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The volatility surprise of leading cryptocurrencies: Transitory and permanent linkages
Finance Research Letters xxx (xxxx) xxx–xxx
Contents lists available at ScienceDirect
Finance Research Letters journal homepage: www.elsevier.com/locate/frl
The volatility surprise of leading cryptocurrencies: Transitory and permanent linkages Elie Bouria, , Brian Luceyb, David Roubaudc ⁎
a b c
USEK Business School, Holy Spirit University of Kaslik, Jounieh, Lebanon Trinity Business School, Trinity College Dublin, Dublin 2, Ireland Montpellier Business School, Montpellier, France
ARTICLE INFO
ABSTRACT
JEL classifications: C13 G15
There is scarce literature examining the volatility linkages among leading cryptocurrencies, and none exists on the linkages among unexpected volatility, called ‘volatility surprise’. To address this literature gap, we build on the concept of volatility surprise and examine the causal linkages among the volatility of leading cryptocurrencies via the frequency-domain test of Bodart and Candelon (2009), discriminating between transitory and permanent causalities. Permanent shocks are more important in explaining the Granger-causality that does not necessarily emanate from the largest, Bitcoin, over short horizons, whereas transitory shocks dominate the causality across smaller cryptocurrencies over long horizons.
Keywords: Bitcoin Cryptocurrency Volatility surprise frequency-domain Granger-causality
1. Introduction The importance of volatility to asset pricing, risk management, and portfolio allocations cannot be overrated. However, what can also be important is the ‘volatility surprise’ that represents unexpected volatility, i.e. the difference between squared residuals and the conditional variance (Hamo et al., 1990; Engle, 1993). Previous studies examine the relationships among the volatility surprises of conventional assets such as equities, bonds, currencies, and commodities (e.g., Aboura and Chevallier, 2014). Such an examination is unprecedented for newly emerged digital assets (Corbet et al., 2018a) that exhibit extreme price volatility and continue to fascinate the financial press, policy-makers, and financial community. Scarce studies (e.g., Koutmos, 2018; Ji et al., 2018) examine the connectedness among leading cryptocurrencies, focusing on predictable variance (volatility) such as the conditional variance (volatility). However, they don't consider the linkages among volatility surprises that are crucial, given they represent the unexpected volatility (Engle, 1993; Aboura, and Chevallier, 2014). Koutmos (2018) and Ji et al. (2018) overlook the possibility that the linkages might not be the same at different frequencies or scales. This is important, as crypto-traders and investors may exhibit heterogeneity in terms of their trading and investment horizons. In this paper, we uncover the causal relationship across the volatility surprise of Bitcoin and seven other leading cryptocurrencies (Ethereum, Ripple, Stellar, Litecoin, Monero, Nem, and Dash) via the Granger-causality in the frequency-domain of Breitung and Candelon (2009). By decomposing the causality at different frequencies, i.e. time scales, we discriminate between transitory and permanent causal relations1 among the unexpected volatility in the cryptocurrency, which might help short-term speculators and
Corresponding author. E-mail addresses: [email protected] (E. Bouri), [email protected] (B. Lucey), [email protected] (D. Roubaud). 1 According to Breitung and Candelon (2009), this test leads to robust causality results because it can deal with the presence of outliers in the data and is unaffected by the presence of multivariate GARCH. ⁎
https://doi.org/10.1016/j.frl.2019.05.006 Received 28 March 2019; Received in revised form 30 April 2019; Accepted 7 May 2019 1544-6123/ © 2019 Elsevier Inc. All rights reserved.
Please cite this article as: Elie Bouri, Brian Lucey and David Roubaud, Finance Research Letters, https://doi.org/10.1016/j.frl.2019.05.006
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long-term investors with different investment horizons make investment and risk management decisions that match their respective investment time horizons (Chauveau and Subbotin, 2013). The current paper relates to the embryonic literature that looks beyond the Bitcoin market to consider issues of other leading cryptocurrencies.2 Such issues include price discovery (Bouri et al., 2018a; Brauneis and Mestel, 2018), returns behaviour (Gkillas and Katsiampa, 2018), volatility modelling (Chu et al., 2017), co-bubbling (Bouri et al., 2018b), market efficiency (Brauneis and Mestel, 2018), and diversification abilities (Platanakis et al., 2018). Our paper complements these strands of research, but differs in both research scope and method. Firstly, in contrast to previous studies considering the return relationships between Bitcoin and other conventional assets (Bouri et al., 2017; Baur et al., 2018; Corbet et al., 2018b; Klein et al., 2018), or among the volatility of cryptocurrencies (Koutmos, 2018; Ji et al., 2018), we concentrate on the Granger-causality in the volatility surprise across leading cryptocurrencies. As argued by Hamo et al. (1990), Engle (1993) and Aboura and Chevallier (2014), volatility surprise represents the unexpected volatility that considerably matches the need of market participants. Secondly, we consider the frequencydomain approach that allows us to differentiate between permanent and transitory causal relations in volatility surprise among leading cryptocurrencies. As such, we provide important insights to both short-term speculators and long-term investors who try to match their investment choices to the right timescale (Chauveau and Subbotin, 2013). We go some way to discovering whether the cryptocurrency market is a heterogeneous market in terms of the volatility causal relationships across its main components. Thirdly, our paper complements Koutmos (2018) and Ji et al. (2018), who, by focusing on the interdependencies among major cryptocurrencies via connectedness measures, overlook the possibility that the relationships might not be the same at different frequencies or scales. Furthermore, they consider the linkages among volatility, whereas our focus is on unexpected volatility (i.e. volatility surprise). Our results provide evidence of causal relationships, and thus predictability, in the volatility surprise across cryptocurrencies that seems to differ across timescales. This suggests the need to differentiate between permanent and transitory causal relations in the volatility surprise of leading cryptocurrencies. Another implication relates to the fact that market participants should not treat leading cryptocurrencies as a heterogeneous group in terms of causal relationships among volatility surprise. The paper continues as follows. Section 2 provides the empirical methods. Section 3 describes the data and presents the empirical results. Section 4 concludes. 2. Research methods According to Granger (1969), a variable x is said to ‘Granger-cause’ another variable y if lagged values of x contain information that helps predict y. Conducted based on the Wald test within a VAR model, this time-domain Granger-causality test relies on a single statistic measure of predictability, ignoring the possibility that Granger-causality may not be the same at different frequencies. Breitung and Candelon (2006) propose a Granger-causality test in the frequency-domain that considers all frequencies. In this paper, we apply the extended version of this test, taken from Bodart and Candelon (2009), which removes all outliers3 and replaces them with a 10-day average centred around the abnormal observation. As argued by Bodart and Candelon (2009), this makes the residuals free from autocorrelation and outliers, leading to a correctly specified model. If not dealt with properly, the presence of outliers in the cryptocurrency data can lead to fallaciously overestimated Granger-causality. Furthermore, Bodart and Candelon (2009) indicate that their test is unaffected by the presence of multivariate GARCH. Consider Zt = [xt, yt] to be a two-dimensional vector of endogenous variables observed at time t = 1, …., T, with a finite order VAR representation, such as: (1)
(L) Zt = vt p
k
where, θ(L) = 1-θ1L− … −θpL is a 2 × 2 lag polynomial with L Zt = Zt − k. The residual vt is assumed to be white noise with zero mean and covariance matrix E( t t ) = Σ defined as positive. The matrix Σ is decomposed as G′G = Σ−1, where G is the inferior triangular matrix of the Cholesky decomposition, such that E(ηtη′t) = I, and ηt = Gεt. If the system is stationary, the VAR process has a moving average:
Zt =
(L ) t = −1
where, φ (L) = θ(L)
1 fx ( ) = { 2
11 (L)
12 (L)
1t
21 (L)
22 (L)
2t
and φ(L)= ∅(L)G
11 e
i
)2 +
i
12 e
= −1
(L )
t
=
11 (L)
12 (L)
1t
21 (L)
22 (L)
2t
(2)
The spectral density of Xt is given by:
) 2}
(3)
The measure of causality in Eq. (4) is used to test the null that yt does not Granger-cause xt at frequency ω [H0 : My→x(ω) = 0].
