Table 7 Open in new tab Tests for...

	Frequency	${LM}_{T}^{HAC}$	p-value	${LM}_{T, Adj}^{HAC}$	p-value	S	T
USD/EUR	Hourly	5.6	0.7832	4.7	0.8575	10	518
	30-minute	51.6	0.0000	45.5	0.0004	19	518
	15-minute	72.9	0.0003	67.5	0.0011	37	518
	5-minute(*)	89.3	0.0000	82.7	0.0000	37	518
	5-minute	168.7	0.0002	168.7	0.0002	109	518
	1-minute(*)	207.6	0.0295	207.6	0.0295	172	518
	1-minute(**)	158.2	0.0000	158.2	0.0000	86	518
USD/JPY	Hourly	18.8	0.0268	16.0	0.0670	10	518
	30-minute	29.8	0.0394	26.3	0.0938	19	518
	15-minute	75.1	0.0001	69.5	0.0007	37	518
	5-minute(*)	106.9	0.0000	98.9	0.0000	37	518
	5-minute	189.4	0.0000	189.4	0.0000	109	518
	1-minute(*)	225.8	0.0031	225.8	0.0031	172	518
	1-minute(**)	194.0	0.0000	193.9	0.0000	86	518
GBP/USD	Hourly	18.0	0.0356	15.3	0.0837	10	518
	30-minute	34.3	0.0115	30.3	0.0350	19	518
	15-minute	44.7	0.1511	41.4	0.2465	37	518
	5-minute(*)	73.1	0.0003	67.6	0.0011	37	518
	5-minute	134.2	0.0446	134.2	0.0446	109	518
	1-minute(*)	203.0	0.0476	203.0	0.0476	172	518
	1-minute(**)	157.4	0.0000	157.4	0.0000	86	518
USD/CAD	Hourly	25.3	0.0027	21.5	0.0106	10	518
	30-minute	62.7	0.0000	55.3	0.0000	19	518
	15-minute	87.3	0.0000	80.9	0.0000	37	518
	5-minute(*)	130.3	0.0000	120.6	0.0000	37	518
	5-minute	188.9	0.0000	188.9	0.0000	109	518
	1-minute(*)	230.7	0.0016	230.7	0.0016	172	518
	1-minute(**)	204.1	0.0000	204.1	0.0000	86	518
AUD/USD	Hourly	15.8	0.0705	13.5	0.1428	10	518
	30-minute	38.7	0.0032	34.1	0.0123	19	518
	15-minute	69.5	0.0007	64.3	0.0026	37	518
	5-minute(*)	98.1	0.0000	90.8	0.0000	37	518
	5-minute	176.4	0.0000	176.4	0.0000	109	518
	1-minute(*)	226.7	0.0028	226.7	0.0028	172	518
	1-minute(**)	188.4	0.0000	188.4	0.0000	86	518
USD/CHF	Hourly	21.1	0.0120	18.0	0.0354	10	518
	30-minute	30.7	0.0313	27.0	0.0781	19	518
	15-minute	77.1	0.0001	71.4	0.0004	37	518
	5-minute(*)	108.9	0.0000	100.8	0.0000	37	518
	5-minute	170.9	0.0001	170.9	0.0001	109	518
	1-minute(*)	226.0	0.0031	226.0	0.0031	172	518
	1-minute(**)	191.8	0.0000	191.8	0.0000	86	518

Frequency

{LM}_{T}^{HAC}

p-value

{LM}_{T, Adj}^{HAC}

p-value

USD/EUR

Hourly

5.6

0.7832

4.7

0.8575

518

30-minute

51.6

0.0000

45.5

0.0004

518

15-minute

72.9

0.0003

67.5

0.0011

518

5-minute(*)

89.3

0.0000

82.7

0.0000

518

5-minute

168.7

0.0002

168.7

0.0002

109

518

1-minute(*)

207.6

0.0295

207.6

0.0295

172

518

1-minute(**)

158.2

0.0000

158.2

0.0000

518

USD/JPY

Hourly

18.8

0.0268

16.0

0.0670

518

30-minute

29.8

0.0394

26.3

0.0938

518

15-minute

75.1

0.0001

69.5

0.0007

518

5-minute(*)

106.9

0.0000

98.9

0.0000

518

5-minute

189.4

0.0000

189.4

0.0000

109

518

1-minute(*)

