1. Sampling Frequency and Jump Detection
Mike Schwert
ECON201FS
4/16/08

2. Background and Motivation
Various tests exist to identify jumps in asset price movements
These tests use high frequency financial data which must be sampled from its highest frequencies to eliminate problems like market microstructure noise
Earlier this semester, I found that it is appropriate to sample prices from approximately 5 to 15 minute frequencies, based on volatility signature plots
Sampling at different frequencies causes jump tests to identify different “jump days,” bringing into question the viability of these tests
Part of the problem might be the stochastic jump diffusion model behind these jump detection tests (Poisson might not be appropriate)
Jump tests used:
Barndorff-Nielsen Shephard ZQP-max and ZTP-max tests
Jiang-Oomen “Swap Variance” Difference and Logarithmic tests
Microstructure Noise Robust Jiang-Oomen Difference and Log tests
Lee-Mykland test
Ait-Sahalia Jacod test

3. Simulated Data
Continuous process with jumps from a tempered stable distribution
Tempered stable is just classical stable with finite variance
α ranges from 0 to 2 and describes the asymptotic behavior of the distribution, essentially the heaviness of the tails
α = 2 is the Gaussian distribution
Classical model of jumps, which is used in much of the jump detection literature, has rare large jumps
Tempered stable distribution has many more medium sized jumps
Jump detection tests could be getting confused by these moderate sized jumps which might not be as prevalent between samples as larger ones
Most tests seem to be more sample robust with this data than with actual asset price data

4. Simulated Data
α = 0.40
α = 0.90
α = 1.50
α = 1.90

5. Contingency Tables – BN-S ZQP-max Statistic
α = 0.40
α = 0.90
freq
5-min
10-min
15-min
20-min
643
281
220
191
500
218
165
440
172
387
freq
5-min
10-min
15-min
20-min
577
234
195
161
448
191
143
417
159
358
α = 1.50
α = 1.90
freq
5-min
10-min
15-min
20-min
499
181
149
112
414
152
103
372
129
318
freq
5-min
10-min
15-min
20-min
390
106 87 69
363
113 73
330 91
288

6. Contingency Tables – BN-S ZTP-max Statistic
α = 0.40
α = 0.90
freq
5-min
10-min
15-min
20-min
637
276
216
190
493
217
163
442
174
383
freq
5-min
10-min
15-min
20-min
578
227
195
162
444
188
140
415
157
356
α = 1.50
α = 1.90
freq
5-min
10-min
15-min
20-min
498
178
145
110
408
151
102
371
128
317
freq
5-min
10-min
15-min
20-min
382
105 85 66
360
113 70
325 86
283

7. Ait-Sahalia Jacod Test
Introduced in 2008 article by Yacine Ait-Sahalia and Jean Jacod

8. Contingency Tables – Ait-Sahalia Jacod Test
S&P 500 k 2 3 4
149 24 80 7 17 k 2 3 4
282 72 46
737
262
396
Exxon Mobil
AT&T k 2 3 4
110 17 46 6 14 k 2 3 4
165 63 37
104 55

9. Contingency Tables – Ait-Sahalia Jacod Test
α = 0.40
α = 0.90 k 2 3 4 49 7 6 22 16 k 2 3 4 63 9 31 6 18
α = 1.50
α = 1.90 k 2 3 4 67 7 40 16 k 2 3 4 69 6 5 40 24

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

11. Microstructure Noise Robust Jiang-Oomen Test
Similar to Jiang-Oomen Swap Variance test, but robust to microstructure noise which often contaminates high-frequency data

12. Microstructure Noise Robust Jiang-Oomen Test
Difference Test:
Logarithmic Test:
Ratio Test:

13. Summary of Sample Robustness
Calculated average ratio of jump days to all days for several jump tests
Calculated average percentage of common jumps between sampling frequencies for each test, using the lower number of jumps as the denominator
Minute-by-minute asset price data:
ExxonMobil (2026 days)
General Electric (2670 days)
Microsoft (2683 days)
AT&T (2680 days)
Procter & Gamble (2686 days)
Chevron (1566 days)
Johnson & Johnson (2685 days)
Bank of America (2685 days)
Cisco Systems (2683 days)
Altria Group (2685 days)

14. Summary of Sample Robustness - Assets
ZTP-max
ZQP-max
JO-Diff
JO-Log
MNR-Diff
MNR-Log LM
ASJ
5-min jump ratio
.0347
.0416
.0303
.0231
.1830
.1909
.0744
N/A
10-min jump ratio
.0243
.0278
.0400
.0283
.1974
.1996
.0433
.0534
15-min jump ratio
.0213
.0244
.0518
.0365
.2050
.2072
.0307
.0268
20-min jump ratio
.0173
.0602
.0408
.2158
.2172
.0249
.0092
5 vs. 10 ratio
.1285
.1519
.2516
.2025
.4692
.4724
.6233
5 vs. 15 ratio
.0881
.1026
.2467
.1850
.4524
.4570
.5920
5 vs. 20 ratio
.0672
.1000
.2500
.1785
.4547
.4599
.5852
10 vs. 15 ratio
.1354
.1414
.2828
.2482
.4804
.4851
.5873
.3498
10 vs. 20 ratio
.1369
.1503
.2998
.2165
.4884
.4992
.6143
.4111
15 vs. 20 ratio
.1760
.1882
.3249
.2548
.4967
.5059
.5514
.4399

15. Summary of Sample Robustness - Simulations
ZTP-max
ZQP-max
JO-Diff
JO-Log
MNR-Diff
MNR-Log LM
ASJ
5-min jump ratio
.2095
.2109
.0486
.0402
.1571
.1568
.0966
N/A
10-min jump ratio
.1705
.1725
.0384
.0305
.1414
.1402
.0524
.0248
15-min jump ratio
.1553
.1559
.0391
.0279
.1388
.1397
.0377
.0133
20-min jump ratio
.1339
.1351
.0363
.0255
.1357
.0270
.0074
5 vs. 10 ratio
.4498
.4534
.4173
.3748
.3581
.3604
.9739
5 vs. 15 ratio
.4027
.4079
.3686
.3131
.3363
.3410
.9770
5 vs. 20 ratio
.3828
.3838
.3418
.3236
.3238
.3298
.9857
10 vs. 15 ratio
.4247
.4261
.3528
.3184
.3356
.3391
.9308
.2334
10 vs. 20 ratio
.3470
.3508
.3571
.2972
.3287
.3310
.9752
.2326
15 vs. 20 ratio
.4007
.4026
.3542
.2934
.3381
.3380
.9427
.2656

16. Conclusions
Microstructure Noise Robust Jiang-Oomen tests detect far more jumps in asset data than the other tests – possible problem with implementation of microstructure noise bias correction?
Jiang-Oomen tests detect more jumps as sampling frequency decreases, while all other tests detect fewer jumps
For asset price data, Barndorff-Nielsen Shephard appears to be least sample robust, whereas Lee-Mykland has nearly four times as many common jump days
Jiang-Oomen and BN-S are more sample robust for simulations, while Ait-Sahalia Jacod and Microstructure Noise Robust Jiang-Oomen are more robust for asset price data
Lee-Mykland test is extremely consistent between samples of simulated data

17. Possible Extensions
Improve implementation of microstructure noise bias for MNR-JO test
Regress test statistics on changes in daily volume to see if high volume days correspond to jump days and common jump days between samples
Different ways to examine sample robustness?
Currently use lower number of days as denominator
Correlations might be more appropriate
Examine jump diffusion models other than the Poisson process used in most jump detection literature