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Cryptocurrencies have become increasingly popular in recent years attracting the attention of the media, academia, investors, speculators, regulators, and governments worldwide. This paper focuses on modelling the volatility dynamics of eight most popular cryptocurrencies in terms of their market capitalization for the period starting from 7th August 2015 to 1st August 2018. In particular, we consider the following cryptocurrencies; Bitcoin, Ethereum, Litecoin, Ripple, Moreno, Dash, Stellar and NEM. The GARCH-type models assuming different distributions for the innovations term are fitted to cryptocurrencies data and their adequacy is evaluated using diagnostic tests. The selected optimal GARCH-type models are then used to simulate out-of-sample volatility forecasts which are in turn utilized to estimate the one-day-ahead VaR forecasts. The empirical results demonstrate that the optimal in-sample GARCH-type specifications vary from the selected out-of-sample VaR forecasts models for all cryptocurrencies. Whilst the empirical results do not guarantee a straightforward preference among GARCH-type models, the asymmetric GARCH models with long memory property and heavy-tailed innovations distributions overall perform better for all cryptocurrencies.

The cryptocurrency market has experienced exponential growth in recent years within a short period of its existence. Cryptocurrencies have become increasingly popular attracting wide coverage from the media and drawing the attention of academia, investors, speculators, regulators, and governments worldwide. A cryptocurrency is a digital asset initially designed to work as a medium of exchange using cryptography [

For the period from January 2017 to December 2017, the market capitalization of the cryptocurrency market increased exponentially. The cryptocurrency market crossed the $100 billion market capitalization for the first time in June 2017, following months of consistent growth [

The cryptocurrency market also experienced its fair share of ups and downs in the year 2018 with events like exchange hacks, market surges and major developments on networks. The hacking of Japan’s largest cryptocurrency OTC market on 26th of January 2018 and the subsequent loss of 530 million US dollars worth of the NEM is the largest ever event of cryptocurrency theft in the history of cryptocurrency markets. The price of Bitcoin lost about 65 percent of its price in a month reaching about 6000 US dollars between January 26, 2018 and February 6, 2018. In March 2018, Coinbase launched the Coinbase Index Fund which tracks the overall performance of the digital assets listed by Coinbase weighted by market capitalization. Late in March 2018, social media giants Facebook, Twitter and search engine Google banned all advertisements related to cryptocurrencies and for initial coin offerings (ICO) and token sales. By the end of the first quarter of 2018, the cryptocurrencies’ market lost about 342 billion US dollars [

By September 2018, cryptocurrencies collapsed 80% of their market capitalization from their highest point in January 2018. This cryptocurrency crash (also known as the Bitcoin Crash) is the worst in the history of cryptocurrencies. By November 15, 2018, Bitcoin’s market capitalization recorded less than 100 billion dollars for the first time since October 2017. Bitcoin being the world’s most widely traded cryptocurrency reflects mounting investor uncertainty over the future of digital currencies. As of 22 December 2018, there were 2067 cryptocurrencies with market value and actively traded in 16,055 cryptocurrency markets and OTC trading desks across the world that are listed on coinmarketcap3. The market capitalization of all the cryptocurrencies stands at $128 billions according to figures from CoinMarketcap.com. The top ten cryptocurrencies represent approximately 85% of the total market value, with Bitcoin dominating with about 53% of the market capitalization. Bitcoin is currently the largest blockchain network, followed by Ripple, Ethereum and Bitcoin cash respectively [

The rest of the paper is structured as follows: Section 2 reviews the cryptocurrency literature. Section 3 presents the GARCH modelling framework including the Maximum Likelihood (ML) estimation of the models with the selected innovations distribution assumptions, VaR estimation and backtesting procedures. Section 4 presents data and some preliminary summary descriptive statistics. Section 5 provides estimation results and empirical results of the VaR backtesting tests and Section 6 concludes the paper.

Cryptocurrencies are generally characterized by high volatility dynamics and extremely erratic price jumps. The cryptocurrency markets still remains a potential source of financial instability and the impact of the unprecedented growth of cryptocurrencies to the financial markets still remains uncertain. Unlike the financial securities like stocks and commodities with regulators and conventional currencies with central banks, cryptocurrencies are completely decentralized and also lack any formal regulation of their markets. There is also limited understanding of the cryptocurrencies as investments assets. Governments and financial market regulatory bodies are particularly concerned about the lack of a formal regulatory framework to regulate the creation of new cryptocurrencies, as well as trading mechanisms in the cryptocurrency markets.

Empirical evidence suggests that cryptocurrencies share most of the stylized facts with financial time series, such as stocks and currencies returns. For example, just like stock prices, cryptocurrency prices also exhibit; time-varying volatility, volatility clustering, asymmetric response to the sign of historical observations of the volatility process (i.e. leverage effects), heavy-tailed distributions and long memory. Cryptocurrencies are also known to be highly volatile and exhibit extreme price jumps compared to traditional financial securities like currencies and are leptokurtic. Osterrieder and Lorenz [

Over the last few years, there has been increased interest in Bitcoin and other cryptocurrencies generally. With the ever increasing interest in cryptocurrencies and their importance in the financial world, there is need for a comprehensive analysis to study volatility dynamics and out-of-sample forecasting behaviour of the cryptocurrencies. However, despite the growing interest, acceptance and integration of cryptocurrencies to the global financial markets, there is limited research on modelling cryptocurrencies’ volatility dynamics. Most of the previous studies have mostly focussed on the Bitcoin market (see e.g. [

However, there have also been several studies on modelling volatility dynamics of the cryptocurrency market recently, for instance, Dyhrberg [

Unfortunately, the majority of recent studies have focused entirely on the Bitcoin behaviour or a few other cryptocurrencies and specifically on the in-sample modelling framework. Trucios [

This paper focusses on analyzing conditional volatility dynamics over eight most popular cryptocurrencies, i.e. Bitcoin, Ethereum, Litecoin, Ripple, Moreno, Dash, Stellar and NEM by market capitalization. The aim is to determine the most appropriate GARCH-type model as well as the best fitting distribution to model the volatility of the major cryptocurrencies returns. This study contributes and extends existing literature on modelling cryptocurrencies volatility dynamics by employing a wider range of GARCH-type models, nine different innovations term distributions and a longer time period to try and fill a gap in the literature. First, a comprehensive in-sample volatility modelling is implemented and their goodness of fit is checked in terms of information selection criteria. The most appropriate GARCH-type models are used to estimate the out-of-sample Value at Risk (VaR) forecasts. The conditional and unconditional coverage tests are used to backtest the accuracy of VaR forecasts. Finally, a comprehensive out-of-sample comparison is implemented to investigate the effects of long memory in the volatility process as well as the asymmetric responses to historical values of the return series to forecast volatility.

This section illustrates the theoretical GARCH modelling framework. First, we outline the alternative Generalized Autoregressive Conditionally Heteroscedastic (GARCH)-type specifications that are used to model time-varying volatility in cryptocurrencies return series and also provide an overview of the set of innovations distributions. Secondly, the selection criteria that will be used to determine the most appropriate GARCH-type specifications are also described. Finally, we describe the estimation of one-day-ahead Value-at-Risk (VaR) forecasts and backtesting procedures.

The GARCH-type models are commonly employed in modelling conditional volatility often present in financial time series. Let P t denote the price of an asset (i.e. cryptocurrency exchange rates) at time t, r t = ln ( P t / P t − 1 ) is the continuously compounded return series, for t = 1 , ⋯ , n . The return series of interest, r t , can be decomposed as follows;

r t = μ t + ε t ,

ε t = σ t z t , (1)

where μ t = E ( r t | F t − 1 ) is the conditional mean given the information set F t − 1 , { ε t } are the return innovations, σ t 2 = Var ( r t | F t − 1 ) = E [ ( r t − μ t ) 2 | F t − 1 ] is the conditional variance of the process { ε t } and { z t } are independent and identically distributed (i.i.d.) innovations with zero mean and unit variance.

The conditional variance equation for standard GARCH (1, 1) model introduced by Bollerslev [

σ t 2 = ω + α ε t − 1 2 + β σ t − 1 2 , (2)

where ω > 0 , α ≥ 0 and β ≥ 0 are unknown parameters. The restrictions on parameters ensure that the conditional variance is always positive. The necessary and sufficient condition for 2 to be uniquely stationary is α + β < 1 and the unconditional variance is given by ω / ( 1 − ( α + β ) ) , thus higher order moments exist. If the GARCH model is correctly specified it will converge to this long term variance as the forecast horizon is increased.

