Forecasting Bitcoin risk measures: A robust approach

International Journal of Forecasting 35 (2019) 836–847

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International Journal of Forecasting journal homepage: www.elsevier.com/locate/ijforecast

Forecasting Bitcoin risk measures: A robust approach Carlos Trucíos São Paulo School of Economics, Fundação Getulio Vargas (FGV), Brazil

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Keywords: Cryptocurrency GARCH Model confidence set Outliers Realised volatility Value-at-Risk

a b s t r a c t Over the last few years, Bitcoin and other cryptocurrencies have attracted the interest of many investors, practitioners and researchers. However, little attention has been paid to the predictability of their risk measures. This paper compares the predictability of the one-step-ahead volatility and Value-at-Risk of Bitcoin using several volatility models. We also include procedures that take into account the presence of outliers and estimate the volatility and Value-at-Risk in a robust fashion. Our results show that robust procedures outperform non-robust ones when forecasting the volatility and estimating the Valueat-Risk. These results suggest that the presence of outliers plays an important role in the modelling and forecasting of Bitcoin risk measures. © 2019 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved.

1. Introduction Bitcoin has attracted the interest of many investors, practitioners and researchers since its creation in 2008. This interest has grown rapidly over recent years, probably due to its decentralised nature and its large profits; see Nakamoto (2008). Previous studies, such as those by Baek and Elbeck (2015), Briere, Oosterlinck, and Szafarz (2015) and Sapuric and Kokkinaki (2014), have observed that Bitcoin is highly volatile. Thus, forecasting the volatility and estimating the Value-at-Risk(VaR) as well as possible is crucial in helping to enable investors and practitioners to make good decisions. Bitcoin’s daily volatility has been studied by Balcilar, Bouri, Gupta, and Roubaud (2017), Catania, Grassi, and Ravazzolo (2018), Chu, Chan, Nadarajah, and Osterrieder (2017), Conrad, Custovic, and Ghysels (2018), Dyhrberg (2016), Katsiampa (2017), Liu, Shao, Wei, and Wang (2017), Naimy and Hayek (2018) and Pichl and Kaizoji (2017) among others. However, most of these studies have focused on in-sample analysis, and the comparisons have been made based on information criteria. In this context, Katsiampa (2017) estimates the E-mail address: [email protected].

Bitcoin volatility using several GARCH-type models assuming Gaussian errors, and concludes that the best model for estimating the volatility is the AR(1)-CGARCH(1,1). Chu et al. (2017) analyse Bitcoin and six other cryptocurrencies using GARCH-type models with different error distributions and conclude that the best models for estimating the Bitcoin volatility are the IGARCH and GJRGARCH models with Gaussian distributions. Liu et al. (2017) compare the GARCH model assuming the normal reciprocal inverse Gaussian (NRIG) distribution with the Gaussian and Student-t error distributions and conclude that the GARCH model with Student-t errors estimates the volatility better. Charles and Darné (2018) replicate the study of Katsiampa (2017) and additionally take into account the presence of extreme observations. They find that, when using the jump-filtered returns as per Laurent, Lecourt, and Palm (2016), the AR(1)-GARCH(1,1) model reports smaller information criteria than the models considered by Katsiampa (2017). On the other hand, Conrad et al. (2018) compare the classical GARCH(1,1) with the GARCH-MIDAS model of Engle, Ghysels, and Sohn (2013) and conclude that the latter model outperforms the former. In an out-of-sample context, Naimy and Hayek (2018) compare the one-step-ahead volatility forecasts estimated by GARCH and EGARCH models with Gaussian, Studentt and generalised error distributions. The authors compare the predicted volatility with the realised volatility

https://doi.org/10.1016/j.ijforecast.2019.01.003 0169-2070/© 2019 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved.

