4.1. Characterization of the intraday returns at the various frequencies

Summary statistics for the foreign exchange market are provided in Table la

for all seventeen possible intraday returns with a 24-hour periodicity. The returns

are continuously compounded i.e. the nth return on day t for the series at

(k. 5)-minute intervals is defined by Rtk, -- 52i=~n_ l)k+ l.nkRt,i, t = 1, 2 . . . . . 260, n = 1, 2 . . . . . K where K- 288/k refers to the number of returns per day. Note

that while the 5-minute return series consists of 74,880 observations, the hourly

series contains only 6,240 observations and the 1/2-day return series has a mere

520 observations. These differences should be kept in mind when interpreting the

evidence.

The standard deviations in Table l a grow at a rate almost proportional to the

square root of the sampling frequency. This is consistent with the 5-minute returns

being approximately uncorrelated, although there is a small, but highly significant,

negative first order autocorrelation coefficients at the higher frequencies. As

mentioned, the weak negative correlation may be the result of spread positioning

by dealers causing mean reversion in the quote midpoints; an effect similar to a

bid-ask bounce in transactions data 24. In line with this explanation, the pj

coefficients generally turn insignificant at the 40-minute and lower frequencies.

Further corroborating evidence along these lines is provided by the variance ratio

statistics,

vR = x. VarT(R ,n)

Varr (Eft= iRtk,n ) ' (6)

where Va r r (R~, ) and Varr(~2,=l, k R,k, n) denote the sample variances for the

intraday and daily returns, respectively. Expanding the daily variance estimate in

the denominator demonstrates that a value of the VR-statistic below unity will

result from positive autocorrelation between adjacent return components, while a

statistic above one is indicative of predominantly negatively correlated intraday

returns zs. Finally, it is worth noting from Table la, that the kurtosis of the DM-$

returns increases almost monotonically with the sampling frequency.

The first order autocorrelations of the absolute returns, pA are, not surprisingly,

all highly significant for the shorter intervals. However, beyond the 2-hour

sampling frequency the autocorrelations drop off very sharply and in fact turn

negative at the 8 and 12 hourly frequencies (k = 96, 144). This is, of course,

consistent with the negative region of the 5-minute absolute return correlogram in

Fig. 4a. The VRA-statistic reported in the final column of Table la is calculated by

replacing Rtk,. with ]R~,.] in the definition of VR in Eq. (6) 26 The statistic starts

out at 0.05 for the 5-minute returns and rises almost monotonically to 0.69 for the

