Expected lifetime range ratio to find mean reversion: Evidence from Indian stock market

Figures & data

Table 1. ELR ratio for $\mu =0,\left|\mu \right|=1,\left|\mu \right|=2$ for different $k$-days

ELR ratio	k-day	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
	$\left|\mu \right|=\mathbf{2}$	1.00	1.02	1.04	1.05	1.06	1.07	1.08	1.09	1.09	1.10	1.10	1.11	1.11	1.11	1.12	1.12	1.12	1.13	1.13	1.13
	$\mu =\mathbf{0}$	1.00	1.21	1.32	1.39	1.45	1.49	1.52	1.55	1.57	1.59	1.60	1.62	1.63	1.65	1.66	1.67	1.68	1.68	1.69	1.70
	$\left|\mu \right|=\mathbf{1}$	1.00	1.10	1.16	1.20	1.23	1.26	1.27	1.29	1.30	1.32	1.33	1.34	1.35	1.35	1.36	1.37	1.37	1.38	1.38	1.39

Figure 1. ELR ratio in the Gaussian random walk as per Spitzer’s identity.

Table 2. LM statistic and the ELR ratio in the MA (1) model as a function of fixed parameter $\theta$ and N with m = ${\mathbf{10}}^{\mathbf{5}}$ simulations

	LM statistic											ELR ratio
θ	−1	−0.8	−0.6	−0.4	−0.2	0	0.2	0.4	0.6	0.8	1	−1	−0.8	−0.6	−0.4	−0.2	0	0.2	0.4	0.6	0.8	1
N = 10	0.10	0.12	0.21	0.38	0.66	1.00	1.35	1.63	1.80	1.88	1.91	0.89	0.92	1.02	1.18	1.38	1.59	1.76	1.88	1.95	1.98	1.99
N = 20	0.05	0.07	0.16	0.35	0.64	1.01	1.37	1.67	1.85	1.94	1.96	0.75	0.80	0.95	1.18	1.44	1.70	1.92	2.07	2.16	2.20	2.21
N = 30	0.03	0.06	0.15	0.33	0.63	1.00	1.38	1.68	1.86	1.96	1.98	0.67	0.74	0.92	1.18	1.47	1.76	1.99	2.16	2.25	2.30	2.31
N = 40	0.03	0.05	0.14	0.33	0.63	1.00	1.38	1.67	1.86	1.95	1.97	0.61	0.69	0.90	1.17	1.49	1.79	2.04	2.21	2.31	2.36	2.37
N = 50	0.02	0.04	0.14	0.32	0.62	1.00	1.37	1.67	1.86	1.95	1.98	0.57	0.65	0.87	1.16	1.48	1.80	2.06	2.24	2.35	2.40	2.41
N = 60	0.02	0.04	0.13	0.32	0.62	1.00	1.38	1.67	1.86	1.95	1.98	0.53	0.63	0.86	1.16	1.49	1.82	2.09	2.27	2.38	2.43	2.44
N = 70	0.01	0.04	0.13	0.32	0.62	1.00	1.38	1.68	1.87	1.97	1.99	0.50	0.61	0.85	1.15	1.50	1.83	2.10	2.29	2.41	2.46	2.47
N = 80	0.01	0.04	0.13	0.32	0.63	1.01	1.40	1.70	1.89	1.98	2.01	0.48	0.59	0.84	1.16	1.51	1.85	2.13	2.32	2.43	2.49	2.50
N = 90	0.01	0.04	0.13	0.32	0.62	1.00	1.38	1.68	1.87	1.96	1.99	0.46	0.58	0.83	1.15	1.51	1.85	2.14	2.34	2.45	2.51	2.52
N = 100	0.01	0.03	0.13	0.32	0.62	1.00	1.39	1.69	1.88	1.97	2.00	0.44	0.57	0.82	1.15	1.51	1.86	2.15	2.34	2.46	2.52	2.53
N = ∞	0.00	0.02	0.12	0.31	0.62	1.00	1.38	1.69	1.88	1.98	2.00	0.00	0.31	0.69	1.11	1.57	2.00	2.35	2.60	2.74	2.81	2.83

Figure 2. LM statistic and ELR ratio in the MA (1) model for different values of N and fixed parameter $\theta$.

