1. Random Walk Tests and Variance Ratios Fin250f: Lecture 4.1 Fall 2005 Reading: Taylor, chapter 5.1-5.6

2. Outline  Variance ratios  Autocorrelation sampling theory

3. Types of Random Walks  e(t) IID: No volatility persistence  e(t): expectation zero, zero correlation No linear predictors Might be nonlinear predictors Allows for volatility prediction

4. Random Walks and Market Efficiency  Classic implications Price forecasting hopeless Technical analysis useless  Modern thoughts/reminders Dynamic strategies that  Increase expected returns  Increase risk  Still consistent with market efficiency

5. Variance Ratio Tests

7. Variance Ratios: Random Walk Test  Test: VR(N)=1  Two problems Distribution of VR? Which N?

8. Distribution of VR(N): Asymptotic

9. Matlab Examples  vratio  vratiotest  nmcvratio  bsvratio

10. Longer Time Horizons  Weekly Lo and Mackinlay(1988) Strong rejections on weekly equal weighted index (not value weighted) Few rejections for individual stocks Stale prices and nontrading?

11. Longer Time Horizons  Monthly Poterba and Summers(1988) Weak positive correlations (not sig)  Annual Weak negative long range correlations (not sig)

12. Autocorrelations  For r(t) IID  Autocorrelations are asymptotically distributed N(0, 1/n)  n=sample size  95% confidence bands +-1.96/sqrt(n) pacf.m

13. Autocorrelations: Small sample issues

14. Autocorrelations: Variance