Modelling returns volatility: mixed-frequency model based on momentum of predictability

Figures & data

Table 1. Descriptive statistics of returns and realized measure.

	Mean	Standard deviation	Skewness	kurtosis	JB test	ADF
Low-frequency return	0.0246	1.5534	−1.0180	6.3441	2.25 × 10^4*	−10.6413*
High-frequency return	0.0005	0.2153	−1.7434	82.9178	1.67 × 10^7*	−35.4364*
RV	2.2257	5.1205	6.9144	62.5583	2.08 × 10^5*	−6.7736*

Note: * in the JB test means that the null hypothesis of normal distribution is rejected at the 1% significance level. In the ADF test, * indicates that the null hypothesis of unit root is rejected at the 1% significance level.

Source: authors.

Figure 1. Closing price sequence of the CSI 300 index.

Source: authors.

Figure 2. Daily return series of the CSI 300 index.

Source: authors.

Figure 3. Realized volatility.

Source: authors.

Table 2. Estimation results of GARCH model parameters.

Residual distribution	$\omega$	$\gamma$	$\beta$	$n$	$\lambda$
Normal distribution	0.0076 (0.0304)	0.0688 (0.0000)	0.9302 (0.0000)	/	/
Skewed-t distribution	0.0122 (0.0223)	0.0739 (0.0000)	0.9251 (0.0000)	4.5667 (0.0000)	0.9998 (0.0000)

Note: the numbers in brackets in the table refer to the p-value of each parameter.

Source: authors.

Table 3. Estimation results of Realized GARCH model parameters.

Residual cdistribution	$\omega$	$\gamma$	$\beta$	$\xi$	$\phi$	${\tau }_1$	${\tau }_2$	${\sigma }_u^2$	$n$	$\lambda$
Normal distribution	0.1308 (0.0000)	0.2932 (0.0000)	0.7033 (0.0000)	−0.4223 (0.0000)	0.9539 (0.0000)	−0.0458 (0.0079)	0.1385 (0.0000)	0.5568 (0.0000)	/	/
Skewed-t distribution	0.1506 (0.0000)	0.3070 (0.0000)	0.6859 (0.0000)	−0.4573 (0.0000)	0.9608 (0.0000)	−0.0454 (0.0086)	0.1429 (0.0000)	0.5565 (0.0000)	4.6654 (0.0000)	0.9650 (0.0000)

Note: the numbers in brackets in the table refer to the p-value of each parameter.

Source: authors.

Table 4. The results of the PT test.

	Normal distribution	Skewed-t distribution
Statistic	4.6507	4.9684
p-value	0.0000	0.0000

Source: authors.

Table 5. Loss function value of volatility prediction.

Model	Distribution	MAE	MSPE	MAPE	R2LOG
GARCH	Normal	0.8808	2.2323	1.0793	0.7042
	Skewed-t	0.8447	1.9115	0.9784	0.6699
Realized GARCH	Normal	0.8234	2.1825	1.0409	0.6375
	Skewed-t	0.7740	1.8274	0.9400	0.5797
MF-MoP	Normal	0.8078	1.9406	0.9577	0.6026
	Skewed-t	0.7636	1.6018	0.8660	0.5560

Source: authors.

Table 6. The results of MCS test.

Model	Distribution	MAE		MSPE		MAPE		R2LOG
		$T_{SQ}$	$T_R$	$T_{SQ}$	$T_R$	$T_{SQ}$	$T_R$	$T_{SQ}$	$T_R$
GARCH	Normal	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	Skewed-t	0.4912	0.0000	0.8152	0.0000	0.6532	0.0000	0.2786	0.0000
Realized GARCH	Normal	0.4060	0.0000	0.0000	0.0000	0.0000	0.0000	0.3874	0.0000
	Skewed-t	1.0000	0.5328	0.9992	0.0000	0.9952	0.0000	1.0000	0.1674
MF-MoP	Normal	0.9564	0.0000	0.2076	0.0000	0.4996	0.0000	1.0000	0.0000
	Skewed-t	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000

Note: bold font indicates the best volatility model, and p-value less than 0.1 indicates that the model is rejected by MCS test at 10% significance level.

Source: authors.

Table 7. The results of PT test for different realized measures.

	RRV		MedRV
	Normal distribution	Skewed-t distribution	Normal distribution	Skewed-t distribution
Statistic	9.6135	9.2637	8.6095	8.4814
p-value	0.0000	0.0000	0.0000	0.0000

Source: authors.

Table 8. MCS test results of different realized measures.

Model	Distribution	MAE		MSPE		MAPE		R2LOG
		$T_{SQ}$	$T_R$	$T_{SQ}$	$T_R$	$T_{SQ}$	$T_R$	$T_{SQ}$	$T_R$
RRV
GARCH	Normal	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	Skewed-t	0.3676	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
Realized GARCH	Normal	0.1878	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	Skewed-t	0.9998	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
MF-MoP	Normal	1.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	Skewed-t	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
MedRV
GARCH	Normal	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	Skewed-t	0.0000	0.0000	0.2892	0.0000	0.0000	0.0000	0.1634	0.0000
Realized GARCH	Normal	0.0000	0.0000	0.4110	0.0000	0.0000	0.0000	0.1108	0.0000
	Skewed-t	0.0000	0.0000	1.0000	0.0000	0.0000	0.0000	1.0000	0.0000
MF-MoP	Normal	0.0000	0.0000	1.0000	0.0000	0.0000	0.0000	1.0000	0.0000
	Skewed-t	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000

Note: bold font indicates the best volatility model, and p-value less than 0.1 indicates that the model is rejected by MCS test at 10% significance level.

