Table 3 The effect of the factors of pandemic severity 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

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

0.0024

0.0014

0.0006

0.0027***

0.0009

− 0.0001***

0.0013**

0.0007

[1.2724]

[0.8122]

[1.6875]

[2.7367]

[1.3047]

[− 97.8743]

[2.2876]

[1.5079]

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

0.0139

− 0.0212

0.0338

− 0.0639

0.0592

− 0.0112***

0.0250

− 0.0305

[0.3160]

[− 0.5319]

[0.8073]

[− 1.3953]

[1.4601]

[− 3.0975]

[0.5858]

[− 0.6969]

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

− 0.0312

− 0.0154

0.0070

− 0.0017

0.0120

− 0.0067***

0.0034

− 0.0098

[− 0.5735]

[− 0.3472]

[0.5041]

[− 0.0573]

[0.3856]

[− 31.0628]

[0.1994]

[− 0.4729]

\(c\)

0.0001

0.0002

 < 0.0000***

0.0002***

 < 0.0000***

0.0001***

 < 0.0000***

 < 0.0000***

[1.9436]

[1.7864]

[2.9899]

[3.5583]

[2.7908]

[55.811]

[2.7366]

[2.9691]

\(a\)

0.1099***

0.0443**

0.1719***

0.7114***

0.0597***

0.0083***

0.1018***

0.1653***

[4.0417]

[2.3538]

[4.4991]

[6.8166]

[2.6894]

[103.6327]

[3.6657]

[4.3255]

\(\beta\)

0.8435***

0.8302***

0.7820***

0.3861***

0.7914***

− 0.2953***

0.8435***

0.7628***

[22.0483]

[12.3274]

[19.1468]

[5.2647]

[14.8168]

[− 1689.9098]

[21.8262]

[16.3102]

\(\gamma\)

0.0023**

0.0030**

0.0002

0.0018

0.0020***

0.0018***

0.0003**

0.0005**

[2.5021]

[2.3286]

[1.7478]

[1.4287]

[2.6451]

[71.2663]

[2.1613]

[2.2087]

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

0.0021

0.0011

0.0005

0.0025***

0.0010

− 0.0001

0.0013**

0.0006

[1.0747]

[0.6289]

[1.3801]

[2.6583]

[1.2871]

[− 0.3484]

[2.3142]

[1.2569]

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

0.0141

− 0.0278

0.0311

− 0.0642

0.0589

0.0372

0.0218

− 0.0285

[0.3011]

[− 0.6833]

[0.7449]

[− 1.5212]

[1.3693]

[0.9102]

[0.4803]

[− 0.6782]

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

0.0188

0.0261

0.0259

0.0232

0.0222

0.0053

− 0.0047

0.0053

[0.2942]

[0.4919]

[1.377]

[0.4175]

[0.5336]

[0.4989]

[− 0.1722]

[0.1519]

\(c\)

0.0002

0.0004**

 < 0.0000***

0.0002***

 < 0.0000***

 < 0.0000***

 < 0.0000***

 < 0.0000***

[1.8128]

[2.0338]

[2.8450]

[3.3421]

[2.6379]

[2.8227]

[2.8650]

[2.9743]

\(a\)

0.1277***

0.0415

0.1858***

0.6823***

0.0698***

0.0688**

0.1087***

0.1733***

[3.6876]

[1.6825]

[4.6423]

[6.1911]

[2.8918]

[2.5658]

[3.9968]

[4.6216]

\(\beta\)

0.8137***

0.7211***

0.7703***

0.3902***

0.7607***

0.7338***

0.8232***

0.7472***

[15.2335]

[6.7015]

[17.9557]

[4.7155]

[12.5501]

[9.4051]

[20.7982]

[15.6587]

\(\gamma\)

0.0038**

0.0087**

0.0003

0.0038

0.0032***

0.0002

0.0006**

0.0009**

[2.0393]

[2.1886]

[1.4849]

[1.3772]

[2.6896]

[1.7899]

[2.3685]

[2.1232]