Semivariance Significance in the S&P500 Baishi Wu, 4/7/08

Outline  Motivation  Background Math  Data Information  Summary Statistics  Regressions  Appendix

Introduction  Want to examine predictive regressions for realized variance by using realized semi-variance as a regressor  Test significance of realized semi-variance and realized up- variance by correlation with daily open-close returns  Regressions are of the HAR-RV form from Corsi (2003)  Semi-variance from Barndorff-Nielsen, Kinnebrock, and Shephard (2008)

Equations  Realized Volatility (RV)  Bipower Variance (BV)

Equations  Realized Semivariance (RS)  Realized upVariance (upRV) upRV = RV - RS  Bipower Downard Variance (BPDV)

Equations  Daily open to close returns (r i ) r i = log(price close ) – log(price open )  The daily open to close returns are correlated with the RV, upRV, and RS to determine whether market volatility is dependent on direction  This statistic is also squared to determine if the size of the open to close price shift correlates with the magnitude of realized volatility

Equations  Heterogenous Auto-Regressive Realized Volatility (HAR-RV) from Corsi, 2003:  Multi-period normalized realized variation is defined as the average of one-period measures. The model is using rough daily, weekly, monthly periods.

Equations  Extensions of HAR-RV  Created different regressions using lagged RS and lagged upRV in predicting RV creating HAR-RS and HAR-upRV  Compared to original HAR-RV model  Created combined regressions of a combination of both RS and upRV to predict RV using HAR-RS-upRV

Equations  Tri-Power Quarticity  Relative Jump

Equations  Max Version z-Statistic (Tri-Power)  The max version Tri-Power z-Statistic is used to measure jumps in the data in this case  Take one sided significance at.999 level, or z = 3.09

Data Preparation  Collected at five minute intervals  S&P 500 Data Set  1985 to late 2007 (5751 Observations) – Included large spike in RV/BV, less sampling days in this data set  1990 to late 2007 (4487 Observations) – Largely influenced by upward trend of S&P 500 in the 1990s  2000 to late 2007 (1959 Observations) – Possibly examines a period of the greatest market volatility  Chose different sample lengths in order to test the consistency in correlations and regressions

Data Preparation S&P500,

Summary Statistics Mean (x 1e -4 ) StdMean (x 1e -4 ) StdMean (x 1e - 4) Std riri ri2ri RV upRV RS BV BPDV  Numbers are similar except for daily returns

Correlation  Semi-variance correlates the highest with squared daily returns; is this indicative of higher volatility in a down market?  Realized up-variance is not higher than Realized Variance S&P500,

Correlation  This segment has the lowest correlation of semi-variance with realized up-variance  Semi-variance does not have a higher correlation with squared daily returns than either RV or upRV S&P500,

Correlation S&P500,  Only segment where daily squared returns are positively correlated (though slightly) with daily returns

Correlation Summary  Anticipate positive correlations of realized up-variance with daily returns, negative correlations of semi-variance  Both semi-variance statistics ought to have a higher correlation with the daily returns than the realized variance (found untrue in dataset)  Expected to see a higher correlation with semi-variance and daily squared returns in order to indicate higher volatility in a down market (not the case)  Bipower Downward Variation is a combination of Bipower Variation and Semivariance; correlates very negatively with daily returns (why?)

HAR-RV  R 2 =  Monthly regressor not significant, very low correlation S&P500,

HAR-RV  R 2 =  Daily lag not significant S&P500,

HAR-RV  R 2 =  Daily, monthly not significant S&P500,

HAR-RS  R 2 =  Weekly lag very insignificant, monthly lag also insignificant S&P500,

HAR-RS  R 2 =  Daily lag not significant S&P500,

HAR-RS  R 2 =  Daily, monthly not significant S&P500,

HAR-upRV  R 2 =  Very low R 2 value, monthly regressor very insignificant, daily insignificant S&P500,

HAR-upRV  R 2 =  Daily insignificant S&P500,

HAR-upRV  R 2 =  Daily, weekly (slightly) insignificant S&P500,

Normal Regressions Summary  Low R 2 coefficient in S&P 500 dataset seems largely caused by the realized up-variance. This is also the only dataset that has the R 2 value of the RV greater than the average of its parts  Observe similar levels of correlation, similar significant variables despite specific statistic (RV, RS, or upRV)  Generally there do not seem to be any noticeable trends that are specific to any individual test statistic; the significances of the regressors seem to be a function of the data set and not the test statistic

RV Regressed with RS  R 2 =  Only monthy lags not significant S&P500,

RV Regressed with RS  R 2 =  Daily lags are not as significant S&P500,

RV Regressed with RS  R 2 =  Monthly lags not significant S&P500,

RV Regressed with upRV  R 2 =  Very low correlation, monthly lags not significant S&P500,

RV Regressed with upRV  R 2 =  Daily lags not significant S&P500,

RV Regressed with upRV  R 2 =  Daily and weekly (to a lesser extent) are not significant S&P500,

RV Regressed with RS and upRV  R 2 =  Both monthly lags in general not significant S&P500,

RV Regressed with RS and upRV  R 2 =  Semi-variance statistics much more significant than realized up-variance statistics S&P500,

RV Regressed with RS and upRV  R 2 =  Semi-variance statistics much more significant than realized up-variance statistics S&P500,

Combined Regressors Summary  Highest R 2 values were found for the HAR-RS-upRV regression combination of using both the semi-variances and the realized-upvariances (could this be the zeros?)  In general, semi-variance is a better predictor of RV than realized up-variance and even RV itself; does this indicate that the down market predicts overall volatility best? (or am I over interpreting the value of R 2 ?)  For the combined regression, the semi-variance coefficients were found to be much more significant

Summary Statistics R 2 values HAR-RV HAR-RS HAR-upRV HAR-RV/RS HAR-RV/upRV HAR-RV/RS/upRV

Summary Statistics Test Statistics L1L5L22L1L5L22L1L5L22 HAR-RV HAR-RS HAR-upRV HAR-RV/RS HAR-RV/upRV HAR-RV/RS/upRV HAR-RV/RS/upRV

Appendix  Graphs for S&P 500 Data Set  Realized Variance and Bipower Variation  Z-Scores with Significance  Semivariance, Realized upVariance  Bipower Variation and Bipower Downward Variation  Autocorrelation Plots for  Realized Variance  Semivariance  Realized upVariance

Realized and Bipower Variance S&P500, StatisticValue mean(RV)8.1299e-05 std(RV)1.2352e-04 mean(BV)7.6804e-05 std(BV)1.1303e-04

Z-Scores S&P500, StatisticValue days4509 mean(z) std(z) jump days166 Jump %3.68%

Semivariance, Realized upVariance S&P500, StatisticValue mean(RS)4.0894e-05 std(RS)7.1114e-05 mean(upRV)4.0405e-05 std(upRV)6.3970e-05

Bipower Downward Variation S&P500, StatisticValue mean(BV)7.6804e-05 std(BV)1.1303e-04 mean(BPDV)2.4916e-06 std(BPDV)2.7787e-05

Correlogram – Realized Variance S&P500,

Correlogram – Realized Semivariance S&P500,

Correlogram – Realized upVariance S&P500,