In this paper, twelve GARCH-type specifications are employed in modelling the volatility behaviour of cryptocurrencies, namely: SGARCH, IGARCH, EGARCH, GJR-GARCH, TGARCH, APARCH, CSGARCH, AVGARCH, NGARCH, NAGARCH, FGARCH, and FIGARCH models. All the GARCH-type models implemented follow the same specification in Equation (1); however, in each case, the models are distinguished by the evolution of the volatility process σ t 2 over time. The GARCH extensions involve different specifications for the conditional variance component. For brevity we consider only the first order lags in all GARCH models, i.e. p = q = 1 , since empirical evidence suggests that higher order models rarely performed better than the lower order models in the out-of-sample analysis [

Additionally, for all GARCH-type models, the innovation term { z t } follow one of the nine distributions; Normal distribution, Skew-Normal distribution, (Skew)-Student’s t distribution, (Skew)-GED, (Skew)-Student (GH), Normal Inverse Gaussian (NIG), Generalized Hyperbolic (GH) and the Johnson’s reparametrized SU (JSU) distribution; see Ghalanos [

The parameters of all GARCH-type models are generally estimated using the Quasi-maximum likelihood estimation (QMLE) method. The Quasi-maximum likelihood estimator (QMLE) is preferred since, according to Bollerslev and Wooldridge [

Model | Conditional variance equation | Proposed by |
---|---|---|

IGARCH | σ t 2 = ω + α ε t − 1 2 + ( 1 − α ) σ t − 1 2 | Engle and Bollerslev [ |

EGARCH | ln ( σ t 2 ) = ω + α ε t − 1 2 + γ ( | ε t − 1 | − E ( | ε t − 1 | ) ) + β ln ( σ t − 1 2 ) | Nelson [ |

GJR | σ t 2 = ω + α ε t − 1 2 + γ I ( ε t − 1 < 0 ) ε t − 1 2 + β σ t − 1 2 | Glosten et al. [ |

APARCH | σ t δ = ω + α ( | ε t − 1 | − γ ε t − 1 ) δ + β σ t − 1 δ | Ding et al. [ |

CSGARCH | σ t 2 = q t + α ( r t − 1 2 + q t − 1 ) + β ( σ t − 1 2 + q t − 1 ) | |

q t = ω + ρ q t − 1 + ϕ ε t − 1 2 − σ t − 1 2 | Engle and Lee [ | |

TGARCH | σ t = ω + α σ t − 1 ( | ε t − 1 | − η 1 ε t − 1 ) + β σ t − 1 | Zakoian [ |

AVGARCH | σ t = ω + α σ t − 1 ( | ε t − 1 − η 2 | − η 1 ( ε t − 1 − η 2 ) ) + β σ t − 1 | Schwert and Seguin [ |

NGARCH | σ t δ = ω + α σ t − 1 δ ( | ε t − 1 | ) δ + β σ t − 1 δ | Higgins and Bera [ |

NAGARCH | σ t 2 = ω + α σ t − 1 2 ( | ε t − 1 − η 2 | ) 2 + β σ t − 1 2 | Engle and Ng [ |

FGARCH | σ t δ = ω + α σ t δ ( | ε t − 1 − η 2 | − η 1 ( ε t − 1 − η 2 ) ) δ + β σ t − 1 δ | Hentschel et al. [ |

FIGARCH | ϕ ( L ) ( 1 − L ) d ε t 2 = α 0 + [ 1 − β ( L ) ] ν t | Baillie et al. [ |

Value-at-Risk (or VaR) is a standard risk measure that is commonly used in risk management which summarizes the downside risk into a single value. It is defined as the maximum loss expected due to a change in the investment position with a given probability over a specific period of time. The VaR forecast for the GARCH-type models relies on the one-day-ahead conditional variance forecast, σ t + 1 2 of the volatility model. To this extent, one-step ahead forecasts of the conditional variance of returns is recursively obtained as:

σ ^ t + 1 2 = E ( σ t + 1 2 | F t ) , (3)

where F t is the information set at time t, and σ t 2 is defined as in

For each GARCH-type model, under the assumption of different innovations term distribution the one-day-ahead VaR forecast at α % confidence level is obtained as:

VaR ^ t + 1 ( α ) = μ ^ t + 1 + F − 1 ( α ) σ ^ t + 1 (4)

where F − 1 ( α ) is the α-quantile of the cumulative distribution function of the innovations distribution. All the twelve GARCH-type models proposed in the previous section are used calculate the econometric VaR assuming the nine innovations distributions for all the cryptocurrencies.

Bitcoin | Ethereum | Monero | Litecoin | Dash | Ripple | Stellar | NEM | |
---|---|---|---|---|---|---|---|---|

Nobs | 1090.000 | 1090.000 | 1090.000 | 1090.000 | 1090.000 | 1090.000 | 1090.000 | 1090.000 |

Min | −0.202077 | −1.373989 | −0.291734 | −0.391050 | −0.243432 | −0.601706 | −0.333422 | −0.430828 |

Max | 0.223513 | 0.403457 | 0.567670 | 0.518452 | 0.383096 | 1.010963 | 0.704038 | 1.068486 |

Mean | 0.003053 | 0.004617 | 0.004699 | 0.002739 | 0.003789 | 0.003665 | 0.004314 | 0.006469 |

Std.Dev | 0.040218 | 0.080835 | 0.072959 | 0.058599 | 0.059552 | 0.077085 | 0.086585 | 0.094524 |

Skewness | −0.160668 | −4.207377 | 1.023870 | 1.351149 | 0.875008 | 3.091153 | 2.083472 | 2.179137 |

Kurtosis | 4.568298 | 78.519090 | 7.087234 | 13.339778 | 5.097968 | 38.634223 | 14.509124 | 19.372076 |

JB | 958.2979 | 284,299.01 | 2484.144 | 8450.867 | 1326.721 | 69,798.432 | 10,394.084 | 17,981.302 |

p−value | (0.0000) | (0.0000) | (0.0000) | (0.0000) | (0.0000) | (0.0000) | (0.0000) | (0.0000) |

Skewness Test statistic | ||||||||

Statistic | −3.756248 | −98.22221 | 23.90249 | 31.54289 | 20.42727 | 72.16369 | 48.63914 | 50.87247 |

p-value | (0.00009) | (0.0000) | (0.0000) | (0.0000) | (0.0000) | (0.0000) | (0.0000) | (0.0000) |

ADF Test statistics | ||||||||

Statistic | −2.7535 | −9.2397 | −9.6014 | −9.36 | −8.8872 | −8.343 | −8.827 | −9.2518 |

P-value | (0.0000) | (0.0000) | (0.0000) | (0.0000) | (0.0000) | (0.0000) | (0.0000) | (0.0000) |

Ljung-Box Test statistics at various lags | ||||||||

Q(5) | 3.9483 | 17.086 | 14.409 | 4.0676 | 7.5975 | 28.37 | 18.857 | 22.819 |

p-value | (0.5569) | (0.004339) | (0.01321) | (0.5397) | (0.1799) | (0.00001) | (0.002044) | (0.000366) |

Q(10) | 9.4171 | 23.716 | 34.91 | 24.166 | 16.638 | 46.097 | 29.805 | 29.725 |

p-value | (0.493) | (0.008391) | (0.0001294) | (0.007172) | (0.08277) | (1.377e-06) | (0.0009219) | (0.0009501) |

Q(15) | 11.065 | 27.348 | 37.383 | 31.254 | 21.286 | 51.782 | 32.263 | 29.725 |

p-value | (0.748) | (0.02602) | (0.001112) | (0.008123) | (0.128) | (6.138e-06) | (0.005928) | (0.0009501) |

Q(20) | 24.583 | 43.69 | 43.322 | 37.097 | 38.457 | 67.26 | 39.917 | 45.247 |

p-value | (0.2178) | (0.001655) | (0.001853) | (0.01139) | (0.007783) | (5.069e-07) | (0.005118) | (0.001022) |

