1. Further Random Walk Tests Fin250f: Lecture 4.2 Fall 2005 Reading: Taylor, chapter 6.1, 6.2, 6.5, 6.6, 6.7

2. Outline  Size and power  More RW tests Multiple tests Runs tests BDS tests and chaos/nonlinearity  Size and power revisited  Sources of minor dependence

3. Size and Power  Type I error Probability of rejecting RW null when RW is true  Type II error Probability of accepting RW null when RW is false

4. Size  Significance level = type I error probability  5% sig level Prob of rejecting RW walk given it is true is 0.05 Most tests adjusted for correct size

5. Power  Power = 0.90 against x  Probability of rejecting RW when true process is x = 0.90  Depends on x  Problem for RW tests Power might be low for some alternatives x

6. Small Samples  Many RW tests are asymptotic meaning the size levels are only true for very large samples  Might be different for small samples

7. Multiple Tests  Use some of the tests we’ve used and design them for multiple stats  Examples Autocorrelations Variances ratios  Need to use Monte-carlo (or bootstrap) to determine test size level  multiacf  Try this with a variance ratio test  Could join many tests together (If you are interested see 6.3)

8. Runs Tests

12. BDS Test and Chaos  BDS test Test for dependence of any kind in a time series This is a plus and a minus  Inspired by nonlinear dynamics and chaos

13. Chaotic Time Series  Deterministic (no noise) processes which are quite complicated, and difficult to forecast  Properties Few easy patterns Difficult to forecast far into the future(weather) Sensitive dependence to initial conditions

14. Example: Tent Map

15. Matlab Tent Example (tent.m)  Completely deterministic process  All correlations are zero  Appears to be white noise to linear tests

16. Brock/Dechert/Scheinkman Test

17. Simple Intuition  Probability x(t) is close to x(s) AND x(t+1) is close to x(s+1)  If x(t) is IID then Prob(A and B) = Prob(A)Prob(B)

18. BDS Test Statistic

19. Matlab Examples  BDS  Distributions Asymptotic Bootstrap/monte-carlo  Matlab code:  Advantages/disadvantages

20. Size and Power Revisited Taylor alternative model

21. Taylor Rejections  See table 6.3, columns 3 and 4  Often pretty low power (<50 percent)  Best 20 lag variance ratio Trend test  BDS test Picks up volatility Rescaled returns -> low power

22. Sources of Minor Dependence  Changes in expected returns due to volatility changes  Bid/ask spreads  Price limits  Random data errors  (All not that big.)