In ope-l 5560 and 5571, respectively, Paul Cockshott and Allin Cottrell offer
similar counterarguments against my attempt to demonstrate, sans a sectoral
size variable, that aggregate sectoral price-value correlations are
meaningless. Both say that my assumption of uniformly distributed random
price-value ratios implies a "rectangular distribution." Paul then alleges
that the assumption of a "rectangular distribution" assumes that prices and
values are positively correlated. Allin seems to hint at the same thing, and
suggests that the "rectangular distribution" implies an unreasonably low upper
bound on the price-value ratio.
I am unable to respond to most of this, because I don't know what is meant by
a "rectangular distribution." I therefore also don't understand why the
assumption of a rectangular distribution implies that prices and values are
correlated. Would you be able to explain these things in simple English for
me (and perhaps some other folks who also don't know this terminology?)
But, in the meantime, there is still something I can respond to. Paul gave me
an interesting idea. He suggested thinking of the price-value ratio, not as a
single number, but as "the ratio of two random variables." This gave me the
following idea: compute the price-value ratio (r) from two sets of numbers
that are perfectly NEGATIVELY correlated.
Thus, instead of perhaps unknowingly assuming that price and value are
positively correlated (the error I've allegedly committed), I have knowingly
assumed that price and value are negatively correlated.
The procedure was this. I generated 50 values for one variable, X, by
assuming it is an evenly distributed random number between 0 and 1. Then I
constructed a second variable, Y, according to the following rules:
Y(n+1) = Y(n) - [X(n+1) - X(n)]; (n = 1, 2,..., 50);
Y(1) = 1.
For instance, if the first X "observation," X(1), is .42, and if X(2) = .52,
then Y(2) = 1 - (.52 - .42) = .9. The function of the second rule is to
ensure that each Y(n) is positive. The function of the first rule is to yield
a linear correlation between X and Y of -1, perfect negative correlation.
(The funny thing is, it took me about 5 times as long to explain this as it
did to generate the series.)
I then constructed 50 values for the price-value ratio r(n) = X(n)/Y(n).
Finally, as in my earlier simulation, I constructed a series of 50 sectoral
aggregate values ranging randomly from 1 to 50, and then computed each
sectoral aggregate price by multiplying r(n) against the corresponding
sectoral aggregate value. I did 20 runs of this, and computed the correlation
between sectoral aggregate value and sectoral aggregate price each time.
The mean of the 20 trials' coefficients of variation (ratio of standard
deviation to mean) of r was 96.0%. The 20 aggregate price-value correlations
ranged from a low of +.18 to a high of +.66, with a mean of +.46.
Andrew Kliman