Volatility analysis based on GARCH-type models: Evidence from the Chinese stock market

Figures & data

Figure 1. The daily indexes.

Source: statistical software EViews.

Figure 2. Descriptive statistics of index returns.

Source: statistical software EViews.

Table 1. Descriptive statistics.

	Sample Size	Mean	Std.	Skewness	Kurtosis	J–B Statistics	P-value
SH	4883	0.000156	0.0156	−0.3741	7.9809	5161.656	0.0000
SCF	4883	0.000244	0.0176	−0.3509	6.6087	2749.914	0.0000

Source: statistical results.

Figure 3. Q-Q plots of return series.

Source: statistical software EViews.

Table 9. Summary statistics in SCF.

	SCF-GARCH	AIC	SC	Mean
Normal distribution	GARCH (1,1)	−5.468396	−5.461746	−5.465071
	GARCH (1, 2)	−5.467996	−5.460016	−5.464006
	GARCH (2,1)	−5.466064	−5.458084	−5.462074
Student’s t-distribution	GARCH (1,1)	−5.530729	−5.522748	−5.5267385
	GARCH (1, 2)	−5.53053	−5.521219	−5.5258745
	GARCH (2,1)	−5.526506	−5.517196	−5.521851
Generalised error distribution	GARCH (1,1)	−5.52948	−5.5215	−5.52549
	GARCH (1, 2)	−5.529154	−5.519844	−5.524499
	GARCH (2,1)	−5.52922	−5.51991	−5.524565

Source: statistical results.

Table 2. ADF test results.

	T-statistics	The Significance Level of 1%	The Significance Level of 5%	The Significance Level of 10%	P-value
SH	−68.3937	−3.4315	−2.8619	−2.5670	0.0001
SCF	−66.7596	−3.4315	−2.8619	−2.5670	0.0001

Source: statistical results.

Table 3. The autocorrelation test results in SH.

Autocorrelation	Partial Autocorrelation		AC	PAC	Q-statistics	P-value
		1	0.021	0.021	2.1861	0.139
		2	−0.022	−0.022	4.5307	0.104
		3	0.028	0.029	8.2587	0.041
		4	0.048	0.047	19.684	0.001
		5	−0.004	−0.005	19.771	0.001
		6	−0.051	−0.050	32.638	0.000
		7	0.031	0.031	37.358	0.000
		8	0.009	0.004	37.758	0.000
		9	0.004	0.008	37.830	0.000
		10	−0.002	0.002	37.842	0.000

Source: statistical results.

Table 10. Estimation for ARMA (4,4)-GARCH (1,1) in SH.

SH	Variables	Coefficient	Std.	Z-statistics	P-value
Mean equation	AR (1(1) $$y_t={\alpha }_0+\sum _{i=1}^p{\alpha }_i{y}_{t-i}+{\epsilon }_t+\sum _{i=1}^q{\beta }_i{\epsilon }_{t-i}$$(1) )	0.677042	0.382763	1.768827	0.0769
(ARMA)	AR (2(2) $$y_t=\varphi x_t+{\mu }_t$$(2) )	0.124923	0.258004	0.484192	0.6282
	AR (3(3) $${\sigma }_{\mathrm{t}}^2=E({\mu }_{\mathrm{t}}^2|{\mu }_{t-1},{\mu }_{t-2}\dots )={\alpha }_0+{\alpha }_1{\mu }_{t-1}^2+\dots \dots +{\alpha }_p{\mu }_{t-p}^2=\sum _{i=1}^p{\alpha }_i{\mu }_{t-i}^2$$(3) )	−0.550004	0.216142	−2.544645	0.0109
	AR (4(4) $$y_t=\varphi x_t+{\mu }_t,\mu \sim N(0,{\sigma }_t^2)$$(4) )	0.134562	0.256962	0.523665	0.6005
	MA (1(1) $$y_t={\alpha }_0+\sum _{i=1}^p{\alpha }_i{y}_{t-i}+{\epsilon }_t+\sum _{i=1}^q{\beta }_i{\epsilon }_{t-i}$$(1) )	−0.662400	0.381017	−1.738506	0.0821
	MA (2(2) $$y_t=\varphi x_t+{\mu }_t$$(2) )	0.128068	0.248170	−0.516050	0.6058
	MA (3(3) $${\sigma }_{\mathrm{t}}^2=E({\mu }_{\mathrm{t}}^2|{\mu }_{t-1},{\mu }_{t-2}\dots )={\alpha }_0+{\alpha }_1{\mu }_{t-1}^2+\dots \dots +{\alpha }_p{\mu }_{t-p}^2=\sum _{i=1}^p{\alpha }_i{\mu }_{t-i}^2$$(3) )	0.583425	0.208923	2.792540	0.0052
	MA (4(4) $$y_t=\varphi x_t+{\mu }_t,\mu \sim N(0,{\sigma }_t^2)$$(4) )	−0.165291	0.261328	−0.632503	0.5271
Variance equation	$\omega$	1.25E-06	3.59E-07	3.497888	0.0005
(GARCH)	ARCH parameter	0.064365	0.004988	9.210643	0.0000
	GARCH parameter	0.935054	0.006344	147.3881	0.0000

Source: statistical results.

