Beta Estimate of High Frequency Data

Angela Ryu

Economics 201FSHonors Junior Workshop: Finance

Duke University

March 3, 2010

Data

• XOM (Exxon Mobile) – Dec 1 1999 – Jan 7 2009 (2264 days)

• GOOG (Google)– Aug 20 2004 – Jan 7 2009 (1093 days)

• WMT (Wal-Mart)– Apr 9 1997 – Jan 7 2009 (2921 days)

FOR ALL 3 stocks

Motivation

• Multivariate Measures: Beta• Problem of balancing bias/precision– High frequency sampling:

biased, due to microstructure noise– Low frequency sampling:

imprecise• Theoretical approach requires more

background knowledge approach empirically!

Preparation• Interday returns are excluded• Beta calculated from: (for βX = Y, X,Y stock

prices) • Sampling intervals: 1 to 20 minutes• Beta Calculation intervals: 1 to 50 days• Mean Squared Error calculated for each Beta interval

– MSE of GOOG(X) vs. XOM(Y) , 30 days interval?= Average of Squared Errors of each days predicted by using β

i.e. ypre_day31 = βday1_30 * xact_day31 SEday31 = (ypre_day31 – yact_day31 )2

ypre_day32 = βday2_31 * xact_day32 SEday32 = (ypre_day32 – yact_day32 )2

…

MSE30 = avg(SEday31 , SEday32 , ... SEday1093 )

WMT vs. XOM (2 min.)

WMT vs. XOM (5 min.)

WMT vs. XOM (10 min.)

WMT vs. XOM (15 min.)

WMT vs. XOM (20 min.)

XOM vs. WMT (2 min.)

XOM vs. WMT (5 min.)

XOM vs. WMT (10 min.)

XOM vs. WMT (15 min.)

XOM vs. WMT (20 min.)

GOOG vs. WMT (2 min.)

GOOG vs. WMT (5 min.)

GOOG vs. WMT (10 min)

GOOG vs. WMT (15 min.)

GOOG vs. WMT (20 min.)

Results

• 5 – 15 days interval for Beta gave least MSE for many stock pairs, for most sampling intervals

• As the sampling interval increased, MSE for shorter Beta intervals increased rapidly

• For 20 min. sampling interval, there is less increase of MSE as increase in Beta interval compared to shorter sampling intervals

Analysis

• Against our intuition: why would more information harm prediction of the price?

• Possible interpretation– Given a sampling interval, after a certain range of

“information” gather for Beta estimation, say 5 – 15 days, more information distorts the prediction

– On the other hand, some short Beta intervals (e.g. 1 day, 2 days) for longer sampling intervals may be insufficient and result in high MSE

Questions & Further Steps

• Theoretical evidence? Any relevant papers?

• Is the estimator biased? Why?

• What is the role of Microstructure noise?

• Check calculations. Try with other stocks or possibly portfolios (industry/macroeconomic factors)

• Use Realized Beta and compare the results

Andersen, Bollerslev, Diebold and Wu (2003)