenson 1972, Barndorff-Nielsen 1977) with the best fitted criterion of Maximum Log Likelihood (Akaike 1974, Bozdogan 1987, minimum the Akaike Information Criterion (AIC) and The Bayesian Information Criterion (BIC) by (Schwarz 1978) respectively.  (Azzalini 1985) Student's t distribution by (Fernández and Steel 1998)

Selection criteria
Equations of Log likelihood by Akaike (1974) and the Bayesian Information Criterion by (Schwarz 1978) presented below: Table 2 comprises the results of descriptive statistics for the daily closing return prices of 8 cryptocurrencies the daily average return of BTC (0.003635), ETH (-0.00362), XRP (-0.00227), XLM (-0.00295), LTC (-0.00123), XMR (-0.00258), DASH (-0.0035) and NEO (-0.00429) with positive standard deviation. Except for Ethereum (ETH), the skewness value of all cryptocurrencies is negative which indicate a long left tail, and the excess kurtosis value from 3 shows the leptokurtic behavior. The Jarque-Bera (JB) test is significant at 1% level, so the statistics value of JB depicts departure from the normality as (Ané 2006, Miron and Tudor 2010, Drachal 2017, Katsiampa 2017. ARCH (5) test for conditional heteroskedasticity rejected the null hypothesis and confirmed the occurrence of ARCH affect in returns of cryptocurrencies which indicates that the GARCH techniques can perform with different specifications (Diebold 2004, Omolo 2014

Description of GARCH type Models
The manifestation of ARCH effect allows to applied the GARCH type models on sample data (Shaw 2018   Under student's t distribution only Power GARCH is significant at 10% level with a negative sign. The results of both distributions indicate the absence of leverage effect, but previous positive shocks or good news has a stronger effect on the subsequent volatility of Bitcoin (Chu, Chan et al. 2017). The Power GARCH is significant at 1% level under both error distribution techniques. The convergence of is which direct inconsistency of error term. The selection criterion (maximum LL, minimum AIC, SIC) indicate the Power GARCH model with Student's t distribution is better to fit as (Watanabe, 2010). The ARCH (5) governed Bitcoin Price Index free from the serial correlation and aberrant distribution of error is observed by the significance of Jarque-Bera test at 1% level as (Ané 2006  1%, 5%, 10% significance level are represented with *, **, *** respectively The results of Ethereum are composed of in table 4 in which easily see the mean constant is significant in EGARCH at 1%, TGARCH at 10% and in PGARCH at 5% level with Normal distribution as well as insignificant with student's t distribution. The constant of variance equation is significant at 5% level under PGARCH model with normal distribution and at 1% level rest of models with both error distribution techniques except PGARCH with student's t distribution. The ARCH and GARCH term are significant at 1% level of confidence in a specific set of GARCH type models with both distribution techniques. The sign of negativity with leverage value has assured the presence of leverage effect in Ethereum price index. Leverage effect is significant under EGARCH with normal distribution at 1% level indicates the negative shocks are having a greater impact on the volatility of Ethereum as compare to positive shocks. TGARCH and PGARCH also showed a positive significance toward leverage effect which also supports the negative shock impact on volatility. With student's t distribution, leverage effect is insignificant in EGARCH with a negative sign and TGARCH & PGARCH with a positive sign. The coefficient of Power GARCH is significant at 1% level with both distributions. The convergence of is which direct error terms are not persistence (Abdullah, Siddiqua et al. 2017). The selection criterion (maximum LL, minimum AIC, SIC) indicate the Exponential GARCH model with Student's t distribution is better to fit as (Bozdogan 1987). The ARCH (5)   1%, 5%, 10% significance level are represented with *, **, *** respectively Table 5 consists the results of Lite coin. The constant of mean is significant under EGARCH, and PGARCH of both distributions and rest of models show an insignificant trend toward mean equation. The constant of variance and ARCH term are significant at 1% level in all except GARCH and TGARCH of student's t distribution. The GARCH term is significant at 10% level in PGARCH with student's t distribution and 1% level under remaining models with both distributions. Leverage effect is positively significant under PGARCH of normal error distribution and EGARCH of student's t distribution. The Power GARCH is significant at 1% level in both error distribution techniques. Maximizing log likelihood and minimizing AIC & SIC governs power GARCH model is the best model for Lite coin from a selected group of GARCH family models. The insignificance of ARCH (5) indicates Lite coin price index has no more serial correlation and the Jarque-Bera's significance shows error is beyond to normal distribution.  Positive significance of EGARCH and negative significance of PGARCH directs the leverage effect is not present in returns but the positive news having an impact on the volatility of Ripple.
Power GARCH is significant at 1% level with a positive sign. Selection criterion leads to PGARCH with student's t distribution as a better-fitted model for Stellar. No further ARCH effect observed and non-normality of returns distribution by whom significance of Jarque-Bera test at 1% level.  1%, 5%, 10% significance level are represented with *, **, *** respectively Table 8 shows the results of Monero which indicate the constant of mean is insignificant in almost all cases except PGARCH with student's t distribution. The constant of variance, ARCH and GARCH term in PGARCH with student's t distribution presents the significance at 5% level and under a remaining set of models the significance level of constant is 1%. In above table, it can be observed that the leverage of EGARCH having positive value or absence of leverage effect in Monero price index but in term of volatility Monero effected by positive shocks. The significance of leverage effect governs leverage effect magnitude and sign of leverage value indicates its direction (Miron and Tudor 2010). No more ARCH (5) effect observed in return series. Significance of Jarque-Bera test pointed toward not normal distribution of errors term. The power GARCH is better-fitted model according to a selection criterion. 1%, 5%, 10% significance level are represented with *, **, *** respectively