Table 3. LM statistic and the ELR ratio in the MA (1) model as functions of stochastic parameter $\tilde{\theta }$ and $N$ with $m={10}^5$ simulations

	LM statistic						ELR ratio
θ	0	0.2	0.4	0.6	0.8	1	0	0.2	0.4	0.6	0.8	1
N = 10	1.00	1.00	1.00	1.00	1.00	1.00	1.59	1.57	1.53	1.49	1.45	1.44
N = 20	1.01	1.01	1.01	1.01	1.01	1.01	1.70	1.68	1.62	1.55	1.50	1.48
N = 30	1.00	1.00	1.00	1.01	1.01	1.01	1.76	1.73	1.67	1.59	1.52	1.49
N = 40	1.00	1.00	1.00	1.00	1.00	1.00	1.79	1.76	1.69	1.60	1.53	1.49
N = 50	1.00	1.00	1.00	1.00	1.00	1.00	1.80	1.77	1.70	1.61	1.53	1.49
N = 60	1.00	1.00	1.00	1.00	1.00	1.00	1.82	1.79	1.72	1.62	1.53	1.49
N = 70	1.00	1.00	1.00	1.00	1.00	1.00	1.83	1.80	1.72	1.63	1.53	1.49
N = 80	1.01	1.01	1.01	1.01	1.01	1.01	1.85	1.82	1.74	1.64	1.54	1.49
N = 90	1.00	1.00	1.00	1.00	1.00	1.00	1.85	1.82	1.74	1.64	1.54	1.49
N = 100	1.00	1.00	1.00	1.00	1.00	1.00	1.86	1.83	1.75	1.64	1.54	1.49
N = ∞	1.00	1.00	1.00	1.00	1.00	1.00	2.00	1.96	1.86	1.71	1.56	1.41

Figure 3. LM statistic and ELR ratio in the MA (1) model for different values of $N$ and stochastic parameter $\tilde{\theta }$.

Figure 4. Deviation of LM statistic and the ELR ratio from their respective bootmean in the MA (1) model as a function of fixed/stochastic parameter for different values of N, i.e., for N = 10, N = 100 and $\mathit{N}=$∞.

Table 4. Deviation of LM statistic and the ELR ratio from their respective bootmean in the MA (1) model as a function of fixed/stochastic parameter for different values of N

	a) Fixed parameter								b) Stochastic parameter
	N = 10		N = 100		N = ∞			$\tilde{\theta }$	N = 10		N = 100		N = ∞
θ	LM	ELR	LM	ELR	LM	ELR			LM	ELR	LM	ELR	LM	ELR
−1	−0.90	−0.70	−0.99	−1.41	−1.00	−2.00		0	0.00	0.00	0.00	0.00	0.00	0.00
−0.8	−0.88	−0.67	−0.97	−1.29	−0.98	−1.69		0.2	0.00	−0.02	0.00	−0.03	0.00	−0.04
−0.6	−0.80	−0.57	−0.88	−1.04	−0.88	−1.31		0.4	0.00	−0.06	0.00	−0.11	0.00	−0.14
−0.4	−0.62	−0.40	−0.69	−0.71	−0.69	−0.89		0.6	0.00	−0.10	0.00	−0.22	0.00	−0.29
−0.2	−0.35	−0.20	−0.38	−0.35	−0.38	−0.43		0.8	0.00	−0.14	0.00	−0.32	0.00	−0.44
0	0.00	0.00	0.00	0.00	0.00	0.00		1	0.00	−0.15	0.00	−0.37	0.00	−0.59
0.2	0.35	0.17	0.38	0.29	0.38	0.35
0.4	0.62	0.29	0.69	0.49	0.69	0.60
0.6	0.80	0.36	0.88	0.60	0.88	0.74
0.8	0.88	0.39	0.97	0.66	0.98	0.81
1	0.90	0.40	0.99	0.67	1.00	0.83

The bold values represents the fixed/stochastic parameter.

