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"An Essay on Aggregation Theory and Practice"
,
by
Edwin
Number 43
r
^.
Ktih
I
September 1969
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
MASS. 1N3T. TECH.
SEP 151959
DEWEY LIBRARY
"An Essay on Aggregation Theory and Practice"
by
Edwin Kuh
Number A3
This research was supported
Foundation. The views expressed
sponsibility, and do not reflect
the Department of Economics, nor
Technology.
September 1969
in part by funds from the National Science
in this paper are the author's sole rethose of the National Science Foundation,
of the Massachusetts Institute of
Summary: An Essay on Aggregation Theory and Practice
Edwin Kuh
This paper shows how aggregation can improve the efficiency of
econometric estimation.
The new theorems depend on the random coef-
ficient regression model, Theil's aggregation analysis, and assumptions
about the stability of shares held by individual economic entities in
macroeconomic totals.
Section
1
summarizes existing views held by econ-
ometricians about aggregation problems and their solution.
Sections
2
and 3 offer an exposition of existing aggregation analysis and the basic
new
theorems which show precisely how the macro parameter variances will
tend to zero as the number of elements in the aggregate increase.
A
Section
presents a simple investment model estimated from data for 105 manufac-
turing firms divided among four industries, to test underlying assumptions
and also to test directly predictions of aggregation gain generated by the
theory.
The test outcomes are sufficiently favorable so that the exist-
ence of aggregation gain and its measurability seem now to be clearly
established.
gains
Section
5
takes up the question of potential aggregation
for all Census four digit industries.
Section 6 discusses
the implications for research strategy when micro coefficients differ,
contrary to a basic assumption of the random coefficient model employed
throughout this paper.
534233
August 8, 1969
An Essay on Aggregation Theory and Practice
Edwin Kuh
1.
Introduction
One of the most perplexing problems in the design of econometric
research, and one that has remained largely unsolved is how aggregated
data can be in order to obtain valid estimates.
The main purpose of
this paper will be to show that aggregation gains not only exist, but how
one can provide operational tests of their extent.
Theil's aggregation theorems
[21] show that behavior as reflected
in postulated population regression coefficients must be nearly identical
for all members in a given population if estimates based on aggregated
data are to reflect behavior in a meaningful sense.
These implications
of Theil's analysis have recently been reinforced by Orcutt, Watts and Ed-
wards [15] whose simulation results suggest that estimates from aggregate
data generated from one particular simplified macro-economic model can be
inefficient and sometimes inconsistent, even when the micro parameters are
identical.
There exists a conflicting position, composed of three strands,
that is favorable to greater aggregation.
First, Griliches and Grunfeld [6]
showed that error variances for aggregate regression equations will be less
than the sum of the micro variances if negative error covariance terms are
sufficiently great; furthermore, costs are normally less for estimates
*
Certain special data configurations permit heterogeneous coefficients,
yet avoid what Theil calls contradictions between the micro hypothesis
and the macro hypothesis.
-2-
based on macro data.
They restricted themselves to a simplified aspect
of forecasting, neglecting altogether the problem of aggregation bias. Second
there is a metaphysical view most vigorously espoused by Friedman and Meiselman [5]
that the relevant behavior of the economy can be most efficiently rep-
resented by a single aggregated model; "truth will out" even in the
aggregates, so that complex, disaggregated models of behavior are in-
effectual ways to test macro economic propositions.
matter of faith unrelated to statistical hypotheses.
This view is a
The third prag-
matic view of perhaps a majority of practicing econometricians is that
micro data probably arS,better, but that it is either unavailable or
excessively expensive, so that all one can do practically is
use macro
data.
From a somewhat different aspect, many cross-sectional studies based
on micro data (individual firms,
families or individuals) have proven so
disappointing that many researchers prefer to avoid this data base.
The
major source of failure is the high error variances i.e. low correlations,
even though the
statistical significance tests may be highly affirm-
ative in the sense of decisively rejecting the no-relationship null
hypothesis.
