Jump Detection and Analysis

Investigation of Media/Telecomm Industry

Prad Nadakuduty

4/9/08

Outline• Introduction• Mathematical Background• Data Preparation and Graphs• Summary Statistics• Correlation• HAR Regressions• Conclusion• Appendix

– Jump Statistics

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)

Mathematical Background• Realized Variation (IV with jump contribution)

• Bipower Variation (robust to jumps)

Mathematical Background• Previous equations used to estimate

integrated quarticity

• Relative Jump (measure of jump contribution to total price variance)

Mathematical Background• Realized Semi-Variance (RS)

• Realized Upward Semi-Variance (upRV)

upRV = RV - RS

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

Mathematical Background• Tri-Power Quarticity

• Z Tri-Power Max Statistic

– Significance Value .999 z > 3.09

Mathematical Background• Heterogeneous autoregressive realized

variance (HAR-RV) Model with daily, weekly, and monthly periods:

• Daily open-close log returns (ri)

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

S&P 5005 min Price Data

High: 1443.7• 12/15/2006

Low: 768• 10/10/2002

Verizon Communications (VZ)5 min Price Data

High: 57.40• 7/19/2001

Low: 26.16• 7/24/2002

AT&T (T)5 min Price Data

High: 43.95• 7/12/2001

Low: 13.50• 4/16/2003

Walt Disney Inc. (DIS)5 min Price Data

High: 34.88• 12/19/2006

Low: 13.15• 8/8/2002

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

Summary StatisticsS&P 500 Verizon AT&T Walt Disney M&T Average

Mean

(x 1e-4)

Std. Dev.

Mean

(x 1e-4)

Std. Dev.

Mean

(x 1e-4)

Std. Dev.

Mean

(x 1e-4)

Std. Dev.

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 .0006 2.320 .0005

RV .8724 .0001 2.398 .0003 3.076 .0004 3.267 .0005 2.914 .0004

RS .4147 .0001 1.211 .0003 1.504 .0003 1.489 .0004 1.401 .0004

upRS .4587 .0000 1.187 .0002 1.572 .0003 1.777 .0004 1.512 .0003

BV .8222 .0001 2.259 .0003 2.918 .0004 3.065 .0004 2.747 .0004

• 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?

Summary Statistics

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

BV_SP 0.0409 0.5562 0.9710 0.5546 0.6593 1.0000 upRS_SP 0.5562 0.4506 0.7014 -0.2156 1.0000 RS_SP -0.5681 0.2564 0.5449 1.0000 RV_SP 0.0629 0.5742 1.0000 r2_SP 0.1274 1.0000 r_SP 1.0000 r_SP r2_SP RV_SP RS_SP upRS_SP BV_SP

(obs=1514). corr r_SP r2_SP RV_SP RS_SP upRS_SP BV_SP

Correlation - Verizon

• upRS highest correlation with daily returns amongst all coefficients for all firms

BV_VZ 0.0354 0.6084 0.9850 0.6313 0.5882 1.0000 upRS_VZ 0.6263 0.4133 0.6082 -0.2330 1.0000 RS_VZ -0.5630 0.3612 0.6302 1.0000 RV_VZ 0.0406 0.6248 1.0000 r2_VZ 0.1054 1.0000 r_VZ 1.0000 r_VZ r2_VZ RV_VZ RS_VZ upRS_VZ BV_VZ

(obs=1491). corr r_VZ r2_VZ RV_VZ RS_VZ upRS_VZ BV_VZ

Correlation - AT&T

BV_ATT 0.0228 0.4478 0.9845 0.5665 0.6564 1.0000 upRS_ATT 0.5352 0.2783 0.6731 -0.2259 1.0000 RS_ATT -0.5589 0.2994 0.5683 1.0000 RV_ATT 0.0277 0.4623 1.0000 r2_ATT -0.0121 1.0000 r_ATT 1.0000 r_ATT r2_ATT RV_ATT RS_ATT upRS_ATT BV_ATT

(obs=1486)corr r_ATT r2_ATT RV_ATT RS_ATT upRS_ATT BV_ATT

• 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

Correlation - Disney

• Largest correlation magnitude discrepancy between returns and RS, upRS

BV_DIS 0.0551 0.5959 0.9763 0.6010 0.6574 1.0000 upRS_DIS 0.5732 0.3896 0.5945 -0.1761 1.0000 RS_DIS -0.4935 0.3884 0.6869 1.0000 RV_DIS 0.0200 0.6049 1.0000 r2_DIS 0.1343 1.0000 r_DIS 1.0000 r_DIS r2_DIS RV_DIS RS_DIS upRS_DIS BV_DIS

