Forecasting Bitcoin volatility: The role of leverage effect and uncertainty

Physica A 533 (2019) 120707

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Physica A journal homepage: www.elsevier.com/locate/physa

Forecasting Bitcoin volatility: The role of leverage effect and uncertainty Miao Yu

∗

Business School, China University of Political Science and Law, Beijing, China

highlights • • • • •

First investigate the impacts of leverage effect and EPU on Bitcoin volatility. The leverage effect significantly impacts future volatility. The leverage effect is more powerful in forecasting Bitcoin volatility. Using the common information can improve the models’ predictive ability. Our conclusions are robust and reliable.

article

info

Article history: Received 10 January 2019 Received in revised form 16 March 2019 Available online 27 March 2019 Keywords: Volatility forecasting Bitcoin Realized volatility Leverage effect EPU

a b s t r a c t In this study, we first investigate the impacts of leverage effect and economic policy uncertainty (EPU) on one-step-ahead Bitcoin volatility using high-frequency data. We find that the leverage effect can impacts on future volatility significantly. However, the jumps and EPU seem not to impact future volatility during in-sample period. The MCS test results show that the leverage effect is more powerful than jump components in forecasting Bitcoin volatility. Moreover, using the common information of the leverage effect and EPU can improve the models’ predictive ability. Finally, our robust tests are supported to these conclusions. © 2019 Elsevier B.V. All rights reserved.

1. Introduction As we know, volatility plays a critical role in many important fields, such as asset pricing, portfolio allocation and risk management. With the availability of high-frequency data, research on financial market volatility has taken new avenues. Since the seminal works of Andersen and Bollerslev [1], financial econometricians have made significant contributions to model and forecast volatility using high-frequency data. To the best of our knowledge, there are many papers to investigate the dynamic volatility using high-frequency data, such as Andersen et al. [2], Wang et al. [3], Corsi [4], Busch et al. [5], Patton and Sheppard [6], Wen et al. [7], Wen et al. [7], Bentes [8], Ma et al. [9,10], Gong and Lin [11]. However, these works are mainly focused on financial markets (e.g., stock and exchange markets). Different from abovementioned studies, we find that there are very few works to investigate the dynamic Bitcoin volatility. Interestingly, cryptocurrencies (e.g., Bitcoin) are a relatively new phenomenon of the 21st century. Feng et al. [12] wrote, ‘‘Bitcoin is the first, largest-capped, and most famous cryptocurrency, which is now accepted as an alternative payment method by many merchants like Subway and Microsoft. Bitcoin is available for trading in many cryptocurrency exchanges across the world’’. As we known, Bitcoin has received some attentions, for example, Baur and Dimpfl [13], Aysan ∗ Corresponding author. E-mail address: [email protected]. https://doi.org/10.1016/j.physa.2019.03.072 0378-4371/© 2019 Elsevier B.V. All rights reserved.

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M. Yu / Physica A 533 (2019) 120707

