**4. Methodology**

Besides assessing the overall error variation in an asset "*k*" due to shock arising in other asset "*j*" or leading to shocks to other asset classes, we are also interested in assessing shares of forecast error variation in an asset "*k*" due to shock to an asset "*j*" at a specific frequency band. To achieve this, we follow the generalized forecast error variance decomposition methodology by Diebold and Yilmaz (2012) and the Baruník and Kˇrehlík (2017).

Let us describe the n-variate stationary process Yt = (yt,1,...,yt,n) by structural VAR(p) at t = 1, ... , T as:

$$
\Phi(\mathcal{L})\mathcal{Y}\_t = \varepsilon\_t \tag{1}
$$

In this equation and below, we define the asset volatility as yt,n = |lnPt,n − ln Pt−1,n|, where Pt,n is the daily closing value of the nth asset in the system (e.g., in intra-cryptocurrency market connectedness n = 4) on day t.<sup>12</sup>

<sup>12</sup> For other advantages of absolute returns one can see Forsberg and Ghysels (2007), Antonakakis and Vergos (2013) and Wang et al. (2016). Indeed, it is well documented in the literature that the use of absolute returns in modeling volatility has some advantages. First of all, absolute returns are more robust than the standard-deviation in the presence of large movements (Davidian and Carroll 1987). In this framework, the standard-deviation may not be investors' most appropriate measure of risk because it rewards the desirable upside movements as hard as it punishes the undesirable downside movements. Furthermore, absolute return modeling is more reliable than the standard-deviation for the non-existence of a fourth moment commonly associated with financial returns (Mikosch 2000).

*J. Risk Financial Manag.* **2018**, *11*, 66

We assume that the roots of |Φ(z)| lie outside the unit-circle. Under this assumption the VAR process has following MA(∞) representation:

$$\mathbf{Y}\_{\mathbf{t}} = \Psi(\mathbf{L})\boldsymbol{\varepsilon}\_{\mathbf{t}} \tag{2}$$

where Ψ(L) is an n × n infinite lag polynomial matrix of coefficients.

Let us define the own variance shares as the fractions of the *H*-step-ahead error variances in forecasting y*j* that are due to shocks to y*j*, for *j* = 1, 2 ... ,n, and across variance shares, or spill over, as the fractions of the *H*-step-ahead error variances in forecasting y*j* that are due to shocks to y*k*, for *k* = 1,2, . . . , n, such that *j* = *k*. This can be written in the form:

$$(\left(\Theta\_{H}\right)\_{j,k} = \left(\left(\Sigma\right)\_{k,k}\right)^{-1} \sum\_{h=0}^{H-1} \left(\left(\Psi\_{h}\Sigma\right)\_{j,k}\right)^{2} / \sum\_{h=0}^{H-1} \left(\Psi\_{h}\Sigma\Psi\_{h}^{\prime}\right)\_{j,j} \tag{3}$$

where Ψ*h* is an n × n matrix of coefficients corresponding to lag *h*, and <sup>σ</sup>*kk* = (Σ)*<sup>k</sup>*,*<sup>k</sup>* .

The (<sup>θ</sup>*H*)*j*,*<sup>k</sup>* captures the Pearson–Shin GFEVD partial contribution from asset class k to asset class *j*.

Given that the effect does not add up to one (∑*Hh*=<sup>0</sup> <sup>θ</sup>*j*,*<sup>k</sup>* = 1) within columns by definition in generalized VAR process of FEVDs, we propose measuring pairwise-directional connectedness *<sup>j</sup>*←*<sup>k</sup>*(*H*), to standardize the effects (<sup>θ</sup>*H*)*j*,*<sup>k</sup>* by:

$$(\left(\tilde{\Theta}\_{H}\right)\_{j,k} = (\Theta\_{H})\_{j,k} / \sum\_{k} (\Theta\_{H})\_{j,k} \tag{4}$$

The total directional connectedness from a variable *k* to the other variables is then defined as:

$$\mathbb{C}\_{j \leftarrow}(H) = 100 \times \sum\_{j \neq k, j=1}^{n} \mathbb{C}\_{j,k}(H) / \sum\_{j,k=1}^{n} \mathbb{C}\_{j,k}(H) \tag{5}$$

Similarly, the total directional connectedness of other variables to *j* is given by:

$$\mathbb{C}\_{\leftarrow k}(H) = 100 \times \sum\_{j:\neq k,k=1}^{n} \mathbb{C}\_{j,k}(H) / \ \sum\_{j,k=1}^{n} \mathbb{C}\_{j,k}(H) \tag{6}$$

The connectedness of the whole system is then defined as the share variances in the forecasts contributed by other than own errors, or equally as the ratio of the sum of the diagonal elements to the sum of the entire matrix:

$$\mathbb{C}\_{H} = 100 \times \frac{\sum\_{j \neq k} (\tilde{\Theta}\_{H})\_{j,k}}{\sum (\tilde{\Theta}\_{H})\_{j,k}} = 100 \times \left( 1 - \frac{\text{Tr}\left\{\tilde{\Theta}\_{h}\right\}}{\sum \tilde{\Theta}\_{h}} \right) \tag{7}$$

Tr{.} is the trace operator.

