Table 5 The effect of the factors of vaccination-related variables on the extreme risks in assets: AR(1)-GARCH(1,1)-X model \(\Delta {y}_{t}={\theta }_{0}+{\theta }_{1}\Delta {y}_{t-1}+{\theta }_{2}\Delta {x}_{t-1}+{\varepsilon }_{t}\), \({\varepsilon }_{t}\sim N\left(0,{\sigma }_{t}^{2}\right)\), \({\sigma }_{t}^{2}=c+a{\varepsilon }_{t-1}^{2}+\beta {\sigma }_{t-1}^{2}+\gamma \Delta {x}_{t-1}\), \(y\) denotes asset price, \(x\) denotes risk factor

\({\theta }_{0}\)

0.0016***

0.0011

0.0006**

0.0026***

0.0015

− 0.0001

0.0013**

0.0006

[27.4399]

[0.6593]

[1.9631]

[62.0257]

[1.8582]

[− 0.3211]

[2.4152]

[1.3733]

\({\theta }_{1}\)

− 0.0288

− 0.0997***

0.0458

− 0.0569***

0.0575

0.0378

0.0276

− 0.0212

[− 1.7839]

[− 2.9274]

[1.0603]

[− 188.3148]

[1.2824]

[0.8988]

[0.6172]

[− 0.4672]

\({\theta }_{2}\)

0.0206***

0.0716***

− 0.0005

0.0088***

− 0.0078

0.0018

0.0048

− 0.0023

[289.5012]

[3.0278]

[− 0.1906]

[48.7046]

[− 0.7662]

[0.3805]

[0.8878]

[− 0.4984]

\(c\)

0.0017***

 < 0.0000

 < 0.0000***

0.0003***

 < 0.0000**

 < 0.0000**

 < 0.0000***

 < 0.0000***

[392.6338]

[0.3723]

[3.5323]

[170.0061]

[2.4139]

[2.4675]

[2.7889]

[2.8470]

\(a\)

0.0937***

− 0.0026***

0.1794***

0.9490***

0.1062***

0.0893***

0.1104***

0.1799***

[271.5141]

[− 3.7399]

[4.6287]

[15.6035]

[3.1134]

[3.1950]

[3.8058]

[4.5671]

\(\beta\)

0.3623***

0.9997***

0.7885***

0.1596***

0.8490***

0.8103***

0.8546***

0.7784***

[4968.7864]

[510.338]

[21.3027]

[52.0864]

[19.8301]

[14.2861]

[25.1078]

[17.8896]

\(\gamma\)

− 0.0024***

0.0003***

 < − 0.0000**

− 0.0004***

0.0000

 < − 0.0000

 < − 0.0000

 < 0.0000

[− 425.8199]

[4.4472]

[− 1.9813]

[− 203.1139]

[− 0.8311]

[− 0.2314]

[− 0.3346]

[0.0230]

\({\theta }_{0}\)

0.0006***

− 0.0004***

0.0003

0.0022***

0.0015

− 0.0001

0.0013**

0.0006

[236.0007]

[− 322.7389]

[0.3474]

[65.0494]

[1.933]

[− 0.3533]

[2.4296]

[1.4152]

\({\theta }_{1}\)

− 0.0316***

− 0.0772***

− 0.0067

− 0.0650***

0.0571

0.0374

0.0278

− 0.0213

[− 58.7025]

[− 30.2465]

[− 0.0931]

[− 16.7246]

[1.3278]

[0.8794]

[0.6601]

[− 0.5002]

\({\theta }_{2}\)

0.0259***

0.0710***

0.0069***

0.0088***

− 0.0086

0.0022

0.0041

− 0.0021

[257.9519]

[418.9184]

[13.9560]

[23470.0241]

[− 0.8689]

[0.4918]

[0.7507]

[− 0.3356]

\(c\)

0.0030***

0.0018***

0.0003***

0.0005***

 < 0.0000**

 < 0.0000**

 < 0.0000***

 < 0.0000***

[8384.6107]

[183.2014]

[182.3012]

[203.9397]

[2.3964]

[2.4720]

[2.6302]

[3.0773]

\(a\)

0.0770***

0.0035***

0.0500***

0.9600***

0.1067***

0.0899***

0.1107***

0.1796***

[579.2403]

[400.1458]

[174.0538]

[35.5704]

[3.0892]

[3.4103]

[3.9439]

[4.8998]

\(\beta\)

0.1133***

0.2333***

0.3327***

0.0448***

0.8485***

0.8086***

0.8544***

0.7789***

[196.2190]

[15545.2828]

[1923.5196]

[55.5196]

[19.9778]

[14.6755]

[24.5872]

[19.5374]

\(\gamma\)

− 0.0032***

− 0.0019***

− 0.0003***

− 0.0005***

 < − 0.0000

 < − 0.0000

 < − 0.0000

 < 0.0000

[− 10873.5210]

[− 184.1849]

[− 204.0440]

[− 213.3385]

[− 0.8282]

[− 0.3083]

[− 0.2748]

[0.1190]