et al. [14], Chaim and Laurini [15], Eom et al. [16], Shen et al. [17]. However, to the best of our knowledge, few studies investigated the dynamic Bitcoin volatility features. Therefore, our paper is to fill this gap to explore Bitcoin volatility, and can enrich the related literature on modeling and forecasting Bitcoin volatility. Therefore, we investigate the dynamic volatility of Bitcoin, which is one of our contributions in this paper. As we known, volatility tends to increase more after a negative shock than after a positive shock of the same magnitude. In this paper, we investigate the effects of leverage effect on Bitcoin volatility using high-frequency data. Corsi and Renò [18] first consider the leverage effect of return to the heterogeneous autoregressive model of realized volatility (HAR-RV) model proposed by Corsi [4], and find that the leverage effect is helpful for predicting stock market volatility. To date, some studies have supported to those conclusions, such as Souček and Todorova [19], Bekaert and Hoerova [20], Wen et al. [7]. However, it is still an open question that the leverage effect is effective in forecasting Bitcoin volatility. To our knowledge, there is no study to investigate the links between Bitcoin volatility and leverage effect. In this paper, we fill this gap, which is our second contribution. Moreover, in this paper, we investigate the links between the economic policy uncertainty (EPU) and Bitcoin volatility. Baker et al. [21] construct a new index for measuring EPU, which is based on newspaper coverage frequency. They use this index to investigate the effects of policy uncertainty on stock price volatility. To date, the EPU index has received many attentions of scholars, such as Liu and Zhang [22], Aastveit et al. [23], Bariviera et al. [24], Liu et al. [25], Alvarez-Ramirez et al. [26], Duan et al. [27], Kristoufek [28], Takaishi [29], Zhang et al. [30] and among other. These papers empirically found that the EPU can impacts on stock market volatility. For example, Liu and Zhang [22] indicate that high EPU leads to high market volatility, and incorporating EPU as an additional predictive variable into the existing volatility prediction models significantly improves forecasting ability of these models. Noteworthy, few studies have investigated the impacts of EPU on Bitcoin volatility. Why should EPU measures be related to Bitcoin volatility?1 Demir et al. [31] considered that Bitcoin was created after the global financial crisis of 2008–2009, and Bitcoin questions the effectiveness of standard economic and financial structures and the digital currencies are decentralized secure alternatives to the fiat currencies, especially during the times of economic and geopolitical unrest. Moreover, Wang et al. [32] found that especially during the 2010–2013 European sovereign debt crisis and the 2012–2013 Cypriot banking crisis, many people resorted to Bitcoin as a safe-heaven or hedging asset to avoid risk and market uncertainty. Therefore, the changes in the EPU index can possibly affect Bitcoin volatility. In this paper, we contribute to fill this gap, which is our third contribution. In this paper, we first investigate the impacts of leverage effect and uncertainty on Bitcoin volatility using highfrequency data. As we known, there is very few studies to investigate the dynamic volatility of Bitcoin, especially using the high-frequency data. This paper can enrich this related research and fill this gap. We find that the leverage effect significantly impacts future volatility. However, the jumps and EPU seem not to impact the future Bitcoin volatility during in-sample period. The goodness-of-fit values of the five models are larger than 0.3. The MCS test results show that the leverage effect is more power than jumps components in forecasting Bitcoin volatility. Therefore, the leverage effect contains the predictive information to predict Bitcoin volatility. Moreover, the model including the leverage effect and EPU variables can improve the models’ predictive ability. Therefore, considering the effects of both EPU and leverage effect can help to forecast Bitcoin volatility. Our robust tests are supported to these conclusions. The remainder of the paper is organized as follows. Section 2 presents key variables, such as realized volatility, jumps, and competing models. Section 3 is the data description. Our empirical results, such as the in-sample estimated results and out-of-sample forecasting evaluation, and various robustness tests (e.g., different forecasting windows and different measures) are exhibited in Section 4. In the last section, we provide our main conclusions. 2. Realized volatility and prediction models 2.1. Realized volatility Andersen and Bollerslev [1] proposed a new volatility measurement based on high-frequency data, named realized volatility (RV). We first divide the time interval into equal subintervals for a given day t, and have M observations in a day, where M = 1/1 and 1 is the sampling frequency. RV can be defined as the sum of all available intraday high-frequency squared returns, RVt =

M ∑

rt2,j ,

(1)

j=1

where rt ,j represents the jth intraday return of day t. Barndorff-Nielsen and Shephard [33] prove that when1 → 0, RV converges to, t

∫

σ 2 (s)ds +

RVt → 0

∑

κ 2 (s),

0
1 Regarding this issue, we are very thankful to the reviewer.

(2)

M. Yu / Physica A 533 (2019) 120707

where

∫t 0

3

σ 2 (s)ds is the integrated variance and can be calculated by realized bi-power variation (BPV) as follows:

2 BPVt = u− 1

M ∑ ⏐ ⏐⏐ ⏐ ⏐rt ,j ⏐ ⏐rt ,j−1 ⏐ ,

(3)

j=2

where u1 =

√

(2/π ), and

∑

0
κ 2 (s) is the discontinuous jump segment of the quadratic variation process.