ARCH-LM Test statistics at various lags | ||||||||

LM (5) | 76.774 | 252.35 | 43.329 | 40.299 | 42.939 | 109.96 | 189.39 | 29.864 |

p-value | (0.0000) | (0.0000) | (0.0000) | (0.0000) | (0.0000) | (0.0000) | (0.0000) | (0.00002) |

LM (10) | 85.385 | 144.71 | 107.36 | 67.757 | 67.394 | 120.83 | 191.84 | 32.942 |

p-value | (0.0000) | (0.0000) | (0.0000) | (0.0000) | (0.0000) | (0.0000) | (0.0000) | (0.00028) |

LM(20) | 106.63 | 123.8 | 124.14 | 81.364 | 78.707 | 123.99 | 197.94 | 33.56 |

p-value | (0.0000) | (0.0000) | (0.0000) | (0.0000) | (0.0000) | (0.0000) | (0.0000) | (0.02926) |

The accuracy of the volatility models in predicting VaR is assessed using statistical backtesting methods. The starting point is normally to compare the out-of-sample VaR forecasts with the actual realized returns in the next time period and this is summarized in terms of a hit ratio. { I t } is a sequence of violations, where it takes the value one if the ex-post loss exceeds the VaR predicted at time t + 1 and the value zero otherwise. Mathematically, the hit function which is also referred to as the indicator function is defined as:

I t ( α ) = ( 1 if r t + 1 < VaR t + 1 | t ( α ) 0 if r t + 1 ≥ VaR t + 1 | t ( α ) (5)

where { r 1 , r 2 , ⋯ , r N } is the sequence of daily return, α is the quantile level of coverage defined by its confidence level.

For a VaR model to be accurate in its predictions, then the average hit ratio or the failure rate over the full sample should be equal α for the ( 1 − α ) th quantile VaR (i.e., for 97.5% VaR, α = 2.5 % ). As expected, the closer the hit ratio is to the expected value ( 1 − α ) , the better the forecasts of the risk model. If the hit ratio is greater than the expectation, then the model underestimates the risk; with a hit ratio smaller than ( 1 − α ) , the model overestimates risk. In this study, two accuracy measure tests: Kupiec [

Kupiec [

LR uc = − 2 ln [ p N ( 1 − p ) T − N ( N T ) N ( 1 − N T ) T − N ] (6)

The Kupiec’s unconditional coverage test has a chi-square distribution, asymptotically, with one degree of freedom. The test can be employed to test whether the sample point estimate is statistically consistent with the VaR model’s prescribed confidence level. This can reject a model that either overestimates or underestimates the true but unobservable VaR, however, it cannot scrutinize whether the exceptions are randomly distributed.

According to Christoffersen [_{cc}) to jointly test the correct unconditional coverage and serial independence. The LR_{cc} test is a joint test of these two properties and the corresponding test statistic is the sum of the individual test statistics for the properties; i.e., LR cc = LR uc + LR ind when conditioned on the first observation. The LR_{ind} test denotes the likelihood ratio statistic that tests whether exceptions are independent, and the LR_{uc} is defined in the previous subsection. Thus, under the null hypothesis of the expected proportion of exceptions equals p and the failure process is independent, the appropriate likelihood ratio test statistic is expressed as follows:

LR cc = − 2 ln [ p N ( 1 − p ) T − N π ^ 01 n 01 ( 1 − π ^ 01 ) n 00 π ^ 11 n 11 ( 1 − π ^ 11 ) n 10 ] (7)

where n i j denotes the number of observations with value i followed by value j ( i , j = 0 , 1 ) , π i j = P { I t = j | T t − 1 = i } ( i , j = 0 , 1 ) , π ^ 01 = n 01 / ( n 00 + n 01 ) , and π ^ 11 = n 11 / ( n 10 + n 11 ) . The Christoffersen’s conditional coverage test has an asymptotically chi-square distribution, with two degrees of freedom.

The sample data used in this empirical study was extracted from http://www.investing.com/. Specifically, the data consists of the daily closing prices of cryptocurrencies starting from 7th August 2015 until the 1st August 2018. The full sample data yields a total of 1091 daily observations, including weekends since trading in cryptocurrencies is not restricted to business days or the trading hours of stock exchanges. A starts date of 7th August 2015 was purposely chosen so that we can analyze eight of the top fifteen cryptocurrencies, ranked according to their market capitalization, as of 7th August 2018 (see [

The summary descriptive statistics and statistical tests results for the daily returns of each cryptocurrency are presented in

and exhibit leptokurtic behaviour beyond that of the normal distribution, with the most peaked being those of Ethereum and Ripple. Moreover, only Bitcoin and Ethereum are negatively skewed while other cryptocurrencies are positively skewed. Additionally, the Jarque-Bera statistic confirms that all cryptocurrencies are not normally distributed. The Augmented Dickey Fuller (ADF) test results reject unit root hypothesis for all cryptocurrencies series, implying that the series are assumed to be stationary. Ljung-Box (Q) statistic for raw returns series reject the null hypothesis that all correlation coefficients up to lag 20 are equal to zero in the majority of cases, except for Bitcoin. Therefore we conclude that some return series present some linear dependence. The significant serial correlations reported in the squared returns imply that there is non-linear dependence in the return series. Finally, the ARCH-LM test rejects the no ARCH effect hypothesis, thus indicating the presence of volatility clustering, long memory and a GARCH-type specification should be considered in the modelling of cryptocurrencies.

In this study, twelve GARCH-type models: the SGARCH, IGARCH, EGARCH, GJR-GARCH, TGARCH, APARCH, CSGARCH, AVGARCH, NGARCH, NAGARCH, FGARCH, and FIGARCH models are utilized to model the conditional volatility and estimate one-step-ahead VaR forecast of the eight cryptocurrencies. Further, two backtesting measures: the conditional and unconditional coverage tests are used to evaluate the out-of-sample VaR forecasts performance of the twelve GARCH models. Prior to implementing the comparative performance of VaR forecast for the above twelve GARCH models, the fitting of the implemented twelve models is explored via the empirical results of the parameter estimates for the competing models.

First, the best fitting ARMA models for the mean components are selected via the information criteria: the Akaike Information Criterion (AIC). The ARMA (p, q) specification for Bitcoin, Ethereum, Monero, Litecoin, and Dash are assumed to be equal to zero, based on the BIC. This indicates that even the AR (1) model is not necessary since there is no significant degree of serial autocorrelation in cryptocurrencies returns. The most appropriate models for Stellar and NEM are MA (1) and MA (2) respectively, while AR (1) is selected for Ripple. For brevity in modelling and forecasting the cryptocurrencies volatility, we assume that mean component is not significant for all the cryptocurrencies return series.

The distribution of the innovations term is also an important component in modelling a GARCH process. For purposes of selecting the most appropriate innovations distribution for all cryptocurrencies, the GARCH (1, 1) model is utilized. The information criteria and log-likelihood results for the fitted GARCH (1, 1) model assuming the nine different innovations distributions are reported in

The most appropriate GARCH-type model is selected from the different specifications (GARCH, IGARCH, EGARCH, GJRGARCH, APARCH, TGARCH, NGARCH, NAGARCH, AVGARCH, FIGARCH and HGARCH) fitted to the eight cryptocurrencies with their respective best fitting innovations distribution.