Table 5. Descriptive statistics of the 22 individual BSE SENSEX stocks

S.no	Stock	Mean		Standard Error	Median	Standard Deviation	Kurtosis	Skewness	Range	Minimum	Maximum
1	AXIS	0.0011	0.0	0.0005	0.0000	0.0292	4.1621	0.3360	0.3640	−0.1788	0.1852
2	BHEL	0.0007	0.0	0.0004	0.0003	0.0260	6.0517	−0.1240	0.4231	−0.2272	0.1959
3	CIPLA	0.0006	0.0	0.0003	0.0001	0.0193	5.0529	−0.2418	0.2581	−0.1540	0.1040
4	DRRD	0.0007	0.0	0.0003	0.0007	0.0201	8.1642	−0.2534	0.3617	−0.2008	0.1610
5	GAIL	0.0006	0.0	0.0004	0.0001	0.0234	13.8683	−0.0735	0.5234	−0.2395	0.2839
6	HDFC	0.0008	0.0	0.0004	0.0002	0.0229	4.8453	0.3963	0.3075	−0.1173	0.1903
7	HDFCB	0.0009	0.0	0.0003	0.0003	0.0196	4.0726	0.2424	0.2713	−0.1195	0.1518
8	HMCL	0.0007	0.0	0.0004	0.0000	0.0220	3.6912	0.4363	0.2577	−0.0922	0.1654
9	HNDL	0.0000	0.0	0.0004	0.0004	0.0267	3.5992	−0.1522	0.3638	−0.1963	0.1676
10	HUVR	0.0004	0.0	0.0003	−0.0001	0.0187	5.1130	0.2374	0.3221	−0.1627	0.1594
11	ICICI	0.0006	0.0	0.0005	0.0000	0.0276	5.1616	0.0193	0.4267	−0.2196	0.2071
12	INFO	0.0005	0.0	0.0004	0.0004	0.0241	15.5132	−1.0062	0.4679	−0.3117	0.1562
13	ITC	0.0006	0.0	0.0003	0.0004	0.0187	3.3187	−0.0378	0.2159	−0.1106	0.1052
14	MM	0.0009	0.0	0.0004	0.0007	0.0247	4.6540	0.0275	0.3826	−0.1748	0.2078
15	ONGC	0.0007	0.0	0.0004	0.0000	0.0229	6.0129	0.2263	0.3450	−0.1626	0.1823
16	RIL	0.0006	0.0	0.0004	0.0006	0.0228	29.0544	1.2279	0.5722	−0.1796	0.3926
17	SBIN	0.0007	0.0	0.0004	0.0008	0.0234	3.9049	−0.0647	0.3440	−0.1597	0.1843
18	SUNP	0.0012	0.0	0.0003	0.0005	0.0203	4.0477	−0.1025	0.2592	−0.1619	0.0973
19	TATA	0.0003	0.0	0.0005	0.0004	0.0281	3.4028	−0.2836	0.3167	−0.1641	0.1526
20	TTMT	0.0008	0.0	0.0005	0.0006	0.0275	3.3502	−0.1111	0.3560	−0.1810	0.1750
21	VEDL	0.0011	0.0	0.0006	0.0000	0.0334	3.4067	0.2880	0.4392	−0.2508	0.1884
22	WPRP	0.0003	0.0	0.0004	0.0004	0.0268	8.3386	−0.2622	0.4045	−0.2029	0.2016

Total daily observations of each stock is 3671 from January 01, 2001 to September 24, 2015.

