In ope-l 5556, Allin wrote,
"As a footnote, it seems to me that if Andrew wants to go on referring to
sectoral price-value correlations as "meaningless" (on the alleged grounds of
spurious correlation) -- rather than arguing merely that they are not the
_best_ measure of association -- then he owes us a response to the
counter-argument that I've posted to ope a couple of times now."
I'll be glad to respond. Thought you'd never ask.
In ope-l 5497, Allin first tells it like it is:
"Suppose we take three vectors of non-negative random numbers, x, y and z. The
elements of each vector are drawn independently, and the expected correlation
between any pair is zero. Now we form two new vectors, u and v, as follows:
the kth element of u is the product of the kth element of x and the kth
element of z, while v(k) = y(k)*z(k). Now, what is the expected value of the
correlation between u and v? It is positive, due to the common factor
provided by the action of z on both x and y. If x and y started out
positively correlated, this correlation will be magnified by the
transformation."
In other words, assume that unit prices (x), unit values (y) and industry
sizes (z) are not correlated. Still, aggregate price (u = xz) and aggregate
value (v = yz) will be positively correlated. And in general, the positive
correlation between aggregate price and aggregate value (xz and yz) will be
greater than that between x (unit price) and y (unit value).
Yet, Allin then objects, industry size does not exist. "The trouble with this
notion is that the z vector doesn't exist" because the units of measurement
are arbitrary -- "bottles, fluid ounces, cases, gallons."
Here's my response:
(1) I'm take it that Allin and Paul Cockshott agree that sectoral aggregate
price and aggregate value do exist.
(2) So, divide each sector's aggregate price by its aggregate value, to obtain
the price/value ratio. That is, divide u(k) by v(k) to obtain r(k) =
u(k)/v(k).
(3) Next, assume that the price/value ratios (r) are wholly random. In any
individual case, price can be very high relative to value or very low. To
approximate this, assume that r is a random number evenly distributed between,
say, 0 and 1,000,000.
(4) Now, note that the aggregate price in sector k is equal to some random
number r(k) times aggregate value. That is, u(k) = r(k)*v(k).
(5) Note also that each aggregate value, v(k) can be written as 1*v(k).
(6) If we now measure the correlation between aggregate prices u(k) and
aggregate values v(k), we are measuring the correlation between r(k)*v(k) and
1*v(k).
(7) Obviously, the vectors r and 1 are uncorrelated. Yet "what is the
expected value of the correlation between u and v? It is positive, due to the
common factor provided by the action of [v] on both [r] and [1]."
(8) I've used Allin's own words, and made the appropriate substitution of
terms, in order to call attention to the fact that what we now have is
precisely analogous to what we had before. (Only now, instead of the
offending z variable, which Allin says does not exist, we have the aggregate
value vector v, which does exist.)
That is, we have two products, the first terms of which are uncorrelated, but,
since the second term of each product is one and the same variable, the
products themselves are positively correlated. In general, the larger v(k)
is, the larger u(k) is, and the smaller v(k) is, the smaller u(k) is -- EVEN
THOUGH ANY GIVEN SECTOR'S AGGREGATE PRICE VARIES RANDOMLY FROM 0 TO 1 MILLION
TIMES ITS AGGREGATE VALUE!!
(9) To illustrate the point, I constructed 50 sectors. The aggregate value
v(k) of each is an evenly distributed random number between 1 and 50. I also
assigned to each sector the random price/value ratio r(k), which again ranges
from 0 to 1 million. Multiplying r(k) by v(k) gave me the sector's aggregate
price. I then computed the correlation between aggregate value and aggregate
price for the 50 sectors, running 20 trials. The lowest correlation
coefficient of the 20 was 0.47 and the highest was 0.81. The mean correlation
coefficient was 0.66.
(10) I think the foregoing allows me to "go on referring to sectoral
price-value correlations as 'meaningless'" and that it should cause everyone
else to do the same.
Andrew Kliman
P.S. I hope to get to the remaining points in Allin's post, and the other
responses to me, in due course.