C. Trucíos / International Journal of Forecasting 35 (2019) 836–847

using the root mean square error (RMSE), the mean absolute error (MAE) and the mean absolute percentage error (MAPE), and conclude that the EGARCH models present the best performances. Catania et al. (2018) compare the Gaussian GARCH model with the generalised autoregressive score (GAS) models of Creal, Koopman, and Lucas (2013) and Harvey (2013). Their comparison is based on the quasi-like (QLIKE) loss function, and the predicted volatility is compared with the squared observed returns. They conclude that the GARCH model is outperformed by the GAS models. Peng, Albuquerque, de Sá, Padula, and Montenegro (2018) compare the volatility forecasts obtained by GARCH, EGARCH and GJR models assuming symmetric and asymmetric Gaussian and Student-t errors against the support vector regression GARCH model of Bezerra and Albuquerque (2017), and find that the latter yields more accurate forecasts. Although out-ofsample comparisons are available in the literature, most of them are restrictive since they do not consider models that presented better performances in previous studies, and they leave out several error distributions and models of the GARCH family. In this sense, a comprehensive out-of-sample comparison is needed. Nevertheless, most of the papers available in the literature do not consider the presence of outliers, which can affect volatility forecasting and VaR estimation drastically, as was mentioned by Boudt, Danielsson, and Laurent (2013), Carnero, Peña, and Ruiz (2012) and Trucíos and Hotta (2016), for instance. As far as we know, only Catania and Grassi (2017), Charles and Darné (2018) and Catania et al. (2018) have taken the presence of outliers into account when estimating the Bitcoin volatility, and only Catania et al. (2018) have assessed the Bitcoin volatility in an out-of-sample context, though they only compare a Gaussian GARCH model with GAS models. In a VaR context, Chan, Chu, Nadarajah, and Osterrieder (2017), Chu et al. (2017), Gkillas and Katsiampa (2018), Osterrieder and Lorenz (2017) and Stavroyiannis (2018) estimated the VaR of Bitcoin. However, none of these works have considered the presence of outliers in the context of conditional heteroscedastic models. The contribution of this paper is threefold. First, we carry out an extensive out-of-sample comparison of the Bitcoin volatility forecast using several classical GARCHtype models with different error distributions, thus filling a gap in the literature, and also summarising the GARCHtype results found in previous work. Second, we address the presence of outliers and show that a better volatility forecast is obtained using robust procedures, thus suggesting that outliers play a crucial role in the modelling and forecasting of the volatility of Bitcoin. Finally, we compare the performance of the usual VaR estimation with the robust bootstrap procedure proposed by Trucíos, Hotta, and Ruiz (2017). The rest of the paper is organised as follows. Section 2 describes the models, realised measures and loss functions that are used in the out-of-sample comparison. Section 3 describes the data and reports the main findings. Finally, Section 4 presents the conclusions and discusses opportunities for future research.

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2. Methodology This section briefly describes the volatility models that we use to forecast the Bitcoin volatility, as well as the realised measures that we use as volatility proxies. At the end of the section, we also describe the loss functions that we use to evaluate the out-of-sample performance and the robust bootstrap procedure of Trucíos et al. (2017) for estimating the VaR. 2.1. Volatility models Several approaches to the modeling and forecasting of the volatility have been proposed in the literature; see for instance Bauwens, Hafner, and Laurent (2012), Francq and Zakoian (2011) and Harvey (2013). This section briefly introduces the procedures used in this paper for forecasting the daily Bitcoin volatility. Section 2.1.1 introduces the classical GARCH-type models, while the subsequent subsections introduce alternative models. 2.1.1. GARCH models The GARCH model has been used widely by researchers and practitioners for modeling and forecasting the secondorder moments of the returns in economic and financial time series. Several extensions of the GARCH model have been proposed in the literature since its introduction by Bollerslev (1986) and Engle (1982), differing as to the way in which the volatility equation is defined. Let rt be the observed return at time t and ϵt the error term which follows a white noise process. The GARCH(1,1) model is defined as rt = σt ϵt ,

σt2 = ω + α rt2−1 + βσt2−1 , with σt2 being the conditional variance (or squared volatility) at time t, and ω, α and β being parameters that satisfy some stationary conditions. Sufficient conditions for stationarity are given by ω > 0, α , β ≥ 0 and α+β < 1. As was mentioned above, different extensions of the GARCH model involve different specifications of the volatility equation (σt2 ). Table 1 describes the volatility equations of the GARCH-type models used in this work. For good reviews of univariate GARCH-type models, see for instance Bollerslev (2010), Engle (1995), Rodríguez and Ruiz (2012) and Teräsvirta (2009). In all non-robust cases, several error distributions are assumed, namely Normal, Skew Normal, Student-t, Skew Student-t, GED, Skew GED, Normal Inverse Gaussian, Generalized Hyperbolic and Johnson’s reparametrized SU innovation distribution; see Ghalanos (2018) for details of the parameterisations of the distributions and GARCHtype models used in this paper. 2.1.2. GARCH-MIDAS model Engle et al. (2013) extended the work of Ghysels, Santa-Clara, and Valkanov (2006) and proposed a procedure which decomposes the conditional variance into two components that distinguish short-run from longrun movements. In that model, called GARCH-MIDAS, the short-run component evolves as a GARCH(1,1) process

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C. Trucíos / International Journal of Forecasting 35 (2019) 836–847 Table 1 GARCH-type models. Model

Volatility equation

GARCH

σt2+1 = ω + α rt2 + βσt2

IGARCH

σ

EGARCH

log(σt2+1 ) = ω + αϵt2 + γ (|ϵt |−E(|ϵt |)) + β log(σt2 ) Nelson (1991)

GJR

σt2+1 = ω + α rt2 + γ I(rt < 0)rt2 + βσt2

Glosten, Jagannathan, and Runkle (1993)