2.2. Jumps From Eq. (2), we have that jump size is non-negative and can be defined as follows: Jt = max(RVt − BPVt , 0).

(4)

To check whether the jumps occur significantly, we employed the Z test of Barndorff-Nielsen and Shephard [33] to test the significance of jumps. Instead of using Z test statistics, we used the Z -ratio test statistic, which has a better power property in Huang and Tauchen [34]. The Z-ratio test is defined as 1 (RVt − BPVt ) ∗ RV− t

1

Zt = 1− 2 √ (

π2 4

3 where TQt = 1−1 u− 4/3

{

+ π − 5) max 1,

TQt (BPVt )2

},

(5)

⏐ ⏐⏐ ∑M ⏐⏐ ⏐⏐ ⏐⏐ rt ,j rt ,j−1 ⏐ ⏐rt ,j−2 ⏐ and it is realized tripower quarticity. j=2

Following Andersen et al. [2], we have a significant jump size with the significance level, α , CJt = max(RVt − BPVt , 0) ∗ I(Zt > 8α ),

(6)

where I(·) is an indicator function, 8α is the cumulative distribution function of the normal distribution with confidence level α . 2.3. The HAR-RV and HAR-CJ models To the best of our knowledge, the HAR model proposed by Corsi [4] has gained popularity due to its simplicity. The HAR formulation is based on a straightforward extension of the so-called heterogeneous ARCH, or HARCH, class of models analyzed by Müller et al. [35]. Under the HAR framework, the conditional variance of the discretely sampled returns is parameterized as a linear function of the lagged squared returns over the identical horizon, together with the squared returns over longer and/or shorter horizons. The original HAR-RV model specifies RV as a function of daily, weekly, and monthly RV components and is expressed as RVt +1 = β0 + βd RVt + βw RVt −4,t + βm RVt −21,t + ωt +1

(7)

where RVt −4,t = (RVt −4 + RVt −3 + · · · + RVt )/5, RVt −21,t = (RVt −21 + RVt −20 + · · · + RVt )/22, and ωt +1 is the disturbance error. To separately test the impact of continuous variation and jumps on RV forecasting, Andersen et al. [2] extended this model by decomposing RV explicitly into its continuous sample path variability and the jump variation and developed the HAR-CJ model, which is expressed as follows: RVt +1 = β0 + βd Ct + βw Ct −4,t + βm Ct −21,t + γd CJt + γw CJt −4,t + γm CJt −21,t + ωt +1 w

where Ct

=

(Ctd

+

Ctd−1

+ ··· +

(Jtd−4 + Jtd−1 + · · · + Jtd )/5, and Jtm =

/

Ctd ) 5, (Jtd−21

Ctm

(Ctd−21 Jtd ) 22.

and = + Jtd−20 + · · · +

+

Ctd−20

+ ··· +

Ctd )

(8)

/22.

Jtd

w

is the significant jump, Jt

=

/

In this paper, we investigate the impacts of leverage effect on future volatility, so we consider an existing model, named LHAR-CJ. The LHAR-CJ model of Corsi and Renò [18] adds a leverage effect to the HAR-CJ model in Eq. (8), and this model can be seen as, RVt +1 = β0 + βd Ct + βw Ct −4,t + βm Ct −21,t + γd CJt + γw CJt −4,t + γm CJt −21,t ,

+ δd rt− + δw rt−−4,t + δm rt−−21,t + ωt +1

(9)

where rt− = min (rt ,0). In this paper, we also study the effects of the EPU on Bitcoin volatility, so we construct two new models, named HAR-CJ-EPU and LHAR-CJ-EPU, respectively, HAR-CJ-EPU model: RVt +1 = β0 + βd Ct + βw Ct −4,t + βm Ct −21,t + γd CJt + γw CJt −4,t + γm CJt −21,t + ηEPUt + ωt +1 , LHAR-CJ-EPU model: RVt +1 = β0 + βd Ct + βw Ct −4,t + βm Ct −21,t + γd CJt + γw CJt −4,t + γm CJt −21,t ,

