Sampling Frequency and Jump Detection Mike Schwert ECON201FS 3/19/08

This Week’s Approach and Data Last week, found different jump days for different sampling frequencies using the Barndorff-Nielsen Shephard jump test ZQP-max statistic This week, trying other jump tests to see if they are sample robust BN-S test using ZQP-max and ZTP-max statistics Jiang-Oomen jump test Lee-Mykland jump test 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)

Barndorff-Nielsen Shephard Tests

Barndorff-Nielsen Shephard Tests

Contingency Tables – ZQP-max Statistic S&P 500 freq 5-min 10-min 15-min 20-min 84 5 1 60 4 42 40 freq 5-min 10-min 15-min 20-min 151 10 5 8 95 92 6 76 Exxon Mobil AT&T freq 5-min 10-min 15-min 20-min 48 6 1 34 5 4 31 28 freq 5-min 10-min 15-min 20-min 185 22 8 7 113 11 94 16 76

Contingency Tables – ZTP-max Statistic S&P 500 freq 5-min 10-min 15-min 20-min 69 3 2 1 51 4 35 5 34 freq 5-min 10-min 15-min 20-min 136 9 3 4 82 6 2 79 64 Exxon Mobil AT&T freq 5-min 10-min 15-min 20-min 44 5 2 34 4 30 3 20 freq 5-min 10-min 15-min 20-min 153 19 5 3 96 9 7 81 63

Jiang-Oomen Swap Variance Tests Introduced by George Jiang and Roel Oomen in a 2005 paper Tests for daily jumps, similar to BN-S, but uses “Swap Variance” measure instead of Bipower Variation to form a test statistic Test is called this because it is “directly related to the profit/loss function of a variance swap replication strategy using a log contract”

Jiang-Oomen Swap Variance Tests Difference Test: Logarithmic Test: Ratio Test:

Contingency Tables – SwapVar Difference Test S&P 500 freq 5-min 10-min 15-min 20-min 74 21 15 94 22 33 112 36 145 freq 5-min 10-min 15-min 20-min 160 50 43 47 234 65 60 295 89 317 Exxon Mobil AT&T freq 5-min 10-min 15-min 20-min 48 13 12 6 58 90 20 98 freq 5-min 10-min 15-min 20-min 95 26 31 120 36 35 159 45 191

Contingency Tables – SwapVar Log Test S&P 500 freq 5-min 10-min 15-min 20-min 51 12 8 10 65 15 79 18 99 freq 5-min 10-min 15-min 20-min 130 34 23 31 172 33 39 197 48 209 Exxon Mobil AT&T freq 5-min 10-min 15-min 20-min 43 9 8 3 10 59 73 freq 5-min 10-min 15-min 20-min 73 18 17 88 25 23 122 27 134

Contingency Tables – SwapVar Ratio Test S&P 500 freq 5-min 10-min 15-min 20-min 51 12 8 10 65 15 79 18 99 freq 5-min 10-min 15-min 20-min 130 34 23 31 172 33 39 197 48 209 Exxon Mobil AT&T freq 5-min 10-min 15-min 20-min 43 9 8 3 10 59 73 freq 5-min 10-min 15-min 20-min 72 18 17 88 25 23 122 27 134

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

Lee-Mykland Test – Summary Statistics GE price data, sampled at 5 minute frequency Something wrong with code…critical value for jumps is 4.6001 GE - Daily Contingencies freq 5-min 10-min 15-min 20-min 2667 2665 2664 2663 Mean 1284 Std. Dev. 1465 Min -7.4 Max 36202 Total Jumps 173146 “Jump Days” 2667 “Jump Hours” 15999 GE – Hourly Contingencies freq 5-min 10-min 15-min 20-min 15999 15986 15952 15817 15951 15816 15790

Possible Extensions Other ways to formally analyze effects of sampling frequency on jump detection? Identify problem with current implementation of Lee-Mykland test Best way to compare Lee-Mykland results between samples? Look at more samples, i.e. {1, 2, …, 20} instead of {5, 10, 15, 20} Experiment with randomly generated data to examine effect of dependence on contingency tables Regress z-statistics on changes in daily volume to see if days with high volume correspond to jump days, common jump days between samples