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

Panel A. Mean equation

\(\mu_{1}\)

 − 0.000

0.000

\(\emptyset_{11}\)

 − 0.171^a

 − 0.093^b

\(\emptyset_{12}\)

 − 0.018

0.076^c

\(\emptyset_{13}\)

0.102^c

0.083

\(\mu_{2}\)

 − 0.000

0.000^c

\(\emptyset_{21}\)

0.067 ^a

0.018

\(\emptyset_{22}\)

 − 0.081^a

 − 0.221^a

\(\emptyset_{23}\)

0.182^a

0.132^c

\(\mu_{3}\)

 − 0.000

0.000

\(\emptyset_{31}\)

0.017^c

0.015

\(\emptyset_{32}\)

0.026^c

0.043

\(\emptyset_{33}\)

 − 0.225^a

 − 0.188^a

Panel B. Variance equation

\(c_{1}\)

0.000^a

0.000^b

\(c_{2}\)

0.001^a

0.001^a

\(c_{3}\)

0.000^a

0.001^a

\(a_{11}\)

0.073^b

 − 0.029^c

\(a_{12}\)

0.014^b

0.042^a

\(a_{13}\)

0.027^a

 − 0.002^c

\(a_{21}\)

0.011^c

0.061^a

\(a_{22}\)

0.045^a

0.150^a

\(a_{23}\)

0.060^a

0.015^c

\(a_{31}\)

0.002

 − 0.065^a

\(a_{32}\)

0.117^a

0.083^a

\(a_{33}\)

0.181^a

0.063^a

\(b_{11}\)

0.880^a

0.995^a

\(b_{12}\)

 − 0.144^a

0.037^a

\(b_{13}\)

 − 0.129^a

0.025^b

\(b_{21}\)

 − 0.058^b

 − 0.035^c

\(b_{22}\)

1.256^a

0.874^a

\(b_{23}\)

 − 0.269^a

 − 0.019^c

\(b_{31}\)

 − 0.153

0.027^c

\(b_{32}\)

1.028^a

0.069^a

\(b_{33}\)

0.860^a

0.938^a

\(d_{1}\)

 − 0.035^b

0.041^b

\(d_{2}\)

 − 0.016^c

0.047^a

\(d_{3}\)

 − 0.017^c

0.034^a

Panel C: Constant correlations

\(p_{21}\)

0.783^a

0.860^a

\(p_{31}\)

0.644^a

0.802^a

\(p_{32}\)

0.691^a

0.831^a

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)

42.173^a

36.776^b

\(Q_{2}\)(20)

45.954^a

36.444^b

\(Q_{3}\)(20)

22.238

25.542

\(Q_{1}^{2}\)(20)

2.610

18.776

\(Q_{2}^{2}\)(20)

4.859

14.444

\(Q_{3}^{2}\)(20)

5.166

10.542