(10)

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M. Yu / Physica A 533 (2019) 120707

Table 1 Descriptive statistics. Mean St.dev. Skewness Kurtosis Jarque–Bera Q(5) ADF

RV

C

CJ

ret

EPU

0.0041 0.0115 10.4323*** 155.9806*** 2089569.7259*** 0.0268*** −5.0329***

0.0040 0.0113 10.6548*** 162.4775*** 2,265,724*** 0.0335*** −5.8500***

0.0001 0.0008 22.7897*** 611.7629*** 31,752,959*** −0.0130*** −15.4691***

0.0031 0.0459 0.0017*** 7.3730*** 4586*** 0.0700*** −10.9116***

84.8254 46.6884 2.2015*** 10.7496*** 11,385*** 0.0960*** −5.2597***

Notes: The Jarque–Bera statistic tests are for the null hypothesis of normality for the distribution of the series. Q(5) is the Ljung–Box statistics for serial correlation. ADF is the t-statistics for Augmented Dickey–Fuller test. ***Asterisk denotes rejections of the null hypothesis at the 1% significance level.

Fig. 1. The 5-min sample frequency prices of Bitcoin over sample period.

+ δd rt− + δw rt−−4,t + δm rt−−21,t + +ηEPUt + ωt +1

(11)

Therefore, we have five volatility models, HAR-RV, HAR-CJ, LHAR-CJ, HAR-CJ-EPU and LHAR-CJ-EPU.

3. Data description

In this paper, our Bitcoin data are collected from https://bitcoincharts.com/. The data obtained from Bitcoincharts website includes open, high, low, close, volume (BTC), volume (currency) and weighted price of all active Bitcoin market. Bitcoin exchange rate displayed on Bitcoinchats are United States Dollar (USD) and other currencies. Considering the availability of high-frequency data, we chose the Bitcoin data that exchanged by dollars. Our data spans from March 1, 2003 from September 30, 2018. The trade time is 24 h. Liu and Zhang [22] found that among a set of 400 volatility estimators for broad-asset classes, none significantly outperforms the 5-min RV. Moreover, in the existing literatures (e.g., [2,6,7,9,18]), the 5-min sample frequency is used as a rule-of-thumb. Therefore, in this paper, we choose 5-min sample frequency as our main frequency to investigate our topic. Table 1 give the descriptive statistics of RV, C, CJ, ret and EPU. They are not satisfied with the Gaussian distribution, which is supported by Jarque–Bera statistic test, Skewness and Kurtosis at the 1% significance level. The Ljung–Box test for correlation shows that the null hypotheses of no autocorrelation up to the 5th order are rejected for most of the series, indicating the existence of correlation. The augmented Dickey–Fuller test (ADF) supports the rejection of the null hypothesis of a unit root at the 1% significance level, thus indicating that all series are stationary and can be modeled directly without further transformations. Fig. 1 displays the 5-min oil futures prices. The Bitcoin prices increase remarkably around 2018. Fig. 2 depicts the magnitudes of the Bitcoin RV in whole sample period, which shows that the Bitcoin market has large fluctuations.

M. Yu / Physica A 533 (2019) 120707

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Fig. 2. RV series in whole sample period.