225.8

0.0031

225.8

0.0031

172

518

1-minute(**)

194.0

0.0000

193.9

0.0000

518

GBP/USD

Hourly

18.0

0.0356

15.3

0.0837

518

30-minute

34.3

0.0115

30.3

0.0350

518

15-minute

44.7

0.1511

41.4

0.2465

518

5-minute(*)

73.1

0.0003

67.6

0.0011

518

5-minute

134.2

0.0446

134.2

0.0446

109

518

1-minute(*)

203.0

0.0476

203.0

0.0476

172

518

1-minute(**)

157.4

0.0000

157.4

0.0000

518

USD/CAD

Hourly

25.3

0.0027

21.5

0.0106

518

30-minute

62.7

0.0000

55.3

0.0000

518

15-minute

87.3

0.0000

80.9

0.0000

518

5-minute(*)

130.3

0.0000

120.6

0.0000

518

5-minute

188.9

0.0000

188.9

0.0000

109

518

1-minute(*)

230.7

0.0016

230.7

0.0016

172

518

1-minute(**)

204.1

0.0000

204.1

0.0000

518

AUD/USD

Hourly

15.8

0.0705

13.5

0.1428

518

30-minute

38.7

0.0032

34.1

0.0123

518

15-minute

69.5

0.0007

64.3

0.0026

518

5-minute(*)

98.1

0.0000

90.8

0.0000

518

5-minute

176.4

0.0000

176.4

0.0000

109

518

1-minute(*)

226.7

0.0028

226.7

0.0028

172

518

1-minute(**)

188.4

0.0000

188.4

0.0000

518

USD/CHF

Hourly

21.1

0.0120

18.0

0.0354

518

30-minute

30.7

0.0313

27.0

0.0781

518

15-minute

77.1

0.0001

71.4

0.0004

518

5-minute(*)

108.9

0.0000

100.8

0.0000

518

5-minute

170.9

0.0001

170.9

0.0001

109

518

1-minute(*)

226.0

0.0031

226.0

0.0031

172

518

1-minute(**)

191.8

0.0000

191.8

0.0000

518

	Frequency	${LM}_{T}^{HAC}$	p-value	${LM}_{T, Adj}^{HAC}$	p-value	S	T
USD/EUR	Hourly	5.6	0.7832	4.7	0.8575	10	518
	30-minute	51.6	0.0000	45.5	0.0004	19	518
	15-minute	72.9	0.0003	67.5	0.0011	37	518
	5-minute(*)	89.3	0.0000	82.7	0.0000	37	518
	5-minute	168.7	0.0002	168.7	0.0002	109	518
	1-minute(*)	207.6	0.0295	207.6	0.0295	172	518
	1-minute(**)	158.2	0.0000	158.2	0.0000	86	518
USD/JPY	Hourly	18.8	0.0268	16.0	0.0670	10	518
	30-minute	29.8	0.0394	26.3	0.0938	19	518
	15-minute	75.1	0.0001	69.5	0.0007	37	518
	5-minute(*)	106.9	0.0000	98.9	0.0000	37	518
	5-minute	189.4	0.0000	189.4	0.0000	109	518
	1-minute(*)	225.8	0.0031	225.8	0.0031	172	518
	1-minute(**)	194.0	0.0000	193.9	0.0000	86	518
GBP/USD	Hourly	18.0	0.0356	15.3	0.0837	10	518
	30-minute	34.3	0.0115	30.3	0.0350	19	518
	15-minute	44.7	0.1511	41.4	0.2465	37	518
	5-minute(*)	73.1	0.0003	67.6	0.0011	37	518
	5-minute	134.2	0.0446	134.2	0.0446	109	518
	1-minute(*)	203.0	0.0476	203.0	0.0476	172	518
	1-minute(**)	157.4	0.0000	157.4	0.0000	86	518
USD/CAD	Hourly	25.3	0.0027	21.5	0.0106	10	518
	30-minute	62.7	0.0000	55.3	0.0000	19	518
	15-minute	87.3	0.0000	80.9	0.0000	37	518
	5-minute(*)	130.3	0.0000	120.6	0.0000	37	518
	5-minute	188.9	0.0000	188.9	0.0000	109	518
	1-minute(*)	230.7	0.0016	230.7	0.0016	172	518
	1-minute(**)	204.1	0.0000	204.1	0.0000	86	518
AUD/USD	Hourly	15.8	0.0705	13.5	0.1428	10	518
	30-minute	38.7	0.0032	34.1	0.0123	19	518
	15-minute	69.5	0.0007	64.3	0.0026	37	518
	5-minute(*)	98.1	0.0000	90.8	0.0000	37	518
	5-minute	176.4	0.0000	176.4	0.0000	109	518
	1-minute(*)	226.7	0.0028	226.7	0.0028	172	518
	1-minute(**)	188.4	0.0000	188.4	0.0000	86	518
USD/CHF	Hourly	21.1	0.0120	18.0	0.0354	10	518
	30-minute	30.7	0.0313	27.0	0.0781	19	518
	15-minute	77.1	0.0001	71.4	0.0004	37	518
	5-minute(*)	108.9	0.0000	100.8	0.0000	37	518
	5-minute	170.9	0.0001	170.9	0.0001	109	518
	1-minute(*)	226.0	0.0031	226.0	0.0031	172	518
	1-minute(**)	191.8	0.0000	191.8	0.0000	86	518