My
x(
) = log 1 +
12 (e
i
11 (e
i
)
2
)
2
(4)
2 Bitcoin's success and underlying technology have inspired the release of many cryptocurrencies such as Ethereum, Ripple, and Stellar, which rapidly grow in market size and become crucial players in the cryptocurrency market and alternative digital assets (Bouri et al., 2018a). 3 Bodart and Candelon (2009) detect outliers via the median absolute deviation (MAD) of Hotta and Tsay (2012).
2
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This measure is zero if |φ12 (e−iω)|2 = 0, implying no causality at frequency ω. To obtain the statistic My→x(ω), we replace |φ11(e−iω)| and |φ12(e−iω)| in Eq. (4) with the estimated values of the fitted VAR representation.4 The significance of the causal relationship between two time series is assessed by comparing the Granger-causality measure for the frequency ω to the 5% critical value of a χ2 distribution with 2 degrees of freedom (5.99). 3. Empirical results 3.1. Data The daily closing prices of eight large cryptocurrencies (Bitcoin, Ethereum, Ripple, Stellar, Litecoin, Monero, Nem, and Dash) are gathered from https://coinmarketcap.com/. Dictated by the availability of price data for some cryptocurrencies (e.g. Ethereum), the sample period spans August 8, 2015 to February 18, 2019 (1291 observations), covering a rich period of booms and busts. Specifically, the eight cryptocurrencies are selected from the largest 20 cryptocurrencies by market value in order to make the analyses richer by covering emerging and large cryptocurrencies, which are of interest to crypto-traders. Our sample period is suitable and rich enough because it includes the years 2016-2017-2018 when Bitcoin and other leading cryptocurrencies started to attract huge interest among investors and thus large trading activities. Using the log returns of each of the eight cryptocurrencies, we estimate the mean equation (Eq. (1)) and the variance equation (Eq. (2)) as:
rt = µ + 2 t
=
(5)
t
+
2 t 1
+
(6)
2 t 1
We follow Aboura, and Chevallier (2014), and compute the mean-zero volatility surprise (˜) as the volatility surprise (ς) (i.e. the difference between the squared residuals (ε2) and the conditional variance (σ2)) normalized by the conditional variance σ2 as:
˜t =
(
2 t
2 t
2 t )
(7)
Summary statistics of the normalized volatility surprise are given in Table 1. The most (least) volatile is Litecoin (Ethereum). Again, Litecoin has the highest kurtosis, whereas Ethereum has the lowest. Skweness is positive for four cases and negative for the rest.5 Table 2 Panel A shows that the pairwise correlation between volatility surprises is weekly positive. It ranges from 0.054 (Stellar Litecoin) to 0.365 (Bitcoin - Ethereum), with evidence that Bitcoin is more correlated with other cryptocurrencies. 3.2. Results of the time-domain Granger-causality test Before conducting the causality tests in the frequency-domain following Bodart and Candelon (2009), we conduct the timedomain Granger-causality test (Granger, 1969) within the framework of VAR models6 and report the results in Table 3. Considering the 5% significance level, the results are summarized as follows. A unidirectional causality runs from Bitcoin to Ethereum and Nem. Ethereum Granger-causes Monero. Ripple Granger-causes Litecoin and Nem Granger-causes Ripple. A unidirectional causality runs from Litecoin to Stellar, and from Nem to Litecoin. Finally, Monero Granger-causes Dash. 3.3. Results of the frequency-domain Granger-causality test Following Bodart and Candelon (2009), we apply the frequency-domain Granger-causality test, after taking care of the outliers, discriminating between temporary and permanent causality. Bodart and Candelon (2009) indicate that the components at low frequencies are more persistent than at high frequencies. Causalities are thus investigated at two frequencies between 0 and π. Permanent causality is instigated for long-run horizons representing cycles of 63 – 6 days (i.e., ω ∈ 0.0–1.5), whereas temporary causality is investigated for short-run horizons representing cycles of 2–6 days (i.e., ω ∈ 1.5 − π).7 For every frequency considered, the null of no Granger-causality is rejected if the test statistic is above the 5% critical value and vice-versa. The results in Table 3 show a transitory Granger-causality from Bitcoin to Ethereum at high frequencies, whereas there is Granger-causality from Bitcoin to Nem at both low and high frequencies. Ethereum Granger-causes Monero at low frequencies, and Nem Granger-causes Ethereum at low frequencies indicating a permanent interdependence. Ripple Granger-causes Litecoin at low and high frequencies. Ripple also Granger-causes Nem at low frequencies, and there is a feedback effect at both low and high frequencies. Litecoin Granger-causes Stellar at low frequencies. Monero Granger-causes Nem at high frequencies and Dash at low frequencies. The results point to the importance of some cryptocurrencies other than Bitcoin to the network structure of causalities of volatility surprise, whereas other cryptocurrencies, such as Stellar and Dash, are peripheral. This finding is not fully in line with previous 4
Notably, the optimal lag of the VAR model is selected based on SIC. Unreported results from unit root tests show that all volatility surprise series are stationary at the 1% level. 6 We select the optimal lag of the VAR model that yields the lowest SIC. 7 The frequency parameter omega (ω) is used to compute the length of cycles (T = 2π /ω). 5
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Table 1 Summary statistics of normalized volatility surprise.