Table 11. Estimation for ARMA (1,1)-GARCH (1,1) in SCF.

SCF	Variables	Coefficient	Std.	Z-statistics	P-value
Mean equation	AR (1(1) $$y_t={\alpha }_0+\sum _{i=1}^p{\alpha }_i{y}_{t-i}+{\epsilon }_t+\sum _{i=1}^q{\beta }_i{\epsilon }_{t-i}$$(1) )	−0.777718	0.156431	−4.971648	0.0000
(ARMA)	MA (1(1) $$y_t={\alpha }_0+\sum _{i=1}^p{\alpha }_i{y}_{t-i}+{\epsilon }_t+\sum _{i=1}^q{\beta }_i{\epsilon }_{t-i}$$(1) )	0.798818	0.149967	5.326618	0.0000
Variance equation	$\omega$	2.95E-06	7.13E-07	4.137136	0.0000
(GARCH)	ARCH parameter	0.066557	0.007421	8.969182	0.0000
	GARCH parameter	0.926246	0.007579	122.2111	0.0000

Source: statistical results.

Table 4. The autocorrelation test results in SCF.

Autocorrelation	Partial Autocorrelation		AC	PAC	Q-statistics	P-value
		1	0.045	0.045	10.038	0.002
		2	−0.023	−0.025	12.598	0.002
		3	0.028	0.030	16.429	0.001
		4	0.034	0.030	21.970	0.000
		5	−0.007	−0.009	22.222	0.000
		6	−0.025	−0.024	25.348	0.000
		7	0.042	0.043	34.095	0.000
		8	0.011	0.005	34.666	0.000
		9	0.001	0.005	34.675	0.000
		10	−0.004	−0.004	34.742	0.000

Source: statistical results.

Table 5. ARMA model estimation results of SH and SCF return series.

SH	AIC	SC	Mean
ARMA (4,4)	−5.4893	−5.4787	−5.4840
ARMA (4, 6)	−5.4899	−5.4766	−5.4833
ARMA (6,4)	−5.4887	−5.4783	−5.4835
ARMA (6,6)	−5.4885	−5.4725	−5.4805
SH	AIC	SC	Mean
ARMA (1,1)	−5.2388	−5.2362	−5.2375
ARMA (1, 7)	−5.2410	−5.2304	−5.2357
ARMA (7,1)	−5.2427	−5.2313	−5.2370
ARMA (7,7)	−5.2450	−5.2263	−5.2356

Source: statistical results.

Table 13. Estimation for PARCH (1,1) in SH.

SH	Variables	Coefficient	Std.	Z-statistics	P-value
Variance equation	$\omega$	9.81E-05	8.01E-05	1.223889	0.2210
(PARCH)	$\alpha$	0.080390	0.007609	10.56515	0.0000
	$\gamma$	0.183320	0.054283	3.377141	0.0007
	$\beta$	0.933875	0.006341	147.2709	0.0000
	$\delta$	0.044826	0.179298	5.827307	0.0000

Source: statistical results.

Table 14. Estimation for PARCH (1,1) in SCF.

SCF	Variables	Coefficient	Std.	Z-statistics	P-value
Variance equation	$\omega$	0.000253	0.000200	1.266037	0.2055
(PARCH)	$\alpha$	0.079024	0.007573	10.43472	0.0000
	$\gamma$	0.201582	0.054292	3.712925	0.0002
	$\beta$	0.930792	0.006829	136.3005	0.0000
	$\delta$	0.922191	0.181949	5.068395	0.0000

Source: statistical results.

Table 8. Summary statistics in SH.

	SH-GARCH	AIC	SC	Mean
Normal distribution	GARCH (1,1)	−5.775053	−5.760415	−5.767734
	GARCH (1, 2)	−5.774783	−5.758814	−5.7667985
	GARCH (2,1)	−5.773185	−5.757216	−5.7652005
Student’s t-distribution	GARCH (1,1)	−5.859664	−5.843695	−5.8516795
	GARCH (1, 2)	−5.859736	−5.842437	−5.8510865
	GARCH (2,1)	−5.860116	−5.842817	−5.8514665
Generalised error distribution	GARCH (1,1)	−5.859201	−5.843233	−5.851217
	GARCH (1, 2)	−5.858948	−5.841648	−5.850298
	GARCH (2,1)	−5.859036	−5.841736	−5.850386

Source: statistical results.

Table 15. Results of the ARCH-LM test for the PARCH model.

