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Table 7 Estimates of tri-variate VAR-AGARCH for the Bitcoin, Ethereum, and Litecoin (Robustness check)

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

  Pre COVID-19 (01 December 2018 to 31 December 2020) COVID-19 (13 January 2020 to 01 April 2020)
Coefficient P value Coefficient P value
Panel A. Mean equation
\(\mu_{1}\)  − 0.000 0.406 0.000 0.142
\(\emptyset_{11}\)  − 0.134a 0.000  − 0.115b 0.028
\(\emptyset_{12}\)  − 0.001 0.908 0.074c 0.097
\(\emptyset_{13}\) 0.113c 0.073 0.037 0.569
\(\mu_{2}\)  − 0.001 0.170 0.000b 0.012
\(\emptyset_{21}\) 0.043b 0.016 0.024 0.505
\(\emptyset_{22}\)  − 0.089a 0.001  − 0.182a 0.001
\(\emptyset_{23}\) 0.148a 0.000 0.112b 0.048
\(\mu_{3}\)  − 0.000 0.305 0.000 0.317
\(\emptyset_{31}\) 0.019 0.103 0.026 0.236
\(\emptyset_{32}\) 0.032c 0.060 0.011 0.745
\(\emptyset_{33}\)  − 0.196a 0.000  − 0.258a 0.000
Panel B. Variance equation
\(c_{1}\) 0.001a 0.000 0.000a 0.000
\(c_{2}\) 0.000b 0.016 0.000a 0.000
\(c_{3}\) 0.000b 0.032 0.001a 0.001
\(a_{11}\) 0.124a 0.001  − 0.022c 0.094
\(a_{12}\) 0.011b 0.032 0.079a 0.000
\(a_{13}\) 0.019b 0.013  − 0.003c 0.086
\(a_{21}\) 0.019 0.115  − 0.072a 0.000
\(a_{22}\) 0.031c 0.051 0.186a 0.000
\(a_{23}\) 0.014b 0.033 0.012c 0.053
\(a_{31}\) 0.018 0.548  − 0.108a 0.002
\(a_{32}\) 0.001 0.736 0.172a 0.003
\(a_{33}\) 0.155a 0.001 0.059b 0.050
\(b_{11}\) 0.864a 0.000 0.986a 0.000
\(b_{12}\)  − 0.036b 0.027 0.016b 0.042
\(b_{13}\)  − 0.022 0.290 0.075a 0.000
\(b_{21}\)  − 0.024c 0.075 0.028b 0.033
\(b_{22}\) 0.953a 0.000 0.885a 0.000
\(b_{23}\)  − 0.010 0.135  − 0.060a 0.000
\(b_{31}\)  − 0.035 0.443 0.028 0.169
\(b_{32}\) 0.277c 0.099 0.115c 0.084
\(b_{33}\) 0.636a 0.000 0.522a 0.000
\(d_{1}\)  − 0.052b 0.050 0.041 0.070
\(d_{2}\)  − 0.018b 0.025 0.030c 0.093
\(d_{3}\)  − 0.103a 0.000 0.064b 0.032
Panel C: Constant correlations
\(p_{21}\) 0.783a 0.000 0.869a 0.000
\(p_{31}\) 0.658a 0.000 0.803a 0.000
\(p_{32}\) 0.710a 0.000 0.831a 0.000
Panel D: Robustness tests
Log L 10,711.7   18,493.2  
AIC  − 21.416    − 20.046  
SIC  − 21.268    − 19.467  
\(Q_{1}\)(20) 31.675b 0.046 33.334b 0.030
\(Q_{2}\)(20) 39.436a 0.005 33.455b 0.030
\(Q_{3}\)(20) 25.676 0.176 22.340 0.322
\(Q_{1}^{2}\)(20) 2.916 0.997 14.670 0.672
\(Q_{2}^{2}\)(20) 4.055 0.989 11.778 0.740
\(Q_{3}^{2}\)(20) 6.238 0.988 16.621 0.802
  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 Bitocoin
  2. a,b,cIndicate the statistical significance at 1%, 5% and 10% respectively