[OPE-L:5610] In defence of correlation

Allin Cottrell (cottrell@wfu.edu)
Wed, 15 Oct 1997 16:55:32 -0400 (EDT)

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My thinking on the following is not "set in stone", and it
is offered in a spirit of inquiry. It represents a slightly
oblique response to points raised by both Andrew and Alan a
couple of weeks ago; I still hope to find time to respond
more directly to some of the specific points they made.

Let's take it that the basic measurement obtained from
input-output studies is the price-to-value ratio for each of
the sectors of the economy, i.e. the number of pounds
(dollars, etc.) that one unit of embodied labour time
exchanges for. Multiplying up the total value product of
each sector by its price-to-value ratio one obtains the
aggregate sectoral price. The correlation under discussion
-- that between aggregate sectoral prices and aggregate
sectoral values -- can therefore be represented as


where r denotes the coefficient of correlation, V denotes
the sectoral values, and X denotes the sectoral
price-to-value ratios.

>From first principles, this correlation can be expressed as

r(XV,V) = Cov(XV,V)/[Var(XV)*Var(V)]^(1/2) (1)

To get a first sense of the properties of this quantity,
I'll entertain the assumption that X and V are distributed
independently. That is, there is no tendency for sectors
that are "bigger" (in value terms) to have a price-to-value
ratio that differs systematically from that of "smaller"
sectors. While I'm at it, I'll assume that X^2 and V^2 are
independent, and that X and V^2 are independent.

In that case (1) resolves to the following (where "<.>" is
used to indicate a mathematical expectation):

r(XV,V) = [(<V^2>-<V>^2)/(<X^2>*<V^2>-<V>^2)]^(1/2) (2)

The square of the right hand side of (2) is of the form

(x - k)/(b*x - k)

where: x = <V^2>, the expectation of the square of sector
size, in value terms; k = <V>^2, the square of the
expectation of sector size; and b = <X^2>, the expectation
of the square of the price-to-value ratio. The variance of
sector size is x - k. The variance of the price-to-value
ratios is

Var(X) = <X^2> - <X>^2 = b - <X>^2.

Using the standard normalization whereby total price equals
total value, <X> = 1, so <X>^2 = 1, and b = 1 + Var(X).

Now consider the behavior of (x - k)/(b*x - k) -- i.e. the
square of the correlation coefficient at issue -- in
consequence of changes in (a) the variance or dispersion of
price-to-value ratios across the sectors, and (b) the
variance or dispersion of sectoral "sizes".

(a) Given x and k, the correlation coefficient will be
smaller, the larger is b, i.e. the larger is the dispersion
of price-to-value ratios. As b tends to its minimum of 1
(degenerate distribution of price-to-value ratios), the
correlation tends to 1 (provided Var(X) > 0, else the
correlation is undefined). As b tends to infinity
(infinitely great dispersion of price-to-value ratios), the
correlation tends to zero.

(b) Given b, the correlation coefficient will be larger, the
greater is the dispersion of sector sizes (i.e. the greater
is x for given k)... up to a point: in the limit, as x goes
to infinity (infinitely wide dispersion of sector sizes),
the correlation coefficient tends to the square root of 1/b.
For example, if the variance of the price-to-value ratios is
2, then b = 3 and the correlation coefficient has an
asymptotic maximum of (1/3)^(1/2) = 0.577, as Var(V) goes to

Point (a) above means that so long as Var(V) is given -- as
it is in any given input-output table -- the sectoral
price-value correlation is monotonically related to the
dispersion of sectoral price-to-value ratios. Point (b) is,
I think, what troubles Alan. It means that r(XV,V) does not
_just_ reflect the dispersion of price-to-value ratios; it
is "contaminated" by the dispersion of sector sizes.

Is this really a case of contamination? I assumed above
that X and V were distributed independently. This
assumption simplified the math, but it was not "innocent".
Think about it: to assume independence -- specifically, to
suppose that there is no tendency for sectors that are
bigger in value terms to have a lower price-to-value ratio
-- is precisely to suppose that a larger denominator in a
sector's price-to-value ratio will tend to be associated
with a larger numerator, or in other words that price and
value are correlated across the sectors.

What does this mean? Well, imagine, if you will, two
economies A and B: each has N sectors, and the distribution
of the price-to-value ratios for these sectors is identical
in A and B. On Alan's reckoning, the economies should then
be reckoned equally "good" (or equally "bad" as the case may
be) as exemplars of price-to-value correspondence. Suppose
now that the distribution of sector sizes (by value) differs
across the economies: in A the sectors are all of similar
size; in B there is a wide dispersion of sector sizes. As a
matter of arithmetic, the sectoral price-value correlation
will be greater for B. Alan holds that this is spurious: it
would be quite wrong to infer that B is a "better" case in
point for the labour theory of value. My objection is this:
if one drops the prior assumption of independence in the
distributions of sector sizes and price-to-value ratios
there is a reasonable case for saying that economy B (with
its larger price-value correlation) offers stronger
confirmation for the labour theory of value than does A. If
_sectoral values and prices_ were distributed independently,
then one would expect to see the greater dispersion of
sector sizes in B reflected in a greater dispersion of
sectoral price-to-value ratios. That one does _not_ see
this suggests the presence of stronger forces limiting
price-value dispersion in B.

[Suppose one saw two similar masses both accelerating at the
same rate (cf. two economies with the same variance of
price-to-value ratios). Other things equal one would
suppose they were subject to the forces of the same
magnitude. But if one were moving through air and the other
through treacle one would have to revise that opinion.]

I'm not claiming that correlation is a preferred alternative
to looking directly at the distribution of price-to-value
ratios (Paul Cockshott and I have done a lot of the latter
in some of our papers). But I do want to argue that
correlation analysis has a legitimate place in the
price-value discussion.

Allin Cottrell
Department of Economics
Wake Forest University