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

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 α = 1.50 α = 1.90

6. Contingency Tables – BN-S ZTP-max Statistic α = 0.40 α = 0.90 α = 1.50 α = 1.90

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

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

9. Contingency Tables – Ait-Sahalia Jacod Test α = 0.40 α = 0.90 α = 1.50 α = 1.90

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 1999-2008 (2026 days) • General Electric 1997-2007 (2670 days) • Microsoft 1997-2008 (2683 days) • AT&T 1997-2008 (2680 days) • Procter & Gamble 1997-2008 (2686 days) • Chevron 2001-2008 (1566 days) • Johnson & Johnson 1997-2008 (2685 days) • Bank of America 1997-2008 (2685 days) • Cisco Systems 1997-2008 (2683 days) • Altria Group 1997-2008 (2685 days)

14. Summary of Sample Robustness - Assets

15. Summary of Sample Robustness - Simulations

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 teststatistics 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