The selected GARCH-type models

Let y_t denote the daily simple returns of the respective cryptocurrencies and exchange rates data series at time t for t = 1, …, n., calculated as the difference between prices at the end of day t and at the end of the preceding day t-1 (P_t − P_t-1). Then, GARCH models can be specified as: (1) where y_t is the return, μ_t denotes the conditional mean and σ_t denotes the volatility process, ( being the conditional variance). z_t, the innovations, are independent and follow a Gaussian distribution with zero mean and unit variance. For brevity, all models are restricted to a maximum order of one (p = q = 1), since they tend to be more flexible, efficient and significant than higher order models in the out-of-sample analysis [14].

All of the GARCH-type selected models follow the specification depicted in Eq (1); however, they differ in the conditional variance specification.

The conditional variance for the Standard GARCH (SGARCH) (1,1) process [15] is given by: (2) (3) where is the estimate of the variance for day t, and represents the associated squared error and the conditional variance on the previous day, respectively, with α and β being their respective weights. The long run variance V_L is an average level towards which variances revert to through a principle called mean reversion, with γ being the weight assigned to such an average level. The main feature of this model is that it captures volatility clustering in the data through the persistence parameter α + β with restrictions ω ≥ 0, α ≥ 0, β ≥ 0 and α + β < 1 to ensure a uniquely stationary process and positivity of the conditional variance. However, if the persistence parameter α + β equals 1, the GARCH model converges to the Integrated GARCH model, where the long term volatility bears an explosive process.

The Integrated GARCH model [16], denoted by IGARCH, is a particular version of SGARCH (1,1) model where, as advanced above, the persistence parameter (α + β) is equal to 1 and typically imports a unit root under the GARCH process. Thus, the conditional variance in the IGARCH (1,1) is expressed in Eq (4), given that β is set equal to (1 − α) with restrictions ω ≥ 0, α ≥ 0 and 1 − α ≥ 0: (4)

In the SGARCH and IGARCH models, the impact of positive and negative news on the conditional variance are symmetrical. These models restrict all coefficients to be greater than zero and thus cannot explain the negative correlation between return and volatility. Current return and future volatility might have a negative correlation and the impact of positive and negative shocks on the conditional variance is rather asymmetrical [17]. This came to be known as the “leverage effect” after which more advanced models were developed to incorporate its effect. It is important to distinguish the leverage effect from volatility feedback. The former explains why a negative return causes an increase in volatility, while the latter explains why an increasing volatility results in a negative return.

The Exponential GARCH model, denoted by EGARCH (p, q) [18], incorporates the asymmetric impact of positive and negative shocks on volatility whereby the latter is believed to produce greater levels of volatility, despite having the same magnitude. This model is specified in logarithmic form, which suggests that parameters are unrestricted, and are thereby allowed to take negative values while ensuring a positive conditional variance. In addition, the conditional variance is written as a function of past standardized innovations, instead of past innovations. Formally, the volatility dynamics of an EGARCH (1,1) can be written as: (5) where β represents the persistence parameter, and α and γ capture the size and the sign (leverage) effect, respectively. The above specification exhibits an asymmetric effect when γ ≠ 0. More specifically, if the leverage parameter “γ” is negative, this means that negative news affect volatility more than positive news. Conversely, if returns and volatility are positively correlated, γ will be positive thereby positive shocks will have a higher impact on volatility than negative shocks, which is irregularly the case. In this specification, if γ ≠ 0 and significant, (α + γ) is the effect on volatility of a previous positive return, whereas γ − α is the corresponding effect when the previous return has been negative.

The Glosten-Jagannathan-Runkle GARCH (GJR-GARCH) model [19] is similar to EGARCH (1,1) in incorporating the asymmetric impact of positive and negative shocks. However, the volatility equation of a GJRGARCH is given by: (6) where I_t−1 = 1 if ε_t−1 < 0 and I_t−1 = 0 if ε_t−1 ≥ 0. A defining feature of this model is that a positive shock will increase volatility by α_t, whereas a negative shock will increase volatility by α_t + γ_t at t. However, in contrast to the EGARCH model, the leverage effect exists when γ > 0, indicating that past “bad news” have stronger impact on current volatility than past “good news”. If γ < 0, then past positive returns increase current volatility more than past negative returns. The persistence in this model relies on α, β, and γk with k representing the average value of standardized errors. Parametric restrictions are similar to the Standard GARCH whereby ω ≥ 0, α ≥ 0, and β ≥ 0.

