An Independence Test Based on Symbolic Time Series
Table 9
SRS, BDS, and runs tests on the residuals of a GARCH(1, 1) model applied to assets, stock indices, and exchange rates.
Generator process
(a)
SRS(2,4)
SRS(3,3)
SRS(3,4)
SRS(4,3) (b)
BDS (c)
runs test (d)
Citigroup (daily)
9170
1.98
99.98*
210.67*
426.33*
0.00*
No rejection
Citigroup (weekly)
1895
19.83
31.22
93.44
279.40*
0.43
No rejection
Citigroup (monthly)
436
7.06
20.15
76.76
51.16
0.18
No rejection
Coca Cola (daily)
12926
11512.60*
694785.94*
23577565.97*
23719.11*
0.00*
Rejection
Coca Cola (weekly)
2678
797.78*
82.32*
168.36*
715.33*
0.39
No rejection
Coca Cola (monthly)
616
8.96
29.07
88.52
48.51
0.26
No rejection
IBM (daily)
12926
207.11*
27.22
88.28
230.90*
0.02*
No rejection
IBM (weekly)
2678
25.24*
28.80
107.19*
62.37*
0.96
No rejection
IBM (monthly)
616
4.37
23.80
91.42
66.03
0.93
No rejection
Caterpillar (daily)
12927
829.84*
205.44*
384.42*
1350.71*
0.00*
Rejection
Caterpillar (weekly)
2678
86.68*
33.22
85.21
78.68
0.81
No rejection
Caterpillar (monthly)
616
13.82
19.49
65.52
56.64
0.74
No rejection
S&P 500 (daily)
15940
224.88*
402.18*
695.57*
595.81*
0.27
Rejection
S&P 500 (weekly)
3304
9.34
85.07*
187.74*
178.62*
0.80
No rejection
S&P 500 (monthly)
760
11.36
39.53*
101.26*
72.82
0.47
No rejection
FTSE (daily)
7350
6.85
60.06*
152.68*
163.52*
0.00*
No rejection
FTSE (weekly)
1518
10.09
52.96*
123.77*
85.27*
0.18
No rejection
FTSE (monthly)
349
16.73
10.85
54732.62 (N)
62.64
0.47
No rejection
DAX (daily)
5684
11.08
101.52*
197.55*
186.67*
0.00*
No rejection
DAX (weekly)
1171
10.63
59.21*
131.33*
69.67
0.18
No rejection
DAX (monthly)
270
4.16
24.57
42260.90 (N)
50.57
0.91
No rejection
BOVESPA (daily)
4955
15.99
134.38*
279.73*
212.50*
0.10
No rejection
BOVESPA (weekly)
1045
17.62
36.40*
111.90*
85.99*
0.06
No rejection
BOVESPA (monthly)
241
19.88
20.27
75278.34 (N)
58.51
0.42
No rejection
NIKKEI (daily)
7213
11.96
106.49*
245.12*
160.36*
0.00*
No rejection
NIKKEI (weekly)
1520
8.84
41.13*
127.62*
71.46
0.71
No rejection
NIKKEI (monthly)
352
9.46
20.13
27638.60 (N)
58.50
0.59
No rejection
SHANGHAI (daily)
5756
194.96*
206.46*
582.42*
506.30*
0.18
Rejection
SHANGHAI (weekly)
1160
176.95*
62.92*
174.07*
149.97*
0.01*
Rejection
SHANGHAI (monthly)
269
18.03
33.44
84149.32 (N)
16620.42 (N)
0.37
No rejection
Ex. Canada/USA (daily)
10644
40.12*
62.44*
140.98*
109.43*
0.42
Rejection
Ex. Canada/USA (monthly)
507
24.76*
45.10*
39926.89*
74.55
0.03*
Rejection
Ex. China/USA (daily)
8133
14387.35*
3913576.79*
41867540.66*
8623222.42*
0.00*
Rejection
Ex. China/USA (monthly)
387
809.75*
826.35*
366368.25 (N)
503663.60 (N)
0.00*
Rejection
Ex. US dollar/euro (daily)
1680
9.22
24.74
77.71
82.80*
0.00*
No rejection
Ex. US dollar/euro (monthly)
171
22.10
4291.65 (N)
252306.64(N)
104887.07*
0.80
Rejection
Ex. USA/UK (daily)
10644
3.56
344.80*
955.78*
216.87*
0.07
No rejection
Ex. USA/UK (monthly)
507
66.98*
104.24*
119642.14 (N)
31433.72 (N)
0.03*
Rejection
Ex. Japan/USA (daily)
10644
74.14*
689.43*
1832.69*
509.97*
0.29
No rejection
Ex. Japan/USA (monthly)
507
87.75*
86.49*
79840.61 (N)
137.41*
0.05*
Rejection
Note: own calculations. Indicate rejection of the independence null hypothesis at 5% of significance. (a) Number of observations. (b) SRS test applying selected number of symbols and length according to the power of size experiments. (c) value of the BDS test. (d) Rejection or no rejection of randomness null hypothesis. (N) Not enough sample size for the SRS test.