Error Distn | Norm | Skewed Norm | Student t | Skewed Student t | ged | sged | nig | ghyp | jsu |
---|---|---|---|---|---|---|---|---|---|

BitCoin | |||||||||

AIC | −3.9038 | −3.9076 | −4.1193 | −4.1189 | −4.1373 | −4.1380 | −4.1395 | −4.1412 | −4.1316 |

BIC | −3.8763 | −3.8755 | −4.0872 | −4.0823 | −4.1052 | −4.1013 | −4.1029 | −4.1000 | −4.0950 |

LLF | 2133.59 | 2136.63 | 2252.02 | 2252.80 | 2261.81 | 2263.19 | 2264.04 | 2265.95 | 2259.75 |

Ethereum | |||||||||

AIC | −2.4230 | −2.4596 | −2.8341 | −2.8377 | −2.8346 | −2.8348 | −2.8304 | −2.8302 | −2.8381 |

BIC | −2.3955 | −2.4275 | −2.8020 | −2.8010 | −2.8025 | −2.7982 | −2.7938 | −2.7889 | −2.8014 |

LLF | 1326.51 | 1347.48 | 1551.58 | 1554.53 | 1551.87 | 1552.99 | 1550.57 | 1551.43 | 1554.76 |

Monero | |||||||||

AIC | −2.5109 | −2.5304 | −2.7169 | −2.7191 | −2.7029 | −2.7089 | −2.7176 | −2.7166 | −2.7192 |

BIC | −2.4834 | −2.4984 | −2.6848 | −2.6824 | −2.6708 | −2.6723 | −2.6809 | −2.6754 | −2.6825 |

LLF | 1374.42 | 1386.09 | 1487.69 | 1489.89 | 1480.09 | 1484.36 | 1489.07 | 1489.54 | 1489.96 |

Litecoin | |||||||||

AIC | −3.0858 | −2.3136 | −3.7286 | −3.7305 | −3.7298 | −3.7300 | −3.7429 | −3.7410 | −3.7460 |

BIC | −3.0583 | −2.2815 | −3.6965 | −3.6939 | −3.6977 | −3.6934 | −3.7062 | −3.6998 | −3.7094 |

LLF | 1687.77 | 1267.89 | 2039.09 | 2041.13 | 2039.75 | 2040.88 | 2047.87 | 2047.84 | 2049.59 |

Dash | |||||||||

AIC | −2.9812 | −3.0190 | −3.1359 | −3.1458 | −3.1210 | −3.1347 | −3.1464 | −3.1466 | −3.1474 |

BIC | −2.9537 | −2.9869 | −3.1039 | −3.1091 | −3.0889 | −3.0981 | −3.1098 | −3.1054 | −3.1107 |

LLF | 1630.75 | 1652.36 | 1716.08 | 1722.45 | 1707.92 | 1716.41 | 1722.79 | 1723.89 | 1723.32 |

Ripple | |||||||||

AIC | −2.9755 | −3.0230 | −3.4227 | −3.4264 | −3.4151 | −3.4278 | −3.4317 | −3.4370 | - |

BIC | −2.9480 | −2.9909 | −3.3907 | −3.3897 | −3.3831 | −3.4278 | −3.3950 | −3.3958 | - |

LLF | 1627.67 | 1654.51 | 1872.39 | 1875.37 | 1868.25 | 1876.17 | 1878.27 | 1882.18 | - |

Stellar | |||||||||

AIC | −2.4069 | −2.4698 | −2.7620 | −2.7724 | −2.7429 | −2.7526 | −2.7778 | −2.7775 | −2.7784 |

BIC | −2.3794 | −2.4378 | −2.7300 | −2.7357 | −2.7109 | −2.7159 | −2.7412 | −2.7362 | −2.77417 |

LLF | 1317.78 | 1353.06 | 1512.31 | 1518.94 | 1501.90 | 1508.15 | 1521.92 | 1522.71 | 1522.22 |

NEM | |||||||||

AIC | −2.1393 | −2.1532 | −2.3528 | −2.3573 | −2.3390 | −2.3443 | −2.3562 | −2.3568 | −2.3583 |

BIC | −2.1118 | −2.1212 | −2.3208 | −2.3206 | −2.3069 | −2.3076 | −2.3195 | −2.3156 | −2.3217 |

LLF | 1171.94 | 1180.52 | 1289.30 | 1292.71 | 1281.74 | 1285.62 | 1292.12 | 1293.48 | 1293.29 |

GARCH (1, 1) | EGARCH (1, 1) | GJR- GARCH (1, 1) | CSGARCH (1, 1) | APARCH (1, 1) | IGARCH (1, 1) | TGARCH (1, 1) | AVGARCH (1, 1) | NGARCH (1, 1) | NAGARCH (1, 1) | FIGARCH (1, 1) | ALL- GARCH (1, 1) | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

BitCoin | ||||||||||||

AIC | −4.1379 | −4.1480 | −4.1415 | −4.1360 | −4.1458 | −4.1399 | −4.1465 | −4.1471 | −4.1433 | −4.1422 | −4.1417 | −4.1454 |

BIC | −4.1104 | −4.1159 | −4.1095 | −4.0994 | −4.1092 | −4.1170 | −4.1144 | −4.1105 | −4.1112 | −4.1102 | −4.1096 | −4.1042 |

HQC | −4.1275 | −4.1358 | −4.1294 | −4.1222 | −4.1319 | −4.1313 | −4.1344 | −4.1333 | −4.1312 | −4.1301 | −4.1295 | −4.1298 |

Ethereum | ||||||||||||

AIC | −2.8299 | −2.8337 | −2.8283 | −2.8618 | −2.8306 | −2.8315 | −2.8172 | −2.8154 | −2.8324 | −2.8289 | −2.8312 | −2.8300 |

BIC | −2.8024 | −2.8016 | −2.7962 | −2.8252 | −2.7940 | −2.8086 | −2.7852 | −2.7787 | −2.8003 | −2.7968 | −2.7992 | −2.7888 |

HQC | −2.8195 | −2.8215 | −2.8161 | −2.8480 | −2.8168 | −2.8228 | −2.8051 | −2.8015 | −2.8202 | −2.8167 | −2.8191 | −2.8144 |

Monero | ||||||||||||

AIC | −2.7141 | −2.7122 | −2.7150 | −2.7105 | −2.7132 | −2.7158 | −2.7082 | −2.7172 | −2.7124 | −2.7191 | −2.7137 | −2.7176 |

BIC | −2.6866 | −2.6802 | −2.6829 | −2.6738 | −2.6765 | −2.6929 | −2.6761 | −2.6805 | −2.6803 | −2.6871 | −2.6816 | −2.6764 |

HQC | −2.7037 | −2.7001 | −2.7029 | −2.6966 | −2.6993 | −2.7071 | −2.6961 | −2.7033 | −2.7002 | −2.7070 | −2.7015 | −2.7020 |

Litecoin | ||||||||||||

AIC | −3.7133 | −3.7321 | −3.7241 | −3.7289 | −3.7421 | −3.7155 | −3.7438 | −3.7427 | −3.7374 | −3.7170 | −3.7270 | −3.7413 |

BIC | −3.6859 | −3.7001 | −3.6920 | −3.6923 | −3.7054 | −3.6926 | −3.7117 | −3.7060 | −3.7053 | −3.6849 | −3.6949 | −3.7001 |

HQC | −3.7029 | −3.7200 | −3.7120 | −3.7151 | −3.7282 | −3.7069 | −3.7317 | −3.7288 | −3.7253 | −3.7048 | −3.7149 | −3.7257 |

Dash | ||||||||||||

AIC | −3.1483 | −3.1495 | −3.1476 | −3.1553 | −3.1463 | −3.1493 | −3.1456 | −3.1437 | −3.1477 | −3.1484 | −3.1567 | −3.1450 |

BIC | −3.1209 | −3.1174 | −3.1155 | −3.1186 | −3.1097 | −3.1264 | −3.1135 | −3.1071 | −3.1156 | −3.1164 | −3.1246 | −3.1037 |

HQC | −3.1379 | −3.1374 | −3.1355 | −3.1414 | −3.1325 | −3.1406 | −3.1334 | −3.1299 | −3.1356 | −3.1363 | −3.1445 | −3.1294 |

Ripple | ||||||||||||

AIC | −3.4161 | −3.4223 | −3.4131 | −3.4007 | - | −3.4145 | −1.3035 | −1.2016 | −3.4390 | −3.4273 | −3.4629 | −3.4298 |

BIC | −3.3794 | −3.3811 | −3.3719 | −3.3549 | - | −3.3824 | −1.2623 | −1.1558 | −3.3978 | −3.3861 | −3.4217 | −3.3794 |

HQC | −3.4022 | −3.4067 | −3.3975 | −3.3834 | - | −3.4024 | −1.2879 | −1.1843 | −3.4234 | −3.4117 | −3.4473 | −3.4107 |

Stellar | ||||||||||||

AIC | −2.7801 | −2.7847 | −2.7784 | −2.8172 | −2.7864 | −2.7820 | −2.7881 | −2.7883 | −2.7858 | −2.7813 | −2.8104 | −2.7864 |

BIC | −2.7480 | −2.7480 | −2.7417 | −2.7759 | −2.7452 | −2.7545 | −2.7515 | −2.7470 | −2.7492 | −2.7447 | −2.7737 | −2.7406 |

