An Independence Test Based on Symbolic Time Series
Table 5
Power test for a time series size of 500.
Generator process
SRS(2,3)
SRS(2,4)
SRS(3,3)
SRS(4,3)
SRS(5,2)
BDS (a)
BDS (b)
BDS (s)
Runs test
Normal
2.32
3.94
1.98
3.22
0.22
6.32
3.30
5.12
4.72
CHI-2()
2.18
4.28
2.06
3.00
0.30
5.72
3.24
4.70
4.90
-Student()
1.90
3.62
1.84
2.78
0.40
5.84
3.26
4.78
4.12
Truncated norm.
1.64
3.44
1.32
2.84
0.16
8.28
4.46
6.84
4.50
Beta(/2,1/)
1.82
3.64
1.94
3.38
0.32
8.16
4.02
6.84
4.18
Uniform(0,)
1.42
3.30
1.64
2.94
0.32
7.58
4.08
6.50
3.80
AR()
0.56
1.78
1.48
2.30
0.14
6.02
3.14
4.88
1.44
MA()
0.34
1.46
0.92
2.38
0.04
5.84
3.50
4.60
1.08
Logistic
17.84
31.28
100.00
100.00
100
84.90
65.48
84.02
13.84
Henon
100.00
100.00
100.00
100.00
100
100.00
100.00
100.00
99.92
Anosov
3.18
6.22
6.30
93.46
0.82
11.04
6.02
9.16
6.04
Lorenz
100.00
100.00
100.00
100.00
100
100.00
100.00
100.00
100.00
TAR
70.60
72.10
20.32
86.56
97.7
23.08
13.94
21.54
88.74
NLSIGN
11.04
14.56
8.38
17.48
3.84
7.18
4.00
6.18
25.24
Bilinear
99.72
99.94
100.00
100.00
66.48
100.00
100.00
100.00
13.22
NLAR
0.46
1.82
6.24
6.30
1.34
7.16
4.34
5.86
1.34
NLMA
13.68
42.06
95.60
97.22
1.32
8.32
7.56
7.02
14.90
BLMA
61.44
85.64
99.40
99.90
98.98
100.00
100.00
100.00
77.34
Modular
99.98
100.00
100.00
100.00
99.8
94.42
96.52
93.88
100.00
GARCH
14.44
17.06
97.42
99.60
99.2
100.00
100.00
100.00
32.58
Note: 5,000 Monte Carlo simulations were conducted applying the pseudorandom numbers from MatLab R2010a. SRS(a,w) indicates the SRS test for a symbolization of a symbols and a length of w. BDS (a), (b), and (s) applied the Kanzler [46] optimal parameters, the Liu et al. [47] best parameters, and the Kanzler [46] simulated critical values for small sample. Bold values refer to reasonable percentage of rejections considering the process and the critical values.
