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

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)

Simulated Data – α = 0.30

Simulated Data – α = 0.90

Simulated Data – α = 1.50

Simulated Data – α = 1.90

Contingency Tables – BN-S ZQP-max Statistic α = 0.30 α = 0.90 freq 5-min 10-min 15-min 20-min 217 20 12 21 235 37 27 245 47 239 freq 5-min 10-min 15-min 20-min 219 22 13 24 235 36 29 246 46 238 α = 1.50 α = 1.90 freq 5-min 10-min 15-min 20-min 217 21 12 24 241 36 29 243 46 237 freq 5-min 10-min 15-min 20-min 218 20 14 25 239 35 27 249 48 237

Contingency Tables – BN-S ZTP-max Statistic α = 0.30 α = 0.90 freq 5-min 10-min 15-min 20-min 212 22 12 21 233 38 26 247 47 237 freq 5-min 10-min 15-min 20-min 215 22 13 24 232 37 29 248 45 236 α = 1.50 α = 1.90 freq 5-min 10-min 15-min 20-min 216 22 13 24 238 37 29 246 45 235 freq 5-min 10-min 15-min 20-min 216 20 14 25 238 35 27 248 45 235

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 = 0.30 α = 0.90 k 2 3 4 69 11 52 12 k 2 3 4 68 12 52 α = 1.50 α = 1.90 k 2 3 4 69 12 52 k 2 3 4 69 12 53

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

Daily Contingency Tables – Lee-Mykland Test S&P 500 freq 5-min 10-min 15-min 20-min 147 63 35 33 99 36 37 67 30 52 freq 5-min 10-min 15-min 20-min Exxon Mobil AT&T freq 5-min 10-min 15-min 20-min 100 28 19 14 59 16 40 30 freq 5-min 10-min 15-min 20-min 244 110 75 53 149 68 59 106 47 87

Daily Contingency Tables – Lee-Mykland Test α = 0.30 α = 0.90 freq 5-min 10-min 15-min 20-min freq 5-min 10-min 15-min 20-min α = 1.50 α = 1.90 freq 5-min 10-min 15-min 20-min freq 5-min 10-min 15-min 20-min

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

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

Contingency Tables – MNR-JO Difference Test S&P 500 freq 5-min 10-min 15-min 20-min 277 77 66 84 271 81 82 284 92 343 freq 5-min 10-min 15-min 20-min Exxon Mobil AT&T freq 5-min 10-min 15-min 20-min 197 53 47 59 186 48 56 210 69 247 freq 5-min 10-min 15-min 20-min 321 123 112 102 377 146 144 415 162 405

Contingency Tables – MNR-JO Log Test S&P 500 freq 5-min 10-min 15-min 20-min 278 70 82 263 77 79 271 88 325 freq 5-min 10-min 15-min 20-min Exxon Mobil AT&T freq 5-min 10-min 15-min 20-min 190 51 43 53 181 50 48 197 65 227 freq 5-min 10-min 15-min 20-min 367 142 121 115 386 151 150 408 155 398

Contingency Tables – MNR-JO Ratio Test S&P 500 freq 5-min 10-min 15-min 20-min 277 69 82 263 77 79 271 88 324 freq 5-min 10-min 15-min 20-min Exxon Mobil AT&T freq 5-min 10-min 15-min 20-min 190 51 43 53 181 50 48 197 65 227 freq 5-min 10-min 15-min 20-min 367 142 121 115 386 151 150 408 155 398

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

Cross-Test Contingency Tables – 5 min sample S&P 500 test BNS JO MNR ASJ LM test BNS JO MNR ASJ LM XOM α = 1.50 test BNS JO MNR ASJ LM test BNS JO MNR ASJ LM N/A

Cross-Test Contingency Tables – 10 min sample S&P 500 test BNS JO MNR ASJ LM test BNS JO MNR ASJ LM XOM α = 1.50 test BNS JO MNR ASJ LM test BNS JO MNR ASJ LM N/A

Cross-Test Contingency Tables – 15 min sample S&P 500 test BNS JO MNR ASJ LM test BNS JO MNR ASJ LM XOM α = 1.50 test BNS JO MNR ASJ LM test BNS JO MNR ASJ LM N/A

Cross-Test Contingency Tables – 20 min sample S&P 500 test BNS JO MNR ASJ LM test BNS JO MNR ASJ LM XOM α = 1.50 test BNS JO MNR ASJ LM test BNS JO MNR ASJ LM N/A

Summary of Tests’ Sample Robustness ZQP-max ZTP-max JODIFF MNR-JODIFF LM ASJ GE JUMPS 5 MIN 84 69 74 277 147 149 10 vs. 5 asset % 13.994 12.839 28.520 31.741 61.640 N/A 10 vs. 5 sim % 9.528 10.014 15 vs. 5 asset % 4.889 5.588 26.194 27.525 56.831 15 vs. 5 sim % 5.854 6.052 20 vs. 5 asset % 6.452 3.488 22.378 30.683 57.016 20 vs. 5 sim % 10.791 10.938 GE JUMPS 10 MIN 60 51 94 271 99 15 vs. 10 asset % 11.258 10.153 25.890 31.474 55.127 38.266 15 vs. 10 sim % 15.161 15.627 22.487 20 vs. 10 asset % 11.960 11.500 27.651 33.075 64.101 32.451 20 vs. 10 sim % 11.790 11.797 25.000 GE JUMPS 15 MIN 42 35 112 284 67 80 20 vs. 15 asset % 13.933 12.170 28.209 35.084 52.794 54.367 20 vs. 15 sim % 19.023 19.230 29.167 GE JUMPS 20 MIN 40 34 145 343 52 17 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

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