HQC | −2.7679 | −2.7708 | −2.7645 | −2.8016 | −2.7708 | −2.7716 | −2.7743 | −2.7727 | −2.7719 | −2.7675 | −2.7965 | −2.7691 |

NEM | ||||||||||||

AIC | −2.3583 | −2.3686 | −2.3566 | −2.3694 | −2.3718 | −2.3587 | −2.3708 | −2.3700 | −2.3702 | −2.3578 | −2.3716 | −2.3717 |

BIC | −2.3217 | −2.3274 | −2.3154 | −2.3236 | −2.3260 | −2.3266 | −2.3296 | −2.3242 | −2.3289 | −2.3166 | −2.3304 | −2.3213 |

HQC | −2.3445 | −2.3530 | −2.3410 | −2.3520 | −2.3545 | −2.3466 | −2.3552 | −2.3526 | −2.3546 | −2.3422 | −2.3560 | −2.3527 |

The parameter estimates for the most appropriate GARCH-type model selected for each cryptocurrencies together with the specifications tests of residual autocorrelation and conditional heteroscedasticity are given in

Bitcoin | Ethereum | Monero | Litecoin | Dash | Ripple | Stellar | NEM | |
---|---|---|---|---|---|---|---|---|

α | 0.001958 | 0.001771 | 0.001546 | 0.001203 | 0.002788 | 0:001548 | 0:000520 | 0.001481 |

(0.024787) | (0.041057) | (0.378756) | (0.104764) | (0.050867) | (0.011039) | (0.676378) | (0.438475) | |

ω | 0.000013 | 0.000220 | 0.000397 | 0.000580 | 0.000101 | 0.000031 | 0.000000 | 0.000153 |

(0.00028) | (0.03340) | (0.004456) | (0.048486) | (0.026018) | (0.221990) | (0.361084) | (0.150275) | |

α_{1} | 0.145588 | 0.035488 | 0.260066 | 0.209287 | 0.171916 | 0.226872 | 0.26754 | 0.374642 |

(0.00000) | (0.00000) | (0.00000) | (0.000002) | (0.000050) | (0.001339) | (0.000001) | (0.042643) | |

β_{1} | 0.854412 | 0.944190 | 0.739934 | 0.868738 | 0.828084 | 0.103973 | 0.484999 | 0.896461 |

(NA) | (0.00000) | (NA) | (0.00000) | (NA) | (0.000066) | (0.000002) | (0.00000) | |

γ | - | - | - | −0.312124 | - | |||

(0.011762) | ||||||||

d | - | - | - | - | - | 0.302758 | 1.000000 | |

(0.00000) | (0.002696) | |||||||

δ | - | 0.999516 | 0.999936 | |||||

(0.00000) | (0.00000) | |||||||

f | - | 0.271006 | - | - | - | 0.023251 | ||

(0.000014) | (0.00000) | |||||||

α_{1} + β_{1} | 1 | 0.979678 | 1 | 1.078025 | 1 | 0.33079326 | 0.752539 | 1.645745 |

Skew | 0.969862 | 1.092519 | 1.091389 | 0.116474 | 0.372318 | 1.116250 | 0.316974 | 0.176591 |

(0.00000) | (0.00000) | (0.00000) | (0.012925) | (0.000334) | (0.00000) | (0.000023) | (0.023133) | |

Shape | 0.923403 | 0.992872 | 3.336433 | 0.928236 | 1.391832 | 0.807481 | 1.172223 | 1.237510 |

(0.00000) | (0.00000) | (0.00000) | (0.00000) | (0.00000) | (0.00000) | (0.00000) | (0.00000) | |

ARCH-LM test for heteroscedasticity | ||||||||

Statistic | 2.376 | 2.463 | 3.5232 | 0.16112 | 0.40051 | 0.19493 | 0.39728 | 0.7637 |

(0.6381) | (0.6201) | (0.4206) | (0.9982) | (0.9865) | (0.9972) | (0.9867) | (0.9487) |

The accuracy of the different fitted GARCH-type models considered in the study is assessed by using exceedances percentages at 95%, 97.5%, and 99% confidence levels. The exceedances involve counting the number of actual realized returns that exceed the VaR forecast, and comparing this number with the hypothetically expected number of exceedances for a given probability. Obviously, the closer the observed number of exceedances is to the hypothetically expected number, the more preferable the GARCH model is for estimating accurate forecasts.

For the 99% VaR forecasts, the violation rates are all relatively close to the expected exceedances rates for most of the GARCH-type models and all cryptocurrencies. Some of the GARCH-models selected at 95% level still perform well at 99% level. The APARCH (1, 1) model still gives the best fit for Bitcoin and Ethereum; EGARCH (1, 1) model for Moreno; CSGARCH (1, 1) for Dash. Litecoin, Stellar and also Bitcoin have several best fitting models including; GARCH (1, 1), CSGARCH (1, 1), AVGARCH (1, 1), APARCH (1, 1) and TGARCH (1, 1). Finally, EGARCH (1, 1) and CSGARCH (1, 1) give the best fit for NEM and GARCH (1, 1) and GJR (1, 1) for Ripple.

We also decided to backtest the GARCH-type model analyzed, since every model has a different distribution of residuals. The forecasting and backtesting procedure is implemented using a fixed-rolling-window scheme. This approach allows us to perform a rolling estimation and forecasting of the GARCH-type model, returning the VaR at specified levels of significance. Notably, it generates the distributional forecast parameters necessary to compute any required measure on the forecast density. The parameters of the fitted GARCH-type models are estimated over a window of length 700 observations and are used to predict the conditional variance process for the following day. Each time the window is shifted forward, the daily returns of the following day are added, the oldest daily returns are dropped from the observation window and the parameters are re-estimated over the new period in order to compute the next set of forecasts. This procedure is iterated until the end of the dataset for a total of 300 one-step ahead forecasts.

Model | GARCH (1, 1) | EGARCH (1, 1) | GJR-GARCH (1, 1) | CS-GARCH (1, 1) | APARCH (1, 1) | IGARCH (1, 1) | TGARCH (1, 1) | AV- GARCH (1, 1) | NGARCH (1, 1) | NA-GARCH (1, 1) | ALL-GARCH (1, 1) |
---|---|---|---|---|---|---|---|---|---|---|---|

95% level of significance | |||||||||||

Bitcoin | 8.0% | 6.7% | 7.3% | 7.7% | 6.3% | 8.0% | 6.7% | 7.3% | 6.3% | 7.3% | 7.0% |

Ethereum | 7.7% | 7.0% | 7.3% | 8.7% | 7.0% | 7.3% | 7.0% | 7.7% | 7.3% | 7.3% | 8.0% |

Monero | 9.0% | 9.3% | 9.0% | 9.3% | 9.3% | 9.0% | 9.0% | 9.0% | 9.0% | 8.7% | -- |

Litecoin | 10.3% | 7.3% | 10.0% | 9.0% | 7.7% | 10.3% | 8.0% | 7.0% | 4.7% | 10.0% | 7.0% |

Dash | 8.0% | 8.0% | 7.3% | 7.0% | 7.3% | 8.0% | 8.0% | 8.0% | 7.7% | -- | 8.3% |

Ripple | 7.0% | 6.0% | 7.3% | 6.7% | 6.7% | 5.3% | 28% | 28% | 6.0% | 6.7% | 8.3% |

Stellar | 7.0% | 7.0% | 7.3% | -- | 6.0% | 7.0% | 6.7% | 6.7% | 5.7% | 7.3% | 6.0% |

NEM | 9.0% | 9.0% | 8.7% | 8.0% | 9.3% | 8.7% | 9.3% | 9.0% | 9.0% | 9.0% | 10.0% |

97.5% level of significance | |||||||||||

Bitcoin | 3.3% | 2.7% | 4.0% | 4.0% | 2.7% | 3.3% | 2.3% | 2.7% | 2.0% | 4.7% | 2.7% |

Ethereum | 4.0% | 3.7% | 4.0% | 4.3% | 3.7% | 4.0% | 3.3% | 3.7% | 3.3% | 4.0% | 3.3% |

Monero | 5.3% | 5.3% | 5.3% | 5.3% | 5.7% | 5.3% | 5.0% | 6.0% | 5.3% | 5.0% | -- |

