1. Jump Detection and Analysis Investigation of Media/Telecomm Industry
Prad Nadakuduty
4/9/08

2. Outline Introduction Mathematical Background
Data Preparation and Graphs
Summary Statistics
Correlation
HAR Regressions
Conclusion
Appendix
Jump Statistics

3. Motivation
Investigate regressions of realized variance using semi-variance
Analyze correlation of variance of M&T industry with S&P 500
Semi-variance from Barndorff-Nielsen, Kinnebrock, and Shephard (2008)
HAR-RV regressions from Corsi (2003)

4. Mathematical Background
Realized Variation (IV with jump contribution)
Bipower Variation (robust to jumps)

5. Mathematical Background
Previous equations used to estimate integrated quarticity
Relative Jump (measure of jump contribution to total price variance)

6. Mathematical Background
Realized Semi-Variance (RS)
Realized Upward Semi-Variance (upRV)
upRV = RV - RS

7. Mathematical Background
Realized semi-variance converges to half the BPV plus negative squared jumps:
Deviation above/below half implies increased/decreased volatility during down-ward market

8. Mathematical Background
Tri-Power Quarticity
Z Tri-Power Max Statistic
Significance Value .999  z > 3.09

9. Mathematical Background
Heterogeneous autoregressive realized variance (HAR-RV) Model with daily, weekly, and monthly periods:
Daily open-close log returns (ri)

10. Data Preparation Investigate Media/Telecomm Industry
Verizon Telecommunications (VZ)
AT&T Inc. (T)
Walt Disney Inc. (DIS)
Include S&P 500 for comparison
Data taken from 1/2/2001 to 12/29/2006
5 min interval (78 observations per day) to reduce microstructure noise
Over ~100K total observations
Incomplete trading days removed

11. S&P min Price Data
High:
12/15/2006
Low: 768
10/10/2002

12. Verizon Communications (VZ) 5 min Price Data
High: 57.40
7/19/2001
Low: 26.16
7/24/2002

13. AT&T (T) 5 min Price Data
High: 43.95
7/12/2001
Low: 13.50
4/16/2003

14. Walt Disney Inc. (DIS) 5 min Price Data
High: 34.88
12/19/2006
Low: 13.15
8/8/2002

15. Data Trends
Downward market from 2001 thru mid 2002 followed by upward market until end of 2006 to nearly same levels
Industry-wide shock from Sept 11, especially Disney
Expect semi-variance and up-variance to have similar but opposite correlations with daily squared returns

16. Summary Statistics S&P 500 Verizon AT&T Walt Disney M&T Average ri
Mean
(x 1e-4)
Std. Dev. ri
.0524
.0093
.6513
.0138
2.031
.0161
7.425
.0157
3.369
.0152
ri2
.8607
.0002
1.900
.0004
2.600
.0006
2.461
2.320
.0005 RV
.8724
.0001
2.398
.0003
3.076
3.267
2.914 RS
.4147
1.211
1.504
1.489
1.401
upRS
.4587
.0000
1.187
1.572
1.777
1.512 BV
.8222
2.259
2.918
3.065
2.747

17. Summary Statistics M&T Industry has nearly 3x more RV than S&P 500
Slightly more upward-RS for given time range than downward-RS (exception for Verizon)
Suggests more volatility during downward market?

18. Correlation – S&P 500
Nearly equal (but opposite) correlation of RS and upRS with returns, as expected
upRS more correlated with daily squared returns than RS; more volatility during upward market

19. Correlation - Verizon
upRS highest correlation with daily returns amongst all coefficients for all firms

20. Correlation - AT&T
Squared returns weakly negatively correlated with daily returns
RS and upRS have similar correlation with squared returns; contradicts intuition of higher volatility during downward market

21. Correlation - Disney
Largest correlation magnitude discrepancy between returns and RS, upRS

22. Correlation - Summary
Upward semi-variance largest correlation with daily returns (except for S&P 500, RS slightly bigger in magnitude)
Both semi-variances are more correlated with daily returns than realized variance
M&T firms share similar results and trends with each other and S&P

23. Regression Regression with Newey-West standard errors
Newey-West  heteroskedasticity robust standard errors
Will provide consistent estimators even if error term is correlated with its own past
Newey command in STATA
newey RV_ATT l1.RV_ATT l5RV_ATT l22RV_ATT, lag(60)

24. HAR-RV Regression – S&P 500
Monthly lag not significant

25. HAR-RV Regression - Verizon
Monthly and (especially) daily lags not significant

26. HAR-RV Regression - AT&T
Monthly and daily lags not significant

27. HAR-RV Regression - Disney
Monthly and (less so) daily lags not significant

28. HAR-RV Regression Summary
Monthly lags insignificant across M&T industry and S&P 500
Comparable R2 values
Varying results in daily lag suggests that significance function of particular data set and not industry-wide trend

29. Combined Regression
Regress realized variance against HAR semi-variances for each firm
Possible extension: Regress realized variance of S&P 500 against HAR semi-variances for each firm to identify possible predictive measures of market with M&T industry

30. Combined Regression – S&P 500

31. Combined Regression – Verizon

32. Combined Regression – AT&T

33. Combined Regression - Disney

34. Combined Regression Summary
S&P: daily, monthly lag upRS significant
Verizon: daily lag RS significant
AT&T: weekly lag RV significant
Disney: daily lag RS and monthly lag upRS

35. Combined Regression Summary
Collinearity results in upRS statistics being dropped from regression (except for S&P)
No overarching pattern in statistic significance
Extension: Investigate regression of one firm’s lagged semi-variances against market