After assessing overall error variation in asset j due to shock arising in an asset *k*, we now follow Barunik et al. (2017) to evaluate connectedness in the frequency domain. We know that GFEVD is the central part of connectedness, hence we consider a frequency response function <sup>Ψ</sup>(*e*<sup>−</sup>*iω*) = ∑ *h e* <sup>−</sup>*<sup>ω</sup>h*Ψ*h*

that can be obtained from Fourier transform of the coefficients Ψ, with *i* = √−1.

Formally, let us have a frequency band d = (a, b): a, b ∈ (−π, <sup>π</sup>), a < b . The generalized variance decompositions on frequency band "*d*" are defined as:

$$(\Theta\_d)\_{j,k} = \frac{1}{2\pi} \int\_{-\pi}^{\pi} \Gamma(\omega) (f(\omega))\_{j,k} \tag{8}$$

where (*f*(*ω*))*j*,*<sup>k</sup>* , denotes the generalized causation spectrum over frequencies ω ∈ (−π, π) , and <sup>Γ</sup>(*ω*) is a weighting function. In fact, to obtain a natural decomposition of original GFEVD to frequencies, we can simply weight the (*f*(*ω*))*j*,*<sup>k</sup>* by the frequency share of the variance of variable *j*. (Please see Baruník and Kˇrehlík (2017) for more detail).

Denote by "*ds*" an interval on the real line from the set of intervals D that form a partition of the interval (−π, π) , such that <sup>∩</sup>*ds*∈D*ds* = ∅ and <sup>∪</sup>*ds*∈D*ds* = (−π, π) . Due to the linearity of integral and the construction of *ds* , we have:

$$(\Theta\_{\infty})\_{j,k} = \sum\_{d\_s \in \mathcal{D}} (\Theta\_{d\_s})\_{j,k} \tag{9}$$

The natural way to describe the time-varying frequency of the connectedness is to consider the spectral representation of GFEVD.<sup>13</sup> For that, let us define the scaled generalized variance decomposition on the frequency band d = (a, b): a, b ∈ (−π, <sup>π</sup>), a < b as:

$$(\tilde{\Theta}\_d)\_{j,k} = (\Theta\_d)\_{j,k} / \sum\_k (\Theta\_\infty)\_{j,k} \tag{10}$$

The within connectedness on the frequency band *d* can be then defined as:

$$\mathbb{C}\_d^W = 100 \left( \frac{\sum \theta\_d}{\sum \theta\_{\text{vs}}} - \frac{\text{Tr}\left\{ \tilde{\theta}\_d \right\}}{\sum \tilde{\theta}\_d} \right) \tag{11}$$

The frequency connectedness on the frequency band *d* is also defined as:

$$\mathbb{C}\_d^F = 100 \left( 1 - \frac{\text{Tr}\{\tilde{\theta}\_d\}}{\sum (\tilde{\theta}\_d)\_{/k}} \right) = \mathbb{C}\_d^W \cdot \frac{\sum \tilde{\theta}\_d}{\sum \tilde{\theta}\_\infty} \tag{12}$$

It is important to note that the within connectedness gives us the connectedness effect that happens within the frequency band and is weighted by the power of the series on the given frequency band exclusively. Moreover, the frequency connectedness decomposes the overall connectedness defined in Equation (3) into distinct parts that, when summed, give the original connectedness measure -∞.

### **5. Data and Descriptive Statistics**

In this paper, we investigate whether or not cryptocurrencies are in co-movement over time with each other, and with other markets. For the cryptocurrency market, we use data of four successful cryptocurrencies: Bitcoin (BTC), Ethereum (ETH), Ripple (XRP) and Litecoin (LTC). We also use data of the most active Bitcoin–USD index, a simple average of global Bitcoin/USD exchange prices, produced by Coindesk. Moreover, our data covers the daily return of five popular stock indexes (i.e., SP500, NASDAQ, FTSE100, Hang Seng and Nikkei225) and another five major Forex currencies (i.e., EUR/USD, GBP/USD, USD/JPY, USD/CHF and USD/CAD). For commodity markets, we limit our attention to the daily return of gold and Brent futures contracts launched by COMEX and the London International Petroleum Exchange (IPE), respectively. The data spans the period 7 October, 2010 to 8 February, 2018, with a total of 1918 daily observations. For the computation of volatility, we restrict the analysis to daily absolute returns. The summary statistics of the data can be found in Table 1.

