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40 THE HANDBOOK OF PORTFOLIO MATHEMATICS
low) and trade number 5is a turningpoint (high or low, so longasit’s the
opposite of what the last turningpoint was), then the phase lengthis1,
since the difference between 5 and 4is1.
With the phase length test you add up the number of phases of length
1, 2, and 3 or more.Therefore, you will have three categories:1, 2, and 3+.
Thus, phase lengths of 4 or 5, and so on, are all totaled under thegroup
of 3+.It doesn’t matterif a phasegoes from a high turningpoint to a low
turningpoint or from a low turningpoint to a high turningpoint;the only
thingthat mattersis how many trades the phaseis comprised of.To figure
the phase length, simply take the trade number of the latter phase (what
numberitisin sequence from 1 to N, where Nis the total number of trades)
and subtract the trade number of the prior phase.For each of the three
categories you will have the total number of complete phases that occurred
between (but notincluding) the first and the last trades.
Each of these three categories also has an expected number of trades
for that category.The expected number of trades of phase length Dis:E(D)= 2 ∗(N−D−2)∗(D∧ 2 ∗ 3 ∗D+1)/(D+3)! (1.08)where: D=The length of the phase.
E(D)=The expected number of counts.
N=The total number of trades.Once you have calculated the expected number of counts for the three
categories of phase length (1, 2, and 3+), you can perform the chi-square
test.Accordingto Kendall and colleagues,^3 you should use 2.5degrees of
freedom herein determiningthe significance levels, as the lengths of the
phases are notindependent.Remember that the phase length test doesn’t
tell you about the dependence (like begettinglike, etc.), but rather whether
or not thereis dependence or randomness.
Lastly, thisdiscussion of dependence addresses convertinga correla-
tion coefficient to a confidence limit.The technique employs whatis known
asFisher’s Z transformation,which converts a correlation coefficient, r,
to a Normally distributed variable:F=. 5 ∗ln(1+r)/(1−r)) (1.09)where: F=The transformed variable, now Normally distributed.
r=The correlation coefficient of the sample.
ln( )=The natural logarithm function.(^3) Kendall, M.G.,A.Stuart, and J.K.Ord.The Advanced Theory of Statistics,Vol.III.
New York:Hafner Publishing, 1983.