Semivariance Significance

Baishi Wu, 4/16/08

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

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 (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

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: 2000 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

Statistical Summaryx 10-4 ATI XOM PFE INTC SP500

rd -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 0.94

8.92 2.03 2.85 6.45 1.26

upRV 4.15 0.94 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

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

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

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)

Correlations with Daily Returnscorr ATI XOM PFE INTC SP500

RV -0.0773 -0.0242 0.0775 0.0093 0.0020

upRV 0.2053 0.2089 0.2909 0.2284 0.2310

RS -0.3444 -0.2621 -0.1311 -0.2322 -0.2494

PFE is unique: RV magnitude is higher than average RS magnitude is lower than average

ATI is unique: RV magnitude is higher than average RS magnitude is higher than average

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

Regression SummaryR2 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?

F-Test SummaryF stat ATI XOM PFE INTC SP500

RS 44.45 12.07 0.5 26.74 12.31

0 0 0.648 0 0

upRV 7.83 0.13 3.57 3.69 1.68

0 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

A Look into Pfizer