BITCOIN ETHEREUM RIPPLE STELLAR LITECOIN MONERO NEM DASH
Mean
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
−0.0011 −0.0018 0.0000 0.0008 −0.0011 −0.0007 −0.0002 −0.0010
43.2077 28.6333 85.4106 126.7129 177.5432 69.9550 103.6639 85.7745
−42.3909 −28.7041 −85.3888 −126.7093 −177.4740 −69.7006 −105.6492 −85.4343
4.1001 3.4049 5.1735 6.2412 7.8565 4.5839 5.4440 4.7003
0.1581 −0.1250 −0.0258 −0.0087 0.0277 0.1131 −0.3457 0.0455
55.2335 27.0410 126.7531 279.6336 403.9178 102.2392 217.2210 175.2074
Notes: This table provides summary statistics of daily volatility surprise, computed according to Eq. (7). The sample period is August 8, 2015–February 18, 2019.
Table 2 Unconditional correlation matrix.
BITCOIN ETHEREUM RIPPLE STELLAR LITECOIN MONERO NEM DASH
BITCOIN
ETHEREUM
RIPPLE
STELLAR
LITECOIN
MONERO
NEM
DASH
1.0000 0.3657 0.1598 0.1169 0.2044 0.3024 0.1729 0.2126
1.0000 0.2113 0.1446 0.1520 0.2284 0.1368 0.2374
1.0000 0.2030 0.1372 0.1445 0.1461 0.1170
1.0000 0.0541 0.0803 0.0761 0.0748
1.0000 0.1028 0.1230 0.0858
1.0000 0.1300 0.2102
1.0000 0.1002
1.0000
Notes: This table provides pairwise unconditional correlations across the daily volatility surprises of leading cryptocurrencies. The sample period is August 8, 2015–February 18, 2019.
studies that consider volatility, and stress the importance of Bitcoin (e.g., Ji et al., 2018; Koutmos, 2018), pointing to the differences in the causal relationships in volatility and volatility surprise. We find that the Granger-causality structure in some cases is frequencydependent (i.e. it varies between short-run and long-run horizons), with more evidence of permanent causality. This heterogeneity in the causality structure is important given that market participants often have different investment horizons. The findings are useful for crypto-traders and investors who are keen to understand volatility linkages in both the short and long run in order to enhance predictability, which is important to both trading and investment strategies. Specifically, market participants in the cryptocurrency market can gain information by looking into the linkages across the volatility surprise of cryptocurrencies while differentiating, in some cases, between transitory and permanent causality. They can exploit evidence of predictability in studying portfolio implications (Platanakis and Urquhart, 2019) and in enhancing investment strategies along the spectrum (Chaudhuri and Lo, 2016; Faria and Verona, 2018). 4. Conclusion We contribute to the cryptocurrency literature by uncovering the linkages among the volatility surprises of eight leading cryptocurrencies. The results show evidence of causality linkages among the volatility surprises that do not necessarily originate from the largest cryptocurrency, Bitcoin, suggesting that the unexpected volatility of other cryptocurrencies matters in the heterogeneous cryptocurrency market. The fact that causalities differ between the short and long runs for some cryptocurrencies indicates the importance of bearing causalities in mind at low and high frequencies when considering integration. It also indicates the possibility of predicting the volatility surprise at specific frequencies, suggesting that crypto-traders should monitor other cryptocurrencies while trading one cryptocurrency. Conversely, some cryptocurrencies, such as Stellar and Dash, are relatively segmented, offering the potential for diversification. Our analysis complements the existing, if embryonic, findings (Bouri et al., 2018a,b; Ji et al., 2018; Platanakis et al., 2018) and shows that the largest cryptocurrency, Bitcoin, is not the only influential cryptocurrency. This is somewhat inconsistent with Koutmos (2018), who shows that Bitcoin has a stronger influence on other cryptocurrencies, given its older presence and higher market capitalization. It could be that our focus on the unexpected volatility and the high growth in other cryptocurrencies, to the detriment of Bitcoin's dominance, affect our findings. Our analysis provides a foundation for further analyses involving the drivers of the volatility surprise of segmented cryptocurrencies. Future research could consider the implications of our findings using a wavelet-based forecasting method (Faria and Verona, 2018). 4
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Table 3 Results of time and frequency Granger-causality tests.
