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Further Random Walk Tests

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

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Further Random Walk Tests

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

  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

  9. Runs Tests

  10. Runs Tests

  11. 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

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