APARCH

σtδ+1 = ω + α (|rt |−γ rt )δ + βσtδ

Ding, Granger, and Engle (1993)

CGARCH

σt2+1 = qt +1 + α (rt2 − qt ) + β (σt2 − qt )

2 t +1

=ω+α

rt2

+ (1 − α )σ

Proposed by Bollerslev (1986) 2 t

Engle and Bollerslev (1986)

qt +1 = ω + ρ qt + φ (rt2 − σt2 )

Lee and Engle (1999)

σt +1 = ω + ασt (|ϵt |−η1 ϵt ) + βσt

Zakoian (1994)

AVGARCH σt +1 = ω + ασt (|ϵt − η2 |−η1 (ϵt − η2 )) + βσt

Schwert (1990)

TGARCH

NGARCH

σtδ+1 = ω + ασtδ (|ϵt |)δ + βσtδ

Higgins and Bera (1992)

NAGARCH σt2+1 = ω + ασt2 (|ϵt − η2 |)2 + βσt2 FGARCH Robust

GARCH

Engle and Ng (1993)

σtδ+1 = ω + ασtδ (|ϵt − η2 |−η1 (ϵt − η2 ))δ + βσtδ ( 2) rt + βσt2 , σt2+1 = ω + γc αρ 2 σ t ⎧ ⎨1, if x > c , with ρ (x) = ⎩ x, if x ≤ c , and γc being a constant to ensure consistency

while the long-run component is determined by past values of the realised volatility. Let ri,t be the observed return for day i of an arbitrary period t, ϵi,t be a white noise error term, and τt and gi,t be the unobserved long-run and short-run components, respectively. The GARCH(1,1)-MIDAS is defined as ri,t =

√

τt × gi,t ϵt ,

gi,t = (1 − α − β ) + α

τt = m + θ

K ∑

ri2−1,t

τt

+ β gi−1,t ,

ϕk (ω)RVt −k ,

k=1 Nt

RVt =

∑

ri2,t ,

i=1

where ϕk (ω) is a weight function, K stands for the number of lags used in the long-run component and Nt is the number of observations within the arbitrary period t. In this paper, we use K = 10 and the arbitrary period t is equal to a month. As an alternative to RVt , the model proposed by Engle et al. (2013) also allows the use of macroeconomic variables; see for instance Conrad et al. (2018) and Walther and Klein (2018). This paper only uses information that is available in the Bitcoin returns. 2.1.3. Realised GARCH model The realised GARCH model proposed by Hansen, Huang, and Shek (2012) combines a model for realised measures with a GARCH structure for the volatility where the squared return is replaced by a realised measure. This approach is based on the argument that returns offer only weak information about the volatility, and as a consequence, they may lead to poor performances when the

Hentschel (1995) Boudt et al. (2013)

with the modification of

Trucíos et al. (2017)

volatility changes quickly. On the other hand, Andersen and Bollerslev (1998), Christoffersen, Feunou, Jacobs, and Meddahi (2014) and Hansen et al. (2012) argue that realised measures are more informative about the volatility than are the squared returns. Let rt be the observed daily return, σt2 its conditional variance, ϵt a white noise error term, and xt a realised measure of the variance. The realised GARCH(1,1) model is defined as rt = σt ϵt , log(σt2 ) = ω + α log(xt −1 ) + β log(σt2−1 ), log(xt ) = δ + γ log(σt2 ) + τ (ϵt ) + ut , where xt is a realised measure at time t, τ (ϵt ) is a function such that E(τ (ϵt )) = 0, and ut is a white noise process. This paper uses τ (ϵt ) = η1 ϵt + η2 (ϵt2 − 1). Note that this model enables us to forecast the volatility as well as the realised measure. Therefore, we denote it by RealGARCHx when the realised measure ‘‘x" is used as an auxiliary variable for forecasting the volatility and by RealGARCH when the volatility is used for forecasting the realised measure. Another model that uses a similar strategy is the highfrequency-based volatility (HEAVY) model of Shephard and Sheppard (2010). 2.1.4. GAS models Creal et al. (2013) and Harvey (2013) proposed a new class of models called generalised autoregressive score (GAS) models, also known as dynamic conditional score (DCS) or score-driven (SD) models. The general expression of the GAS model is given by ft +1 = ω + β ft + α St

[

] ∂ log p(rt |ft ) , ∂ ft

(1)