Bitcoin-Ethereum Bitcoin-Ripple Bitcoin-Stellar Bitcoin-Litecoin Bitcoin-Monero Bitcoin-Nem Bitcoin-Dash Ethereum-Ripple Ethereum-Stellar Ethereum-Litecoin Ethereum-Monero Ethereum-Nem Ethereum-Dash Ripple-Stellar Ripple-Litecoin Ripple-Monero Ripple-Nem Ripple-Dash Stellar-Litecoin Stellar-Monero Stellar-Nem Stellar-Dash Litecoin-Monero Litecoin-Nem Litecoin-Dash Monero-Nem Monero-Dash Nem-Dash
Null hypothesis
Panel A: Time-domain Granger-causality Chi-sq P values
Panel B: Frequency-domain Granger-causality Permanent (63-6 days) Temporary (6-2) days
Bitcoin ≠› Ethereum Ethereum ≠› Bitcoin Bitcoin ≠› Ripple Ripple ≠› Bitcoin Bitcoin ≠› Stellar Stellar ≠› Bitcoin Bitcoin ≠› Litecoin Litecoin ≠› Bitcoin Bitcoin ≠› Monero Monero ≠› Bitcoin Bitcoin ≠› Nem Nem ≠› Bitcoin Bitcoin ≠› Dash Dash ≠› Bitcoin Ethereum ≠› Ripple Ripple ≠› Ethereum Ethereum ≠› Stellar Stellar ≠› Ethereum Ethereum ≠› Litecoin Litecoin ≠› Ethereum Ethereum ≠› Monero Monero ≠› Ethereum Ethereum ≠› Nem Nem ≠› Ethereum Ethereum ≠› Dash Dash ≠› Ethereum Ripple ≠› Stellar Stellar ≠› Ripple Ripple ≠› Litecoin Litecoin ≠› Ripple Ripple ≠› Monero Monero ≠› Ripple Ripple ≠› Nem Nem ≠› Ripple Ripple ≠› Dash Dash ≠› Ripple Stellar ≠› Litecoin Litecoin ≠› Stellar Stellar ≠› Monero Monero ≠› Stellar Stellar ≠› Nem Nem ≠› Stellar Stellar ≠› Dash Dash ≠› Stellar Litecoin ≠› Monero Monero ≠› Litecoin Litecoin ≠› Nem Nem ≠› Litecoin Litecoin ≠› Dash Dash ≠› Litecoin Monero ≠› Nem Nem ≠› Monero Monero ≠› Dash Dash ≠› Monero Nem ≠› Dash Dash ≠› Nem
17.045 5.902 7.124 4.891 0.034 0.013 6.669 1.190 3.824 0.459 34.843 5.058 9.979 5.100 5.137 3.641 6.290 2.516 2.284 2.211 20.234 3.056 2.387 17.992 3.735 6.991 7.121 1.025 79.718 15.495 2.726 2.957 14.520 62.844 5.864 3.706 2.240 27.300 2.658 5.497 3.587 2.064 3.313 0.755 1.045 1.278 2.271 6.861 4.842 0.612 14.920 7.283 16.886 8.885 9.278 4.598
Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Rejected Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Rejected Accepted Accepted Rejected Accepted Accepted Accepted Accepted Rejected Accepted Accepted Accepted Rejected Rejected Accepted Accepted Accepted Rejected Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Rejected Accepted Accepted Accepted
0.048 0.749 0.624 0.891 0.982 0.993 0.572 0.997 0.797 0.999 0.000 0.829 0.367 0.825 0.822 0.933 0.790 0.990 0.942 0.947 0.005 0.879 0.992 0.055 0.928 0.638 0.624 0.999 0.000 0.502 0.974 0.965 0.099 0.000 0.753 0.929 0.987 0.001 0.953 0.703 0.936 0.990 0.950 0.999 0.994 0.989 0.993 0.122 0.847 0.999 0.093 0.607 0.050 0.447 0.412 0.867
Rejected Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Rejected Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Rejected Accepted Accepted Accepted Accepted Rejected Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Rejected Accepted Accepted Accepted Accepted Accepted
Notes: This table presents the results of the null hypothesis of no Granger-causality in the time-domain and the frequency-domain. The sample period is August 8, 2015–February 18, 2019.
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Supplementary materials Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.frl.2019.05.006. References Aboura, S., Chevallier, J., 2014. Volatility equicorrelation: a cross-market perspective. Econ. Lett. 122 (2), 289–295. Baur, D.G., Hong, K., Lee, A.D., 2018. Bitcoin: medium of exchange or speculative assets? J. Int. Financ. Mark. Inst. Money 54, 177–189. Bodart, V., Candelon, B., 2009. Evidence of interdependence and contagion using a frequency domain framework. Emerg. Mark. Rev. 10 (2), 140–150. Bouri, E., Molnar, P., Azzi, G., Roubaud, D., Hagfors, L.I., 2017. On the hedge and safe haven properties of Bitcoin: is it really more than a diversifier? Finance Res. Lett. 20, 192–198. Bouri, E., Lau, C.K.M., Lucey, B., Roubaud, D., 2018a. Trading volume and the predictability of return and volatility in the cryptocurrency market. Finance Res. Lett. https://doi.org/10.1016/j.frl.2018.08.015. Bouri, E., Shahzad, S.J.H., Roubaud, D., 2018b. Co-explosivity in the cryptocurrency market. Finance Res. Lett. https://doi.org/10.1016/j.frl.2018.07.005. Brauneis, A., Mestel, R., 2018. Price discovery of cryptocurrencies: Bitcoin and beyond. Econ. Lett. 165, 58–61. Breitung, J., Candelon, B., 2006. Testing for short-and long-run causality: a frequency-domain approach. J. Econ. 132 (2), 363–378. Chaudhuri, S.E. and Lo, A.W. (2016). Spectral portfolio theory. http://alo.mit.edu/wp-content/uploads/2016/06/spectral_14_standard.pdf. Chauveau, T., Subbotin, A., 2013. Price dynamics in a market with heterogeneous investment horizons and boundedly rational traders. J. Econ. Dyn. Control 37 (5), 1040–1065. Chu, J., Chan, S., Nadarajah, S., Osterrieder, J., 2017. GARCH modelling of cryptocurrencies. J. Risk Financ. Manag. 10 (4), 17. Corbet, S., Lucey, B.M, Urquhart, A., Yarovaya, L., 2018a. Cryptocurrencies as a financial asset: a systematic analysis. Int. Rev. Financ. Anal. https://doi.org/10.1016/ j.irfa.2018.09.003. Corbet, S., Meegan, A., Larkin, C., Lucey, B., Yarovaya, L., 2018b. Exploring the dynamic relationships between cryptocurrencies and other financial assets. Econ. Lett. 165, 28–34. Engle, R., 1993. Technical note: statistical models for financial volatility. Financ. Anal. J. 49 (1), 72–78. Faria, G., Verona, F., 2018. Forecasting stock market returns by summing the frequency-decomposed parts. J. Empir. Finance 45, 228–242. Gkillas, K., Katsiampa, P., 2018. An application of extreme value theory to cryptocurrencies. Econ. Lett. 164, 109–111. Granger, C.W., 1969. Investigating causal relations by econometric models and cross-spectral methods. Econometrica: J. Econometric Society 424–438. Hamao, Y., Masulis, R.W., Ng, V., 1990. Correlations in price changes and volatility across international stock markets. Rev. Financ. Stud. 3, 281–307. Hotta, L.K., Tsay, R.S., 2012. Outliers in GARCH processes. Econ. Time Series: Model. Seasonality 337–358. Ji, Q., Bouri, E., Lau, M.C.K., Roubaud, D., 2018. Dynamic connectedness and integration in cryptocurrency markets. Int. Rev. Financ. Anal. https://doi.org/10.1016/ j.irfa.2018.12.002. Klein, T., Pham Thu, H., Walther, T., 2018. Bitcoin is not the New Gold: a comparison of volatility, correlation, and portfolio performance. Int. Rev. Financ. Anal. 59, 105–116. Koutmos, D., 2018. Return and volatility spillovers among cryptocurrencies. Econ. Lett. 173, 122–127. Platanakis, E., Sutcliffe, C., Urquhart, A., 2018. Optimal vs naïve diversification in cryptocurrencies. Econ. Lett. 171, 93–96. Platanakis, E., Urquhart, A., 2019. Portfolio management with cryptocurrencies: the role of estimation risk. Econ. Lett. 177, 76–80.
