1. Semivariance Significance
Baishi Wu, 4/16/08

2. Outline Motivation Background Math Data Information Summary Statistics
Correlation Summary
Regression Summary

3. 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)

4. Equations
Realized Volatility (RV)
Bipower Variance (BV)

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

6. ri = log(priceclose) – log(priceopen)
Equations
Daily open to close returns (ri)
ri = log(priceclose) – log(priceopen)
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

7. 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.

8. 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

9. Equations
Tri-Power Quarticity
Relative Jump

10. 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

11. Data Preparation Collected at five minute intervals
S&P 500 Data Set: to late 2007 (1959 Observations)
Exxon Mobile Corp: 2000 to 2008 (1967 Observations)
Intel Corp: 2000 to 2008 (1720 Observations)
Pfizer Inc: 2000 to 2008 (1968 Observations)
Allegheny Technologies Inc: 2000 to 2008 (1964 Observations)
Chose different stocks to view consistency in previous conclusions as well as dissect any errors found this week

12. Statistical Summary x 10-4 ATI XOM PFE INTC SP500 rd rd2 RV upRV RS BV
-11.89
-1.05
-3.11
-14.06
.37
270.99
128.63
141.69
232.13
96.80
rd2
7.35
1.65
2.01
5.41
0.94
14.96
4.74
4.53
10.91
2.27 RV
8.30
1.89
2.37
5.64
8.92
2.03
2.85
6.45
1.26
upRV
4.15
1.18
2.83
0.47
4.88
1.08
1.46
3.52
0.71 RS
4.14
0.95
1.19
2.81
0.46
4.91
1.05
1.56
3.20
0.65 BV
6.82
1.80
2.20
5.02
0.88
7.21
2.00
2.56
6.21
1.20
BPDV
0.70
0.04
0.09
0.15
0.03
2.38
0.34
0.63
1.04
0.24

13. Statistical Summary
When looking at upRV vs. RS, notice that they are approximately same in terms of mean and std
Individual stocks are expectedly less volatile than the market as a whole
Daily returns are negative on average

14. Correlations with Daily Returns
RS and upRV are much more highly correlated with daily returns than RV is upon average - positive daily returns tend to indicate greater positive returns as a whole
Anticipate positive correlations of realized up-variance with daily returns, negative correlations of semi-variance
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) – price movements do not coincide with volatility

15. Correlations with Daily Returns
Semi-variance and realized up-variance are not better correlated with themselves (shown by earlier autocorrelations ran through correlograms)
Larger daily returns in magnitude do not correlate with higher market volatility (if measured through semi- variance)

16. Correlations with Daily Returns
XOM
PFE
INTC
SP500 RV
0.0775
0.0093
0.0020
upRV
0.2053
0.2089
0.2909
0.2284
0.2310 RS
PFE is unique:
RV magnitude is higher than average
RS magnitude is lower than average
ATI is unique:
RS magnitude is higher than average

17. Combined Regressors Summary
Highest R2 values were found for the HAR-RS-upRV regression combination of using both the semi-variances and the realized-upvariances
Much of this is due to the strength of the regression coefficient in the HAR-RS regression
In general, semi-variance is a better predictor of RV than realized up-variance and RV itself; this indicates that the down market predicts overall volatility best

18. Regression Summary
R2 values
ATI
XOM
PFE
INTC
SP500
HAR-RV
0.4426
0.5594
0.3938
0.6783
0.4965
HAR-RS
0.4467
0.5738
0.3705
0.6959
0.5126
HAR-upRV
0.3807
0.5139
0.3952
0.6273
0.4389
HAR-RS-upRV
0.4509
0.5732
0.3976
0.7040
0.5180
PFE – seen as an exception in an earlier circumstance; unreliable low correlations? Boost in predictive power comes from upRV
XOM – HAR-RS is very comparable to HAR-RS- upRV; is this difference negligible?

19. F-Test Summary
F stat
ATI
XOM
PFE
INTC
SP500 RS
44.45
12.07
0.5
26.74
12.31
0.648
upRV
7.83
0.13
3.57
3.69
1.68
0.9417
0.0135
0.0115
0.1698
PFE does not seem to find either RS or upRV predictions significant
Generally, the predictive power arrives from the HAR-RS regression with upRV only stronger in a weak predictive case

20. A Look into Pfizer