C. Trucíos / International Journal of Forecasting 35 (2019) 836–847

with St being a scaling function for the score. St is usually 1 and It is the Fisher inused as It−λ , where [( λ = 0, 1/2 or )2 ] ∂ log p(rt |ft ) formation, Et −1 . The main difference ∂ ft from the classical GARCH model is in the evolution of the volatility equation (ft = σt2 ), which depends on the past values of the score of the conditional distribution instead of the squared returns. Since the score depends on the complete density instead of only second-order moments, the GAS models allow us to exploit the dynamic structure of the data better. Although well-known models such as GARCH and EGARCH can be obtained as special cases of the GAS model by using p(·) as the Gaussian density and appropriate values of St and ft , the main advantage of the GAS model comes from using p(·) as a non-Gaussian density. An interesting and well-studied choice for p(·) is the Student-t density. This choice is particularly attractive due to its flexibility at capturing the volatility dynamics, and also to its robustness to heavy tails and outliers; see for instance Blasques, Koopman, and Lucas (2017), Blazsek and Villatoro (2015), Harvey and Chakravarty (2008) and Harvey and Sucarrat (2014). This paper uses the betatEGARCH and betaSkewtEGARCH models proposed by Harvey and Chakravarty (2008) and Harvey and Sucarrat (2014), as well as the t-GAS model; see Gao and Zhou (2016) for a brief explanation of the differences among these models.

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2.3. Robust loss function The evaluation of the volatility forecast is usually conducted by using volatility proxies such as those described in the previous section. However, these proxies are imperfect, since they are estimates of the integrated variance. Patton (2011) avoided the issue of these volatility proxies achieving misleading results when comparing the volatility forecasts by proposing the use of loss functions that were robust to the microstructure. The general class of the robust loss function is defined by Patton (2011), and is given by L(σˆ 2 , h, b)

⎧ σˆ 2 ⎪ 2 2 ⎪ ), for b = −1, h − σ ˆ + σ ˆ log( ⎪ ⎪ h ⎪ ⎨ 2 σˆ 2 σˆ = − log( ) − 1, for b = −2, ⎪ h h ⎪ 2b+4 2b+4 b+1 2 ⎪ ⎪ (σˆ −h ) h (σˆ − h) ⎪ ⎩ − , otherwise, (b + 1)(b + 2) b+1 where σˆ 2 is the squared forecast volatility and h is the squared volatility proxy. This paper considers the robust loss function of Patton (2011) with three different values of b: b = −2 (QLIKE), b = −1 (MSE) and b = 0 (denoted hereafter by RLF). Observe that all loss functions are equal to zero when h = σˆ 2 . 2.4. VaR estimation

2.2. Realised measures Evaluating the predictive abilities of different approaches to forecasting the volatility is a challenging task, since the volatility is a latent variable, and consequently cannot be observed directly. A common practice in the literature is to use a proxy for the true conditional variance. This paper uses the realised variance (RV; see Andersen, Bollerslev, Diebold, & Labys, 2003) and some other realised measures that are robust to jumps and microstructure noise as proxies of the true squared volatility. Specifically, we use the bipower variation (BV; see Barndorff-Nielsen & Shephard, 2004), MinRV (Andersen, Dobrev, & Schaumburg, 2012) and MedRV (Andersen et al., 2012). We prefer to use realised measures instead of square observed daily returns as per Catania et al. (2018) because realised measures have been shown to be better proxies of the true volatility (Alizadeh, Brandt, & Diebold, 2002; McAleer & Medeiros, 2008; Patton, 2011) and are used most widely nowadays. Let rt (i) be the ith high-frequency return of day t, with the realised measures used in this paper being given in Table 2. The main difference between RV and the other realised measures is that the latter are jump-robust and aim to estimate the integrated variance (IV), while the former estimates the quadratic variation (QV). The relationship between QV and IV is QVt = IVt + JVt , where JVt stands for the jump component. For more details about realised measures, as well as for asymptotic properties, see for instance Andersen, Bollerslev, and Diebold (2007), Andersen et al. (2012), Liu, Patton, and Sheppard (2015) and McAleer and Medeiros (2008).

Assuming that the returns are zero mean, the onestep-ahead VaR is usually estimated as VaRα = Qα σˆ T +1|T , where Qα is the α quantile of the assumed error distribution (scaled to have a unit variance) and σˆ T +1|T is the one-step-ahead volatility forecast. We estimate the VaR in a robust way by using the robust bootstrap procedure proposed recently by Trucíos et al. (2017). This procedure is based on a residual-based bootstrap scheme, combined with the robust GARCH estimator defined previously and robust filters for the volatility. The procedure can be summarized in the following steps:

• Step 1: Estimate the parameters ω, α and β in a robust way and obtain the standardized residuals ϵˆt = σrˆt . Denote the empirical distribution of these t

centred standardised residuals by Fˆϵ . • Step 2: Using ϵt∗ (bootstrap extractions from Fˆϵ ), generate bootstrap series through the following recursion: rt∗ = σt∗ ϵt∗ ,

σt∗+21 = ωˆ + ασ ˆ t∗2 cγ rc

(

rt∗2

σt∗2

)

ˆ t∗2 , + βσ

(2)

where σ1∗2 = σˆ 12 and the filter rc (·) is similar to the filter defined in Table 1 except that large values are replaced by new squared bootstrap extractions from Fˆϵ . Using the bootstrap series obtained in Eq. (2), estimate the parameters ω ˆ ∗ , αˆ ∗ and βˆ ∗ using the same estimator as in Step 1.