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Contents lists available at ScienceDirect
Finance Research Letters journal homepage: www.elsevier.com/locate/frl
The volatility surprise of leading cryptocurrencies: Transitory and permanent linkages Elie Bouria, , Brian Luceyb, David Roubaudc ⁎
a b c
USEK Business School, Holy Spirit University of Kaslik, Jounieh, Lebanon Trinity Business School, Trinity College Dublin, Dublin 2, Ireland Montpellier Business School, Montpellier, France
ARTICLE INFO
ABSTRACT
JEL classifications: C13 G15
There is scarce literature examining the volatility linkages among leading cryptocurrencies, and none exists on the linkages among unexpected volatility, called ‘volatility surprise’. To address this literature gap, we build on the concept of volatility surprise and examine the causal linkages among the volatility of leading cryptocurrencies via the frequency-domain test of Bodart and Candelon (2009), discriminating between transitory and permanent causalities. Permanent shocks are more important in explaining the Granger-causality that does not necessarily emanate from the largest, Bitcoin, over short horizons, whereas transitory shocks dominate the causality across smaller cryptocurrencies over long horizons.
Keywords: Bitcoin Cryptocurrency Volatility surprise frequency-domain Granger-causality
1. Introduction The importance of volatility to asset pricing, risk management, and portfolio allocations cannot be overrated. However, what can also be important is the ‘volatility surprise’ that represents unexpected volatility, i.e. the difference between squared residuals and the conditional variance (Hamo et al., 1990; Engle, 1993). Previous studies examine the relationships among the volatility surprises of conventional assets such as equities, bonds, currencies, and commodities (e.g., Aboura and Chevallier, 2014). Such an examination is unprecedented for newly emerged digital assets (Corbet et al., 2018a) that exhibit extreme price volatility and continue to fascinate the financial press, policy-makers, and financial community. Scarce studies (e.g., Koutmos, 2018; Ji et al., 2018) examine the connectedness among leading cryptocurrencies, focusing on predictable variance (volatility) such as the conditional variance (volatility). However, they don't consider the linkages among volatility surprises that are crucial, given they represent the unexpected volatility (Engle, 1993; Aboura, and Chevallier, 2014). Koutmos (2018) and Ji et al. (2018) overlook the possibility that the linkages might not be the same at different frequencies or scales. This is important, as crypto-traders and investors may exhibit heterogeneity in terms of their trading and investment horizons. In this paper, we uncover the causal relationship across the volatility surprise of Bitcoin and seven other leading cryptocurrencies (Ethereum, Ripple, Stellar, Litecoin, Monero, Nem, and Dash) via the Granger-causality in the frequency-domain of Breitung and Candelon (2009). By decomposing the causality at different frequencies, i.e. time scales, we discriminate between transitory and permanent causal relations1 among the unexpected volatility in the cryptocurrency, which might help short-term speculators and
Corresponding author. E-mail addresses: [email protected] (E. Bouri), [email protected] (B. Lucey), [email protected] (D. Roubaud). 1 According to Breitung and Candelon (2009), this test leads to robust causality results because it can deal with the presence of outliers in the data and is unaffected by the presence of multivariate GARCH. ⁎
https://doi.org/10.1016/j.frl.2019.05.006 Received 28 March 2019; Received in revised form 30 April 2019; Accepted 7 May 2019 1544-6123/ © 2019 Elsevier Inc. All rights reserved.
Please cite this article as: Elie Bouri, Brian Lucey and David Roubaud, Finance Research Letters, https://doi.org/10.1016/j.frl.2019.05.006
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long-term investors with different investment horizons make investment and risk management decisions that match their respective investment time horizons (Chauveau and Subbotin, 2013). The current paper relates to the embryonic literature that looks beyond the Bitcoin market to consider issues of other leading cryptocurrencies.2 Such issues include price discovery (Bouri et al., 2018a; Brauneis and Mestel, 2018), returns behaviour (Gkillas and Katsiampa, 2018), volatility modelling (Chu et al., 2017), co-bubbling (Bouri et al., 2018b), market efficiency (Brauneis and Mestel, 2018), and diversification abilities (Platanakis et al., 2018). Our paper complements these strands of research, but differs in both research scope and method. Firstly, in contrast to previous studies considering the return relationships between Bitcoin and other conventional assets (Bouri et al., 2017; Baur et al., 2018; Corbet et al., 2018b; Klein et al., 2018), or among the volatility of cryptocurrencies (Koutmos, 2018; Ji et al., 2018), we concentrate on the Granger-causality in the volatility surprise across leading cryptocurrencies. As argued by Hamo et al. (1990), Engle (1993) and Aboura and Chevallier (2014), volatility surprise represents the unexpected volatility that considerably matches the need of market participants. Secondly, we consider the frequencydomain approach that allows us to differentiate between permanent and transitory causal relations in volatility surprise among leading cryptocurrencies. As such, we provide important insights to both short-term speculators and long-term investors who try to match their investment choices to the right timescale (Chauveau and Subbotin, 2013). We go some way to discovering whether the cryptocurrency market is a heterogeneous market in terms of the volatility causal relationships across its main components. Thirdly, our paper complements Koutmos (2018) and Ji et al. (2018), who, by focusing on the interdependencies among major cryptocurrencies via connectedness measures, overlook the possibility that the relationships might not be the same at different frequencies or scales. Furthermore, they consider the linkages among volatility, whereas our focus is on unexpected volatility (i.e. volatility surprise). Our results provide evidence of causal relationships, and thus predictability, in the volatility surprise across cryptocurrencies that seems to differ across timescales. This suggests the need to differentiate between permanent and transitory causal relations in the volatility surprise of leading cryptocurrencies. Another implication relates to the fact that market participants should not treat leading cryptocurrencies as a heterogeneous group in terms of causal relationships among volatility surprise. The paper continues as follows. Section 2 provides the empirical methods. Section 3 describes the data and presents the empirical results. Section 4 concludes. 2. Research methods According to Granger (1969), a variable x is said to ‘Granger-cause’ another variable y if lagged values of x contain information that helps predict y. Conducted based on the Wald test within a VAR model, this time-domain Granger-causality test relies on a single statistic measure of predictability, ignoring the possibility that Granger-causality may not be the same at different frequencies. Breitung and Candelon (2006) propose a Granger-causality test in the frequency-domain that considers all frequencies. In this paper, we apply the extended version of this test, taken from Bodart and Candelon (2009), which removes all outliers3 and replaces them with a 10-day average centred around the abnormal observation. As argued by Bodart and Candelon (2009), this makes the residuals free from autocorrelation and outliers, leading to a correctly specified model. If not dealt with properly, the presence of outliers in the cryptocurrency data can lead to fallaciously overestimated Granger-causality. Furthermore, Bodart and Candelon (2009) indicate that their test is unaffected by the presence of multivariate GARCH. Consider Zt = [xt, yt] to be a two-dimensional vector of endogenous variables observed at time t = 1, …., T, with a finite order VAR representation, such as: (1)
(L) Zt = vt p
k
where, θ(L) = 1-θ1L− … −θpL is a 2 × 2 lag polynomial with L Zt = Zt − k. The residual vt is assumed to be white noise with zero mean and covariance matrix E( t t ) = Σ defined as positive. The matrix Σ is decomposed as G′G = Σ−1, where G is the inferior triangular matrix of the Cholesky decomposition, such that E(ηtη′t) = I, and ηt = Gεt. If the system is stationary, the VAR process has a moving average:
Zt =
(L ) t = −1
where, φ (L) = θ(L)
1 fx ( ) = { 2
11 (L)
12 (L)
1t
21 (L)
22 (L)
2t
and φ(L)= ∅(L)G
11 e
i
)2 +
i
12 e
= −1
(L )
t
=
11 (L)
12 (L)
1t
21 (L)
22 (L)
2t
(2)
The spectral density of Xt is given by:
) 2}
(3)
The measure of causality in Eq. (4) is used to test the null that yt does not Granger-cause xt at frequency ω [H0 : My→x(ω) = 0].