Litecoin | 3.7% | 2.7% | 4.7% | 3.7% | 1.7% | 3.7% | 3.0% | 2.3% | 0.7% | 6.7% | 3.0% |

Dash | 5.3% | 4.3% | 5.3% | 5.0% | 5.7% | 5.3% | 4.7% | 6.0% | 5.7% | -- | 6.3% |

Ripple | 3.3% | 3.3% | 3.3% | 3.3% | 3.0% | 2.3% | 24.3% | 24.7% | 2.7% | 3.0% | 4.0% |

Stellar | 3.3% | 3.0% | 3.3% | -- | 2.7% | 3.3% | 2.7% | 2.3% | 2.7% | 4.3% | 2.3% |

NEM | 4.3% | 3.7% | 4.3% | 3.3% | 3.7% | 3.3% | 3.7% | 4.0% | 3.0% | 4.3% | 4.7% |

99% level of significance | |||||||||||

Bitcoin | 1.3% | 1.0% | 1.3% | 1.0% | 1.0% | 1.3% | 1.0% | 1.0% | 1.0% | 1.7% | 1.0% |

Ethereum | 1.7% | 1.7% | 1.7% | 1.7% | 1.3% | 1.7% | 1.3% | 1.7% | 1.7% | 1.7% | 1.7% |

Monero | 1.3% | 1.0% | 1.3% | 1.3% | 1.3% | 1.3% | 0.7% | 2.0% | 1.3% | 1.7% | -- |

Litecoin | 1.0% | 0.7% | 1.7% | 1.0% | 1.0% | 1.0% | 1.0% | 1.0% | 0.3% | 2.7% | 1.7% |

Dash | 1.7% | 2.0% | 1.7% | 1.3% | 2.7% | 1.7% | 2.0% | 2.7% | 3.3% | -- | 4.0% |

Ripple | 1.0% | 0.7% | 1.0% | 0.7% | 0.7% | 0.3% | 19% | 19% | 0.7% | 1.0% | 1.3% |

Stellar | 2.0% | 1.3% | 2.0% | -- | 1.3% | 2.0% | 1.7% | 1.3% | 1.7% | 2.3% | 1.3% |

NEM | 1.7% | 1.0% | 1.7% | 1.0% | 1.3% | 1.7% | 0.7% | 1.7% | 1.7% | 1.7% | 2.0% |

The Kupiec’s unconditional and Christoffersen’s conditional coverage Value-at-Risk exceedances tests are utilized to assess the VaR forecast performance of the twelve GARCH-type models: the SGARCH, IGARCH, EGARCH, GJR-GARCH, TGARCH, APARCH, CSGARCH, AVGARCH, NGARCH, NAGARCH, FGARCH, and the FIGARCH models at 95%, 97.5%, and 99% confidence levels. In principle, the GARCH-type model with the higher number of passes among the two back-testing procedures bear a better performance than the GARCH-type model with the less number that passes. The most appropriate GARCH-type model according to conditional and unconditional coverage tests is defined as the one with the highest p-value amongst all the fitted models for all cryptocurrencies.

_{uc}) test and Christoffersen conditional (LR_{cc}) test for twelve GARCH-type models fitted to the cryptocurrencies under 95%, 97.5%, and 99% confidence level. p-values of the unconditional coverage and coverage tests are presented in parentheses. Besides the hypothetical expected percentage of exceedances for the 5%, 2.5%, and 1% level of significance, the percentage of actual exceedances is presented for selected quantiles associated with the distribution. In addition, regarding a specified GARCH-type model, the total number of times that a cryptocurrency pass the LR_{uc} and LR_{cc} types of back-testing are counted respectively at different levels

α | GARCH (1, 1) | EGARCH (1, 1) | GJR-GARCH (1, 1) | CS-GARCH (1, 1) | APARCH (1, 1) | IGARCH (1, 1) | TGARCH (1, 1) | AV-GARCH (1, 1) | NGARCH (1, 1) | NAGARCH (1, 1) | ALL-GARCH (1, 1) | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Bitcoin | ||||||||||||

1% | LR_{UC} | 0.305 | 0.000 | 0.305 | 0.000 | 0.000 | 0.305 | 0.000 | 0.000 | 0.000 | 1.122 | 0.000 |

p-value | (0.581) | (1.000) | (0.581) | (1.000) | (1.000) | (0.581) | (1.000) | (1.000) | (1.000) | (0.290) | (1.000) | |

LR_{CC} | 0.413 | 0.061 | 0.413 | 0.061 | 0.061 | 0.413 | 0.061 | 0.061 | 0.061 | 1.292 | 0.061 | |

p-value | (0.813) | (0.970) | (0.813) | (0.970) | (0.970) | (0.813) | (0.970) | (0.970) | (0.970) | (0.524) | (0.970) | |

2.5% | LR_{UC} | 0.775 | 0.033 | 2.350 | 2.350 | 0.033 | 0.775 | 0.035 | 0.033 | 0.330 | 4.622 | 0.289 |

p-value | (0.379) | (0.855) | (0.125) | (0.125) | (0.855) | (0.379) | (0.852) | (0.855) | (0.566) | (0.032) | (0.591) | |

LR_{CC} | 1.467 | 0.473 | 3.353 | 3.353 | 0.473 | 1.467 | 0.371 | 0.473 | 0.576 | 4.796 | 0.848 | |

p-value | (0.480) | (0.789) | (0.187) | (0.187) | (0.789) | (0.480) | (0.831) | (0.789) | (0.750) | (0.091) | (0.654) | |

5% | LR_{UC} | 4.847 | 1.596 | 3.025 | 3.889 | 1.039 | 4.847 | 1.596 | 3.025 | 1.039 | 3.025 | 2.259 |

p-value | (0.028) | (0.207) | (0.082) | (0.049) | (0.308) | (0.028) | (0.207) | (0.082) | (0.308) | (0.082) | (0.133) | |

LR_{CC} | 5.583 | 1.702 | 3.1169 | 3.951 | 1.082 | 5.583 | 1.666 | 3.169 | 3.479 | 3.169 | 2.406 | |

p-value | (0.061) | (0.427) | (0.205) | (0.139) | (0.582) | (0.061) | (0.435) | (0.205) | (0.176) | (0.205) | (0.300) | |

Ethereum | ||||||||||||

1 | LR_{UC} | 1.122 | 1.122 | 1.122 | 1.122 | 0.305 | 1.122 | 0.305 | 1.122 | 1.122 | 1.122 | 1.122 |

p-value | (0.290) | (0.290) | (0.290) | (0.290) | (0.581) | (0.290) | (0.581) | (0.290) | (0.290) | (0.290) | (0.290) | |

LR_{CC} | 1.292 | 1.292 | 1.292 | 1.292 | 0.413 | 1.292 | 0.413 | 1.292 | 1.292 | 1.292 | 1.292 | |

p-value | (0.524) | (0.524) | (0.524) | (0.524) | (0.813) | (0.524) | (0.813) | (0.524) | (0.524) | (0.524) | (0.524) | |

2.5% | LR_{UC} | 2.350 | 1.468 | 2.350 | 3.405 | 1.468 | 2.350 | 0.775 | 1.468 | 0.775 | 2.350 | 0.775 |

p-value | (0.125) | (0.226) | (0.125) | (0.065) | (0.226) | (0.125) | (0.379) | (0.226) | (0.379) | (0.125) | (0.379) | |

LR_{CC} | 3.353 | 2.308 | 3.358 | 4.588 | 2.308 | 3.353 | 1.467 | 2.308 | 1.467 | 3.358 | 1.467 | |

p-value | (0.187) | (0.315) | (0.187) | (0.101) | (0.315) | (0.187) | (0.480) | (0.315) | (0.480) | (0.187) | (0.480) | |

5% | LR_{UC} | 3.889 | 2.259 | 3.015 | 7.033 | 2.259 | 3.025 | 2.259 | 3.889 | 3.025 | 3.025 | 4.847 |

p-value | (0.049) | (0.133) | (0.082) | (0.008) | (0.133) | (0.082) | (0.133) | (0.049) | (0.082) | (0.082) | (0.028) | |