The Asymmetric Power ARCH (APARCH) [20] models for both the leverage and the effect that the sample autocorrelation of absolute returns is usually larger than that of squared returns through its “power parameter, δ”; allowing for more flexibility. In this specification is replaced by , which is given by: (7) with δ, α, β, ω ≥ 0 and −1 ≤ γ ≤ 1, where δ is the Taylor (power effect) parameter [21] for the Box-Cox Transformation, γ is the leverage parameter and the persistence parameter is given by β + αk. Signs analysis for the leverage parameter are similar to the GJR-GARCH model, where a leverage effect exists once γ > 0. It is of note that APARCH (1,1) converges to GJR-GARCH (1,1) when δ = 2 and to the SGARCH (1,1) for δ = 2 and γ = 0.

The Threshold GARCH (TGARCH) model [22] is similar to the GJR GARCH model and is a particular case of APARCH (1,1) with δ = 1, which models for the conditional standard deviation instead of the conditional variance with the restraint −1 ≤ γ ≤ 1. The volatility equation of TGARCH (1,1) is typically expressed as follows: (8)

By contrast to the SGARCH (1,1) model, which shows mean reversion to a constant term “ω", the Component GARCH (CGARCH) model [23] allows mean reversion to a varying level “q_t”, known as the time varying long run volatility. CGARCH (1,1) splits the conditional variance into its transient (Eq 9) and permanent components (Eq 10) to examine short and long-term effects on volatility, as presented below: (9) (10)

Similar to GJR-GARCH model, the CGARCH specification in Eq (9) captures asymmetric responses to shocks by introducing the slope dummy variable “I_t−1” to the leverage parameter that takes the value of “1” for ε_t−1< 0, and “0” otherwise. A positive gamma “γ" indicates the presence of transitory leverage effect in the conditional variance. Stationarity of the CGARCH model and non-negativity of the conditional variance are ensured once the following inequality constraints are satisfied: ω ≥ 0, α ≥ 0, ϕ ≥ 0, β ≥ 0, β ≥ ϕ and α + β ≤ ρ ≤ 1. An interesting general revision on GARCH-type volatility modelling can be seen in Racicot [24].

The parameters of all GARCH-type models are estimated using Maximum Likelihood, since it is generally consistent and efficient, and provides asymptotic standard errors that are valid under non-normality. The conditional log-likelihood or support function (LLF) is given by: (11) where f (·) is the conditional probability density function and θ_i, I = 1, …, k, are the model parameters at time t. For each model, the innovation process z_t is allowed to follow one of the following three distributions: the Normal Distribution, the Student’s t Distribution, and the Generalized Error Distribution.

The selection of the optimal GARCH model is based on the MAE, MAPE, and RMSE. Also, the three information criteria specifically Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC) and the Hannan-Quinn Information Criterion (HQC) were used for the selection of the best distribution curve. The purpose of selecting the optimal out-of-sample GARCH model for each currency and cryptocurrency is to forecast the one-day ahead volatility that will be successively used to estimate VaR forecasts.

Value at risk estimation

The VaR forecast for the GARCH-type models relies on the one-day ahead conditional mean, μ_t+1 and the conditional variance forecast of the volatility model. Under each of the innovations term distribution assumptions, the one-day-ahead VaR forecast is calculated as: (12) Where F⁻¹(α) is the α-quantile of the cumulative distribution function of the innovation distribution.

The accuracy of the estimated VaR in forecasting returns is assessed by using the Kupiec’s Unconditional Coverage Test [25]. It is a likelihood ratio test that gauges the level of accuracy in back testing VaR [26]. Effectively, the likelihood ratio, denoted by “LR_K” is given by Kupiec’s test statistic, LR_K, is given by: (13) where p is the specified model probability (in accordance to the VaR confidence level), is the observed failure rate, with X being the number of exceptions/violations, when the actual loss exceeds VaR, and T the number of trials, which is 250 at all times. Specifically, the Kupiec test will reject the model if it overstates/understates the true VaR. Under the null, LR_K distributes as a Chi-squared with 1 degree of freedom.

Parameters analysis of the selected GARCH models

The parameter estimates resulting from the estimation of the GARCH-type specifications considered in this research, in light of the in-sample database, are depicted in S1 Table and used for both in-sample and out-of-sample volatility forecasting. Looking at SGARCH (1,1) and IGARCH (1,1), the ARCH component “α” ranges between 9% and 37% for the cryptocurrencies and between 0% and 9% for the fiat currencies, except for the British Pound, having an α of 16% in SGARCH(1,1). The relatively high disturbance in the British Pound compared to the remaining currencies is due to the Brexit turmoil following the UK-wide referendum in June 2016 and, effectively, its associated repercussions. Predictably, all cryptocurrencies are however, sensitive to disturbances in the market.