My
x(
) = log 1 +
12 (e
i
11 (e
i
)
2
)
2
(4)
2 Bitcoin's success and underlying technology have inspired the release of many cryptocurrencies such as Ethereum, Ripple, and Stellar, which rapidly grow in market size and become crucial players in the cryptocurrency market and alternative digital assets (Bouri et al., 2018a). 3 Bodart and Candelon (2009) detect outliers via the median absolute deviation (MAD) of Hotta and Tsay (2012).
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This measure is zero if |φ12 (e−iω)|2 = 0, implying no causality at frequency ω. To obtain the statistic My→x(ω), we replace |φ11(e−iω)| and |φ12(e−iω)| in Eq. (4) with the estimated values of the fitted VAR representation.4 The significance of the causal relationship between two time series is assessed by comparing the Granger-causality measure for the frequency ω to the 5% critical value of a χ2 distribution with 2 degrees of freedom (5.99). 3. Empirical results 3.1. Data The daily closing prices of eight large cryptocurrencies (Bitcoin, Ethereum, Ripple, Stellar, Litecoin, Monero, Nem, and Dash) are gathered from https://coinmarketcap.com/. Dictated by the availability of price data for some cryptocurrencies (e.g. Ethereum), the sample period spans August 8, 2015 to February 18, 2019 (1291 observations), covering a rich period of booms and busts. Specifically, the eight cryptocurrencies are selected from the largest 20 cryptocurrencies by market value in order to make the analyses richer by covering emerging and large cryptocurrencies, which are of interest to crypto-traders. Our sample period is suitable and rich enough because it includes the years 2016-2017-2018 when Bitcoin and other leading cryptocurrencies started to attract huge interest among investors and thus large trading activities. Using the log returns of each of the eight cryptocurrencies, we estimate the mean equation (Eq. (1)) and the variance equation (Eq. (2)) as:
rt = µ + 2 t
=
(5)
t
+
2 t 1
+
(6)
2 t 1
We follow Aboura, and Chevallier (2014), and compute the mean-zero volatility surprise (˜) as the volatility surprise (ς) (i.e. the difference between the squared residuals (ε2) and the conditional variance (σ2)) normalized by the conditional variance σ2 as:
˜t =
(
2 t
2 t
2 t )
(7)
Summary statistics of the normalized volatility surprise are given in Table 1. The most (least) volatile is Litecoin (Ethereum). Again, Litecoin has the highest kurtosis, whereas Ethereum has the lowest. Skweness is positive for four cases and negative for the rest.5 Table 2 Panel A shows that the pairwise correlation between volatility surprises is weekly positive. It ranges from 0.054 (Stellar Litecoin) to 0.365 (Bitcoin - Ethereum), with evidence that Bitcoin is more correlated with other cryptocurrencies. 3.2. Results of the time-domain Granger-causality test Before conducting the causality tests in the frequency-domain following Bodart and Candelon (2009), we conduct the timedomain Granger-causality test (Granger, 1969) within the framework of VAR models6 and report the results in Table 3. Considering the 5% significance level, the results are summarized as follows. A unidirectional causality runs from Bitcoin to Ethereum and Nem. Ethereum Granger-causes Monero. Ripple Granger-causes Litecoin and Nem Granger-causes Ripple. A unidirectional causality runs from Litecoin to Stellar, and from Nem to Litecoin. Finally, Monero Granger-causes Dash. 3.3. Results of the frequency-domain Granger-causality test Following Bodart and Candelon (2009), we apply the frequency-domain Granger-causality test, after taking care of the outliers, discriminating between temporary and permanent causality. Bodart and Candelon (2009) indicate that the components at low frequencies are more persistent than at high frequencies. Causalities are thus investigated at two frequencies between 0 and π. Permanent causality is instigated for long-run horizons representing cycles of 63 – 6 days (i.e., ω ∈ 0.0–1.5), whereas temporary causality is investigated for short-run horizons representing cycles of 2–6 days (i.e., ω ∈ 1.5 − π).7 For every frequency considered, the null of no Granger-causality is rejected if the test statistic is above the 5% critical value and vice-versa. The results in Table 3 show a transitory Granger-causality from Bitcoin to Ethereum at high frequencies, whereas there is Granger-causality from Bitcoin to Nem at both low and high frequencies. Ethereum Granger-causes Monero at low frequencies, and Nem Granger-causes Ethereum at low frequencies indicating a permanent interdependence. Ripple Granger-causes Litecoin at low and high frequencies. Ripple also Granger-causes Nem at low frequencies, and there is a feedback effect at both low and high frequencies. Litecoin Granger-causes Stellar at low frequencies. Monero Granger-causes Nem at high frequencies and Dash at low frequencies. The results point to the importance of some cryptocurrencies other than Bitcoin to the network structure of causalities of volatility surprise, whereas other cryptocurrencies, such as Stellar and Dash, are peripheral. This finding is not fully in line with previous 4
Notably, the optimal lag of the VAR model is selected based on SIC. Unreported results from unit root tests show that all volatility surprise series are stationary at the 1% level. 6 We select the optimal lag of the VAR model that yields the lowest SIC. 7 The frequency parameter omega (ω) is used to compute the length of cycles (T = 2π /ω). 5
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Table 1 Summary statistics of normalized volatility surprise.
