Further Random Walk Tests Fin250f: Lecture 4.2 Fall 2005 Reading: Taylor, chapter 6.1, 6.2, 6.5, 6.6, 6.7
Outline Size and power More RW tests Multiple tests Runs tests BDS tests and chaos/nonlinearity Size and power revisited Sources of minor dependence
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
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
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
Small Samples Many RW tests are asymptotic meaning the size levels are only true for very large samples Might be different for small samples
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)
Runs Tests
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
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
Example: Tent Map
Matlab Tent Example (tent.m) Completely deterministic process All correlations are zero Appears to be white noise to linear tests
Brock/Dechert/Scheinkman Test
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)
BDS Test Statistic
Matlab Examples BDS Distributions Asymptotic Bootstrap/monte-carlo Matlab code: Advantages/disadvantages