Table 6. Empirical analysis of LM statistic and ELR ratio on cross-sectional average of BSE SENSEX using bootstrap method in the overall sample period (January 2001 to September 2015)

a) LM statistic
k-day	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
Actual LM	1	1.04	1.03	1.01	0.99	0.98	0.96	0.95	0.95	0.94	0.94	0.94	0.94	0.94	0.94	0.95	0.95	0.95	0.95	0.95
Boot LM (Boot s.e)	1	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	0.99	0.99	1.00
	(0.00)	(0.02)	(0.02)	(0.03)	(0.04)	(0.04)	(0.04)	(0.05)	(0.05)	(0.06)	(0.06)	(0.06)	(0.06)	(0.07)	(0.07)	(0.07)	(0.07)	(0.08)	(0.08)	(0.08)
t-stat	N/A	2.45*	1.16	0.26	−0.14	−0.49	−0.80	−0.94	−0.97	−0.96	−0.91	−0.87	−0.85	−0.82	−0.76	−0.69	−0.66	−0.61	−0.56	−0.52
Degrees of freedom	N/A	19	18	17	16	15	14	13	12	11	10	9	8	7	6	5	4	3	2	1
GMM stat (p-value)	N/A	17.89 (0.53)	11.44 (0.87)	10.12 (0.90)	10.08 (0.86)	10.04 (0.82)	9.73 (0.78)	9.18 (0.76)	7.98 (0.79)	6.78 (0.82)	5.67 (0.84)	4.83 (0.85)	4.04 (0.85)	3.46 (0.84)	2.63 (0.85)	2.03 (0.85)	1.45 (0.83)	0.97 (0.81)	0.61 (0.74)	0.26 (0.61)
b) ELR ratio
k-day	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
Actual LM	1	1.04	1.05	1.06	1.07	1.08	1.08	1.08	1.08	1.09	1.09	1.09	1.09	1.09	1.09	1.09	1.09	1.10	1.10	1.10
Boot LM (Boot s.e)	1	1.08	1.11	1.13	1.15	1.16	1.16	1.17	1.18	1.18	1.18	1.19	1.19	1.19	1.19	1.19	1.19	1.20	1.20	1.20
	(0.00)	(0.00)	(0.01)	(0.01)	(0.01)	(0.01)	(0.01)	(0.01)	(0.01)	(0.02)	(0.02)	(0.02)	(0.02)	(0.02)	(0.02)	(0.02)	(0.02)	(0.02)	(0.02)	(0.02)
t-stat	N/A	−10.16**	−9.15**	−8.35**	−7.92**	−7.39**	−6.91**	−6.62**	−6.30**	−6.09**	−5.86**	−5.69**	−5.47**	−5.31**	−5.12**	−4.97**	−4.88**	−4.71**	−4.57**	−4.48**
Degrees of freedom	N/A	19	18	17	16	15	14	13	12	11	10	9	8	7	6	5	4	3	2	1
GMM stat (p-value)	N/A	804.13** (0.00**)	728.98** (0.00**)	626.14** (0.00**)	569.71** (0.00**)	489.16** (0.00**)	441.63** (0.00**)	397.86** (0.00**)	352.32** (0.00**)	314.82** (0.00**)	273.32** (0.00**)	238.43** (0.00**)	203.64** (0.00**)	167.43** (0.00**)	138.93** (0.00**)	112.54** (0.00**)	91.54** (0.00**)	61.41** (0.00**)	38.25** (0.00**)	17.50** (0.00**)

$\ast \ast ,\ast$ represent level of significance at $1\mathrm{\%},5\mathrm{\%}$, respectively.