Yet it is exceedingly cold comfort to find a "very sig-
nificant" multiple correlation which in a large sample may explain no
more than 5% or 10% of the dependent variables fluctuations
,
so that the
systematic component of the explanation, while significant, is negligible.
I
have shown [13] that a strong presumption exists that a component of
cross-sectional error variances can and should be imputed to individual
effects which vary across individuals but are constant over time.
basic problem of explanatory power still remains, however.
The
-3Finally, as with Goldilocks, something that seems "just right" can
be found in sub-aggregates such as two digit industries
of the labor force broken down by age, sex and skill.
,
or sub-catagories
While the last re-
mark is mere obiter dictum at this juncture, an analytical basis for it
will soon be provided: that is the central purpose of this paper.
In short,
"proper" aggregation is not merely neutral, nor just a major cost saving
(which it usually is), but proper aggregation causes major reductions in the
estimated parameter error variances, relative to the underlying, individual
error variances.
2
.
One Resolution - A Sketched Proof
Because the theorems advanced later necessitate an involved notation,
even though the results are straightforward, it is convenient to introduce
the notation here in the context of a simplified example.
analytical tools are twofold.
The basic
The first is the random coefficient model,
that was brought to the attention of practicing econometricians by Klein [10
in the context of cross section models, much as we are doing here.
]
The
complex maximum liklihood estimator proposed by Klein discouraged subsequent
application.
Nevertheless, the random coefficient model itself has much
to recommend it.
Casual introspection (the kind most of us excel in) suggests
that in making our individual consumer expenditure decisions, we do not go
through an exact calculation involving many relative prices and income and
then tack on a random error to this painstaking result; rather, we often
buy haphazardly, on impulse, even though on "average" our purchases do reflect more basic consumer preferences and economic parameters.
This latter
-4-
behavior is more compatible with a random coefficient model than it is
The random coefficient model, whether or
with the standard shock model.
not it is inspired by the motivational scheme above, also helps to rationalize the dilemma posed by the extremely low variance explanation of so
much cross-section analysis.
The basic economic behavior
is^
stable
and explicable on the average, even though it is not in one given time slice
when the variances of the individual coefficients are large.
Indeed, a re-
lated alternative explanation is the errors-in-variables model where the
argument has been made that cross-sectional consxjmption and income data
contain transient i.e. error components which will lead to biased least
The effects on error variances and estimated standard
square estimates.
errors of the regression coefficients have normally been given less attention.
C.
R.
Rao [16] has shown that ordinary least squares is an unbiased,
efficient estimator in the random coefficient case.
That article contains
the statistical basis for the application of ordinary least squares in the
aggregation context, to which we now turn.
Theil's aggregation theory [21], the second analytical device on which this
paper depe-ds, has been translated into compact matrix notation by Kloeck[ll].
I
shall reproduce the main results (not complete proofs) in a slightly extend-
ed notation.
We will suppose that there are N individual behavior equations of the form:
^i
v;here
(T
y^.
"
"^
""i^i
is a column
i = 1,
^i
2,...N
vector with T rows for the
X K) set of exogenous variables for the
parameter vector for the
i*^^
i*^^
(1)
i
individual; x^ is a
individual, Q^ is a (K x 1)
individual, and e^ is the column vector of
errors for the i^h individual.*
Corresponding to these
The time subscript is inessential in what follows, so we have
suppressed it from the outset.
-5-
micro equations Is the macro equation:
Y = X6 + e
(2)
where the variables X and Y are aggregates of the microdata
the context of Theil-type aggregation,
3.
x,
and y..
In
is defined as the mathematical ex-
pectation of the least squares estimator.
When each micro series for each exogenous
variable is regressed on all the macro exogenous variables (a useful step essential to a clear interpretation of the macro parameters in terms of the micro
parameters) the resulting regression coefficients will be called the ag-
gregation weights.
These are presented in (3).
