Comparing Constant and Time-varying Beta

1. Comparing Constant and Time-varying Beta Angela Ryu Economics 201FS Honors Junior Workshop: Finance Duke University April 28, 2010

2. Data • BAC, CPB, IBM, JPM, LOW, WMT, XOM, XRX • SPY • Data from Aug 23 2004 – Jan 7 2009 (1093 days) • Sampling interval: 1 – 20 min. • Beta trailing window: 1 – 200 days Notation: • (s, w) = (Sampling interval, Beta window) • gMin(s,w) = (s,w) that yields global minimum MSE for given stock

3. Summary of Previous Work • Goal: Empirically find an optimal (s,w) that yields minimum MSE. • Found gMin(s,w) of each stocks, w varying from 1 – 50. Relatively short span for total 1093 days of data, so extension of windows is necessary. • Observed that gMin(s,w) is at higher frequency (1,2 min interval) with relatively short trailing window.(Given short interval s, longer w than w of gMin yielded higher MSEs.) • Concluded that there is an optimality in calculating Beta with recent data of high frequency despite Microstructure noise.

4. Additional Computation • Extend the range of w to 1- 200, plot 3D distribution of MSE and find gMin(s,w) • Plot Realized Beta of each stock with s, w of gMin. (Anderson, Bollerslev, Diebold and Wu, 2003) • Compare MSE levels using 1) constant beta and 2) time-varying beta, attained from gMin(s,w)

5. 3D plot

6. Global Minimum of MSE • (sampling frequency, Beta trailing window) • 20 min. sample was used as a standard* *Standard data: for each stock, MSEs were calculated w.r.t. the stock data of a fixed sampling interval** in order to have their levels comparable

7. Time-series plot of Realized Beta BAC CPB IBM JPM LOW WMT XOM XRX • Used 2 min., 15 days • 1078 (=1093 – 15) observations

8. MSE Comparison: Constant and Time-varying Beta • βc: Biannual beta with daily returns; • β: Varying beta of each window: avg Global minMSE point (2, 15) * 10(-3) • MSE with varying beta values are lower for all stocks

9. Results • Unlike our intuition, despite microstructure noise, global minimum of MSEs of all pairs of (s, w) appeared at the highest frequency. • More days in the trailing window did not compensate the loss of data due to lower sampling frequency, even longer than half a year (w >126) • MSE levels was lower in time-varying beta case, but choosing the constant beta was made arbitrarily which remains as a problem

10. Conclusion • Using high-frequency data, we empirically found a time-varying beta of some sampling frequency and trailing window that performs better than a constant beta in terms of the level of MSE • Even in obvious presence of microstructure noise, high frequency with some threshold range of trailing window gave the least MSE, suggesting that high-frequency data gives a relatively better prediction of beta • However, using time-varying beta may hurt the prediction even more than does using a constant beta if misspecified(Ghysels, 1998) ; thus comparison with constant beta should be made more carefully.

11. Final Concerns • Comparing the 95% CI interval bandwidth for Realized Beta with the results from Anderson, Bollerslev, Diebold and Wu, 2003(?) • Beta of Yahoo/Google finance? • AR(p)