BITCOIN ETHEREUM RIPPLE STELLAR LITECOIN MONERO NEM DASH
Mean
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
−0.0011 −0.0018 0.0000 0.0008 −0.0011 −0.0007 −0.0002 −0.0010
43.2077 28.6333 85.4106 126.7129 177.5432 69.9550 103.6639 85.7745
−42.3909 −28.7041 −85.3888 −126.7093 −177.4740 −69.7006 −105.6492 −85.4343
4.1001 3.4049 5.1735 6.2412 7.8565 4.5839 5.4440 4.7003
0.1581 −0.1250 −0.0258 −0.0087 0.0277 0.1131 −0.3457 0.0455
55.2335 27.0410 126.7531 279.6336 403.9178 102.2392 217.2210 175.2074
Notes: This table provides summary statistics of daily volatility surprise, computed according to Eq. (7). The sample period is August 8, 2015–February 18, 2019.
Table 2 Unconditional correlation matrix.
BITCOIN ETHEREUM RIPPLE STELLAR LITECOIN MONERO NEM DASH
BITCOIN
ETHEREUM
RIPPLE
STELLAR
LITECOIN
MONERO
NEM
DASH
1.0000 0.3657 0.1598 0.1169 0.2044 0.3024 0.1729 0.2126
1.0000 0.2113 0.1446 0.1520 0.2284 0.1368 0.2374
1.0000 0.2030 0.1372 0.1445 0.1461 0.1170
1.0000 0.0541 0.0803 0.0761 0.0748
1.0000 0.1028 0.1230 0.0858
1.0000 0.1300 0.2102
1.0000 0.1002
1.0000
Notes: This table provides pairwise unconditional correlations across the daily volatility surprises of leading cryptocurrencies. The sample period is August 8, 2015–February 18, 2019.
studies that consider volatility, and stress the importance of Bitcoin (e.g., Ji et al., 2018; Koutmos, 2018), pointing to the differences in the causal relationships in volatility and volatility surprise. We find that the Granger-causality structure in some cases is frequencydependent (i.e. it varies between short-run and long-run horizons), with more evidence of permanent causality. This heterogeneity in the causality structure is important given that market participants often have different investment horizons. The findings are useful for crypto-traders and investors who are keen to understand volatility linkages in both the short and long run in order to enhance predictability, which is important to both trading and investment strategies. Specifically, market participants in the cryptocurrency market can gain information by looking into the linkages across the volatility surprise of cryptocurrencies while differentiating, in some cases, between transitory and permanent causality. They can exploit evidence of predictability in studying portfolio implications (Platanakis and Urquhart, 2019) and in enhancing investment strategies along the spectrum (Chaudhuri and Lo, 2016; Faria and Verona, 2018). 4. Conclusion We contribute to the cryptocurrency literature by uncovering the linkages among the volatility surprises of eight leading cryptocurrencies. The results show evidence of causality linkages among the volatility surprises that do not necessarily originate from the largest cryptocurrency, Bitcoin, suggesting that the unexpected volatility of other cryptocurrencies matters in the heterogeneous cryptocurrency market. The fact that causalities differ between the short and long runs for some cryptocurrencies indicates the importance of bearing causalities in mind at low and high frequencies when considering integration. It also indicates the possibility of predicting the volatility surprise at specific frequencies, suggesting that crypto-traders should monitor other cryptocurrencies while trading one cryptocurrency. Conversely, some cryptocurrencies, such as Stellar and Dash, are relatively segmented, offering the potential for diversification. Our analysis complements the existing, if embryonic, findings (Bouri et al., 2018a,b; Ji et al., 2018; Platanakis et al., 2018) and shows that the largest cryptocurrency, Bitcoin, is not the only influential cryptocurrency. This is somewhat inconsistent with Koutmos (2018), who shows that Bitcoin has a stronger influence on other cryptocurrencies, given its older presence and higher market capitalization. It could be that our focus on the unexpected volatility and the high growth in other cryptocurrencies, to the detriment of Bitcoin's dominance, affect our findings. Our analysis provides a foundation for further analyses involving the drivers of the volatility surprise of segmented cryptocurrencies. Future research could consider the implications of our findings using a wavelet-based forecasting method (Faria and Verona, 2018). 4
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Table 3 Results of time and frequency Granger-causality tests.
Bitcoin-Ethereum Bitcoin-Ripple Bitcoin-Stellar Bitcoin-Litecoin Bitcoin-Monero Bitcoin-Nem Bitcoin-Dash Ethereum-Ripple Ethereum-Stellar Ethereum-Litecoin Ethereum-Monero Ethereum-Nem Ethereum-Dash Ripple-Stellar Ripple-Litecoin Ripple-Monero Ripple-Nem Ripple-Dash Stellar-Litecoin Stellar-Monero Stellar-Nem Stellar-Dash Litecoin-Monero Litecoin-Nem Litecoin-Dash Monero-Nem Monero-Dash Nem-Dash
Null hypothesis
Panel A: Time-domain Granger-causality Chi-sq P values
Panel B: Frequency-domain Granger-causality Permanent (63-6 days) Temporary (6-2) days
Bitcoin ≠› Ethereum Ethereum ≠› Bitcoin Bitcoin ≠› Ripple Ripple ≠› Bitcoin Bitcoin ≠› Stellar Stellar ≠› Bitcoin Bitcoin ≠› Litecoin Litecoin ≠› Bitcoin Bitcoin ≠› Monero Monero ≠› Bitcoin Bitcoin ≠› Nem Nem ≠› Bitcoin Bitcoin ≠› Dash Dash ≠› Bitcoin Ethereum ≠› Ripple Ripple ≠› Ethereum Ethereum ≠› Stellar Stellar ≠› Ethereum Ethereum ≠› Litecoin Litecoin ≠› Ethereum Ethereum ≠› Monero Monero ≠› Ethereum Ethereum ≠› Nem Nem ≠› Ethereum Ethereum ≠› Dash Dash ≠› Ethereum Ripple ≠› Stellar Stellar ≠› Ripple Ripple ≠› Litecoin Litecoin ≠› Ripple Ripple ≠› Monero Monero ≠› Ripple Ripple ≠› Nem Nem ≠› Ripple Ripple ≠› Dash Dash ≠› Ripple Stellar ≠› Litecoin Litecoin ≠› Stellar Stellar ≠› Monero Monero ≠› Stellar Stellar ≠› Nem Nem ≠› Stellar Stellar ≠› Dash Dash ≠› Stellar Litecoin ≠› Monero Monero ≠› Litecoin Litecoin ≠› Nem Nem ≠› Litecoin Litecoin ≠› Dash Dash ≠› Litecoin Monero ≠› Nem Nem ≠› Monero Monero ≠› Dash Dash ≠› Monero Nem ≠› Dash Dash ≠› Nem