k T/k Mean St. Dev. Skew. Kurtosis Pl Q(10) VR pA QA(10) VR a

(a) Summary statistics for intraday DM-$ exchange rate

l 74,880 0.018 0.047 0.368 21.5 -0.040 281 1.194 0.309 36,680 0.054

2 37,440 0.035 0.066 0.363 16.6 -0.070 263 1.162 0.313 17,563 0.071

3 24,960 0.053 0.079 0.200 13.6 -0.089 220 1.118 0.307 10,710 0.086

4 18,720 0.070 0.089 0.276 14.0 -0.082 154 1.084 0.287 6,296 0.099

6 12,480 0.105 0.107 0.534 12.6 -0.043 36.5 1.023 0.268 2,757 0.127

8 9,360 0.140 0.121 0.135 9.11 -0.023 22.0 0.994 0.272 1,736 0.149

9 8,320 0.158 0.126 0.345 10.1 0.002 18.9 0.948 0.251 1,212 0.161

12 6,240 0.210 0.148 0.326 11.0 -0.001 12.7 0.978 0.229 609 0.193

16 4,680 0.280 0.170 0.318 8.77 0.032 13.1 0.968 0.246 425 0.235

18 4,160 0.315 0.178 0.489 10.6 0.058 21.9 0.947 0.193 219 0.260

24 3,120 0.420 0.212 0.166 8.94 0.011 16.7 1.012 0.164 159 0.311

32 2,340 0.560 0.246 0.326 9.15 0.018 43.3 1.018 0.171 238 0.373

36 2,080 0.630 0.253 0.329 7.89 0.047 37.7 0.954 0.097 109 0.416

48 1,560 0.840 0.300 0.400 5.92 0.002 30.5 1.007 0.075 66.6 0.487

72 1,040 1.261 0.373 0.319 6.59 -0.019 20.6 1.042 0.007 67.60.679

96 780 1.681 0.423 0.389 5.19 -0.022 18.2 1.004 -0.025 53.20.653

144 520 2.521 0.520 0.192 4.31 -(l.021 12.2 1.012 -0.033 28.2 0.692

(b) Summary statistics for intraday S&P 500 returns

1 79,280 0.064 0.104 -0.597 29.3 0.009 87.6 0.774 0.292 32,425 0.099

2 39,640 0.128 0.150 -1.212 30.9 -0.009 81.5 0.801 0.285 12,641 0.128

4 19,820 0.255 0.212 -l.609 33.2 0.014 53.4 0.795 0.232 3,374 0.179

5 15,856 0.319 0.234 -1.755 33.3 0.032 60.6 0.780 0.243 2,323 0.205

8 9,910 0.511 0.299 -1.478 22.2 0.047 44.6 0.793 0.207 1,295 0.261

10 7,928 0.638 0.339 -1.417 21.2 0.039 41.7 0.819 0.211 973 0.300

16 4,955 1.021 0.437 -1.803 26.6 0.040 35.2 0.849 0.135 437 0.405

20 3,964 1.277 0.499 -1.869 27.4 0.016 24.4 0.884 0.114 268 0.463

40 1,982 2.553 0.728 -1.541 16.2 0.026 22.6 0.942 0.148 175 0.673

12 hourly returns. The results for the multiple day returns reported in Andersen

and Bollerslev (1994) continue this near monotone ascent, reaching 1.94 for the

biweekly sampling interval. The smooth increase suggests that a common component

accounts for a substantial part of the positive higher order dependence in all

of the return series. The corresponding p statistics of 0.123 and 0.118 for the

weekly and biweekly sampling frequencies also testify to the importance of the

interday heteroskedasticity 27

The summary statistics for the S&P 500 index futures returns in Table lb

largely parallel those for the DM-$ returns. However, in contrast to the results for

the exchange rates, the first order autocorrelations and the VR-statistics in Table

lb all indicate a slight positive intraday dependence. Moreover, the equity returns

are negatively skewed and display very significant excess kurtosis 28. Finally, the

intraday return periodicity, here depicted in Fig. 2b, again have a strong effect on

the correlations for the absolute intraday returns, although the decay in the pA

coefficients for the lower frequencies is less pronounced than for the exchange

rates.

Summary statistics for the foreign exchange market are provided in Table la

for all seventeen possible intraday returns with a 24-hour periodicity. The returns

are continuously compounded i.e. the nth return on day t for the series at

(k. 5)-minute intervals is defined by Rtk, -- 52i=~n_ l)k+ l.nkRt,i, t = 1, 2 . . . . . 260, n = 1, 2 . . . . . K where K- 288/k refers to the number of returns per day. Note

that while the 5-minute return series consists of 74,880 observations, the hourly

series contains only 6,240 observations and the 1/2-day return series has a mere

520 observations. These differences should be kept in mind when interpreting the

evidence.

The standard deviations in Table l a grow at a rate almost proportional to the

square root of the sampling frequency. This is consistent with the 5-minute returns

being approximately uncorrelated, although there is a small, but highly significant,

negative first order autocorrelation coefficients at the higher frequencies. As

mentioned, the weak negative correlation may be the result of spread positioning

by dealers causing mean reversion in the quote midpoints; an effect similar to a

bid-ask bounce in transactions data 24. In line with this explanation, the pj

coefficients generally turn insignificant at the 40-minute and lower frequencies.

Further corroborating evidence along these lines is provided by the variance ratio

statistics,

vR = x. VarT(R ,n)

Varr (Eft= iRtk,n ) ' (6)

where Va r r (R~, ) and Varr(~2,=l, k R,k, n) denote the sample variances for the

intraday and daily returns, respectively. Expanding the daily variance estimate in

the denominator demonstrates that a value of the VR-statistic below unity will

result from positive autocorrelation between adjacent return components, while a

statistic above one is indicative of predominantly negatively correlated intraday

returns zs. Finally, it is worth noting from Table la, that the kurtosis of the DM-$

returns increases almost monotonically with the sampling frequency.