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C. Trucíos / International Journal of Forecasting 35 (2019) 836–847 Table 2 Realised measures. Realised measure

Formula N ∑

RVt

rt2 (i)

i=1

π

BVt

(

2

N N −1

π π −2

MinRVt

(

i=1

)∑ N −1

N N −1

π √

MedRVt

)∑ N −1 |rt (i)||rt (i + 1)|

(

6−4 3+π

• Step 3: Obtain h-step-ahead forecasts as ϵT∗+h σˆ T∗+h|T ,

rˆT∗+h|T

=

σˆ T∗+2 h|T

= ωˆ + αˆ

( ∗

∗

σˆ T∗+2 h−1|T cγ rc

2 rT∗+ h−1|T

σˆ T∗+2 h−1|T

) + βˆ ∗ σˆ T∗+2 h−1|T , (3)

for h = 1, . . . , H, and where

rˆT∗|T

= rT ,

ϵT∗+h

are

bootstrap extractions from Fˆϵ and σˆ T∗|2T is obtained ˆ ∗ + αˆ ∗ σˆ t∗−21|T cγ rc through the recursion σˆ t∗|T2 = ω

(

rt2−1 σˆ t∗−21|T

)

+ βˆ ∗ σˆ t∗−21|T for t = 2, . . . , T , with σˆ 1∗|2T = σˆ 12 .

• Step 4: Repeat steps 2 and 3 B times to obtain B ∗(1) ∗(B) bootstrap replicates (rˆT +h|T , . . . , rˆT +h|T ), where the α VaR is estimated as the h-step-ahead α empirical quantile of the bootstrap replicates. 3. Data and results We use daily Bitcoin closing prices (in US dollars) traded on Bitstamp from September 13, 2011, to December 31, 2017 (2,280 observations). Since Bitcoin is traded 24/7, the daily closing prices are considered as the last price traded at each day, and were constructed using the tick-by-tick data obtained from bitcoincharts.1 Returns are calculated as rt = log(Pt /Pt −1 ), with Pt being the closing price on day t. Table 3 reports descriptive statistics and Fig. 1 shows the daily returns, as well as the autocorrelations of returns and squared returns. The confidence bands for the returns were constructed using the generalised non-parametric Bartlett’s formula for nonlinear processes (Francq & Zakoïan, 2009), while the confidence bands for the squared returns were constructed using Bartlett’s formula (Bartlett, 1946). We can observe that Bitcoin is highly volatile, with an √ annualised standard deviation of 1.0049 (0.0526 × 365), and its returns also present asymmetry and a large kurtosis. The large kurtosis is probably explained by the presence of extreme returns, as observed in Fig. 1. Because the returns series does not exhibit serial correlation, no ARMA filter is applied to the data, so that the series is only centred to have a zero mean. The largest return (in absolute value) corresponds to April 10, 2013, when Bitcoin prices dropped by around 1 https://api.bitcoincharts.com/v1/csv/bitstampUSD.csv.

min (|rt (i)|, |rt (i + 1)|)2

i=1

N N −2

)∑ N −1

med (|rt (i − 1)|, |rt (i)|, |rt (i + 1)|)2

i=1

60% of their value. This event was attributed to a DDoS (distributed denial of service) attack.2 The largest value in 2012 corresponds to August 19. According to Forbes,3 Bitcoin prices doubled between July 1 and August 18, 2012, then dropped considerably. More than 25% of the loss between June 2014 and November 2016 happened on January 13, 2015, a week after the temporary suspension of the services of Bitstamp and following the news that Russia had started banning websites related to Bitcoin.4 We evaluate the out-of-sample performances of the models by using a rolling window scheme with a window size of 1000 days. The one-step-ahead conditional variance is estimated in each window. As the daily volatility is not observed, we use the realised measures described in Section 2.2 as proxies (using five-minute high-frequency data). Table 4 reports the descriptive statistics of those realised measures in the out-of-sample period and Fig. 2 shows their evolution over time.5 As expected, the RV is larger than the other realised measures. The MinRV, MedRV and BV are all robust to jumps and microstructure noise, which are desirable properties in volatility proxies, as was pointed out by Andersen et al. (2012) and Barndorff-Nielsen and Shephard (2004). This makes these measures preferred to RV. The extreme value observed when the RV is computed can explain the large values of the skewness and kurtosis. Unlike Charles and Darné (2018), who take into account the presence of outliers using jump-filtered returns as per Laurent et al. (2016), we estimate the conditional variance in a robust way using the robust estimator of Boudt et al. (2013) with the modification introduced by Trucíos et al. (2017). Alternative robust procedures such as those proposed by Carnero et al. (2012) and Muler and Yohai (2008) could also be used, although the results of Trucíos, Hotta, and Ruiz (2015) show that the procedure of Boudt et al. (2013) with the modification introduced by Trucíos et al. (2017) performs the best in a forecasting context. Table 5 reports the MSE, QLIKE and RLF values between the σˆ T2+1 obtained using the GARCH-type models 2 https://bitcoinmagazine.com/articles/the-bitcoin-crash-anexamination-1365911041/. 3 https://www.forbes.com/sites/timothylee/2013/04/11/anillustrated-history-of-bitcoin-crashes/#640b1c940397. 4 https://www.cnbc.com/2015/01/14/bitcoin-falls-below-200making-some-investors-worry-about-downward-spiral.html. 5 To assist in the visualisation of the results, the values were cut off at 0.0023. However, there is a large RV (0.0151) on April 17, 2016.