	F-statistics	Probability	$T\times R^2$ Statistics	Probability
SH	0.726645	0.3940	0.726834	0.3939
SCF	0.289720	0.5904	0.289822	0.5903

Source: statistical results.

Table 16. Estimation results of CARCH (1,1) for SH.

SH	Variables	Coefficient	Std.	Z-statistics	P-value
Variance equation	$\omega$	0.004638	0.011617	0.399226	0.6897
(CARCH)	$\rho$	0.999722	0.000710	1407.496	0.0000
	$\varphi$	0.068794	0.006517	10.55607	0.0000
	$\alpha$	−0.062890	0.020310	−3.096562	0.0020
	$\gamma$	0.073117	0.028678	2.549594	0.0108
	$\beta$	0.640358	0.196195	3.263879	0.0011

Source: statistical results.

Table 17. Estimation results of CARCH (1,1) for SCF.

SCF	Variables	Coefficient	Std.	Z-statistics	P-value
Variance equation	$\omega$	0.000412	0.000173	2.386235	0.0170
(CARCH)	$\rho$	0.992354	0.004110	241.4322	0.0000
	$\varphi$	0.069251	0.007643	9.060927	0.0000
	$\alpha$	−0.039013	0.024711	−1.578804	0.1144
	$\gamma$	0.014143	0.032789	0.431318	0.6662
	$\beta$	−0.232817	0.561740	−0.414458	0.6785

Source: statistical results.

Table 18. Results of the ARCH-LM test for the CARCH model.

	F-statistics	Probability	$T\times R^2$ Statistics	Probability
SH	1.520640	0.2176	1.520790	0.2175
SCF	1.997023	0.1577	1.997024	0.1576

Source: statistical results.

Table 19. Estimation results of TGARCH (1,1) for SH.

SH	Variables	Coefficient	Std.	Z-statistics	P-value
Variance equation	$\omega$	1.44E-06	3.81E-07	3.770795	0.0002
(TGARCH)	$\alpha$	0.052649	0.008181	6.435696	0.0000
	$\lambda$	0.028738	0.010537	2.727310	0.0064
	$\beta$	0.931614	0.006491	143.5330	0.0000

Source: statistical results.

Table 20. Estimation results of TGARCH (1,1) for SCF.

SCF	Variables	Coefficient	Std.	Z-statistics	P-value
Variance equation	$\omega$	3.23E-06	7.40E-07	4.362482	0.0000
(TGARCH)	$\alpha$	0.050050	0.008197	6.105689	0.0000
	$\lambda$	0.036598	0.010735	3.409107	0.0007
	$\beta$	0.923446	0.007620	121.1859	0.0000

Source: statistical results.

Table 21. Results of the ARCH-LM test for the TGARCH model.

	F-statistics	Probability	$T\times R^2$ Statistics	Probability
SH	0.532270	0.7120	2.130335	0.7118
SCF	0.094907	0.7580	0.094944	0.7580

Source: statistical results.

Table 22. Estimates of EGARCH (1,1) for SH.

SH	Variables	Coefficient	Std.	Z-statistics	P-value
Variance equation	$\omega$	−0.194159	0.025892	−7.498868	0.0000
(EGARCH)	$\alpha$	0.153050	0.013938	10.98043	0.0000
	$\lambda$	−0.028658	0.007799	−3.674773	0.0002
	$\beta$	0.990301	0.002467	401.4211	0.0000

Source: statistical results.

Table 23. Estimates of EGARCH (1,1) for SCF.

SCF	Variables	Coefficient	Std.	Z-statistics	P-value
Variance equation	$\omega$	−0.207422	0.027986	−7.411757	0.0000
(EGARCH)	$\alpha$	0.150245	0.013806	10.88240	0.0000
	$\lambda$	−0.028715	0.007515	−3.821092	0.0001
	$\beta$	0.988419	0.002804	352.5051	0.0000

Source: statistical results.

Table 24. Results of the ARCH-LM test for the EGARCH model.

	F-statistics	Probability	$T\times R^2$ Statistics	Probability
SH	0.975060	0.4198	3.901122	0.4196
SCF	0.236844	0.6265	0.236929	0.6264

Source: statistical results.

Table 25. Forecast values of the SH series.