Table 7. Empirical analysis of LM statistic and ELR ratio on cross-sectional average of BSE SENSEX using bootstrap method in the pre-crisis sample period

	Pre-crisis period (January 2001 to December 2007)									a) LM statistic
k-day	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
Actual LM	1	1.03	1.00	0.98	0.97	0.96	0.95	0.94	0.93	0.93	0.93	0.93	0.93	0.93	0.93	0.93	0.93	0.93	0.93	0.93
Boot LM (Boot s.e)	1	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	0.99	0.99	0.99	0.99	0.99	0.99	0.99	0.99	0.99	0.99	0.99
	(0.00)	(0.02)	(0.04)	(0.04)	(0.05)	(0.06)	(0.07)	(0.07)	(0.08)	(0.08)	(0.08)	(0.09)	(0.09)	(0.10)	(0.10)	(0.10)	(0.11)	(0.11)	(0.11)	(0.12)
t-stat	N/A	1.44	0.17	−0.40	−0.51	−0.58	−0.69	−0.80	−0.87	−0.83	−0.76	−0.71	−0.71	−0.67	−0.62	−0.59	−0.57	−0.55	−0.52	−0.50
Degrees of freedom	N/A	19	18	17	16	15	14	13	12	11	10	9	8	7	6	5	4	3	2	1
GMM		10.40	7.57	7.50	7.27	7.07	6.80	6.27	5.58	4.75	4.16	3.41	2.84	2.28	1.82	1.39	1.13	0.81	0.51	0.25
		(0.94)	(0.98)	(0.98)	(0.97)	(0.96)	(0.94)	(0.94)	(0.94)	(0.94)	(0.94)	(0.95)	(0.94)	(0.94)	(0.94)	(0.93)	(0.89)	(0.85)	(0.77)	(0.61)
	Pre-crisis period (January 2001 to December 2007)									b) ELR ratio
k-day	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
Actual ELR	1	1.04	1.06	1.07	1.07	1.08	1.08	1.09	1.09	1.09	1.09	1.09	1.10	1.10	1.10	1.10	1.10	1.10	1.10	1.10
Boot ELR (Boot s.e)	1 (0.00)	1.08 (0.01)	1.12 (0.01)	1.14 (0.01)	1.16 (0.01)	1.17 (0.02)	1.18 (0.02)	1.18 (0.02)	1.19 (0.02)	1.19 (0.02)	1.20 (0.02)	1.21 (0.02)	1.21 (0.03)	1.21 (0.03)	1.21 (0.03)	1.21 (0.03)	1.21 (0.03)	1.21 (0.03)	1.21 (0.03)	1.21 (0.03
t-stat	N/A	−7.50**	−6.95**	−6.38**	−6.01**	−5.72**	−5.40**	−5.20**	−4.98**	−4.78**	−4.63**	−4.44**	−4.24**	−4.09**	−4.00**	−3.87**	−3.73**	−3.60**	−3.52**	−3.45**
Degrees of freedom	N/A	19	18	17	16	15	14	13	12	11	10	9	8	7	6	5	4	3	2	1
GMM	N/A	487.77** (0.00**)	432.78** (0.00**)	387.04** (0.00**)	336.34** (0.00**)	296.7** (0.00**)	264.59** (0.00**)	229.99** (0.00**)	202.42** (0.00**)	175.24** (0.00**)	152.56** (0.00**)	132.24** (0.00**)	114.47** (0.00**)	98.82** (0.00**)	83.29** (0.00**)	64.72** (0.00**)	48.96** (0.00**)	35.15** (0.00**)	23.81** (0.00**)	11.61** (0.00**)

**,* represent the level of significance at 1% and 5% respectively.

Table 8. Empirical analysis of LM statistic and ELR ratio on cross-sectional average of BSE SENSEX using bootstrap method in the post-crisis sample period