With the exception of the British Pound, all fiat currencies exhibit a relatively larger β compared to cryptocurrencies suggesting that exchange rates are more explicable and less “spiky” as illustrated in Figs 1 and 2. Noticeably, the "ω" term for fiat currencies is relentlessly insignificant and close to zero. This provides further verification that the IGARCH model provides a very good fit for the fiat currencies, while at the same time, drawing attention towards advanced GARCH models as they provide better explanation to cryptocurrencies’ volatility. Also, it is important to note that in the case of SGARCH (1,1), the persistence parameter “α + β” equals 1 for Dogecoin, thereby indicating that the conditional variance is strictly stationary with an unattainable long-term variance. As for the Bitcoin, Ripple, Litecoin, Monero and Dash, the series are stationary and mean reverting with long-term volatilities surpassing the 100% mark. Specifically, Bitcoin and Ripple reported the highest long-term volatilities with respective values of 213% and 164%, which further underlines cryptocurrencies’ “intensifying” levels of volatility.

Regarding the EGARCH estimation results, the positive sign of γ ranges between 1% and 19%. Consequently, none of the cryptocurrencies and fiat currencies exhibits a leverage effect and hence positive shocks have a greater impact on their volatility than negative shocks, particularly for Ripple (19%) and the Swiss Franc (15%). The GARCH term “β” is quite remarkable for all cryptocurrencies and fiat currencies except for the Euro, Canadian Dollar and the Swiss Franc, revealing that one distinctive feature in cryptocurrencies is persistence in their volatility. The long-term volatility “V_L” of fiat currencies ranges between 7% and 10%, although it reaches 55% exceptionally for the Japanese Yen. Nevertheless, cryptocurrencies’ long term volatility ranges between 96% and 182%. For instance, Ripple’s long-term volatility is 17 times larger than the Swiss Franc’s long term volatility, further emphasizing the increased volatility in cryptocurrencies with respect to fiat currencies.

As for GJR-GARCH (1,1) results, we notice that the volatility of cryptocurrencies tends to cluster in response to market shocks, unlike fiat currencies. The larger beta in the case of fiat currencies evidences that they are relatively more explicable and are subject to less ‘spikes’ than cryptocurrencies. Unlike EGARCH (1,1), the leverage coefficient "γ" for GJR-GARCH (1,1) ranges between 0% for Australian Dollar and -96% for Ripple. However, the low values attained for the leverage coefficient for most of the fiat currencies (namely: Australian Dollar, Canadian Dollar, Euro, and Japanese Yen) in conjunction with the absence of the constant term "ω" give additional support to the hypothesis that the IGARCH alternative provides the best fit for fiat currencies.

Under the APGARCH (1,1) specification, the estimates of α, β and ω show consistency in the behavior of cryptocurrencies and exchange rates. Estimates for the leverage parameter “γ” exhibit notable differences with those obtained for the GJR-GARCH model, with γ ranging between 0% and -72%. Curiously, the lowest percentage (in absolute value) was for Bitcoin (0%) which reveals that its volatility is affected symmetrically by positive and negative shocks. The leverage parameter appears to be the largest (in absolute value) for Monero (72%) and the Canadian Dollar (57%), with only the Euro having a remarkable positive leverage parameter (10%). These results are inconsistent with the previous models where some fiat currencies showed insignificant asymmetry effects.

By looking at TGARCH (1,1) parameter estimates, we notice that the Bitcoin and Japanese Yen have a γ of 0.33%, which implies that the impact of returns on their volatility is symmetrical and thereby, they do not exhibit an asymmetric effect. All remaining cryptocurrencies and fiat currencies (except Euro) have a significant negative leverage parameter, and therefore further emphasize the results attained earlier (except for EGARCH).

Finally, the CGARCH (1,1) results show that the high value attained for the trend intercept “ω” in the case of Ripple and Dogecoin, points towards the relative significance of their permanent component and thus suggests that the CGARCH model may provide a good fit to both cryptocurrencies. This is further supported by the fact that Ripple and Dogecoin are the only cryptocurrencies that present shocks of transitory nature (sum of alpha and beta coefficients “α+ β” are close to “ρ”). It is also of note that the Euro, British Pound and Canadian Dollar also reveal that their volatilities are highly prone to short term effects. The AR coefficient of the permanent volatility “ρ” is almost 1 for all cryptocurrencies and fiat currencies and its size exceeds the coefficients of the transitory component in all cases, implying that the CGARCH model is quite stable for all cryptocurrencies and fiat currencies. The forecasting error term “Ø” is positive but insignificant for most cryptocurrencies and fiat currencies, which implies that actual and estimated volatilities are close. In contrast to all remaining models, the CGARCH is the only model that reports the presence of leverage effect in most cryptocurrencies, particularly for Litecoin (γ = 23%).

