Further Random Walk Tests

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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

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Outline

• Size and power
• More RW tests
• Multiple tests
• Runs tests
• BDS tests and chaos/nonlinearity
• Size and power revisited
• Sources of minor dependence

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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

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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

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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

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Small Samples

• Many RW tests are asymptotic meaning the size
  levels are only true for very large samples
• Might be different for small samples

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Multiple Tests

• Use some of the tests weve 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)

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Runs Tests
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Runs Tests
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Runs Tests
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Runs Tests
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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

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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

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Example Tent Map
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Matlab Tent Example (tent.m)

• Completely deterministic process
• All correlations are zero
• Appears to be white noise to linear tests

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Brock/Dechert/Scheinkman Test
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Simple Intuition

• Probability x(t) is close to x(s) AND x(t1) is
  close to x(s1)
• If x(t) is IID then Prob(A and B) Prob(A)Prob(B)

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BDS Test Statistic
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Matlab Examples

• BDS
• Distributions
• Asymptotic
• Bootstrap/monte-carlo
• Matlab code
• Advantages/disadvantages