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2/1/2005
January Effect Cont. Part Whole Correlation, by Alex Castaldo
PWC is a dangerous fallacy which often affects
people at this time of the year.
Suppose x is the amount by which the S&P rises in January
and y is the amount by which it rises in the remaining 11
months of the year. Then there will be a correlation
between x and x+y, that is between January and the whole
year. And this will be "discovered" empirically and
reported in newsletters and research reports.
Let us estimate this spurious correlation:
Corr(x, x+y) = Covar(x,x+y) / SQRT[ Var(x) Var(x+y) ] /*
by definition */
Assuming that x and y are not related, i.e. Covar(x,y) = 0
this reduces to
Var(x) / SQRT[ Var(x) (Var(x)+Var(y)) ]
If all months are equally volatile and serially independent,
then the monthly variances add and so Var(y) = 11 Var (x).
So we get
1 / SQRT[ 1 + 11 ] = 0.288675
So we expect a nearly 30% correlation even if January has no
predictive value for the rest of the year.