The precise nature of
these weights will be made clear in the example which follows immediately.
ip
This
ip
p = 1, 2,...K
ip
(3)
vector obviously has K components and for each individual there
exist K such vectors.
Theil has shown (using a modification of Kloeck's notation) that a
typical macro coefficient &g is a B-weighted sum of the micro parameters,
where the aggregation weights have certain properties, e.g. a typical
macro parameter is:
6, -
j^\
WAD
- (b;«b;^>.
\2
^N l\<^)
\
^i:
N
2
I
^Jk,
\,
\6
NK
/
There would be K such equations, one for each macro parameter.
The B aggregation weights satisfy the following conditions as a direct
implication of least squares:
-6-
^
.Z.
a)
= 1 for p = £
-'
B,*-
=
In words:
for p ^ Z
the weights for corresponding parameters sum to unity, and sum
to zero for non-corresponding parameters.
Having set out much of the general notation,
I
main point of this article with a simple example.
wish to illustrate the
We wish to estimate
the parameters of the following macro equation:
Y = (XiX2)/6i\
:iX2)/6i\ + e
(6)
(7)
N
N
^2 "iSl?i
h
N
^ iSl^ii
^ii"*"
i=i^i2 ^±:
Suppose now that the micro parameters are mutually independent identically
distributed random variables.
Then the expected values of the macro
parameters:
^
(1)
E(gi) = (i^iB^i
)
E(6^j) =
Bj, and similarly for Ba-
(8)
This result follows immediately from the fact that the aggregation
weights are fixed numbers which sum to unity for corresponding parameters
and zero for noncorresponding parameters.*
Theil, in a related context,
has indicated how critical the assumption is that the x's are fixed
*
Zellner [22] proved this same proposition. A concluding section relates the material of this paper to papers by Theil [20] and Zellner [22].
-7-
variates.
It is worth quoting his observations on this matter in full:
These results indicate that there are no problems
of aggregation bias if one works with a random-coefficients
micromodel.
In fact, the macrocoef ficients contrary to
the microcoef ficients are not random at all when N is sufficiently large. Note, however, that the assumption of nonIf we select
stochastic x's is not at all innocent.
households at random, not only their 6's but also their
x's become stochasticThis does not mean that we cannot
But it does mean that
treat the x's as if they are fixed.
if we do so, we operate conditionally on the x's and we assume
implicitly that the conditional distribution of the 6's
given the x's is independent of these x's.
Thus, the analysis
is based on the condition that over the set of individuals who
are aggregated, there is stochastic independence between the
factors determining their behavior (the x's) and the way in which
they react given these factors (the 3's).*
The great convenience of the random coefficient model in the aggregation
context (initially recognized by Zellner [22]
>
see the last footnote) is
that it allows individual parameters to be different at each point of
time, which seems realistic, yet be the same on average
a necessary
condition for successful aggregation which should be realizable through
appropriate grouping of the data.
The main contribution of this paper is to show that the variance
of the estimate'! macro coefficient goes to zero as the number of individuals in the
aggregate increases, although this result would appear to be counterintuitive
since the variance of the sum of independently distributed random variables
tends to infinity as the number of items in the sum tends to infinity.
Thus the efficiency of estimation increases as the aggregate grows
^
In
one liiiiting case, the variance diminishes as 1/N, i.e. it is as if the
parameters were averaged instead of being made up of aggregated data for
which the underlying error variances cumulate toward an indefinitely large
sum as they are subsuir.ed
*Theil,[20], pp. 5-6.
in the aggregation.
1
-8-
The variance of
ft
can be written by inspection of the first row in (7)
as
where
(T,
is the variance of the first micro parameter,
the variance
CT *'is
of the second micro parameter and O12 their covariance assuming the micr«> parar
meters to be identically and indeoendently distributed for all N individuals.
We must now evaluate the nature of the B weights more closely, for
it is their
behavior that basically determines what happens to the
variance of the macro parameter as N increases.
