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

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

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

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

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

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

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

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

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

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

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

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

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)

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

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

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

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