Source: authors.

Table 9. The results of PT test in different look-back periods.

	k = 1		k = 3		k = 6
	Normal distribution	Skewed-t distribution	Normal distribution	Skewed-t distribution	Normal distribution	Skewed-t distribution
Statistic	6.3503	7.6864	6.9398	7.7582	5.9361	6.8244
p-value	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000

Source: authors.

Table 10. The results of MCS test in different look-back periods.

Model	Distribution	MAE		MSPE		MAPE		R2LOG
		$T_{SQ}$	$T_R$	$T_{SQ}$	$T_R$	$T_{SQ}$	$T_R$	$T_{SQ}$	$T_R$
k = 1
GARCH	Normal	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	Skewed-t	0.0000	0.0000	0.0000	0.0000	0.2662	0.0000	0.1538	0.0000
Realized GARCH	Normal	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.1708	0.0000
	Skewed-t	0.0000	0.0000	0.0000	0.0000	0.4810	0.0000	1.0000	0.0000
MF-MoP	Normal	0.0000	0.0000	0.0000	0.0000	1.0000	0.0000	1.0000	0.0000
	Skewed-t	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
k = 3
GARCH	Normal	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	Skewed-t	0.0000	0.0000	0.6270	0.0000	0.2062	0.0000	0.1634	0.0000
Realized GARCH	Normal	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.1108	0.0000
	Skewed-t	0.3250	0.0000	0.7164	0.0000	0.5560	0.0000	1.0000	0.0000
MF-MoP	Normal	0.1238	0.0000	0.6290	0.0000	0.9886	0.0000	1.0000	0.0000
	Skewed-t	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
k = 6
GARCH	Normal	0.0000	0.0000	0.1028	0.0000	0.0000	0.0000	0.0000	0.0000
	Skewed-t	0.2436	0.0000	1.0000	0.0000	0.1728	0.0000	0.2200	0.0000
Realized GARCH	Normal	0.1668	0.0000	0.1112	0.0000	0.0000	0.0000	0.2632	0.0000
	Skewed-t	1.0000	0.1134	1.0000	0.0000	0.8918	0.0000	1.0000	0.0000
MF-MoP	Normal	1.0000	0.0000	1.0000	0.0000	0.4638	0.0000	1.0000	0.0000
	Skewed-t	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000

Note: bold font indicates the best volatility model, and p-value less than 0.1 indicates that the model is rejected by MCS test at the 10% significance level.

Source: authors.

Table 11. The results of PT test of different data.

	KS11		N225		CSI 300
	Normal distribution	Skewed-t distribution	Normal distribution	Skewed-t distribution	Normal distribution	Skewed-t distribution
Statistic	2.3331	5.9391	6.8122	5.6643	3.2784	4.1794
p-value	0.0098	0.0000	0.0000	0.0000	0.0000	0.0000

Source: authors.

Table 12. MCS test results of different data.

Model	Distribution	MAE		MSPE		MAPE		R2LOG
		$T_{SQ}$	$T_R$	$T_{SQ}$	$T_R$	$T_{SQ}$	$T_R$	$T_{SQ}$	$T_R$
KS11
GARCH	Normal	0.0000	0.0000	0.9978	0.0000	0.0000	0.0000	0.0000	0.0000
	Skewed-t	0.0000	0.0000	0.3490	0.2198	0.0000	0.0000	0.0000	0.0000
Realized GARCH	Normal	0.0000	0.0000	1.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	Skewed-t	0.9226	0.3856	1.0000	1.0000	0.3570	0.3536	1.0000	0.4226
MF-MoP	Normal	0.4010	0.0000	1.0000	1.0000	0.0000	0.0000	0.1794	0.0000
	Skewed-t	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
N225
GARCH	Normal	0.0000	0.0000	0.2008	0.0000	0.2120	0.0000	0.2938	0.0000
	Skewed-t	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
Realized GARCH	Normal	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	Skewed-t	0.0000	0.0000	0.2008	0.0000	0.7384	0.0000	0.5518	0.0000
MF-MoP	Normal	1.0000	1.0000	0.5742	0.0000	1.0000	0.2406	0.9260	0.4658
	Skewed-t	0.9928	0.745	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
CSI 300
GARCH	Normal	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.2650	0.0000
	Skewed-t	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.3010	0.0000
Realized GARCH	Normal	0.8010	0.7974	0.0000	0.0000	0.4544	0.4714	1.0000	1.0000
	Skewed-t	0.9890	0.9872	0.8986	0.9066	1.0000	1.0000	0.8674	0.5014
MF-MoP	Normal	0.9958	0.9934	0.0000	0.0000	0.1964	0.1986	1.0000	1.0000
	Skewed-t	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	0.8792	0.5252

Note: bold font indicates the best volatility model, and p-value less than 0.1 indicates that the model is rejected by MCS test at 10% significance level.

Source: authors.