The
which sum to
B.
one by the aggregation weight rules, are the partial regression coefficients
of the
i
micro series
x.
1
on the macro series X
12^*'
,
= B
in x.
11
1
,
,
ii
+
X
1
B,
ii
X
are the partial regression coefficients of micro series x.
and the B.„
on macro series X,
in the i
1
12
regression x.^ =B.
multiple
f
6
X2
12
+
X
1
X
B.
i2
2
.
While it is true, as Theil and Kloeck have emphasized, that these are
arbitrary weights, it is nevertheless reasonable to suppose that under a
wide variety of circumstances that the B.j will be approximately the
proportion of
X^
that originates with x.
.
One rigid assumption with
many interesting implications is that these are exact proportions, a matter to which we return in later empirical sections.
would suppose that the B
than the B.
.
Furthermore,
one
weights would tend to be small, much smaller
The arbitrary nature of the entire estimation procedure
for the B's leads one to suppose that the net regression coefficient of
X.
on X^ can be expected to be small term by term.
""
know that
.Z,
1=1
il) =
BT,
1 and
ii
Furthermore, we
""
(1)
S^b)/
i=l i2
= 0.
While in theory the individual B's could be anything, we shall
suppose that in most empirical applications that the
B^-*:^
are positive
2
-9fractions, while the B.
s
erably smaller than the
B,
11
may be positive or negative, but are consids.
From this assumption the following highly significant and equally
simple assertion emerges:
The share of each component in an aggregate will remain unchanged
or decrease when the aggregate Increases^and the corresponding B
weights will behave in corresponding fashion.
The sum of squares,
"
(1)2
.Z.B.
.will thus O
gradually
decline as the squared fractions and
J
1=1 11 J
their sum decrease.
Consider the case where the following simplified (but illuminating)
conditions are correct:
weights represent shares which are equally divided
(a)The B.
among members in the aggregate,
(1)
are individually negligible o£ the variance of 3.
(b)The B.2
negligible relative to that of
variance a
6.
is
and similarly for the co-
among the micro parameters g.j,
Bj,'
Then,
^
\<^^>
^
(V\2
^iSl^i'
K,,
In this illustrative case,
" 1
' ^iil<l>'^
1
\,
^=\%,
(10)
the variance of the macro parameter approx-
imately obeys the law of large numbers, decreasing as
—
,
i.e. as if we
were averaging parameters of individually distributed variables rather
than taking weighted sums of independent random variables.
The more
unequal the weight distribution, the more slowly does the variance of
the macro parameter decrease, but it will decrease.
Empirical results
N
'
-10-
confirm that, even with skewed distributions that exist in many manufacturing industries, the sums of the squared shares to which corresponding B weights are close analogues tend rapidly to very small magnitudes
in most four digit industries.
A general proof
The remainder of this paper proceeds as follows.
for the full regression model is presented.
Then several sets of empirical
results will be examined to validate or refute the basic postulates which,
after all, are assertions about measurable reality.
3.
General Proof
Let for
i
= 1, 2
N individuals and p = 1, 2,.
.
.
,!i,.
.
.
,
K parameters:
(11)
^i= (^i^i.-'-V
b(^)= [B(^>Bf^>....Bf^>],
—
iK
11
1
12
(12)
^
Bf^>
^'\
h
B =
(13)
;
:
also
and
,(K)
Ji
1<^>
—Bf>|
2
W,
6, = .l^ll 'B.
,
(14)
= (B^63....6„)'^
(15)
.
(K x 1) vector of the micro parameters for individual i,
Here
j3.
(£)
B.
is the (1 X K) vector of the B-weights for individual i and macro parameter
£,
is the
B is the (K x KN) matrix of all B-weights, 6p is the typical macro
parameter and finally,
B
is the
(K x 1) vector of macro parameters.