LR_{CC} | 4.747 | 3.723 | 4.163 | 7.300 | 3.723 | 4.163 | 3.723 | 4.884 | 4.163 | 4.163 | 5.583 | |

p-value | (0.093) | (0.155) | (0.125) | (0.026) | (0.155) | (0.125) | (0.155) | (0.087) | (0.125) | (0.125) | (0.061) |

Monero | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

1% | LR_{UC} | 0.305 | 0.000 | 0.305 | 0.305 | 0.305 | 0.305 | 0.382 | 2.348 | 0.305 | 1.122 | - |

p-value | (0.581) | (1.000) | (0.581) | (0.581) | (0.581) | (0.581) | (0.537) | (0.125) | (0.581) | (0.290) | - | |

LR_{CC} | 0.413 | 0.061 | 0.413 | 0.413 | 0.413 | 0.413 | 0.408 | 2.594 | 0.413 | 1.292 | - | |

p-value | (-813) | (0.970) | (0.813) | (0.813) | (0.813) | (0.813) | (0.815) | (0.273) | (0.813) | (0.524) | - | |

2.5% | LR_{UC} | 7.495 | 7.495 | 7.495 | 7.495 | 9.134 | 7.495 | 5.988 | 10.898 | 7.495 | 5.988 | - |

p-value | (0.006) | (0.006) | (0.006) | (0.006) | (0.003) | (0.006) | (0.014) | (0.001) | (0.006) | (0.014) | - | |

LR_{CC} | 7.521 | 7.521 | 7.521 | 7.521 | 9.136 | 7.521 | 7.524 | 10.906 | 7.521 | 6.071 | - | |

p-value | (0.023) | (0.023) | (0.023) | (0.023) | (0.010) | (0.023) | (0.023) | (0.004) | (0.023) | (0.048) | - | |

5% | LR_{UC} | 8.253 | 9.555 | 8.253 | 9.555 | 9.555 | 8.253 | 8.253 | 8.253 | 8.253 | 7.033 | - |

p-value | (0.004) | (0.002) | (0.004) | (0.002) | (0.002) | (0.004) | (0.004) | (0.004) | (0.004) | (0.008) | ||

LR_{CC} | 9.397 | 10.949 | 9.397 | 10.949 | 10.949 | 9.397 | 9.397 | 9.397 | 9.397 | 7.949 | - | |

p-value | (0.009) | (0.004) | (0.009) | (0.004) | (0.004) | (0.009) | (0.009) | (0.009) | (0.009) | (0.019) | - | |

Litecoin | ||||||||||||

1% | LR_{UC} | 0.000 | 0.382 | 1.122 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 1.816 | 5.778 | 1.122 |

p-value | (1.000) | (0.537) | (0.290) | (1.000) | (1.000) | (1.000) | (1.000) | (1.000) | (0.178) | (0.016) | (0.290) | |

LR_{CC} | 0.061 | 0.408 | 1.292 | 0.061 | 0.061 | 0.061 | 0.061 | 0.061 | 0.1.823 | 7.456 | 4.660 | |

p-value | (0.970) | (0.815) | (0.524) | (0.970) | (0.970) | (0.970) | (0.970) | (0.970) | (0.402) | (0.024) | (0.099) | |

2.5% | LR_{UC} | 1.468 | 0.033 | 4.622 | 1.468 | 0.967 | 1.468 | 0.289 | 0.035 | 5.816 | 14.775 | 0.289 |

p-value | (.226) | (0.855) | (0.032) | (0.226) | (0.566) | (0.315) | (0.591) | (0.852) | (0.016) | (0.000) | (0.591) | |

5% | LR_{UC} | 13.924 | 3.025 | 13.393 | 8.253 | 3.889 | 13.924 | 4.847 | 2.259 | 0.072 | 12.393 | 2.259 |

p-value | (0.000) | (0.082) | (0.000) | (0.004) | (0.049) | (0.000) | (0.028) | (0.000) | (0.789) | (0.000) | (0.133) | |

LR_{CC} | 14.565 | 4.163 | 12.856 | 8.353 | 4.747 | 14.565 | 5.469 | 3.723 | 0.246 | 12.765 | 5.898 | |

p-value | (0.001) | (0.125) | (0.002) | (0.015) | (0.093) | (0.001) | (0.065) | (0.155) | (0.884) | (0.002) | (0.052) | |

Dash | ||||||||||||

1% | LR_{UC} | 1.122 | 2.348 | 1.122 | 0.305 | 5.778 | 1.122 | 2.348 | 5.778 | 10.246 | - | 15.547 |

p-value | (0.290) | (0.125) | (0.290) | (0.581) | (0.016) | (0.290) | (0.125) | (0.016) | (0.001) | - | (0.000) | |

LR_{CC} | 1.292 | 2.594 | 1.292 | 0.386 | 7.456 | 1.292 | 2.594 | 6.218 | 11.201 | - | 16.019 | |

p-value | (0.525) | (0,273) | (0.524) | (0.824) | (0.024) | (0.524) | (0.273) | (0.045) | (0.004) | - | (0.000) | |

2.5% | LR_{UC} | 7.495 | 3.405 | 7.495 | 5.988 | 9.134 | 7.495 | 4.622 | 10.88 | 9.134 | - | 12.781 |

p-value | (0.006) | (0.065) | (0.006) | (0.014) | (0.003) | (0.006) | (0.032) | (0.001) | (0.003) | - | (0.000) | |

LR_{CC} | 7.546 | 3.490 | 7.546 | 6.113 | 9.144 | 7.546 | 4.857 | 10.899 | 9.44 | - | 12.802 | |

p-value | (0.023) | (0.150) | (0.023) | (0.047) | (0.010) | (0.023) | (0.088) | (0.004) | (0.010) | - | (0.002) | |

5% | LR_{UC} | 4.847 | 4.847 | 3.025 | 2.259 | 3.025 | 4.847 | 4.847 | 4.847 | 3.889 | - | 5.896 |

p-value | (0.028) | (0.028) | (0.082) | (0.133) | (0.082) | (0.028) | (0.028) | (0.028) | (0.049) | - | (0.015) | |

LRCC | 4.862 | 4.862 | 3.169 | 2.406 | 3.276 | 4.862 | 4.862 | 4.862 | 3.951 | - | 5.896 | |

p-value | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | - | (0.052) |

Ripple | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

1% | LR_{UC} | 0.000 | 0.382 | 0.000 | 0.382 | 0.382 | 1.816 | 238.14 | 238.14 | 0.382 | 0.000 | 0.305 |

p-value | (1.000) | (0.537) | (1.000) | (0.537) | (0.537) | (0.178) | (0.000) | (0.000) | (0.537) | (1.000) | (0.581) | |

LR_{CC} | 0.061 | 0.408 | 0.061 | 0.408 | 0.408 | 1.823 | 238.16 | 238.16 | 0.408 | 0.061 | 0.413 | |

p-value | (0.970) | (0.815) | (0.970) | (0.815) | (0.402) | (0.000) | (0.000) | (0.815) | (0.970) | (0.813) | ||

2.5% | LR_{UC} | 0.775 | 0.775 | 0.775 | 0.775 | 0.289 | 0.035 | 217.14 | 222.21 | 0.033 | 0.289 | 2.350 |

p-value | (0.379) | (0.379) | (0.379) | (0.379) | (0.591) | (0.852) | (0.000) | (0.000) | (0.855) | (0.591) | (0.125) | |

LR_{CC} | 1.467 | 1.467 | 1.467 | 1.467 | 0.848 | 0.371 | 217.71 | 223.03 | 0.473 | 0.848 | 2.822 | |

p-value | (0.480) | (0.480) | (0.480) | (0.480) | (0.654) | (0.831) | (0.000) | (0.000) | (0.789) | (0.654) | (0.244) | |

5 | LR_{UC} | 2.259 | 0.595 | 3.025 | 1.596 | 1.596 | 0.069 | 169.67 | 169.67 | 0.595 | 1.596 | 5.896 |

p-value | (0.133) | (0.440) | (0.082) | (0.207) | (0.207) | (0.793) | (0.000) | (0.000) | (0.440) | (0.207) | (0.015) | |

LR_{CC} | 2.455 | 2.903 | 3.338 | 4.465 | 4.465 | 1.879 | 169.71 | 169.71 | 2.903 | 4.465 | 6.709 | |

p-value | (0.293) | (0.234) | (0.188) | (0.107) | (0.107) | (0.391) | (0.000) | (0.000) | (0.234) | (0.107) | (0.035) | |