4. Empirical results 4.1. The rolling forecasts and comparative method Following Patton and Sheppard [6], Wen et al. [7], Liu et al. [36], Ma et al. [10], we use popular method, rolling window, to obtain future volatility of Bitcoin. The data begins from the March 1, 2003 to September 31, 2018, and then we divide it into two subgroups: (1) in-sample data for volatility modeling, covering 800 trading days; (2) out-of-sample data for model evaluation, covering the residual trading days of the total data sample. The estimation period is then rolled forward by adding one new day and dropping the most distant day. In this way, the sample size used to estimate the models remains at a fixed length and the forecasts do not overlap, thereby allowing daily (one-day-ahead) out-of-sample volatility forecasts to be obtained. The volatility forecasts obtained by the five models are indicated by σˆ t2 , and the scaled RV measurement denoted as RVt is taken as a proxy for actual daily volatility (the forecasting benchmark). Various forecasting criteria or loss functions can be considered in assessing the predictive accuracy of a volatility model. To assess the differences between models, we use the following two loss functions: HMSE = M −1

M ∑

(1 − σˆ m2 /RVm )2 ,

(12)

M ∑ ⏐ ⏐ ⏐1 − σˆ 2 /RVm ⏐ ,

(13)

m=1

HMAE = M −1

m

m=1

where HMSE and HMAE represent the heteroscedasticity-adjusted mean absolute error and the heteroscedasticityadjusted mean squared error, respectively. M is the length of the out-of-sample period. When a loss function is smaller for model A than it is for model B, it can be concluded that the forecasting performance of the former is superior to that of the latter. Such a conclusion cannot be used based on a single loss function and a single sample. Recent work has focused on a testing framework that can determine whether one model is outperformed by another. As discussed in the Introduction, The Superior Predictive Ability (SPA) test, an extension of the White framework proposed by Hansen et al. [37], has been shown to possess good power properties and to be more robust than previous approaches. However, SPA test is designed to address whether a particular benchmark is significantly outperformed by any of the alternatives used in the comparison. Recently, Hansen et al. [37] proposed a new test method to compare the volatility models, which named Model Confidence Set (MCS). Compared with SPA test, the MCS test has many attractive features. Firstly, MCS test does not require a benchmark to be specified, which is very useful in applications without an obvious benchmark. Secondly, MCS test acknowledges the limitations of the data. Thirdly, the MCS procedure is that it allows for the possibility that more than one model can be the ‘‘best’’. To save space, this paper includes no further technical details of the SPA test, but more in-depth discussions can be found in the studies of Hansen et al. [37]. In this paper, we take the MCS test in comparing volatility models to robust our conclusions.

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M. Yu / Physica A 533 (2019) 120707

Consider a set, M0 , that contains a finite number of models, indexed by i = 1, . . . , m0 . The objects are evaluated over the sample t = 1, . . . , n, in terms of a loss function and we denote the loss that is associated with object i in period t as Li,t . Define the relative performance variables dij,t ≡ Li,t -Lj,t , t for all i, j ∈ M0 . Then the set of superior objects is defined by M ∗ ≡ {u ∈ M0 : E(di,uv,m ) ≤ 0

for all v ∈ M0 }

(14)

This is done through a sequence of significance tests, where objects that are found to be significantly inferior to other elements of M0 are eliminated. The hypotheses that are being tested take the form, H0,M : E(di.uv,m ) = 0

for all u, v ∈ M ⊂ M0

(15)

The MCS procedure is based on an equivalence test δM and elimination rule eM . The equivalence test, δM , is used to test the hypothesis H0 , M for any M ⊂M0 , and eM identifies the object of M that is to be removed from M, in the event that H0 , M is rejected. Hence, the MCS Algorithm is based on the following three steps: Step 1: Initially set M = M 0 . Step 2: Test H0 , M using δM at level α . Step 3: If H0 , M is ‘‘accepted’’ we define the, otherwise we use eM to eliminate an object from M and repeat the procedure beginning with Step 2. ˆ ∗ , which consists of the set of ‘‘surviving’’ objects (those that survived all tests without being eliminated) The set, M 1−α is referred to as the model confidence set. Following Hansen et al. [37], the significant level α is equal to 0.1. And if p-value is larger than 0.1, the corresponding model is ‘‘surviving’’ model, which means that the performance of forecasting is good. The higher the p-value, the better the ability forecasting of the model. Moreover, we use the Range Statistic and Semi-quadratic Statistic, defined that, TR = max √ u,v∈M

