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Dr. Alex Castaldo

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.