k-day	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
Actual LM	1.00	1.04	1.04	1.03	1.01	0.98	0.96	0.95	0.95	0.94	0.94	0.94	0.94	0.94	0.94	0.94	0.94	0.94	0.94	0.94
Boot LM (Boot s.e)	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	0.99	0.99	0.99	0.99	0.99	0.99	0.99	0.99	0.99	0.99
	(0.00)	(0.02)	(0.03)	(0.04)	(0.05)	(0.06)	(0.06)	(0.07)	(0.07)	(0.08)	(0.08)	(0.09)	(0.09)	(0.09)	(0.10)	(0.10)	(0.10)	(0.11)	(0.11)	(0.11)
t-stat	N/A	1.87	1.30	0.64	0.17	−0.24	−0.57	−0.66	−0.65	−0.66	−0.65	−0.64	−0.60	−0.60	−0.57	−0.57	−0.53	−0.50	−0.46	−0.43
Degrees of freedom	N/A	19	18	17	16	15	14	13	12	11	10	9	8	7	6	5	4	3	2	1
GMM		10.64	7.20	5.39	4.94	4.91	4.83	4.39	3.96	3.57	3.13	2.79	2.34	2.07	1.59	1.26	0.90	0.63	0.39	0.19
	N/A	(0.94)	(0.99)	(0.99)	(0.99)	(0.99)	(0.99)	(0.99)	(0.98)	(0.98)	(0.98)	(0.97)	(0.97)	(0.96)	(0.95)	(0.94)	(0.92)	(0.89)	(0.82)	(0.66)
k-day	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
Actual ELR	1.00	1.03	1.05	1.06	1.07	1.07	1.07	1.08	1.08	1.08	1.08	1.08	1.08	1.09	1.09	1.09	1.09	1.09	1.09	1.09
Boot ELR (Boot s.e)	1.00	1.07	1.09	1.11	1.12	1.13	1.13	1.14	1.14	1.14	1.15	1.15	1.15	1.15	1.15	1.15	1.15	1.15	1.15	1.15
	(0.00)	(0.01)	(0.01)	(0.01)	(0.01)	(0.01)	(0.02)	(0.02)	(0.02)	(0.02)	(0.02)	(0.02)	(0.02)	(0.03)	(0.03)	(0.03)	(0.03)	(0.03)	(0.03)	(0.03)
t-stat	N/A	−5.91**	−5.06**	−4.50**	−4.12**	−3.82**	−3.54**	−3.33**	−3.16**	−3.04**	−2.89**	−2.79**	−2.67**	−2.56**	−2.50**	−2.42**	−2.38**	−2.32**	−2.29**	−2.27**
Degrees of freedom	N/A	19	18	17	16	15	14	13	12	11	10	9	8	7	6	5	4	3	2	1
GMM		246.22**	197.50**	168.50**	142.50**	120.62**	107.79**	96.21**	82.09**	68.72**	60.47**	52.93**	43.44**	37.73**	31.10**	26.14**	20.37**	15.10**	10.15**	5.04*
		(0.00**)	(0.00**)	(0.00**)	(0.00**)	(0.00**)	(0.00**)	(0.00**)	(0.00**)	(0.00**)	(0.00**)	(0.00**)	(0.00**)	(0.00**)	(0.00**)	(0.00**)	(0.00**)	(0.00**)	(0.01**)	(0.02*)

**,* represent the level of significance at 1% and 5% respectively.

Table 9. Scatterplot of the deviation of CS average of LM statistic and ELR ratio from its bootmean in the pre-crisis period and post-crisis period for k-day = 10 and k-day = 20 {fig a, fig b, fig c, fig d}. Scatterplot of the deviation of actual ELR and actual theta from the CS average of BSE SENSEX in the pre-crisis and post-crisis period {fig e, fig f}.

Figure 5. LM statistic and ELR ratio on cross-sectional (CS) average of BSE SENSEX with $\mathbf{95}\mathbf{\%}$ confidence interval in the overall sample period (January 2001 to September 2015).

Figure 6. Scatterplot of LM atatistic and ELR ratio from its bootmean for k-day = 10 and k-day = 20 in the combined sample.

Figure 7. Scatterplot of ELR ratio and $\theta$ for k-day = 10 and k-day = 20 in the combined sample.

Figure 8. LM statistic and ELR ratio on CS average of BSE SENSEX with $95\mathbf{\%}$ confidence interval in the pre-crisis period (January 2001 to December 2007).

Figure 9. LM statistic and ELR ratio on CS average of BSE SENSEX with $\mathbf{95}\mathbf{\%}$ confidence interval in the post-crisis period (January 2008 to September 2015).