(obs=1492). corr r_DIS r2_DIS RV_DIS RS_DIS upRS_DIS BV_DIS

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

• 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)

Regression

HAR-RV Regression – S&P 500

• R2 = .6631

• Monthly lag not significant

_cons 4.77e-06 2.19e-06 2.18 0.030 4.68e-07 9.06e-06 l22RV_SP .0198499 .0578903 0.34 0.732 -.0937041 .1334039 l5RV_SP .7230236 .1465312 4.93 0.000 .4355971 1.01045 L1. .2080566 .1017968 2.04 0.041 .0083784 .4077347 RV_SP RV_SP Coef. Std. Err. t P>|t| [95% Conf. Interval] Newey-West

Prob > F = 0.0000maximum lag: 60 F( 3, 1509) = 746.70Regression with Newey-West standard errors Number of obs = 1513

. newey RV_SP l1.RV_SP l5RV_SP l22RV_SP, lag(60)

HAR-RV Regression - Verizon

_cons 7.75e-06 4.20e-06 1.85 0.065 -4.86e-07 .000016 l22RV_VZ -.0436835 .03895 -1.12 0.262 -.1200862 .0327193 l5RV_VZ .9943557 .1209524 8.22 0.000 .7571002 1.231611 L1. .0193481 .0989888 0.20 0.845 -.1748246 .2135207 RV_VZ RV_VZ Coef. Std. Err. t P>|t| [95% Conf. Interval] Newey-West

Prob > F = 0.0000maximum lag: 60 F( 3, 1486) = 1607.43Regression with Newey-West standard errors Number of obs = 1490

. newey RV_VZ l1.RV_VZ l5RV_VZ l22RV_VZ, lag(60)

• R2 = .7441

• Monthly and (especially) daily lags not significant

HAR-RV Regression - AT&T

_cons .000018 9.18e-06 1.96 0.050 -3.17e-08 .000036 l22RV_ATT .0405617 .0485951 0.83 0.404 -.0547608 .1358842 l5RV_ATT .7426145 .1922035 3.86 0.000 .3655944 1.119635 L1. .1635275 .1437578 1.14 0.256 -.1184632 .4455181 RV_ATT RV_ATT Coef. Std. Err. t P>|t| [95% Conf. Interval] Newey-West

Prob > F = 0.0000maximum lag: 60 F( 3, 1481) = 529.03Regression with Newey-West standard errors Number of obs = 1485

. newey RV_ATT l1.RV_ATT l5RV_ATT l22RV_ATT, lag(60)

• R2 = .5606

• Monthly and daily lags not significant

HAR-RV Regression - Disney

_cons .0000225 .0000122 1.84 0.065 -1.43e-06 .0000463 l22RV_DIS -.0237027 .0633382 -0.37 0.708 -.1479444 .1005389 l5RV_DIS .8151953 .1570954 5.19 0.000 .5070432 1.123347 L1. .1425726 .0743647 1.92 0.055 -.0032981 .2884434 RV_DIS RV_DIS Coef. Std. Err. t P>|t| [95% Conf. Interval] Newey-West

Prob > F = 0.0000maximum lag: 60 F( 3, 1487) = 1211.38Regression with Newey-West standard errors Number of obs = 1491

. newey RV_DIS l1.RV_DIS l5RV_DIS l22RV_DIS, lag(60)

• R2 = .6637

• Monthly and (less so) daily lags not significant

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

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

Combined Regression – S&P 500

_cons 3.12e-19 1.30e-19 2.39 0.017 5.61e-20 5.67e-19 l22upRS_SP 3.64e-14 1.59e-14 2.28 0.023 5.09e-15 6.77e-14 L1. -1.86e-14 7.14e-15 -2.60 0.009 -3.26e-14 -4.55e-15 --. 1 2.83e-14 . 0.000 1 1 upRS_SP RS_SP 1 2.94e-14 . 0.000 1 1 l22RV_SP -2.68e-14 1.34e-14 -1.99 0.047 -5.32e-14 -4.04e-16 l5RV_SP 5.52e-14 3.16e-14 1.75 0.081 -6.83e-15 1.17e-13 L1. 2.51e-14 9.38e-15 2.68 0.008 6.70e-15 4.35e-14 RV_SP RV_SP Coef. Std. Err. t P>|t| [95% Conf. Interval] Newey-West

Prob > F = .maximum lag: 60 F( 0, 1505) = .Regression with Newey-West standard errors Number of obs = 1513

note: l5upRS_SP dropped because of collinearitynote: l22RS_SP dropped because of collinearitynote: l5RS_SP dropped because of collinearitynote: L.RS_SP dropped because of collinearity> P l1.upRS_SP l5upRS_SP l22upRS_SP, lag(60). newey RV_SP l1.RV_SP l5RV_SP l22RV_SP RS_SP l1.RS_SP l5RS_SP l22RS_SP upRS_S