Table 7

Open in new tab

Tests for non-stationary periodicity in the zero-process of intraday exchange rate returns: Main trading hours, that is, from 08:00 CET to 17:00 CET (see Section 3.1)

	Frequency	${LM}_{T}^{HAC}$	p-value	${LM}_{T, Adj}^{HAC}$	p-value	S	T
USD/EUR	Hourly	5.6	0.7832	4.7	0.8575	10	518
	30-minute	51.6	0.0000	45.5	0.0004	19	518
	15-minute	72.9	0.0003	67.5	0.0011	37	518
	5-minute(*)	89.3	0.0000	82.7	0.0000	37	518
	5-minute	168.7	0.0002	168.7	0.0002	109	518
	1-minute(*)	207.6	0.0295	207.6	0.0295	172	518
	1-minute(**)	158.2	0.0000	158.2	0.0000	86	518
USD/JPY	Hourly	18.8	0.0268	16.0	0.0670	10	518
	30-minute	29.8	0.0394	26.3	0.0938	19	518
	15-minute	75.1	0.0001	69.5	0.0007	37	518
	5-minute(*)	106.9	0.0000	98.9	0.0000	37	518
	5-minute	189.4	0.0000	189.4	0.0000	109	518
	1-minute(*)	225.8	0.0031	225.8	0.0031	172	518
	1-minute(**)	194.0	0.0000	193.9	0.0000	86	518
GBP/USD	Hourly	18.0	0.0356	15.3	0.0837	10	518
	30-minute	34.3	0.0115	30.3	0.0350	19	518
	15-minute	44.7	0.1511	41.4	0.2465	37	518
	5-minute(*)	73.1	0.0003	67.6	0.0011	37	518
	5-minute	134.2	0.0446	134.2	0.0446	109	518
	1-minute(*)	203.0	0.0476	203.0	0.0476	172	518
	1-minute(**)	157.4	0.0000	157.4	0.0000	86	518
USD/CAD	Hourly	25.3	0.0027	21.5	0.0106	10	518
	30-minute	62.7	0.0000	55.3	0.0000	19	518
	15-minute	87.3	0.0000	80.9	0.0000	37	518
	5-minute(*)	130.3	0.0000	120.6	0.0000	37	518
	5-minute	188.9	0.0000	188.9	0.0000	109	518
	1-minute(*)	230.7	0.0016	230.7	0.0016	172	518
	1-minute(**)	204.1	0.0000	204.1	0.0000	86	518
AUD/USD	Hourly	15.8	0.0705	13.5	0.1428	10	518
	30-minute	38.7	0.0032	34.1	0.0123	19	518
	15-minute	69.5	0.0007	64.3	0.0026	37	518
	5-minute(*)	98.1	0.0000	90.8	0.0000	37	518
	5-minute	176.4	0.0000	176.4	0.0000	109	518
	1-minute(*)	226.7	0.0028	226.7	0.0028	172	518
	1-minute(**)	188.4	0.0000	188.4	0.0000	86	518
USD/CHF	Hourly	21.1	0.0120	18.0	0.0354	10	518
	30-minute	30.7	0.0313	27.0	0.0781	19	518
	15-minute	77.1	0.0001	71.4	0.0004	37	518
	5-minute(*)	108.9	0.0000	100.8	0.0000	37	518
	5-minute	170.9	0.0001	170.9	0.0001	109	518
	1-minute(*)	226.0	0.0031	226.0	0.0031	172	518
	1-minute(**)	191.8	0.0000	191.8	0.0000	86	518

