Download - 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 powerMore RW tests
Multiple tests Runs tests BDS tests and chaos/nonlinearity
Size and power revisitedSources of minor dependence
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Size and Power
Type I error Probability of rejecting RW null when RW is
trueType 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 xProbability of rejecting RW when true
process is x = 0.90Depends on xProblem 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 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)
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Runs Testsq t =sign(rt)
ct =1:qt ≠qt+1
0 :qt =qt+1
⎧⎨⎩
C =1+ ctt=1
n−1
∑
E(C) =n+1−1n
nj2
j=1
3
∑
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Runs Tests
Example :n1 =n2 =12
n
E(C) =n+1−1n
(n2)2
j=1
2
∑ =n2+1
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Runs Tests
E(C) =n+1−1n
nj2
j=1
3
∑
var(C) = nj2
j=1
3
∑ (n+n2 +⎧⎨⎩
nj2
j=1
3
∑ )−
n3 −2n nj3
j=1
3
∑⎫⎬⎭/ (n3 −n)
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Runs Tests
K =
(C−E(C))var(C)
: N(0,1)
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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
⎩⎨⎧
≥−<
=5.0:22
5.0:2
tt
tt
t xx
xxx
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Matlab Tent Example (tent.m)
Completely deterministic processAll correlations are zeroAppears to be white noise to linear tests
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Brock/Dechert/Scheinkman Test
€
I(xs , xt ,ε ) =1:| xs − xt |< ε
0 : otherwise ⎧ ⎨ ⎩
Im (xs , xt ,ε ) = I(xs+k , xt+k ,ε )k=0
m−1
∏
Cm (xs , xt ,ε ) =2
(n −m)(n −m +1)Im (xs , xt ,ε
t=s+1
n−m+1
∑s=1
n−m
∑ )
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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)
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BDS Test Statistic
€
dm = max j=0,m−1 | x t+ j − x s+ j |
Cm (n,ε ) = Pr(dm (x t , x s ) < ε ))
Cm (n,ε ) =C1(n,ε )m
W =Cm (n,ε ) −C1(n,ε )m
Vm /n
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Matlab Examples
BDSDistributions
Asymptotic Bootstrap/monte-carlo
Matlab code:Advantages/disadvantages
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Size and Power RevisitedTaylor alternative model
€
rt = μ t +σ tε t ,1μ t = 0.95μ t−1 + ε t ,2log(σ t ) = α + 0.973(log(σ t−1) −α ) + ε t ,3var(μ t ) = 0.02 var(rt )
var(log(σ t )) = 0.422
α = −5.15
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Taylor Rejections
See table 6.3, columns 3 and 4Often pretty low power (<50 percent)Best
20 lag variance ratio Trend test
BDS test Picks up volatility Rescaled returns -> low power
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Sources of Minor Dependence
Changes in expected returns due to volatility changes
Bid/ask spreadsPrice limitsRandom data errors(All not that big.)