C. Trucíos / International Journal of Forecasting 35 (2019) 836–847

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Fig. 1. Daily returns (top panel), sample autocorrelation function of returns with their corresponding generalised non-parametric Bartlett 95% confidence bands (left bottom panel) and sample autocorrelations of squared returns with their corresponding Bartlett 95% confidence bands (right bottom panel).

Fig. 2. Realised measures in the out-of-sample period. BV (left top panel), MedRV (right top panel), MinRV (left bottom panel) and RV (right bottom panel).

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C. Trucíos / International Journal of Forecasting 35 (2019) 836–847 Table 3 Descriptive statistics of Bitcoin daily returns. Mean

Std. Dev.

Min

Q1

Med.

Q3

Max

Skewness

Kurtosis

0.0034

0.0526

−0.6639

−0.0111

0.0024

0.0200

0.4455

−1.4011

28.4802

Table 4 Descriptive statistics (multiplied by 100) of the realised measures in the out-of-sample period. RV MinRV MedRV BV

Mean

Std. Dev.

Min

Q1

Med.

Q3

Max

Skewness

Kurtosis

0.0333 0.0171 0.0203 0.0154

0.0479 0.0151 0.0163 0.0124

0.0011 0.0002 0.0001 0.0006

0.0150 0.0070 0.0088 0.0068

0.0255 0.0128 0.0156 0.0122

0.0434 0.0226 0.0271 0.0205

1.5133 0.1485 0.1155 0.1184

23.3025 2.3828 1.7340 2.1705

714.2308 12.6226 7.1631 11.4692

and the realised measures RV, MinRV, MedRV and BV used as proxies of the true value σT2+1 . The shadowed cells indicate the set of models with the best out-ofsample performance obtained using the model confidence set (MCS) approach (Hansen, Lunde, & Nason, 2011) at the 75% significance level.6 The best model (first position in the MCS rank) in each case is shown in bold. The MSE selects a large set of models, reflecting a low power of distinguishing between different models. Indeed, when looking at the MSE, only 15 of the 100 models (16 when considering the MinRV as the volatility proxy) are left out. Other works have also found that the MSE has a low power to distinguish between different models; see for instance Liu et al. (2015) and Patton and Sheppard (2009). The QLIKE and RLF loss functions generally select small numbers of models as best models.7 In general, we can observe that the MCS methodology selects the same models for a given loss function regardless of the realised measure used. An exception is observed when the QLIKE measure is used, in which case the CGARCH model belongs to the group of models with the best performance only when the RV is used. Considering all loss functions and realised measures, only five models are in the set of best models in all cases, namely TARCH and AVGARCH, both assuming GED (with both the standard and skew versions), and the robust GARCH model. The CGARCH model, which was the best model according to Katsiampa (2017), now only appears in the set when using the MSE and RLF loss functions and/or when using the QLIKE loss function with the RV as the volatility proxy. The IGARCH model, which was the best model in the in-sample analyses of Chu et al. (2017), only appears in the set when considering the MSE loss function. In all cases, the GARCH model estimated in a robust way reports the best results, regardless of the loss function and realised measure used. This result is extremely important, since most of the studies available in the literature about Bitcoin do not take into account the presence of outliers. As we can see in Table 5, the forecasts can be improved substantially by using a robust 6 We used the R package MCS of Bernardi (2017). 7 It is important to note that the TARCH model with SGED innovations distribution is extremely large, even when considering the average MSE. However, the MCS still selects this model in the set of best models due to the fact that this high average value is given by a single extreme case, while the remaining cases reported small values of the MSE.