<sup>13</sup> The spectral representation of Yt at frequency d can be defined as a Fourier transform of MA(∞) filtered series as: Sy(*d*) = ∑∞*<sup>h</sup>*<sup>=</sup>−<sup>∞</sup> <sup>E</sup>(ytyt−<sup>h</sup>)*eidh* = <sup>Ψ</sup>(*e*<sup>−</sup>*id*)ΣΨ(*e*+*id*). The power spectrum *<sup>S</sup>*y(*d*) is a key quantity for understanding frequency dynamics, since it describes how the variance of the Yt is distributed over the frequency components d.



12

Ljung-Box statistics for serial correlation in the model residuals, computed with 5 and 20 lags, respectively .\*, \*\*, \*\*\* denotes the rejection of the null hypothesis at the 1, 5 and 10% significance levels, respectively.

### *J. Risk Financial Manag.* **2018**, *11*, 66

Following Table 1, the daily mean return is 0.55% for BPI and thus it is the largest average daily return. At the same time, it is observed that its volatility (represented by the standard deviation) is also higher than the volatility of traditional currencies, equities, or commodities. Therefore, this high volatility may prevent the use of cryptocurrency as a currency (Yermack 2015). The direction of skewness is given by the sign. The positive sign is perceived to have a long tail to the right of the distribution, while the negative sign is perceived to have a long tail to the left. From Table 1, the larger positive value of skewness is associated with CAD, followed by BPI. However, the larger negative value is associated with CHF, followed by GBP. This means that there is a grea<sup>t</sup> chance that BPI return goes ups more than downs, during our sample period. Note that rational investors value this positively skewed return, which comes from BPI. On the other hand, all volatility series showed a coefficient of kurtosis significantly in excess of the normal distribution reference value. This means that data has a heavy tail. This departure from normality is confirmed by the Jarque–Bera (J–B) test.

The Augmented Dickey–Fuller (ADF) test indicates that the time series of returns are stationary because we reject the null hypothesis of a unit root at 1% level of significance in all cases. The Phillips–Perron (PP) and Kwiatkowski–Phillips–Schmidt–Shin (KPSS) tests also confirm this stationarity. Furthermore, the results of the Lagrange multiplier (LM) test for autoregressive conditional heteroskedastic (ARCH) errors (LM-ARCH test) prove that there is an ARCH effect for almost all series, except the CHF series. Finally, the Q statistics and the *p*-value reveal the absence of the autocorrelation in the residuals from an AR(1) regression model for BPI and US stock indices.

The descriptive statistics in Table 2 show that the average daily return of the four top cryptocurrencies is positive. The XRP gives the greatest average of daily return. It has also the very high average of daily volatility measured by standard deviation. The XRP followed by LTC seems the riskier digital currencies. More interestingly, we can observe that the skewness is positive for all cryptocurrencies. This led to our precedent notification on BPI that a "rational investor" always prefers a positively skewed distribution to negative distribution. In another term, this result can be probably part of the reason for the latest trend involving using cryptocurrencies to raise money.


**Table 2.** Summary statistics of daily returns related to four top cryptocurrencies.

Note: J-B: Jarque–Bera statistic, ADF statistic: Augmented Dickey–Fuller, PP test: Phillips–Perron test and KPSS test: Kwiatkowski–Phillips–Schmidt–Shin test. Q(5) and Q(20) are the Ljung-Box statistics for serial correlation in the model residuals, computed with 5 and 20 lags, respectively. \*, \*\* denotes the rejection of the null hypothesis at the 1 and 5% significance levels, respectively.

Notice that the Kurtosis value of all cryptocurrencies is above 3. This is consistent with the presence of fat tails in data series and all are not normally distributed.

The results for the ADF test indicate that series are stationary. The PP and KPSS tests confirm also this stationarity. Additionally, the ARCH (LM) test indicates that there is an ARCH effect in almost all series, except the XRP series. Finally, the Q statistics and the *p*-value show that there is no autocorrelation in the residuals from an AR(1) model for BTC and ETH series.