Date	Real Return	GARCH (1,1)	TGARCH (1,1)	EGARCH (1,1)	PARCH (1,1)	CARCH (1,1)
2020-03-05	−0.012173	0.002141	0.002155	0.001054	0.001337	0.001316
2020-03-06	−0.030522	0.000218	0.000264	0.001264	0.000604	0.000438
2020-03-09	0.018004	0.000934	0.001006	0.000544	0.000920	0.000811
2020-03-10	−0.009470	−0.000839	−0.000803	−0.000406	−0.000764	−0.000807
2020-03-11	−0.015286	−0.000284	−0.000248	−0.000486	−0.000160	−0.000092
2020-03-12	−0.012411	−0.000781	−0.000777	−0.000134	−0.000239	−0.000186
2020-03-13	−0.034592	0.000023	0.000042	0.000229	0.000543	0.000563
2020-03-16	−0.003452	−0.000039	−0.000044	0.000161	−0.000023	−0.000071
2020-03-17	−0.018476	0.000368	0.000369	0.000026	0.000022	−0.000015
2020-03-18	−0.009805	0.000126	0.000105	−0.000118	−0.000297	−0.000301

Source: statistical results.

Table 26. Forecast values of the SCF series.

Date	Real Return	ARMA (1,1)- GARCH (1,1)	ARMA (1,1)- TGARCH (1,1)	ARMA (1,1)- EGARCH (1,1)	ARMA (1,1)- PARCH (1,1)	ARMA (1,1)- CARCH (1,1)
2020-03-05	−0.011038	0.001045	0.001110	0.001074	0.001051	0.000991
2020-03-06	−0.041808	−0.000813	−0.000861	−0.000834	−0.000814	−0.000781
2020-03-09	0.026203	0.000632	0.000667	0.000647	0.000631	0.000615
2020-03-10	−0.017999	−0.000492	−0.000517	−0.000502	−0.000489	−0.000484
2020-03-11	−0.023400	0.000382	0.000401	0.000390	0.000379	0.000381
2020-03-12	−0.010094	−0.000297	−0.000311	−0.000303	−0.000293	−0.000300
2020-03-13	−0.054826	0.000231	0.000241	0.000235	0.000227	0.000237
2020-03-16	−0.004941	−0.000180	−0.000187	−0.000182	−0.000176	−0.000186
2020-03-17	−0.017120	0.000140	0.000145	0.000141	0.000136	0.000147
2020-03-18	−0.000969	−0.000109	−0.000112	−0.000110	−0.000106	−0.000116

Source: statistical results.

Table 27. Loss function values of the forecasting of SH return series.

	ARMA (4,4)- GARCH (1,1)	ARMA (4,4)- TGARCH (1,1)	ARMA (4,4)- EGARCH (1,1)	ARMA (4,4)- PARCH (1,1)	ARMA (4,4)- CARCH (1,1)
MSE	0.000354	0.000354	0.000360	0.000357	0.000357
MAE	0.016419	0.016425	0.016524	0.016429	0.016423
RMSE	0.018814	0.018823	0.018984	0.018907	0.018894
MEAN	0.011862	0.011867	0.011956	0.011898	0.011891

Source: statistical results.

Table 28. Loss function values of the forecasting of SCF return series.

	ARMA (1,1)- GARCH (1,1)	ARMA (1,1)- TGARCH (1,1)	ARMA (1,1)- EGARCH (1,1)	ARMA (1,1)- PARCH (1,1)	ARMA (1,1)- CARCH (1,1)
MSE	0.000680	0.000680	0.000680	0.000680	0.000680
MAE	0.020767	0.020764	0.020766	0.020768	0.020767
RMSE	0.026079	0.026073	0.026076	0.026078	0.026085
MEAN	0.015842	0.015839	0.015841	0.015842	0.015844

Source: statistical results.

Table 6. The autocorrelation test results of SH’s squared residual.

Autocorrelation	Partial Autocorrelation		AC	PAC	Q-statistics	P-value
		1	0.160	0.160	124.52	0.000
		2	0.141	0.119	221.83	0.000
		3	0.180	0.147	380.18	0.000
		4	0.151	0.098	492.11	0.000
		5	0.130	0.068	574.76	0.000
		6	0.120	0.051	644.76	0.000
		7	0.131	0.064	729.12	0.000
		8	0.119	0.048	798.73	0.000
		9	0.090	0.016	838.09	0.000
		10	0.146	0.081	942.65	0.000

Source: statistical results.

Table 7. The autocorrelation test results of SCF’s squared residual.

Autocorrelation	Partial Autocorrelation		AC	PAC	Q-statistics	P-value
		1	0.152	0.152	112.70	0.000
		2	0.168	0.148	250.80	0.000
		3	0.173	0.134	396.31	0.000
		4	0.157	0.102	516.67	0.000
		5	0.127	0.060	595.90	0.000
		6	0.128	0.058	676.05	0.000
		7	0.138	0.069	769.67	0.000
		8	0.148	0.076	876.71	0.000
		9	0.091	0.009	917.53	0.000
		10	0.150	0.075	1027.1	0.000

Source: statistical results.

Table 12. Results of ARCH-LM test.

	F-statistics	Probability	$T\times R^2$ Statistics	Probability
SH	0.829600	0.5061	3.319546	0.5058
SCF	0.053695	0.8168	0.053716	0.8167

Source: statistical results.