approach, so that a simple model such as the robust GARCH model considered in this paper outperforms more sophisticated models. These results are in accordance with those of Charles and Darné (2018), who also find, in an in-sample context, that sophisticated models are outperformed by a GARCH model when the presence of outliers is considered in the analysis. The results in Table 5 compare most (if not all) of the classical GARCH-type models used in the literature for modeling and forecasting Bitcoin’s volatility. Furthermore, we go one step further and forecast the volatility using the alternative approaches described in Sections 2.1.2 to 2.1.4, the results of which are reported in Table 6. For the sake of comparison, Table 6 also includes the results obtained using the robust GARCH procedure already presented in Table 5. As expected, the GARCH-MIDAS model and the realised GARCH model (regardless of the realised measures used as an auxiliary variable) outperform the Gaussian GARCH model (Table 5). However, they are outperformed by the robust GARCH model. Table 6 includes additional robust procedures for modeling and forecasting the volatility by reporting results using the betatEGARCH, betaSkewtEGARCH and t-GAS models, which are all particular cases of the general class of GAS models. For the sake of comparison, we also include the leverage effect in the betatEGARCH and betaSkewtEGARCH; see Gao and Zhou (2016) for a brief explanation. The results reveal that, in general, the GAStype models used in this paper are better at capturing the dynamic of Bitcoin volatility than the robust GARCH procedure. Mostly, the best result is achieved using the t-GAS model; in only one case is the best performance achieved by the betatEGARCH model with leverage. These results are not surprising, since the literature about GAS models argues that scores are more informative than the squared returns for capturing the volatility dynamics, and also down-weight extreme observations. Given that high-frequency data are available, Table 7 also compares the predictability of the realised measures. To this end, we use the ARFIMA model suggested by Andersen et al. (2003) and Koopman, Jungbacker, and Hol (2005), the heterogeneous autoregressive (HAR) model of Corsi (2009), the HEAVY model of Shephard and Sheppard (2010) and the realised GARCH model described in Section 2.1.3. ARFIMAL and HARL stand for the ARFIMA and HAR models applied to the logarithm of the realised measure. The results using the logarithm are always better than those using the realised measures directly. Note

C. Trucíos / International Journal of Forecasting 35 (2019) 836–847 Table 5 Average MSE, QLIKE and RLF values of σˆ T2+1 (obtained using GARCH-type models) and the realised measures.

Note: The shadowed cells indicate the set of models selected by the MCS.

843

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C. Trucíos / International Journal of Forecasting 35 (2019) 836–847 Table 6 Average MSE, QLIKE and RLF values of σˆ T2+1 (obtained using alternative models) and the realised measures.

Note: The shadowed cells stand for the set of models selected by the MCS. Table 7 Average MSE, QLIKE and RLF values of the one-step-ahead forecast realised measures and its true value.

Note: The shadowed cells stand for the set of models selected by the MCS. Table 8 Proportion of returns smaller than the 1% VaR, p-values of the UC, CC and DQ tests, and the average quantile loss function (AQLF) (×103 ). Models

Prop. fails

UC

CC

DQ

AQLF

AVGARCHGED AVGARCHSGED Robust bootstrap t-GAS RealGARCH

0.1023 0.1023 0.0063 0.0266 0.0867

0.0000 0.0000 0.1475 0.0000 0.0000

0.0000 0.0000 0.3331 0.0000 0.0000

0.0000 0.0000 0.6602 0.0000 0.0000

2.9776 2.9710 1.4283 1.7676 3.4591

also that, with the exception of the ARFIMA and HAR models,8 the procedures that forecast the realised measures report better performances than the t-GAS model, and the best performance is achieved using the realised GARCH model. These results are not surprising, since these models are designed specifically for forecasting the realised measure directly, and also exploit the information available in the intra-day data fully, whereas the models in Tables 5 and 6 forecast the daily volatility and use (mostly) daily information. The 1% VaR is also computed, and backtesting procedures for the VaR are carried out. The models compared are the AVGARCH (best performance among the non-robust procedures), robust GARCH (best performance among the classical GARCH-type models), t-GAS (best performance for forecasting the daily volatility) and realised GARCH (best performance for forecasting the realised measures). In all cases, the usual VaR estimator was used, except for the robust GARCH model, where the robust bootstrap procedure of Trucíos et al. (2017) was applied. The results are shown in Table 8 and include the proportion of fails (returns smaller than the 1% VaR), p-values of the unconditional coverage (UC; see Kupiec, 1995), conditional coverage (CC; see Christoffersen, 1998) and dynamic quantile (DQ; see Engle & Manganelli, 2004) tests, 8 We exclude the HAR model from the comparison when using MinRV because they were forecasted as negative values on two days.

and the average quantile loss function of González-Rivera, Lee, and Mishra (2004). The results reveal that the non-robust procedures underestimate the VaR and the backtesting tests reject the null hypothesis that the proportion of fails is equal to 0.01. Only the robust bootstrap procedure of Trucíos et al. (2017) showed a good performance in all cases (the tests fail to reject the null hypothesis and the smallest average quantile loss function is observed). The t-GAS reports an improvement with respect to the performances obtained by the AVGARCH and the realised GARCH. However, the tests UC, CC and DQ reject the null hypothesis. Fig. 3 reports the returns on the out-of-sample period and for the 1% VaR estimated in a robust way. 4. Conclusions and future work This paper has carried out a comprehensive out-ofsample comparison in the context of the daily volatility forecasting of Bitcoin using several volatility models, including robust procedures and procedures based on intra-day data. Their VaR performances have also been addressed. The best model for forecasting the volatility among the non-robust classical GARCH-type models was the AVGARCH assuming errors coming from a GED (both the symmetric and skew versions). However, this model was outperformed by the robust procedures.