	Frequency	${LM}_{T}^{HAC}$	p-value	${LM}_{T, Adj}^{HAC}$	p-value	S	T
USD/EUR	Hourly	5.6	0.7832	4.7	0.8575	10	518
	30-minute	51.6	0.0000	45.5	0.0004	19	518
	15-minute	72.9	0.0003	67.5	0.0011	37	518
	5-minute(*)	89.3	0.0000	82.7	0.0000	37	518
	5-minute	168.7	0.0002	168.7	0.0002	109	518
	1-minute(*)	207.6	0.0295	207.6	0.0295	172	518
	1-minute(**)	158.2	0.0000	158.2	0.0000	86	518
USD/JPY	Hourly	18.8	0.0268	16.0	0.0670	10	518
	30-minute	29.8	0.0394	26.3	0.0938	19	518
	15-minute	75.1	0.0001	69.5	0.0007	37	518
	5-minute(*)	106.9	0.0000	98.9	0.0000	37	518
	5-minute	189.4	0.0000	189.4	0.0000	109	518
	1-minute(*)	225.8	0.0031	225.8	0.0031	172	518
	1-minute(**)	194.0	0.0000	193.9	0.0000	86	518
GBP/USD	Hourly	18.0	0.0356	15.3	0.0837	10	518
	30-minute	34.3	0.0115	30.3	0.0350	19	518
	15-minute	44.7	0.1511	41.4	0.2465	37	518
	5-minute(*)	73.1	0.0003	67.6	0.0011	37	518
	5-minute	134.2	0.0446	134.2	0.0446	109	518
	1-minute(*)	203.0	0.0476	203.0	0.0476	172	518
	1-minute(**)	157.4	0.0000	157.4	0.0000	86	518
USD/CAD	Hourly	25.3	0.0027	21.5	0.0106	10	518
	30-minute	62.7	0.0000	55.3	0.0000	19	518
	15-minute	87.3	0.0000	80.9	0.0000	37	518
	5-minute(*)	130.3	0.0000	120.6	0.0000	37	518
	5-minute	188.9	0.0000	188.9	0.0000	109	518
	1-minute(*)	230.7	0.0016	230.7	0.0016	172	518
	1-minute(**)	204.1	0.0000	204.1	0.0000	86	518
AUD/USD	Hourly	15.8	0.0705	13.5	0.1428	10	518
	30-minute	38.7	0.0032	34.1	0.0123	19	518
	15-minute	69.5	0.0007	64.3	0.0026	37	518
	5-minute(*)	98.1	0.0000	90.8	0.0000	37	518
	5-minute	176.4	0.0000	176.4	0.0000	109	518
	1-minute(*)	226.7	0.0028	226.7	0.0028	172	518
	1-minute(**)	188.4	0.0000	188.4	0.0000	86	518
USD/CHF	Hourly	21.1	0.0120	18.0	0.0354	10	518
	30-minute	30.7	0.0313	27.0	0.0781	19	518
	15-minute	77.1	0.0001	71.4	0.0004	37	518
	5-minute(*)	108.9	0.0000	100.8	0.0000	37	518
	5-minute	170.9	0.0001	170.9	0.0001	109	518
	1-minute(*)	226.0	0.0031	226.0	0.0031	172	518
	1-minute(**)	191.8	0.0000	191.8	0.0000	86	518

${LM}_{T}^{HAC}$ and ${LM}_{T, Adj}^{HAC}$ ⁠, the LM statistics in Equations (2.15) and (3.7), respectively, see Section 2. p-value, $P (χ^{2} (S - 1) > x)$ where x is the value on the statistic in question. (*), only every third period included in the vector of seasons representation (after removal of columns to ensure nonsingularity). (**), only every sixth period included in the vector of seasons representation (after removal of columns to ensure nonsingularity). S, number of intraday periods included in test. T, number of days. All computations in R.

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