Realised volatility prediction of high-frequency data with jumps based on machine learning

Figures & data

Figure 1. The architecture of models.

Table 1. Descriptive statistics of return rate of stock data of six different industries.

	PAP	SZP	CG	YTP	WCP	HTH
Mean	−0.0004	0.0001	0.0013	0.0012	0.0006	0.0009
Var	0.3004	0.5675	0.5710	0.4621	0.4839	0.4895
Kurtosis	66.1508	50.3707	52.083	75.6129	59.5715	55.8324
Skewness	1.1347	−0.5713	0.1656	−0.7178	0.9322	−0.4177
Min	−5.6520	−9.5336	−9.5663	−9.6674	−9.5775	−11.8428
Max	10.0441	10.5815	10.5620	10.5360	10.5958	10.5700

Figure 2. 5-min logarithmic Income Fluctuation Trend.

Figure 3. Realised volatility trend.

Table 2. Estimation results of various volatility model parameters of Shanxi coking coal sample population.

	HAR	HAR-J	HARQ	HARQ-J	HARQ-F	HARQ-F-J
${\text{β}}_0$	0.0888${}^{\ast \ast }$	0.0850***	0.0644*	0.0535****	−0.0064	−0.0129
${\text{β}}_1$	0.3733${}^{\ast \ast \ast }$	0.3514***	0.4750***	0.4694***	0.4251***	0.4287***
${\text{β}}_{1Q}$	–	–	−10.7700*	−13.3200*	−6.1580	−9.3680
${\text{β}}_2$	0.3951${}^{\ast \ast \ast }$	0.3906***	0.3857***	0.3773***	0.5850***	0.5284***
${\text{β}}_{2Q}$	–	–	–	–	−2.6460	−1.2970
${\text{β}}_3$	0.1018*	0.1064*	0.0758	0.0759	0.1977	0.2227****
${\text{β}}_{3Q}$	–	–	–	–	−2.1670	−3.2530
${\text{β}}_J$	–	0.2126	–	0.2889*	–	0.2695****
$R^2$	0.5161	0.5167	0.5174	0.5188	0.5183	0.5194

Note: * indicates that the coefficient is significant at the level of 0.05, ** indicates that the coefficient is significant at the level of 0.01, *** indicates that the coefficient is significant at the level of 0.001, **** indicates that the coefficient is significant at the level of 0.1.

Table 3. Parameter estimation results of various volatility models of Angang stock sample population.

	HAR	HAR-J	HARQ	HARQ-J	HARQ-F	HARQ-F-J
${\text{β}}_0$	0.0976${}^{\ast \ast \ast }$	0.1004***	0.0748*	0.0704*	−0.0521	−0.0508
${\text{β}}_1$	0.2844${}^{\ast \ast \ast }$	0.3018***	0.3921***	0.3899***	0.3185***	3.1620***
${\text{β}}_{1Q}$	–	–	−8.0100${}^{\ast }$	−8.8130${}^{\ast }$	−3.5780	−2.0590
${\text{β}}_2$	0.5396***	0.5300***	0.5045***	0.5081***	0.8979***	0.9274***
${\text{β}}_{2Q}$	–	–	–	–	−4.9560	−5.7920
${\text{β}}_3$	0.01997	0.01998	0.00952	0.00847	0.18250	0.17440
${\text{β}}_{3Q}$	–	–	–	–	−3.3060	−2.7790
${\text{β}}_J$	–	−0.0855	–	0.0639	–	−0.0934
$R^2$	0.4791	0.4789	0.4807	0.4804	0.4804	0.4844

Note: * indicates that the coefficient is significant at the level of 0.05, *** indicates that the coefficient is significant at the level of 0.001.

Table 4. Prediction effect of volatility model under each model.