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Fig. 3. Return in the out-of-sample period (black solid line) and the estimated 1% VaR (red dashed line) obtained using the robust bootstrap procedure of Trucíos et al. (2017).

We emphasize the fact that the presence of outliers should not be neglected, and that substantial improvements can be obtained by using robust procedures. The robust GARCH procedure and the GAS model outperformed all of the non-robust procedures for forecasting the volatility, with the best performance being achieved when using the t-GAS model. Procedures for forecasting the realised measures directly were also applied. Procedures applied to the logarithm of the realised measure showed better performances than those applied to the realised measures directly. The best performance for forecasting the realised measures was obtained using the realised GARCH model. In the non-robust procedures, the VaR estimation reported a large proportion of fails and the backtesting tests rejected the null hypothesis that the proportion of fails was equal to 1%, giving a misleading picture of what can be expected in the future. On the other hand, the VaR estimated using the robust bootstrap procedure of Trucíos et al. (2017) showed a good performance and none of the tests rejected the null hypothesis. In the absence of high-frequency data, we suggest using the t-GAS model to forecast the volatility, while the realised GARCH model seems to be a good alternative when using high-frequency data. In the context of VaR estimation, we suggest using the robust bootstrap procedure of Trucíos et al. (2017). Extensions of this procedure to apply it in other volatility models such as t-GAS or realised GARCH models forms an interesting research topic, and could provide a good alternative for better estimating the VaR. Additional comparisons considering different robust procedures, as well as the application of regime-switching models to cryptocurrencies (Ardia, Bluteau, & Rüede, 2019; Caporale & Zekokh, 2018), provide other interesting research topics. In the same spirit, it is important that a more extensive comparison using models to forecast realised measures be conducted. Some papers, such as those by Balcilar et al. (2017), Cermak (2017) and Dyhrberg (2016), have also considered explanatory variables for better estimating the volatility. Thus, a further step in this research would be to analyse the performances of the out-of-sample volatility forecasts when considering explanatory variables and robust procedures. This paper accords with the results of Carnero et al. (2012), Trucíos and Hotta (2016) and Trucíos et al. (2017),

which showed the dramatic effects of additive outliers on the estimation and prediction of the volatility and VaR. For a good review of outliers in GARCH models, we refer to Hotta and Trucíos (2018). In a multivariate framework, Boudt et al. (2013), Grané, Veiga, and Martín-Barragán (2014) and Trucíos, Hotta and Ruiz (2018) showed the dramatic effect of outliers in the estimation and prediction of the volatilities and covolatilities. In this sense, it is important that future studies considering Bitcoin and other cryptocurrencies take into account the presence of outliers. In this context, the procedures proposed by Boudt and Croux (2010), Boudt et al. (2013), Croux, Gelper, and Mahieu (2010), Iqbal (2013), Trucíos, Hotta and Ruiz (2018) and Trucíos, Hotta and Valls Pereira (2018), for example, could be useful. Other recently published papers are likewise in accordance with our findings. For instance, Troster, Tiwari, Shahbaz, and Macedo (2019) also concluded that GAS models perform better than GARCH models in a Bitcoin risk measure context. Finally, Chaim and Laurini (2018) found that the presence of jumps should be taken into account when modelling the Bitcoin volatility. Acknowledgments The author acknowledges financial support from São Paulo Research Foundation (FAPESP), Brazil grants 2016/18599-4 and 2018/03012-3, as well as the support of the Centre of Quantitative Studies in Economics and Finance (CEQEF), Brazil and the Centre for Applied Research on Econometrics, Finance and Statistics (CAREFS), Brazil. The author thanks João H. G. Mazzeu and Luiz K. Hotta, as well as the editors and two anonymous referees, for helpful comments and suggestions. References Alizadeh, S., Brandt, M. W., & Diebold, F. X. (2002). Range-based estimation of stochastic volatility models. The Journal of Finance, 57(3), 1047–1091. Andersen, T. G., & Bollerslev, T. (1998). Answering the skeptics: yes, standard volatility models do provide accurate forecasts. International Economic Review, 39, 885–905. Andersen, T. G., Bollerslev, T., & Diebold, F. X. (2007). Roughing it up: including jump components in the measurement, modeling, and forecasting of return volatility. The Review of Economics and Statistics, 89(4), 701–720.

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