Stellar | ||||||||||||

1% | LR_{UC} | 2.348 | 0.305 | 2.348 | - | 0.305 | 2.348 | 1.122 | 0.305 | 1.122 | 3.916 | 0.305 |

p-value | (0.125) | (0.581) | (0.125) | - | (0.581) | (0.125) | (0.290) | (0.581) | (0.290) | (0.048) | (0.581) | |

LR_{CC} | 2.594 | 0.413 | 2.594 | - | 0.413 | 2.594 | 1.292 | 0.413 | 1.292 | 4.252 | 0.413 | |

p-value | (0.273) | (0.813) | (0.273) | - | (0.813) | (0.273) | (0.524) | (0.813) | (0.524) | (0.119) | (0.813) | |

2.5% | LR_{UC} | 0.775 | 0.289 | 0.775 | - | 0.033 | 0.775 | 0.033 | 0.035 | 0.033 | 3.405 | 0.035 |

p-value | (0.379) | (0.591) | (0.379) | - | (0.855) | (0.379) | (0.855) | (0.852) | (0.855) | (0.065) | (0.852) | |

LR_{CC} | 1.467 | 0.848 | 1.467 | - | 0.473 | 1.467 | 0.473 | 0.371 | 0.473 | 4.588 | 0.371 | |

p-value | (0.480) | (0.654) | (0.480) | - | (0.789) | (0.480) | (0.789) | (0.831) | (0.789) | (0.101) | (0.831) | |

5% | LR_{UC} | 2.259 | 2.259 | 3.205 | - | 0.595 | 2.259 | 1.596 | 1.596 | 0.270 | 3.025 | 0.595 |

p-value | (0.133) | (0.133) | (0.082) | - | (0.440) | (0.133) | (0.207) | (0.207) | (0.604) | (0.082) | (0.440) | |

LR_{CC} | 2.455 | 2.455 | 3.338 | - | 2.903 | 2.455 | 4.465 | 1.702 | 2.321 | 3.123 | 0.603 | |

p-value | (0.293) | (0.293) | (0.188) | - | (0.234) | (0.293) | (0.107) | (0.427) | (0.313) | (0.210) | (0.740) | |

NEM | ||||||||||||

1% | LR_{UC} | 0.000 | 1.122 | 0.000 | 0.305 | 1.122 | 0.382 | 1.122 | 1.122 | 1.122 | 1.122 | 2.348 |

p-value | (1.000) | (0.290) | (1.000) | (0.581) | (0.290) | (0.537) | (0.290) | (0.290) | (0.290) | (0.290) | (0.125) | |

LR_{CC} | 0.06 | 1.292 | 0.061 | 0.413 | 1.292 | 0.408 | 1.292 | 1.292 | 1.292 | 1.292 | 2.594 | |

p-value | (0.970) | (0.524) | (0.970) | (0.813) | (0.524) | (0.815) | (0.524) | (0.524) | (0.524) | (0.524) | (0.273) | |

2.5% | LR_{UC} | 3.405 | 1.468 | 3.405 | 0.775 | 1.468 | 0.775 | 1.468 | 2.350 | 0.289 | 3.405 | 4.622 |

p-value | (0.065) | (0.226) | (0.065) | (0.379) | (0.226) | (0.379) | (0.226) | (0.125) | (0.591) | (0.065) | (0.032) | |

LR_{CC} | 4.588 | 2.308 | 4.588 | 1.467 | 2.308 | 1.467 | 2.308 | 3.353 | 0.848 | 4.588 | 5.998 | |

p-value | (0.101) | (0.315) | (0.101) | (0.480) | (0.315) | (0.480) | (0.315) | (0.187) | (0.654) | (0.101) | (0.050) | |

5% | LR_{UC} | 8.253 | 8.253 | 7.033 | 4.847 | 9.555 | 7.033 | 9.555 | 8.253 | 8.253 | 8.253 | 12.393 |

p-value | (0.004) | (0.004) | (0.008) | (0.028) | (0.002) | (0.008) | (0.002) | (0.004) | (0.004) | (0.004) | (0.000) | |

LR_{CC} | 8.353 | 8.353 | 7.070 | 5.382 | 9.748 | 7.070 | 9.748 | 8.353 | 8.353 | 8.353 | 12.293 | |

p-value | (0.015) | (0.015) | (0.029) | (0.068) | (0.008) | (0.029) | (0.008) | (0.015) | (0.015) | (0.015) | (0.002) |

of significance. For example, regarding the first cryptocurrency (Bitcoin) in _{uc} and LR_{cc} test at 95%, 97.5% and 99% confidence levels. While the EGARCH (1, 1) and AVGARCH (1, 1) pass the LR_{uc} test at only 97.5% and 99% confidence levels. Hence, the APARCH (1, 1) model has the highest number of passes and is therefore considered to be the most appropriate model in forecasting VaR for the Bitcoin. In relation to the other seven cryptocurrencies, the results are summarized as follows; the APARCH (1, 1) and TGARCH (1, 1) models have the highest number of passes for Ethereum; NAGARCH (1, 1) model for Moreno; NGARCH (1, 1) for Litecoin; CSGARCH (1, 1) for Dash, Ripple, and NEM. Finally, APARCH (1, 1) for Stellar. Both the LR_{uc} and LR_{cc} coverage tests recommend the same GARCH-type models in most of the cases. These results demonstrate that the asymmetric GARCH-type models mostly have better VaR forecast performance for all cryptocurrencies especially at 99% level of significance and are also consistent with those found in the failure rate performance. Moreover, the fact that more GARCH-type models pass the LR_{cc} test for 99% VaR than for 95% VaR can be explained by the independence test where a smaller number of exceedances makes it easier not to occur after each other.

Finally, the VaR forecast performance of GARCH-type models is greatly dependent on the GARCH-type specification, with most GARCH models performing fairly better at the 95% level of significance. The p-values for both conditional and unconditional coverage tests are relatively low for most of the GARCH models, with the TGARCH and AVGARCH models showing among the lowest probability values. Generally, the conditional variance component of the GARCH-type specification plays a significant role since it provides models with a long memory and a more flexible lag structure.

Cryptocurrencies are relatively new and innovative investment assets that are characterized by high volatility and are uncorrelated with traditional financial assets such as stocks, currencies and bonds. In this paper, the focus is on modelling the volatility dynamics and out-of-sample forecasting performance of several GARCH-type models for cryptocurrency returns. Specifically, we have considered twelve symmetric and asymmetric GARCH processes, to evaluate the out-of-sample VaR forecasting performance of the eight major cryptocurrencies by market capitalization. This is implemented under the assumption that the innovations distributions of cryptocurrencies returns are skewed, heavy-tailed and leptokurtic. The out-of-sample VaR forecast performance of the GARCH-type specification is evaluated using by means of backtesting using conditional and unconditional coverage tests.

The empirical results of the study can be summarized as follows. Firstly, innovations distributions that capture skewness, kurtosis and heavy tails constitute excellent tools in modelling distribution of cryptocurrencies returns. The skewed versions of Student-t, GED and hyperbolic distributions for return innovations confirm their predominance over the alternatives in terms of better predictive ability. Secondly, the GARCH-type volatility models combined with a skewed distribution of return innovations, like the skewed t-Student or the Skewed-GED, provide acceptable VaR forecasts. While the results do not guarantee a straightforward preference between GARCH-type models, the asymmetric GARCH models with long memory property with skewed and heavy-tailed innovations distributions demonstrate better overall performance for all cryptocurrencies. Finally, regarding the accuracy tests, the VaR forecast performance comparison results vary with the cryptocurrencies. Given the high volatility dynamics present in all the cryptocurrencies, investors need to be cautious about their investments decisions in any cryptocurrency while investment managers should select asymmetric GARCH-type models with a long memory to forecast the VaR of cryptocurrencies.

The authors declare no conflicts of interest regarding the publication of this paper.

Ngunyi, A., Mundia, S. and Omari, C. (2019) Modelling Volatility Dynamics of Cryptocurrencies Using GARCH Models. Journal of Mathematical Finance, 9, 591-615. https://doi.org/10.4236/jmf.2019.94030