⏐ ⏐ ⏐di,uv ⏐ var(di,uv )

, TSQ = max

u,v∈M

(di,uv )2 var(di,uv )

, di,uv =

1 M

H +M

∑

di,uv,m

(16)

m=H +1

The p-value of TR and TSQ is larger than the 0.1, we can get that the null hypothesis cannot be rejected. The asymptotic distributions of the test statistics, TR and TSQ , are non-standard because they depend on nuisance parameters (under both the null and the alternative). However, this poses no obstacle as their distributions are easily estimated using bootstrap methods that implicitly solve the nuisance parameter problem. For details about the implementation of the bootstrap we refer to Hansen et al. [37]. 4.2. In-sample estimations Table 2 presents the estimated results of six volatility models over the in-sample period using ordinary least squares method. From the empirical results of Table 2, we find that the daily RV is significantly positive to one-step-ahead volatility. However, the coefficients of the weekly and monthly are not significant, which is substantially different from the stock and oil markets. In a word, the short-term volatility can lead to high volatility, and the middle- and long-terms are not effective in future volatility from the statistical perspective. The jump components are not significant impacts on future volatility. The leverage effect of return has significant effects on one-step-ahead volatility, and can decrease Bitcoin volatility, especially in short- and long-terms horizons. The economic policy uncertainty has positive effects on future volatility, but not significant during in-sample period. Finally, the R-square of five models are bigger than 0.3, and the LHAR-CJ-EPU model has the bigger values among them. 4.3. Out-of-sample results Following Aït-Sahalia and Mancini [38], Ma et al. [9] and Paye [39], we use the logarithmic RV to model and forecast future volatilities, which is possible that the distribution of the log form can be closer to Gaussian. Table 3 show the empirical results of five models based on the MCS test. We find that the p-values of HAR-RV and HAR-CJ models are less than LHAR-CJ and LHAR-CJ-EPU, which can indicate that the leverage effect has more power than jumps components in forecasting the Bitcoin volatility, which is also supported by the results of HAR-RV and HAR-CJ. Compared with the jump components, the leverage effects are absolutely superior in forecasting Bitcoin volatility. Our conclusions support that jump components do not have effective on future volatility, which is supported by Andersen et al. [2]. In addition, the leverage effect contains the predictive information to predict the Bitcoin volatility, which is also supported the conclusions of Corsi and Renò [18]. Finally, the LHAR-CJ-EPU model has higher MCS p-values, which clearly provides the empirical evidence that this model including the leverage effect and EPU variables can improve the models’ predictive ability. Therefore, the EPU and leverage effect are helpful to forecast Bitcoin volatility.

M. Yu / Physica A 533 (2019) 120707

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Table 2 The estimated results of the individual models over the in-sample period.

β0 βd βw βm

HAR-RV

HAR-CJ

LHAR-CJ

HAR-CJ-EPU

LHAR-CJEPU

0.003* (0.0015) 0.521*** (0.037) 0.035 (0.061) 0.125 (0.081)

0.002 (0.002) 0.521*** (0.037) 0.035 (0.062) 0.100 (0.083) −0.500 (1.611) −1.042 (3.940) 9.353 (6.016)

0.005* (0.028) 0.412*** (0.044) 0.066 (0.095) 0.277** (0.137) −0.206 (1.589) −0.352 (3.891) 7.619 (6.000) −0.213*** (0.045) 0.078 (0.126) 0.355* (0.208)

0.0004 (0.003) 0.520*** (0.038) 0.036 (0.062) 0.100 (0.083) −0.528 (1.611) −1.012 (3.940) 9.259 (6.02)

0.003 (0.003) 0.412*** (0.044) 0.070 (0.096) 0.270** (0.137) −0.233 (1.590) −0.331 (3.893) 7.557 (6.004) −0.214*** (0.045) 0.085 (0.127) 0.345* (0.209) 0.183 (0.324) 0.3380

γd γw γm δd δw δm η R2

0.3239

0.3161

0.3378

0.182 (0.328) 0.3163

Notes: The values in parentheses are the Newey–West Standard errors. The R2 represents the goodness of fit. ***Asterisk denote the rejection of the null hypothesis at the 1% significance level. **Asterisk denote the rejection of the null hypothesis at the 5% significance level. *Asterisk denote the rejection of the null hypothesis at the 10% significance level.