-11We make the
Assumption
;
1
{6^.,
= 1, 2,...5,N}
identically distributed (K x
vector
B_
is a sequence of mutually independent
random variables, each with mean
1)
and positive definite symmetric covariance matrix £.
Then E(6) =
,t
J E[S%
-r-t^t
t,tt= B[p"e'-.
.,B^'^]
-
.
.
so that the estimator is unbiased.
7rt,t
.
.
Furthermore,
Q
=
has covariance matrix
3
Bf'W'^'.
^
h^l^
(16)
=6.
.6"]
Ib^
I
A generic diagonal term of V
l^
1=1—X
""^-1
—1
=^1
(17)
N
i^l
.Q
N
—a
=
-i
lsl<i^'
(B)
^
can be written as;
i=l-^
(18)
—1
Ib^'\^'?
i
iK
i=l
= tr n-
,
tr[g
i=lf\K
.a)
•
^''^J
(19)
^
It must now be shown what restrictions on the B-weights will lead
to a decrease in the macro parameter variances as the aggregate population
grows.
A lemma concerning a sequence of fractions which sum to unity is
-12-
needed at this point:
Lemma
For
:
2,...,N and N = 1, 2,.... let (p.^} be such that
= 1,
i
N
-1 <
.E.p.., = 1.
and
p.., < 1
Then a necessary and sufficient
condition that
1
N
•
„
N
^
=
1=1 P?„
»^iN
oo
^
lim p
Proof:
that
is
.E,
where p
E
max {|p
i =
|,
^1^11
"P^
= iil
.
To prove sufficiency, examine the sum
Necessity is obvious.
i£l ^pJn
2,...N}
1,
^PiN-^N )<PiN
-
N
^•
Clearly,
^
1
Pn
-
^°^ ^^""^ ^' "°
IPin'- N
N
.
F^ ^
iil ^PiN -
Pn
(
i^l
^^N
> 0, so that
By hypothesis lim p
,
^
1
N
,.
lini
,
iim
1 \
2
/
V
"
N - - iSl (PiN
N^^
==
,
r/ V
N -
CO
2
\
[(i£i PiN^
^i
P
^
-
n
°-
N
But the minimum value of
N
.E
iSl
7
pf
--iN
.
(•^i
1=1 P-»t)
^iN
"
;t
N
is
'"
—1
,
N'
N
1.
>
—
0»
whereupon
^
->-
0°
1-1— =
= .,
1=1 p.„
^iN
N
.E,
.,
N
implying for each N that:
This lemma enables us to prove Theorem
Theorem
1.
Provided that:
N
(1)
|B^f
|<
1
and
.E^ b{J^ = 1,
->-
1.
0°
N
0.
+ ^
)
^
= 7a
-13-
When
(2)
^^"^
(3)
.-hen
c
Proof:
,.
j
iB^^^lllB^f
4 1,
i =
max {Bf^^
;'"l.V
1^™
.u
that^ j^ ^
(_B)
i^l
1
-
,
=
Lemma.
follows from condition (3) and the
.
.
.
.
= 0.
Z
Since iB^f IllBjf
.N}
1, 2
N
oo
and
U
TB^'^'^I'
'
^^iK,
implies that
N
fo\
,Z,[B<5>]^
^
<
(9,)
.l^i
,
^^ convergence
all
inequality to zero as N - - implies that
of the right hand side of the
diagonal terms in
^^^
converge to zero as N -
By the Cauchy-Schwartz
«>.
(2,)
to zero as N ^
inequality convergence of all diagonal terms in ^^
implies that
f^ . i^'^
^
=
^^^ ^^"^^
S^h^
N, V (3) is positive definite symmetric, ^
implies that
^''l
oo
^^i^
"^
-
as N
^
I*}(^Jl>
- «.
=
°°
since for each
for
il=
1,
2
,
.
.
.
,
K
= ^^
Some few comments may be in order on conditions
1
- 3 of the theorem.
