An Independence Test Based on Symbolic Time Series

Table 7

Summary of the returns of financial time series.

Series	Frequency	𝑇	Mean	Variance	Skewness	Kurtosis	Normality test statistic
Citigroup	Daily	9170	0.0536%	0.0007	1.2025	53.4066	973018.60*
	Weekly	1895	0.2794%	0.0040	3.7869	89.8884	600633.18*
	Monthly	436	1.0121%	0.0119	0.1130	10.5866	1046.54*
Coca Cola	Daily	12926	0.0570%	0.0003	−2.9897	90.2182	4116265.26*
	Weekly	2678	0.2620%	0.0013	−1.7079	26.7327	64150.50*
	Monthly	616	1.0907%	0.0047	−1.3229	13.1023	2799.09*
IBM	Daily	12926	0.0470%	0.0003	0.0294	13.0155	54027.14*
	Weekly	2678	0.2226%	0.0012	0.0999	6.1514	1112.62*
	Monthly	616	0.9686%	0.0049	0.2140	4.8790	95.32*
Caterpillar	Daily	12927	0.0517%	0.0004	−1.2125	37.7400	653214.89*
	Weekly	2678	0.2534%	0.0019	−0.2678	10.9607	7103.35*
	Monthly	616	1.0656%	0.0076	−0.1546	6.0064	234.43*
S&P 500	Daily	15940	0.0335%	0.0001	−0.6504	24.2393	300735.36*
	Weekly	3304	0.1601%	0.0004	−0.3453	7.9369	3421.03*
	Monthly	760	0.6904%	0.0018	−0.4240	4.7084	115.20*
FTSE	Daily	7350	0.0305%	0.0001	−0.2089	10.9092	19211.26*
	Weekly	1518	0.1467%	0.0006	−0.7059	11.0775	4252.88*
	Monthly	349	0.6105%	0.0021	−0.8005	6.1044	177.41*
DAX	Daily	5684	0.0413%	0.0002	0.0486	7.8679	5614.43*
	Weekly	1171	0.1987%	0.0010	−0.3384	7.1055	844.73*
	Monthly	270	0.8400%	0.0038	−0.5284	4.9958	57.38*
BOVESPA	Daily	4955	0.1856%	0.0006	0.9411	16.2854	37171.74*
	Weekly	1045	0.8837%	0.0029	0.5525	6.6351	628.51*
	Monthly	241	4.1075%	0.0198	2.0642	12.5057	1078.50*
NIKKEI	Daily	7213	0.0155%	0.0002	−0.0492	11.0987	19715.15*
	Weekly	1520	0.0650%	0.0008	−0.4998	7.7195	1473.98*
	Monthly	352	0.2828%	0.0037	−0.2933	3.7348	12.97*
SHANGHAI	Daily	5756	0.0844%	0.0007	12.2275	456.4849	49464803.30*
	Weekly	1160	0.4507%	0.0052	10.5452	189.3924	1700702.47*
	Monthly	269	2.1082%	0.0301	5.6259	52.8034	29219.85*
Ex. Canada/USA	Daily	10644	0.0008%	0.0000	−0.0052	15.2475	66525.07*
	Monthly	507	0.0111%	0.0002	0.8962	13.9378	2595.15*
Ex. China/USA	Daily	8133	0.0190%	0.0000	55.8492	3703.1319	4643757374.76*
	Monthly	387	0.3930%	0.0009	13.0613	211.0026	708653.19*
Ex. US dollar/euro	Daily	1680	−0.0540%	0.0006	−37.1096	1471.7500	151391452.12*
	Monthly	171	0.0994%	0.0006	0.0775	2.9535	0.19
Ex. USA/UK	Daily	10644	−0.0022%	0.0000	−0.1398	7.7514	10046.90*
	Monthly	507	−0.0605%	0.0006	−0.1249	4.5997	55.38*
Ex. Japan/USA	Daily	10644	−0.0099%	0.0000	−0.5942	12.1587	37827.76*
	Monthly	507	−0.2199%	0.0007	−0.3986	3.8298	27.97*

Note: own computations. ∗Denotes rejection of the Jarque-Bera null hypothesis of normality at 5% of significance.