Sampling Frequency and Jump Detection Mike Schwert ECON201FS 4/7/08

1. Sampling Frequency and Jump Detection Mike Schwert ECON201FS 4/7/08

2. This Week’s Approach and Data • Counted common jump days between different sampling frequencies for Ait-Sahalia Jacod, Lee-Mykland and Jiang-Oomen jump tests • Counted common jump days between different tests • Summarized effect of sampling frequency on jump detection • 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) • Simulated Data: • Continuous process plus jumps from a tempered stable distribution • Alphas of 0.30, 0.90, 1.50, 1.90 • 78 daily price observations for 2500 days (195000 total observations)

3. Simulated Data – α = 0.30

4. Simulated Data – α = 0.90

5. Simulated Data – α = 1.50

6. Simulated Data – α = 1.90

7. Contingency Tables – BN-S ZQP-max Statistic α = 0.30 α = 0.90 α = 1.50 α = 1.90

8. Contingency Tables – BN-S ZTP-max Statistic α = 0.30 α = 0.90 α = 1.50 α = 1.90

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

10. Contingency Tables – Ait-Sahalia Jacod Test GE S&P 500 Exxon Mobil AT&T

11. Contingency Tables – Ait-Sahalia Jacod Test α = 0.30 α = 0.90 α = 1.50 α = 1.90

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. Daily Contingency Tables – Lee-Mykland Test GE S&P 500 Exxon Mobil AT&T

14. Daily Contingency Tables – Lee-Mykland Test α = 0.30 α = 0.90 α = 1.50 α = 1.90

15. Microstructure Noise Robust Jiang-Oomen Test • Similar to Jiang-Oomen Swap Variance test, but robust to microstructure noise which often contaminates high-frequency data • NOTE: NO SIMULATED RESULTS YET BECAUSE OF DIFFICULTIES WITH SEXTICITY CALCULATION

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

17. Contingency Tables – MNR-JO Difference Test GE S&P 500 Exxon Mobil AT&T

18. Contingency Tables – MNR-JO Log Test GE S&P 500 Exxon Mobil AT&T

19. Contingency Tables – MNR-JO Ratio Test GE S&P 500 Exxon Mobil AT&T

20. Cross-Test Contingency Tables • Checked to see if different jump tests find jumps on same days • Tests: • Barndorff-Nielsen Shephard ZQP-max test • Jiang-Oomen swap variance test • Microstructure Noise Robust Jiang-Oomen test • Ait-Sahalia Jacod test • Lee-Mykland test

21. Cross-Test Contingency Tables – 5 min sample GE S&P 500 XOM α = 1.50

22. Cross-Test Contingency Tables – 10 min sample GE S&P 500 XOM α = 1.50

23. Cross-Test Contingency Tables – 15 min sample GE S&P 500 XOM α = 1.50

24. Cross-Test Contingency Tables – 20 min sample GE S&P 500 XOM α = 1.50

25. Summary of Tests’ Sample Robustness • Denominator is frequency with lower number of jumps • Lee-Mykland test results based on daily findings – detected jumps are not necessarily at the same time during the day

26. Possible Extensions • Regress Barndorff-Shephard Nielsen ZQP-max statistics on changes in daily volume to see if high volume days correspond to jump days and common jump days between samples • Examine jump diffusion models other than the Poisson process used in most jump detection literature