Table 3 Out-of-sample forecasting performance using the MCS test. Models

HAR-RV HAR-CJ LHAR-CJ HAR-CJ-EPU LHAR-CJ-EPU

HMSE

HMAE

TR

TSQ

TR

TSQ

0.010 0.001 0.047 0.010 1.000

0.014 0.001 0.029 0.004 1.000

0.005 0.001 0.047 0.009 1.000

0.005 0.001 0.029 0.003 1.000

Notes: MCS p-values are calculated according to the test statistics TR and TSQ . The MCS p-values larger than 0.1 are indicated in bold. The formulations of these models can be seen the Eqs. (7)– (11). The HAR-RV is our benchmark. Following Hansen et al. [37], the significant level α is equal to 0.1. And if p-value is larger than 0.1, the corresponding model is ‘‘surviving’’ model, which means that the performance of forecasting is good. The higher the p-value, the better the ability forecasting of the model. From this table, we find that only LHAR-CJ-EPU model has higher pvalues under the HMSE and HMAE loss functions, which implies that this model (LHAR-CJ-EPU) model can achieve higher forecasts, so the EPU and leverage effect are helpful to forecast Bitcoin volatility.

5. Robust tests 5.1. Different windows

In view of the fact that different estimations and forecasting windows could produce different empirical results based on data-bias, the forecasting windows are of importance to evaluate the predictive ability of forecasting models, which is supported by Liu et al. [36], Ma et al. [10] and Zhang et al. [30]. To make our conclusions reliable and robust, we choose two different in-sample windows, 600 and 1000, as different windows. From the empirical results in Table 4, we determine that the LHAR-CJ-EPU model achieves higher forecasting accuracies than those of the other models, and this finding is very stable and reliable. Thus, the EPU and leverage effect are helpful to forecast Bitcoin volatility, this is because only this model can pass the MCS test

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M. Yu / Physica A 533 (2019) 120707

Table 4 Out-of-sample forecasting performance using the MCS test with 5-min sampling frequency. Models

HMSE

HMAE

TR

TSQ

TR

TSQ

HAR-RV HAR-CJ LHAR-CJ HAR-CJ-EPU LHAR-CJ-EPU

0.001 0.001 0.248 0.006 1.000

0.002 0.000 0.115 0.002 1.000

0.003 0.003 0.248 0.021 1.000

0.001 0.001 0.115 0.001 1.000

600 (in-sample) HAR-RV HAR-CJ LHAR-CJ HAR-CJ-EPU LHAR-CJ-EPU

0.005 0.001 0.007 0.005 1.000

0.003 0.001 0.015 0.003 1.000

0.001 0.001 0.007 0.001 1.000

0.002 0.001 0.015 0.002 1.000

1000 (in-sample)

Notes: MCS p-values are calculated according to the test statistics TR and TSQ . The MCS p-values larger than 0.1 are indicated in bold. To make our conclusions reliable and robust, we choose two different in-sample windows, 600 and 1000, as different windows. The formulations of these models can be seen the Eqs. (7)–(11). The HAR-RV is our benchmark. Following Hansen et al. [37], the significant level α is equal to 0.1. And if p-value is larger than 0.1, the corresponding model is ‘‘surviving’’ model, which means that the performance of forecasting is good. The higher the p-value, the better the ability forecasting of the model. From this table, we find that only LHAR-CJ-EPU model has higher p-values under the HMSE and HMAE loss functions, which implies that this model (LHAR-CJ-EPU) model can achieve higher forecasts, so the EPU and leverage effect can contain useful information to forecast Bitcoin volatility. Table 5 Out-of-sample forecasting performance using the MCS test with 5-min sampling frequency. Models