Much of the motivation for them has been provided in the previous section
and empirical support will be forthcoming in subsequent sections.
all B weights are less than
1
That
in absolute value follows from the generally propor-
tion-like character of the corresponding B weights;
SB (£) =1
follows
identically from the least squares restriction.
The second condition, that non-corresponding B weights be smaller
than corresponding B weights is an implication of the greater systematic
correspondence that x.„ has with X„ than with X..
evidence supports this presumption.
Available empirical
Finally, the last and most critical
assumption for the Lemma and hence for the proof, depends on how increasingly large aggregates are made up.
(£)
Suppose that the B.p
are proportions:
lo
-14then
by construction, the addition of another amount decreases the propor-
,
tional share of every single antecedent member of the aggregate.
Unless
the aggregate happened to be constructed systematically to make the N
member a larger fraction than the largest preceding one, the condition
will be satisfied.
Aggregates are usually built up in two ways:
by
combining various sub-populations (e.g. two digit into one digit industries),
or expanding a given sample.
In the first case, one can only suppose that
the share of U.S. Steel in total manufacturing is less than it is in the
steel industry.
In the second case, supposing sampling to be random,
the share of each member can be expected to halve, if the sample is
doubled. irrespective of the shape of the underlying population.
In general,
then, we suppose that the B weights will behave in similar fashion and
that this condition will hold in the vast majority of cases.
The preceding theorem and discussion lead directly to an extension of
the assumptions to permit unequal variance matrices.
When the
3.^
are
mutually independent but do not have common variance matrices, setting
V(3_.)
=
Q.
,
we may rewrite (17) as
r
N
\(6)
,,(l)^(i)g(K)t
—1 =
—1
(20)
i^l
g(K)^(i)3(K)t
—1
—1 -=
The task of also proving that the limiting variances are zero when
the individual parameter variance matrices are correlated has not been
attempted. One may conjecture that when the tedious algebra is done, the
basic results obtained here will not be altered. Highly complex product
chains of fractions under the assumptions I have used throughout will surely
converge to zero in the limit. Swamy [19] derived related results in the
heteroschedastic case to be considered in Theorem 2.
,
-15-
and (18) as
VCB,)N Z
Let
?u
be a generic element of
re
Ry setting 5 =
trl
1=1
=
lBf^>f^(^>B^^>^
= —1
1=1—1
[
6
(21)
6
re
=
max
{Q,
re
,
i
= 1
,
2,
.
.
.
,
N}
]
Bf ^Vf ^>]
hold, the proof that V^(6p) ^
as N
applies in this more general ease.
as N
->
°°,
(22)
tr[Mf ^]
are bounded and that conditions (l)-(3)
Provided that elements of
^-
and
Q,
=
V^(6^) < trlgi^l^
that if $
«(i>Bf^)Sf^>
—1
—1
==
^-
°°
when
g^^^ = g^^^ all
i,
.i
This is a direct consequence of the fact
so does tr
^^
•
Also, the Cauchy-Schwartz
inequality again implies that if all diagonal terms of V
(6^)
approach zero
as N approaches infinity, so do the off-diagonal elements of ^„(6^).
Hence
we have:
Theorem
Let
2
:
i =
(B^.
1
common mean
1,
be mutually independent random vectors with
2,..., N
*
B_,
but not necessarily identical positive definite symmetric
variance matrices
and (3) of Theorem
V(6^.
1
)
=£
,
i = 1,
obtain, then
^^
2,.,., N.
^ ^m^^^
^ £•
If conditions (1),
(2)
4.
Empirical Evidence
—
Firm Investment Relations
Manufacturing investment equations will provide some evidence on
the underlying assumptions and the aggregation properties contained in
the basic theorem.
I
have chosen investment behavior for several reasons,
not the least of which is an earlier interest in its substantive aspects
as well as its aggregation characteristics
[13],
In addition, the earli-
est empirical study of aggregation theory was that of Boot and de Wit [2]
using investment behavior and based on a study by Grunfeld [7].
