Empirical study on the efficiency of the stock index futures market from the information and functional perspectives – empirical evidence from China

Figures & data

Table 1. Results of intermittent regression estimation.

Method	Estimated break dates
1. Sequential L + 1 breaks vs. L	6/13/2011, 1/28/2013, 3/31/2015, 5/19/2016
2. Sequential test all subsets	6/13/2011, 1/28/2013, 3/31/2015, 5/19/2016
3. Global L breaks vs. none	6/13/2011, 1/28/2013, 3/31/2014, 6/08/2015, 7/28/2016
4. L + 1 breaks vs. global L	6/13/2011, 1/30/2013, 3/12/2015, 4/29/2016
5. Global information criteria	6/13/2011, 1/30/2013, 3/12/2015, 4/29/2016

Table 2. Results of Wright and Blaire-Contreras variance ratio test.

Whole sample (4/10/2010–12/13/2017)
Individual tests
	Rank			Rank scores			Signs
Period	VR	Z	Prob	VR	Z	Prob	VR	Z	Prob
2	0.9676	−1.3979	0.1622	0.9751	−1.0748	0.2812	0.9780	−0.9499	0.3388
5	0.9437	−1.1087	0.2680	0.9395	−1.1923	0.2278	0.9648	−0.6937	0.4864
10	0.9083	−1.1728	0.2492	0.9434	−0.7231	0.4794	0.8794	−1.5412	0.1228
30	1.0016	0.0113	0.9904	1.0244	0.1707	0.8720	0.9045	−0.6685	0.5214
Joint tests
	Rank			Rank scores			Signs
	Value	df	Prob	Value	df	Prob	Value	df	Prob
Max|z|	1.3979	1863	0.3776	1.1923	1863	0.5074	1.5412	1863	0.2940
Wald	4.5689	4	0.3336	2.4456	4	0.6696	7.2443	4	0.1202
Group A (4/16/2010 – 6/10/2011)
Individual Tests
	Rank			Rank scores			Signs
Period	VR	Z	Prob	VR	Z	Prob	VR	Z	Prob
2	0.9437	−0.9391	0.3564	0.9250	−1.2505	0.2150	0.9856	−0.2399	0.7694
5	0.9223	−0.5913	0.5766	0.9533	−0.3555	0.7366	0.8993	−0.7665	0.4450
10	0.9878	−0.0601	0.9586	1.0614	0.3032	0.7784	0.8763	−0.6111	0.5640
30	0.8811	−0.3215	0.8032	0.9423	−0.1561	0.9096	0.6034	−1.0725	0.3204
Joint tests
	Rank			Rank scores			Signs
	Value	df	Prob	Value	df	Prob	Value	df	Prob
Max|z|	0.9391	278	0.7236	1.2505	278	0.4876	1.0725	278	0.6182
Wald	2.0287	4	0.7674	3.8945	4	0.4316	2.2440	4	0.7082

Figure 1. Daily yield of CSI 300 futures.

Figure 2. Daily return of CSI 300.

Table 3. ARCH effect test of ${\epsilon }_{st}$ and ${\epsilon }_{ft}$ sequences of ECM model.

${\epsilon }_{st}$
$F$	54.5940	p value.$F$(5, 1851)	0.0000
$T{R}^2$	238.6595	p value.${\chi }^2(5)$	0.0000
${\epsilon }_{ft}$
$F$	93.0210	p value.$F$(5, 1851)	0.0000
$T{R}^2$	372.9107	p value.${\chi }^2(5)$	0.0000

Table 4. Dynamic hedging ratio and effectiveness.

	Mean	Median	Maximum	Minimum	Std.Dev	Skewness	Kurtosis
$h_{t-1}^{*}$	0.8702	0.8765	1.1487	0.3409	0.1192	−0.5887	3.6807
$h_{e,t}^{*}$	0.8611	0.9198	0.9781	−0.3091	0.1521	−2.8135	13.0888

Table 5. Results of linear and nonlinear Granger test.

$l_S=l_F$	$\text{ln}\hspace{0.17em}S\Rightarrow \hspace{0.17em}\text{ln}\hspace{0.17em}F$		$\text{ln}\hspace{0.17em}F\Rightarrow \hspace{0.17em}\text{ln}\hspace{0.17em}S$
	$F$	$T_2$	$F$		$T_2$
Whole sample (4/16/2010–12/13/2017, 1864 obs.）
1	7.407***	1.546*		7.132***		3.530***
2	8.397***	2.572***		20.32***		2.913***
3	6.136***	2.169**		13.44***		2.121**
4	13.11***	2.738***		10.68***		0.937
5	9.615***	3.065***		8.117***		1.450*
6	7.950***	3.026***		6.754***		0.897
7	7.440***	2.930***		6.300***		0.863
8	6.851***	2.992***		6.130***		0.995
Group A: (4/16/2010–6/10/2011, 279 obs.）
1	6.364**	−0.828		0.345		1.362*
2	2.561*	−1.623		1.531		0.667
3	1.745	−0.833		1.093		1.657**
4	2.501**	−0.774		1.589		1.343*
5	2.327**	−0.756		1.551		1.274
Group B (6/13/2011–1/29/2013, 400 obs.)
1	2.019	−1.795		3.235*		0.698
2	1.07	−1.333		1.542		0.68
3	0.68	−2.054		0.995		−0.701
4	0.513	−1.784		0.838		−0.537
5	1.43	−2.09		1.664		−0.971
Group C (1/30/2013–3/11/2015,508 obs.)
1	1.44	0.249		3.596*		2.384***
2	1.684	1.25		4.304**		2.392***
3	1.205	1.211		3.160**		1.985**
4	0.919	1.056		2.441**		1.970**
5	1.344	1.171		2.075*		1.976**
GroupD (3/12/2015–4/28/2016,279 obs.)
1	0.24	1.002		6.818***		1.494*
2	2.186	1.461*		6.243***		1.410*
3	1.338	0.807		5.021***		0.792
4	3.488***	0.549		4.217***		0.847
5	2.060*	0.931		2.878**		0.399
Group E (4/29/2016–12/13/2017,398 obs.)
1	8.828***	1.985**		0.398		−2.135
2	4.497**	0.7		0.357		−2.74
3	3.291**	−0.313		0.466		−1.594
4	3.490***	−0.224		1.186		−1.324
5	2.018*	−0.213		0.985		0.873