Sampling Frequency and Jump Detection Mike Schwert ECON201FS 2/27/08

This Week’s Approach Last time: Counted jump days at various sampling frequencies Volatility signature plots for RV and BV This week: Semivariance calculations for GE price data Volatility signature plots for RJ, RS- and RS+ Counted common jump days between different sampling frequencies Correlation matrices for ZQP-max statistics at different sampling frequencies

Realized Semivariance Introduced by Barndorff-Nielsen, Kinnebrock, and Shephard Separates positive and negative components of realized variance Summary Statistics – GE Price Data, 5-minute sampling frequency Mean Std. Dev Min Max RVolannualized 0.2562 0.3117 0.0529 1.6693 RV 2.6053 x 10-4 3.8553 x 10-4 1.1104 x 10-5 0.0111 RS- 1.2866 x 10-4 1.8166 x 10-4 4.8495 x 10-6 0.0041 RS+ 1.3187 x 10-4 2.1816 x 10-4 4.3544 x 10-6 0.0069

Volatility Signature Plots Introduced by Andersen, Bollerslev, Diebold, and Labys (1999). Calculate RJ, RS-, RS+ at 1, 2, …, 30 minute sampling frequencies Plot relationship between sampling frequency and mean RJ, mean RS RJ and RS are higher for high-frequency samples because returns are distorted by microstructure noise such as bid-ask bounce Must be wary of using too low of a sampling frequency, as sampling variation will affect volatility calculations

Volatility Signature Plot – Relative Jump

Volatility Signature Plot – Negative Semivariance

Volatility Signature Plot – Positive Semivariance

Contingency and Correlation Matrices Calculated ZQP-max statistics and counted jump days for GE, ExxonMobil, AT&T, and S&P 500 at 5, 10, 15 and 20 minute sampling frequencies Counted common jump days between each sampling frequency and organized in contingency matrices Surprisingly few common jump days exist between sampling frequencies Calculation of jump days seems to depend a great deal on sampling choices ZQP-max statistics are relatively uncorrelated between sampling frequencies GE data minute-by-minute for 1997 – 2007 (2670 days) ExxonMobil data minute-by-minute for 1999 – 2008 (2026 days) AT&T data minute-by-minute for 1997 – 2008 (2680 days) S&P data every 5 minutes, 1985 – 2007 (5545 days, excluding short days)

Contingency Tables GE S&P 500 Exxon Mobil AT&T freq 5-min 10-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

Correlation Matrices - Z-Statistics GE S&P 500 – N/A freq 5-min 10-min 15-min 20-min 1.000 .1030 .0150 .0431 .2241 .0969 .2809 freq 5-min 10-min 15-min 20-min 1.000 NaN Exxon Mobil AT&T freq 5-min 10-min 15-min 20-min 1.000 .0849 .0728 .0631 .2274 .1273 .2996 freq 5-min 10-min 15-min 20-min 1.000 .1644 .0844 .0607 .2694 .1753 .3270

Possible Extensions Try other jump test statistics to see if one outperforms the others in detecting jumps consistently between sampling frequencies Regress z-statistics on changes in daily volume to see if days with high volume correspond to jump days, common jump days between samples Other ways to formally analyze effects of sampling frequency on jump detection? Any way to separate negative jumps from positive jumps? Effect of sampling frequency on volatility forecasts?