The first order autocorrelations of the absolute returns, pA are, not surprisingly,

all highly significant for the shorter intervals. However, beyond the 2-hour

sampling frequency the autocorrelations drop off very sharply and in fact turn

negative at the 8 and 12 hourly frequencies (k = 96, 144). This is, of course,

consistent with the negative region of the 5-minute absolute return correlogram in

Fig. 4a. The VRA-statistic reported in the final column of Table la is calculated by

replacing Rtk,. with ]R~,.] in the definition of VR in Eq. (6) 26 The statistic starts

out at 0.05 for the 5-minute returns and rises almost monotonically to 0.69 for the

k T/k Mean St. Dev. Skew. Kurtosis Pl Q(10) VR pA QA(10) VR a

(a) Summary statistics for intraday DM-$ exchange rate

l 74,880 0.018 0.047 0.368 21.5 -0.040 281 1.194 0.309 36,680 0.054

2 37,440 0.035 0.066 0.363 16.6 -0.070 263 1.162 0.313 17,563 0.071

3 24,960 0.053 0.079 0.200 13.6 -0.089 220 1.118 0.307 10,710 0.086

4 18,720 0.070 0.089 0.276 14.0 -0.082 154 1.084 0.287 6,296 0.099

6 12,480 0.105 0.107 0.534 12.6 -0.043 36.5 1.023 0.268 2,757 0.127

8 9,360 0.140 0.121 0.135 9.11 -0.023 22.0 0.994 0.272 1,736 0.149

9 8,320 0.158 0.126 0.345 10.1 0.002 18.9 0.948 0.251 1,212 0.161

12 6,240 0.210 0.148 0.326 11.0 -0.001 12.7 0.978 0.229 609 0.193

16 4,680 0.280 0.170 0.318 8.77 0.032 13.1 0.968 0.246 425 0.235

18 4,160 0.315 0.178 0.489 10.6 0.058 21.9 0.947 0.193 219 0.260

24 3,120 0.420 0.212 0.166 8.94 0.011 16.7 1.012 0.164 159 0.311

32 2,340 0.560 0.246 0.326 9.15 0.018 43.3 1.018 0.171 238 0.373

36 2,080 0.630 0.253 0.329 7.89 0.047 37.7 0.954 0.097 109 0.416

48 1,560 0.840 0.300 0.400 5.92 0.002 30.5 1.007 0.075 66.6 0.487

72 1,040 1.261 0.373 0.319 6.59 -0.019 20.6 1.042 0.007 67.60.679

96 780 1.681 0.423 0.389 5.19 -0.022 18.2 1.004 -0.025 53.20.653

144 520 2.521 0.520 0.192 4.31 -(l.021 12.2 1.012 -0.033 28.2 0.692

(b) Summary statistics for intraday S&P 500 returns

1 79,280 0.064 0.104 -0.597 29.3 0.009 87.6 0.774 0.292 32,425 0.099

2 39,640 0.128 0.150 -1.212 30.9 -0.009 81.5 0.801 0.285 12,641 0.128

4 19,820 0.255 0.212 -l.609 33.2 0.014 53.4 0.795 0.232 3,374 0.179

5 15,856 0.319 0.234 -1.755 33.3 0.032 60.6 0.780 0.243 2,323 0.205

8 9,910 0.511 0.299 -1.478 22.2 0.047 44.6 0.793 0.207 1,295 0.261

10 7,928 0.638 0.339 -1.417 21.2 0.039 41.7 0.819 0.211 973 0.300

16 4,955 1.021 0.437 -1.803 26.6 0.040 35.2 0.849 0.135 437 0.405

20 3,964 1.277 0.499 -1.869 27.4 0.016 24.4 0.884 0.114 268 0.463

40 1,982 2.553 0.728 -1.541 16.2 0.026 22.6 0.942 0.148 175 0.673

12 hourly returns. The results for the multiple day returns reported in Andersen

and Bollerslev (1994) continue this near monotone ascent, reaching 1.94 for the

biweekly sampling interval. The smooth increase suggests that a common component

accounts for a substantial part of the positive higher order dependence in all

of the return series. The corresponding p statistics of 0.123 and 0.118 for the

weekly and biweekly sampling frequencies also testify to the importance of the

interday heteroskedasticity 27

The summary statistics for the S&P 500 index futures returns in Table lb

largely parallel those for the DM-$ returns. However, in contrast to the results for

the exchange rates, the first order autocorrelations and the VR-statistics in Table

lb all indicate a slight positive intraday dependence. Moreover, the equity returns

are negatively skewed and display very significant excess kurtosis 28. Finally, the

intraday return periodicity, here depicted in Fig. 2b, again have a strong effect on

the correlations for the absolute intraday returns, although the decay in the pA

coefficients for the lower frequencies is less pronounced than for the exchange

rates.