	HAR	HAR-J	HARQ	HARQ-J	HARQ-F	HARQ-F-J
In-of-sample MSE	1.8795	1.8778	1.8752	1.8706	1.8730	1.8695
Out-of-sample MSE	0.2192	0.2170	0.2156	0.2113	0.2048	0.2004
Out-of-sample HMSE	3.5345	3.5504	3.1503	3.1111	3.1525	3.1268
Out-of-sample MAE	0.1147	0.1140	0.1040	0.1025	0.0974	0.0965
Out-of-sample HMAE	0.8303	0.8253	0.7211	0.7075	0.6612	0.6534
Out-of-sample QLIKE	0.1642	0.1633	0.1548	0.1523	0.1400	0.1390
Out-of-sample $R^2$ LOG	0.4099	0.4067	0.3631	0.3554	0.3267	0.3231

Table 5. Prediction effect of volatility based on different machine learning models.

	LSTM	KNN	RNN	DT
In-of-sample MSE	2.0728	0.4855	1.9771	1.4600
Out-of-sample MSE	0.1275	0.2548	0.1667	0.1890
Out-of-sample HMSE	3.1478	6.4877	4.9384	5.8689
Out-of-sample MAE	0.0301	0.0340	0.0369	0.0395
Out-of-sample HMAE	1.4300	1.5283	1.8129	1.9544
Out-of-sample QLIKE	0.3170	0.4996	0.3867	0.4074
Out-of-sample $R^2$ LOG	0.8572	1.0370	1.1110	1.2006

Table 6. Prediction effect of mixed models based on LSTM.

	HAR	HAR-J	HARQ	HARQ-J	HARQ-F	HARQ-F-J
In-of-sample MSE	0.9479	0.7722	0.8188	0.7341	0.8966	0.7849
Out-of-sample MSE	0.4182	0.3927	0.4060	0.3899	0.3630	0.3488
Out-of-sample HMSE	0.8389	0.7000	0.7401	0.6950	0.6426	0.6184
Out-of-sample MAE	0.1408	0.1500	0.1364	0.1280	0.1256	0.1247
Out-of-sample HMAE	0.6857	0.6442	0.6488	0.6032	0.5994	0.5923
Out-of-sample QLIKE	0.7326	0.2340	0.2235	0.2205	0.1998	0.1896
Out-of-sample $R^2$ LOG	0.6606	0.4106	0.4052	0.4017	0.3646	0.3527

Table 7. MCS test results of prediction results of traditional models with different volatility.

	HAR	HAR-J	HARQ	HARQ-J	HARQ-F	HARQ-F-J
MSE	0.3082	0.6802	0.3706	1	1	1
HMSE	1	0.0130	1	1	1	1
MAE	0	0	0.0340	0.1056	1	1
HMAE	0	0	0.0148	0.0614	1	1
QLIKE	0.0666	0.0920	0.6412	1	1	1
$R^2$ LOG	0	0	0	0.0194	1	1

Table 8. MCS test results of prediction results of traditional models of the last four volatility.

	HARQ	HARQ-J	HARQ-F	HARQ-F-J
MSE	0	0.0542	1	1
HMSE	0.6942	1	0.6950	1
MAE	0.0336	0.1132	1	1
HMAE	0.0132	0.0636	1	1
QLIKE	0	0	0.4760	1
$R^2$ LOG	0.0138	0.3880	1	1

Table 9. MCS test results based on prediction results of different volatility models under LSTM.

	HAR	HAR-J	HARQ	HARQ-J	HARQ-F	HARQ-F-J
MSE	0.0572	0.9152	0.6662	0.9990	1	1
HMSE	0.0308	1	0.9620	1	1	1
MAE	0.0186	0.7270	0.3536	1	1	1
HMAE	0.0150	0.8866	0.8162	1	1	1
QLIKE	0.5378	1	1	1	1	1
$R^2$ LOG	0.2292	1	1	1	1	1

Table 10. MCS test results based on the prediction results of the last four volatility models under LSTM.

	HARQ	HARQ-J	HARQ-F	HARQ-F-J
MSE	0.2248	0.6240	1	1
HMSE	0.4310	0.9318	1	1
MAE	0.0106	1	1	1
HMAE	0.1214	1	1	1
QLIKE	0.6004	0.4720	1	1
$R^2$ LOG	0.3908	0.5260	1	1