GARCH

CARR

$r_t={\epsilon }_t$

$R_t={\lambda }_t{\mu }_t$

${\epsilon }_t|I_{t-1}\sim N\left(0,h_t\right)$

${\mu }_t|I_{t-1}\sim \exp \left(1;.\right)$

$h_t={\alpha }_0+{\alpha }_1{\epsilon }_{t-1}^2+{\beta }_1{h}_{t-1}$

${\lambda }_t=\omega +\alpha R_{t-1}+\beta {\lambda }_{t-1}$ .

CSI-300

Spot

${\widehat{\alpha }}_0$

${\widehat{\alpha }}_1$

${\widehat{\beta }}_1$

Q(12)

LLF

$\widehat{\omega }$

$\widehat{\alpha }$

$\widehat{\beta }$

Q(12)

LLF

1.063

(1.987)

0.133

(3.765)

0.814

(15.537)

21.319

[0.445]

−2293.301

0.381

(3.415)

0.262

(8.374)

0.676

(16.655)

7.320

[0.831]

−1835.309

CSI-300 Futures

${\widehat{\alpha }}_0$

${\widehat{\alpha }}_1$

${\widehat{\beta }}_1$

Q(12)

LLF

$\widehat{\omega }$

$\widehat{\alpha }$

$\widehat{\beta }$

Q(12)

LLF

1.021

(2.432)

0.211

(4.153)

0.712

(16.021)

23.551

[0.512]

−2107.547

0.372

(2.827)

0.237

(6.593)

0.612

(12.613)

7.412

[0.845]

−1722.232