1. Sampling Frequency and Jump Detection
Mike Schwert
ECON201FS
3/19/08

2. This Week’s Approach and Data
Last week, found different jump days for different sampling frequencies using the Barndorff-Nielsen Shephard jump test ZQP-max statistic
This week, trying other jump tests to see if they are sample robust
BN-S test using ZQP-max and ZTP-max statistics
Jiang-Oomen jump test
Lee-Mykland jump test
Price Data:
GE minute-by-minute 1997 – 2007 (2670 days)
ExxonMobil minute-by-minute 1999 – 2008 (2026 days)
AT&T minute-by-minute 1997 – 2008 (2680 days)
S&P 500 every 5 minutes, 1985 – 2007 (5545 days, excluding short days)

3. Barndorff-Nielsen Shephard Tests

4. Barndorff-Nielsen Shephard Tests

5. Contingency Tables – ZQP-max Statistic
S&P 500
freq
5-min
10-min
15-min
20-min 84 5 1 60 4 42 40
freq
5-min
10-min
15-min
20-min
151 10 5 8 95 92 6 76
Exxon Mobil
AT&T
freq
5-min
10-min
15-min
20-min 48 6 1 34 5 4 31 28
freq
5-min
10-min
15-min
20-min
185 22 8 7
113 11 94 16 76

6. Contingency Tables – ZTP-max Statistic
S&P 500
freq
5-min
10-min
15-min
20-min 69 3 2 1 51 4 35 5 34
freq
5-min
10-min
15-min
20-min
136 9 3 4 82 6 2 79 64
Exxon Mobil
AT&T
freq
5-min
10-min
15-min
20-min 44 5 2 34 4 30 3 20
freq
5-min
10-min
15-min
20-min
153 19 5 3 96 9 7 81 63

7. Jiang-Oomen Swap Variance Tests
Introduced by George Jiang and Roel Oomen in a 2005 paper
Tests for daily jumps, similar to BN-S, but uses “Swap Variance” measure instead of Bipower Variation to form a test statistic
Test is called this because it is “directly related to the profit/loss function of a variance swap replication strategy using a log contract”

8. Jiang-Oomen Swap Variance Tests
Difference Test:
Logarithmic Test:
Ratio Test:

9. Contingency Tables – SwapVar Difference Test
S&P 500
freq
5-min
10-min
15-min
20-min 74 21 15 94 22 33
112 36
145
freq
5-min
10-min
15-min
20-min
160 50 43 47
234 65 60
295 89
317
Exxon Mobil
AT&T
freq
5-min
10-min
15-min
20-min 48 13 12 6 58 90 20 98
freq
5-min
10-min
15-min
20-min 95 26 31
120 36 35
159 45
191

10. Contingency Tables – SwapVar Log Test
S&P 500
freq
5-min
10-min
15-min
20-min 51 12 8 10 65 15 79 18 99
freq
5-min
10-min
15-min
20-min
130 34 23 31
172 33 39
197 48
209
Exxon Mobil
AT&T
freq
5-min
10-min
15-min
20-min 43 9 8 3 10 59 73
freq
5-min
10-min
15-min
20-min 73 18 17 88 25 23
122 27
134

11. Contingency Tables – SwapVar Ratio Test
S&P 500
freq
5-min
10-min
15-min
20-min 51 12 8 10 65 15 79 18 99
freq
5-min
10-min
15-min
20-min
130 34 23 31
172 33 39
197 48
209
Exxon Mobil
AT&T
freq
5-min
10-min
15-min
20-min 43 9 8 3 10 59 73
freq
5-min
10-min
15-min
20-min 72 18 17 88 25 23
122 27
134

12. Lee-Mykland Test
Introduced by Suzanne Lee and Per Mykland in a 2007 paper
Allows identification of jump timing, multiple jumps in a day

13. Lee-Mykland Test – Summary Statistics
GE price data, sampled at 5 minute frequency
Something wrong with code…critical value for jumps is
GE - Daily Contingencies
freq
5-min
10-min
15-min
20-min
2667
2665
2664
2663
Mean
1284
Std. Dev.
1465
Min
-7.4
Max
36202
Total Jumps
173146
“Jump Days”
2667
“Jump Hours”
15999
GE – Hourly Contingencies
freq
5-min
10-min
15-min
20-min
15999
15986
15952
15817
15951
15816
15790

14. Possible Extensions
Other ways to formally analyze effects of sampling frequency on jump detection?
Identify problem with current implementation of Lee-Mykland test
Best way to compare Lee-Mykland results between samples?
Look at more samples, i.e. {1, 2, …, 20} instead of {5, 10, 15, 20}
Experiment with randomly generated data to examine effect of dependence on contingency tables
Regress z-statistics on changes in daily volume to see if days with high volume correspond to jump days, common jump days between samples