ENTREPRENEURSHIP AND SUSTAINABILITY ISSUES
In table 9 the results of Dash are presented which indicated the constant of the mean equation is insignificant under all GARCH models with normal error distribution. The constant mean with student's t distribution is significant at 10% level under all specific models. The constant variance shows significance at 1% and 10% with both distributions and models. The ARCH and GARCH terms are also significant at 1% level. The fluctuation of volatility according to time periods refers the presence of volatility clustering. Except for PGARCH with student's t distribution leverage effect is significant in all cases. The positive significance of EGARCH indicates the absence of leverage effect but the impact of positive events on future future volatility. Power GARCH is significant at 1% level with both distributions. PGARCH with student's t distribution is a best-fitted model for Dash coin. In two cases ARCH (5) is significant at 5% level which indicates the presence of serial correlation in returns and rest of 6 models are shown the elimination of serial correlation. PGARCH model with both distributions indicates the presence of serial correlation and returns not normally distributed observed by the significance of Jarque-Bera's significance at 1% level. (1,1) model and positively significant at 1% level in rest of models. The ARCH term (α) and the GARCH term (β) are significant at 1% level. The greater value of α directed a strong reaction of volatility and β depicts clustering volatility, if the sum of ARCH and GARCH term is less than unity, so data is close to stationary. The leverage effect of EGARCH is positively significant at 5% level with normal distribution and insignificant with a positive sign with student's t ENTREPRENEURSHIP AND SUSTAINABILITY ISSUES ISSN 2345-0282 (online) http://jssidoi.org/jesi/ 2019 Volume 7 Number 3 (March) http://doi.org/10. 9770/jesi.2019.7.3(11) distribution which shows no leverage effect in data. The leverage effect of TGARCH & PGARCH model is negative and significant with normal error distribution and EGARCH, and PGARCH with student's t distribution is insignificant with the positive and negative sign respectively. The negativity of the PGARCH (1,1)'s leverage effect indicates that positive news has more impact on volatility and ratifies the presence of a leverage effect. Power GARCH model is significant at 1% level of confidence. By ARCH (5) test shows that the serial correlation not eliminated observed from NEO prices index. JB explained errors not normally distributed. The maximum value of Log Likelihood (LL) and a minimum value of AIC and SIC governed the PGARCH (1,1) model is a better-fitted model for NEO.

Conclusions
Cryptocurrency and its volatility is a burning issue in the present decade for investors, financial manager, researchers and policy makers. The substantial volatile nature and high growth rate of cryptocurrency increase more interest of investors; because of more fluctuating prices the return rate has keenly effected (Bouoiyour and Selmi 2015). This study not only determined the high rate volatility of volatility in cryptocurrency prices but the better GARCH fitted model with efficient measuring error distribution technique. The findings of Bitcoin, Stellar, Ripple, Monero, Dash, NEO, Lite coin have shown the Power GARCH model with student's t distribution is better fitted model according to selection criterion i.e. LL, AIC & SIC as (Bouri, Azzi et al. 2016, Cermak 2017, Katsiampa 2017. Only Ethereum directs toward Exponential GARCH model with student's t distribution. The smaller and thus asymmetric volatility response to positive shocks explained with contrarian behavior of all digital currencies except Lite coin and Ethereum. The negativity of leverage effect for Lite coin and Ethereum show these both currencies are effected by previous negative shocks in the line of (Baur & Dimpfl, 2018;Chan et al., 2018). Cryptocurrency may suffer the effect of information asymmetry, as its framework is moderately perplexing and in this way may not be effortlessly comprehended by all clients (Ciaian et al. 2014). This study will provide a guideline for investors to check the volatility of cryptocurrency and the effect of both types of shocks (positive & negative) at the fluctuation of cryptocurrency that legitimate safety efforts are winding up more reasonable for general society by guaranteeing that Bitcoin is as protected as would be prudent.