17.045 5.902 7.124 4.891 0.034 0.013 6.669 1.190 3.824 0.459 34.843 5.058 9.979 5.100 5.137 3.641 6.290 2.516 2.284 2.211 20.234 3.056 2.387 17.992 3.735 6.991 7.121 1.025 79.718 15.495 2.726 2.957 14.520 62.844 5.864 3.706 2.240 27.300 2.658 5.497 3.587 2.064 3.313 0.755 1.045 1.278 2.271 6.861 4.842 0.612 14.920 7.283 16.886 8.885 9.278 4.598
Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Rejected Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Rejected Accepted Accepted Rejected Accepted Accepted Accepted Accepted Rejected Accepted Accepted Accepted Rejected Rejected Accepted Accepted Accepted Rejected Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Rejected Accepted Accepted Accepted
0.048 0.749 0.624 0.891 0.982 0.993 0.572 0.997 0.797 0.999 0.000 0.829 0.367 0.825 0.822 0.933 0.790 0.990 0.942 0.947 0.005 0.879 0.992 0.055 0.928 0.638 0.624 0.999 0.000 0.502 0.974 0.965 0.099 0.000 0.753 0.929 0.987 0.001 0.953 0.703 0.936 0.990 0.950 0.999 0.994 0.989 0.993 0.122 0.847 0.999 0.093 0.607 0.050 0.447 0.412 0.867
Rejected Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Rejected Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Rejected Accepted Accepted Accepted Accepted Rejected Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Accepted Rejected Accepted Accepted Accepted Accepted Accepted
Notes: This table presents the results of the null hypothesis of no Granger-causality in the time-domain and the frequency-domain. The sample period is August 8, 2015–February 18, 2019.
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Supplementary materials Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.frl.2019.05.006. References Aboura, S., Chevallier, J., 2014. Volatility equicorrelation: a cross-market perspective. Econ. Lett. 122 (2), 289–295. Baur, D.G., Hong, K., Lee, A.D., 2018. Bitcoin: medium of exchange or speculative assets? J. Int. Financ. Mark. Inst. Money 54, 177–189. Bodart, V., Candelon, B., 2009. Evidence of interdependence and contagion using a frequency domain framework. Emerg. Mark. Rev. 10 (2), 140–150. Bouri, E., Molnar, P., Azzi, G., Roubaud, D., Hagfors, L.I., 2017. On the hedge and safe haven properties of Bitcoin: is it really more than a diversifier? Finance Res. Lett. 20, 192–198. Bouri, E., Lau, C.K.M., Lucey, B., Roubaud, D., 2018a. Trading volume and the predictability of return and volatility in the cryptocurrency market. Finance Res. Lett. https://doi.org/10.1016/j.frl.2018.08.015. Bouri, E., Shahzad, S.J.H., Roubaud, D., 2018b. Co-explosivity in the cryptocurrency market. Finance Res. Lett. https://doi.org/10.1016/j.frl.2018.07.005. Brauneis, A., Mestel, R., 2018. Price discovery of cryptocurrencies: Bitcoin and beyond. Econ. Lett. 165, 58–61. Breitung, J., Candelon, B., 2006. Testing for short-and long-run causality: a frequency-domain approach. J. Econ. 132 (2), 363–378. Chaudhuri, S.E. and Lo, A.W. (2016). Spectral portfolio theory. http://alo.mit.edu/wp-content/uploads/2016/06/spectral_14_standard.pdf. Chauveau, T., Subbotin, A., 2013. Price dynamics in a market with heterogeneous investment horizons and boundedly rational traders. J. Econ. Dyn. Control 37 (5), 1040–1065. Chu, J., Chan, S., Nadarajah, S., Osterrieder, J., 2017. GARCH modelling of cryptocurrencies. J. Risk Financ. Manag. 10 (4), 17. Corbet, S., Lucey, B.M, Urquhart, A., Yarovaya, L., 2018a. Cryptocurrencies as a financial asset: a systematic analysis. Int. Rev. Financ. Anal. https://doi.org/10.1016/ j.irfa.2018.09.003. Corbet, S., Meegan, A., Larkin, C., Lucey, B., Yarovaya, L., 2018b. Exploring the dynamic relationships between cryptocurrencies and other financial assets. Econ. Lett. 165, 28–34. Engle, R., 1993. Technical note: statistical models for financial volatility. Financ. Anal. J. 49 (1), 72–78. Faria, G., Verona, F., 2018. Forecasting stock market returns by summing the frequency-decomposed parts. J. Empir. Finance 45, 228–242. Gkillas, K., Katsiampa, P., 2018. An application of extreme value theory to cryptocurrencies. Econ. Lett. 164, 109–111. Granger, C.W., 1969. Investigating causal relations by econometric models and cross-spectral methods. Econometrica: J. Econometric Society 424–438. Hamao, Y., Masulis, R.W., Ng, V., 1990. Correlations in price changes and volatility across international stock markets. Rev. Financ. Stud. 3, 281–307. Hotta, L.K., Tsay, R.S., 2012. Outliers in GARCH processes. Econ. Time Series: Model. Seasonality 337–358. Ji, Q., Bouri, E., Lau, M.C.K., Roubaud, D., 2018. Dynamic connectedness and integration in cryptocurrency markets. Int. Rev. Financ. Anal. https://doi.org/10.1016/ j.irfa.2018.12.002. Klein, T., Pham Thu, H., Walther, T., 2018. Bitcoin is not the New Gold: a comparison of volatility, correlation, and portfolio performance. Int. Rev. Financ. Anal. 59, 105–116. Koutmos, D., 2018. Return and volatility spillovers among cryptocurrencies. Econ. Lett. 173, 122–127. Platanakis, E., Sutcliffe, C., Urquhart, A., 2018. Optimal vs naïve diversification in cryptocurrencies. Econ. Lett. 171, 93–96. Platanakis, E., Urquhart, A., 2019. Portfolio management with cryptocurrencies: the role of estimation risk. Econ. Lett. 177, 76–80.
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