Combined Regression – Verizon

_cons -4.34e-19 6.04e-19 -0.72 0.473 -1.62e-18 7.51e-19 upRS_VZ 1 1.14e-14 . 0.000 1 1 l22RS_VZ -9.89e-14 3.97e-14 -2.49 0.013 -1.77e-13 -2.11e-14 L1. -9.40e-15 3.51e-15 -2.68 0.007 -1.63e-14 -2.52e-15 --. 1 7.91e-15 . 0.000 1 1 RS_VZ l22RV_VZ 6.14e-14 2.40e-14 2.56 0.011 1.43e-14 1.09e-13 l5RV_VZ -4.25e-14 1.83e-14 -2.33 0.020 -7.83e-14 -6.64e-15 L1. 1.26e-14 5.16e-15 2.45 0.014 2.51e-15 2.28e-14 RV_VZ RV_VZ Coef. Std. Err. t P>|t| [95% Conf. Interval] Newey-West

Prob > F = .maximum lag: 60 F( 0, 1482) = .Regression with Newey-West standard errors Number of obs = 1490

note: l22upRS_VZ dropped because of collinearitynote: l5upRS_VZ dropped because of collinearitynote: L.upRS_VZ dropped because of collinearitynote: l5RS_VZ dropped because of collinearity> VZ l1.upRS_VZ l5upRS_VZ l22upRS_VZ, lag(60). newey RV_VZ l1.RV_VZ l5RV_VZ l22RV_VZ RS_VZ l1.RS_VZ l5RS_VZ l22RS_VZ upRS_

Combined Regression – AT&T

.

_cons .0000142 .0000105 1.35 0.178 -6.47e-06 .0000348 L1. -.0494867 .1187391 -0.42 0.677 -.2824017 .1834283 upRS_ATT l22RS_ATT .2283878 .1186433 1.92 0.054 -.0043393 .4611148 l22RV_ATT -.0554209 .0470517 -1.18 0.239 -.1477161 .0368744 l5RV_ATT .7367683 .1975402 3.73 0.000 .3492795 1.124257 L1. .1901843 .1542099 1.23 0.218 -.1123091 .4926777 RV_ATT RV_ATT Coef. Std. Err. t P>|t| [95% Conf. Interval] Newey-West

Prob > F = 0.0000maximum lag: 60 F( 5, 1479) = 399.87Regression with Newey-West standard errors Number of obs = 1485

note: l22upRS_ATT dropped because of collinearitynote: l5upRS_ATT dropped because of collinearitynote: l5RS_ATT dropped because of collinearitynote: L.RS_ATT dropped because of collinearity> upRS_ATT l5upRS_ATT l22upRS_ATT, lag(60). newey RV_ATT l1.RV_ATT l5RV_ATT l22RV_ATT l1.RS_ATT l5RS_ATT l22RS_ATT l1.

Combined Regression - Disney

_cons -4.34e-19 5.46e-19 -0.79 0.427 -1.50e-18 6.37e-19 upRS_DIS 1 2.94e-15 . 0.000 1 1 l22RS_DIS -4.77e-14 2.19e-14 -2.18 0.029 -9.06e-14 -4.75e-15 L1. -1.01e-14 4.69e-15 -2.15 0.032 -1.93e-14 -8.97e-16 --. 1 6.47e-15 . 0.000 1 1 RS_DIS l22RV_DIS 1.83e-14 7.92e-15 2.31 0.021 2.76e-15 3.38e-14 l5RV_DIS -1.71e-15 2.24e-15 -0.76 0.447 -6.10e-15 2.69e-15 L1. 4.54e-15 1.80e-15 2.52 0.012 1.01e-15 8.07e-15 RV_DIS RV_DIS Coef. Std. Err. t P>|t| [95% Conf. Interval] Newey-West

Prob > F = .maximum lag: 60 F( 0, 1483) = .Regression with Newey-West standard errors Number of obs = 1491

note: l22upRS_DIS dropped because of collinearitynote: l5upRS_DIS dropped because of collinearitynote: L.upRS_DIS dropped because of collinearitynote: l5RS_DIS dropped because of collinearity> _DIS upRS_DIS l1.upRS_DIS l5upRS_DIS l22upRS_DIS, lag(60). newey RV_DIS l1.RV_DIS l5RV_DIS l22RV_DIS RS_DIS l1.RS_DIS l5RS_DIS l22RS

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

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