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 powerMore RW tests

Multiple tests Runs tests BDS tests and chaos/nonlinearity

Size and power revisitedSources of minor dependence

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

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 xProbability of rejecting RW when true

process is x = 0.90Depends on xProblem 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 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

∑

Runs Tests

Example :n1 =n2 =12

n

E(C) =n+1−1n

(n2)2

j=1

2

∑ =n2+1

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)

Runs Tests

K =

(C−E(C))var(C)

: N(0,1)

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

⎩⎨⎧

≥−<

=5.0:22

5.0:2

tt

tt

t xx

xxx

Matlab Tent Example (tent.m)

Completely deterministic processAll correlations are zeroAppears to be white noise to linear tests

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

∑ )

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

€

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

Matlab Examples

BDSDistributions

Asymptotic Bootstrap/monte-carlo

Matlab code:Advantages/disadvantages

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

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

Sources of Minor Dependence

Changes in expected returns due to volatility changes

Bid/ask spreadsPrice limitsRandom data errors(All not that big.)