HAR-RV HAR-CJ LHAR-CJ HAR-CJ-EPU LHAR-CJ-EPU

HMSE

HMAE

TR

TSQ

TR

TSQ

0.007 0.000 0.030 0.007 1.000

0.050 0.000 0.050 0.003 1.000

0.009 0.001 0.030 0.009 1.000

0.028 0.001 0.028 0.005 1.000

Notes: MCS p-values are calculated according to the test statistics TR and TSQ . The MCS p-values larger than 0.1 are indicated in bold. We use the BPV to replace the RV and continuous sample path, and further evaluate their forecasting performance. The formulations of these models can be seen the Eqs. (7)–(11). The HAR-RV is our benchmark. Following Hansen et al. [37], the significant level α is equal to 0.1. And if p-value is larger than 0.1, the corresponding model is ‘‘surviving’’ model, which means that the performance of forecasting is good. The higher the p-value, the better the ability forecasting of the model. From this table, we find that only LHAR-CJ-EPU model has higher p-values under the HMSE and HMAE loss functions, which implies that this model (LHAR-CJ-EPU) model can achieve higher forecasts, so the EPU and leverage effect can contain useful information to forecast Bitcoin volatility.

5.2. Different measurements In this paper, we use the BPV to replace the RV and continuous sample path, and further evaluate their forecasting performance. Table 5 provides the empirical results of five models based on the MCS test. We find that the MCS p-values of the LHAR-CJ and LHAR-CJ-EPU are larger than other models, which indicates that the jump components are not useful in forecasting Bitcoin volatility, and including the leverage effect and EPU can significantly increase the predictive ability. Therefore, our conclusions are robust. 6. Conclusions In this paper, we first investigate the impacts of leverage effect and uncertainty on Bitcoin volatility using 5-min high-frequency data. To the best of our knowledge, there are few studies to investigate the dynamic volatility of Bitcoin, especially using the high-frequency data. Therefore, this paper can enrich this related research to fill this gap. We draw some interesting conclusions. First, we find that the leverage effect has significant impacts on the future Bitcoin volatility. However, the jumps and EPU seem not to impact the future volatility in-sample period. Additionally, the goodness-of-fit values of the five models are all larger than 0.3. In general, these variables are explained the dependent variable (onestep-ahead volatility) about excess 30%. Second, the MCS test results show that the leverage effect has more power than jumps components in forecasting Bitcoin volatility. Obviously, the leverage effect contains useful predictive information to forecast Bitcoin volatility. Moreover, adding the leverage effect and EPU to benchmark model can substantially improve the predictive ability. Hence, considering both the EPU and leverage effect can help to forecast Bitcoin volatility. Finally, our robust tests are supported to these conclusions.

M. Yu / Physica A 533 (2019) 120707

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Acknowledgments Sponsored by ‘‘Basic Tasks of Technical forecasting’’ commissioned by CASTED, Training and Supporting Project for Young or Middle-aged Teachers of China University of Political Science and Law, College Scientific Research Project of China University of Political Science and Law (Grant No. 17ZFG63001) and Natural Science Foundation of China (Grant No. L1422009). References [1] T.G. Andersen, T. Bollerslev, Answering the skeptics: Yes, standard volatility models do provide accurate forecasts, Internat. Econom. Rev. (1998) 885–905. [2] T.G. Andersen, T. Bollerslev, F.X. Diebold, Roughing it up: Including jump components in the measurement, modeling, and forecasting of return volatility, Rev. Econ. Stat. 89 (4) (2007) 701–720. [3] T. Wang, J. Wu, J. Yang, Realized volatility and correlation in energy futures markets, J. Futures Mark.: Futures Options Deriv. Prod. 28 (10) (2008) 993–1011. [4] F. Corsi, A simple approximate long-memory model of realized volatility, J. 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