Realized vs estimated volatility and model optimization

S1 and S2 Figs plot the realized volatility against GARCH volatilities for cryptocurrencies and fiat currencies, respectively, over the in-sample period, while S3 and S4 Figs refer to the out-of-sample comparisons. S2 Table details the in-sample error statistics values along with their rankings for each cryptocurrency and fiat currency for the selected models, while S3 Table depicts the out-of-sample values. Table 4 illustrates the optimality of the GARCH-type models and shows consistency among fiat currencies, whereby the IGARCH has proven to perform best for most of the fiat currencies, particularly the British Pound, Australian Dollar, Swiss Franc and the Japanese Yen. The IGARCH model was also found to be the most accurate model for the Canadian Dollar, but only for the out-of-sample period given that the TGARCH performed better during the in-sample period. However, and quite surprisingly, the CGARCH modeled the Euro almost impeccably. This may be attributable to the distinctive characteristics of the CGARCH specification, which divides the conditional variance into its transitory and permanent components, whereby the long-run component is allowed to be continuously updated rather than held uniform, thereby better capturing and reflecting on volatility clusters and persistence in Euro’s returns. It is important to note that when ω is null the IGARCH model becomes nothing different from the EWMA model, which is the case of all fiat currencies. Therefore, the IGARCH has proven to be the prevailing model when modelling foreign exchange markets. This may be attributable to their low volatile nature, their typical symmetrical behavior to shocks, and ‘persistent variance’ in which current information remains important when forecasting volatility.

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Table 4. Optimal models under the in-sample & out-of-sample periods.

https://doi.org/10.1371/journal.pone.0245904.t004

Exceptionally and among all cryptocurrencies, the IGARCH was also the best performing model for Monero, in both sampled periods. This might be due to the fact that the absence of a long-run average variance in the IGARCH model entails that any disturbance in the market brings an everlasting change in Monero’s volatility structure, which explains the overstated volatility estimates obtained under the IGARCH model (S1–S4 Figs).

As for the remaining cryptocurrencies, the GJR-GARCH specification proved to be superior during the in-sample period while the CGARCH and TGARCH alternatives proved to be the best performers during the out-of-sample interval, which validates the assumption that advanced GARCH models better model asymmetries in cryptocurrencies’ volatilities. Specifically, for the in-sample period, the GJR-GARCH model is the optimal for Bitcoin, Litecoin and Dash, APARCH is the best competing alternative for Ripple, and GARCH for Dogecoin. Regarding the out-of-sample period, TGARCH performed the best for Bitcoin and Dash while CGARCH showed to be the optimal for Ripple and Dogecoin and APARCH for Litecoin. Apparently, it is natural to observe some discrepancies among cryptocurrencies due to their relatively highly volatile feature. But remarkably, however, the EGARCH specification, which was considered superior in [9] was one with the worst performance among all fiat and virtual currencies.

VaR estimation and backtesting results

S5 and S6 Figs compare the VaR estimates with the corresponding returns over the out-of-sample sampled period for cryptocurrencies and fiat currencies. The VaR accuracy is tested by using Eq (13) and the results are listed in Table 5.

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Table 5. VaR results.

https://doi.org/10.1371/journal.pone.0245904.t005

Remarkably, the results from the test of Kupiec show that the VaR provides a very accurate measure for the level of downside risk imperiling fiat currencies, given that the VaR model was only rejected at the 97.5% confidence level for JPY. Dash and Dogecoin provided similar results to fiat currencies, where the VaR results were not rejected at all the confidence level. As for the remaining cryptocurrencies, the results were disparate. In the case of Litecoin, the test of Kupiec displayed increased accuracy as the confidence level augmented, so that, the specification was not rejected at the 95%, 97.5% and 99% confidence levels. Perhaps, the most peculiar results were those for Monero, where the VAR specification in the null was not rejected at the 90% and 99% confidence levels. It is evident that the VaR provides a poor measure for Bitcoin and Ripple whereby the optimal model was rejected at all confidence levels with the exception of 99%. At the 99% confidence level it was not rejected, which implies that precision was attained only at the highest degree of confidence. In fact, in all the rejection cases, the model overstates the risk in cryptocurrencies due to their distinctively highly volatile feature.