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Table 3 Estimates of multivariate VAR-AGARCH model for the Bitcoin, Ethereum, and Litecoin

From: Discovering interlinkages between major cryptocurrencies using high-frequency data: new evidence from COVID-19 pandemic

  Pre COVID-19 COVID-19
Coefficient P value Coefficient P value
Panel A. Mean equation
\(\mu_{1}\)  − 0.000 0.380 0.000 0.349
\(\emptyset_{11}\)  − 0.171a 0.000  − 0.093b 0.019
\(\emptyset_{12}\)  − 0.018 0.501 0.076c 0.052
\(\emptyset_{13}\) 0.102c 0.072 0.083 0.150
\(\mu_{2}\)  − 0.000 0.277 0.000c 0.052
\(\emptyset_{21}\) 0.067 a 0.000 0.018 0.499
\(\emptyset_{22}\)  − 0.081a 0.000  − 0.221a 0.000
\(\emptyset_{23}\) 0.182a 0.000 0.132c 0.094
\(\mu_{3}\)  − 0.000 0.323 0.000 0.249
\(\emptyset_{31}\) 0.017c 0.071 0.015 0.509
\(\emptyset_{32}\) 0.026c 0.052 0.043 0.223
\(\emptyset_{33}\)  − 0.225a 0.000  − 0.188a 0.000
Panel B. Variance equation
\(c_{1}\) 0.000a 0.000 0.000b 0.012
\(c_{2}\) 0.001a 0.000 0.001a 0.000
\(c_{3}\) 0.000a 0.000 0.001a 0.000
\(a_{11}\) 0.073b 0.018  − 0.029c 0.070
\(a_{12}\) 0.014b 0.019 0.042a 0.000
\(a_{13}\) 0.027a 0.000  − 0.002c 0.055
\(a_{21}\) 0.011c 0.068 0.061a 0.000
\(a_{22}\) 0.045a 0.000 0.150a 0.000
\(a_{23}\) 0.060a 0.000 0.015c 0.061
\(a_{31}\) 0.002 0.293  − 0.065a 0.000
\(a_{32}\) 0.117a 0.001 0.083a 0.000
\(a_{33}\) 0.181a 0.000 0.063a 0.000
\(b_{11}\) 0.880a 0.000 0.995a 0.000
\(b_{12}\)  − 0.144a 0.000 0.037a 0.000
\(b_{13}\)  − 0.129a 0.000 0.025b 0.040
\(b_{21}\)  − 0.058b 0.032  − 0.035c 0.052
\(b_{22}\) 1.256a 0.000 0.874a 0.000
\(b_{23}\)  − 0.269a 0.000  − 0.019c 0.094
\(b_{31}\)  − 0.153 0.113 0.027c 0.079
\(b_{32}\) 1.028a 0.000 0.069a 0.000
\(b_{33}\) 0.860a 0.000 0.938a 0.000
\(d_{1}\)  − 0.035b 0.043 0.041b 0.018
\(d_{2}\)  − 0.016c 0.061 0.047a 0.001
\(d_{3}\)  − 0.017c 0.082 0.034a 0.006
Panel C: Constant correlations
\(p_{21}\) 0.783a 0.000 0.860a 0.000
\(p_{31}\) 0.644a 0.000 0.802a 0.000
\(p_{32}\) 0.691a 0.000 0.831a 0.000
Panel D: Robustness tests
Log L 11,912.2   24,025.6  
AIC  − 20.972    − 20.392  
SIC  − 20.850    − 19.918  
\(Q_{1}\)(20) 42.173a 0.003 36.776b 0.012
\(Q_{2}\)(20) 45.954a 0.000 36.444b 0.013
\(Q_{3}\)(20) 22.238 0.245 25.542 0.181
\(Q_{1}^{2}\)(20) 2.610 0.991 18.776 0.352
\(Q_{2}^{2}\)(20) 4.859 0.988 14.444 0.415
\(Q_{3}^{2}\)(20) 5.166 0.981 10.542 0.480
  1. # of lags for VAR is decided using SIC and AIC criteria. JB, Q(20), and Q2(20) indicate the empirical statistics of Jarque–Bera test for normality, Ljung–Box Q statistics of order 20 for autocorrelation applied to the standardized residuals and squared standardized residuals, respectively. BTC, Bitcoin; ETH, Ethereum; LTC, Litecoin. Variable order is the Bitcoin (1), Ethereum (2), and Litecoin (3). In the mean equations, \(\mu\) denotes the constant terms, whereas \(\emptyset_{12}\) denotes the return spillover from Bitcoin to Ethereum. In the variance equation, 'c' denotes the constant terms, 'a' denotes the ARCH terms, and 'b' denotes the GARCH terms. In the variance equation, \(a_{12}\) indicates the shock spillover from Bitcoin to Ethereum, whereas \({\text{b}}_{12}\) denotes the long-term volatility spillover from Bitcoin to Ethereum. \(d_{1}\) is the asymmetric effect of the Bitcoin
  2. a,b,cIndicate the statistical significance at 1%, 5% and 10% respectively