Among
the most recent contributions to testing alternative models of investment
behavior, research by Jorgenson and Siebert [9] placed heavy reliance on
individual firm investment equations.
In nearly all the studies cited, a
basic equation form was the simple Chenery-Koyck distributed lag [3] [12]
in which investment is a linear function of sales or output, and the lag-
ged capital stock.
Since my present aim is to study aggregation,
I
shall adopt the Chenery-
Koyck formulation without theoretical motivation (available however, in the
sources cited) and without a detailed critique of its merits.
The basic
data have similar origins to the previous firm studies, except that none
of the series on Investment, sales and gross fixed assets were price cor-
rected.
The cost of rectifying the data did not seem warranted in light
of the predominant methodological thrust of this study.
There is little
reason to believe that failure to correct for price variations will seriously affect the aggregation characteristics of the underlying series,
although comparability with previous studies is diminished.
-17-
The raw data came from the COMPUSTAT tape which contains financial
information from 868 companies listed on the New York Stock Exchange, American Stock Exchange, and, in a few instances, regional exchanges or those
with securities traded over the counter.
While in most
cases
the num-
bers are identical with corporate records, some disparities exist.
Accord-
ing to the COMPUSTAT manual [18]:
Differences will frequently occur between the financial
statistics included in COMPUSTAT and those in Corporation Records.
These differences may be due to one of the following
considerations.
Some restatement of company-reported information
has been made in COMPUSTAT in accordance with the
definitions outlined in this manual for the purpose
of increasing comparability, both within a single
company and within a single industry.
COMPUSTAT includes some material taken from SEC
and outside sources....
Although almost any item of information in COMPUSTAT may
differ from that in the Annual Reports in accordance with the
definitions given, a few specific notes are listed below in
some of the most frequently used statistics in the system....
Annual data in COMPUSTAT for years prior to a merger are
not stated on a pro forma basis (see page 3-1).
Both
pro forma amd reported data are normally given in
Corporation records.
Employment and capital expenditure data may vary
slightly because of differing sources of information....
From this larger universe, several industries were selected whose
aggregation properties will be reported.
These were chosen on the basis of
size and relative internal homogeneity (two digit industries are often com-
posed of several heterogeneous three-digit industries, each containing only
a few firms)
as well as different production characteristics between industries.
Since computer processing costs exceeded one dollar per firm, calculations
were restricted to four reasonable sized industries that have different production processes, in order to restrain costs within reasonable bounds.
-18-
Table
1
Sample Composition
Years
Industry
Firms
Observations
Steel
1949-1967 (19)
23
437
Retail
19A9-1966 (18)
19
342
Petroleum
1950-1967 (18)
22
396
Machinery
1950-1967 (18)
41
738
105
1913
Total
We thus have data beginning in 1949 and ending in 1967 for 105 firms.
A total of 1,913 observations on investment are to be explained by corresponding data vectors for sales and gross fixed assets.
Since interest may attach to the individual micro results, these,
together with their industry macro regressions, have been reported in
Appendix Table 1-A.
They
bear a strong resemblance to similar firm re-
gressions reported by other investigators.
A.
Model Assumptions - B-Weights and Proportions
One basic assumption of the theorem is that proportions resemble R-
weights.
By calculating proportions as the sample period sum of each firm's
sales or assets divided by the corresponding sample aggregate sum, various
comparisons with estimated B-weights are possible.
One comprehensive com-
parison is shown in Table 2-1 which records the simple regression of the
B-weights on proportions, with and without the intercept.
Since the basic
hypothesis is homogeneous, the most pertinent regression for each industry
pair is the one that has been forced through the origin.
-19-
For some reason, the regressions for asset parameters are very close
or within plausible range of the slope value of unity required by the hy-
pothesis in every case, although the hypothesis is refuted for each sales
coefficient except for steel.
Considering the extent of collinearity in
all but steel (on which a table will be presented shortly) the net outcome
is mived,
but not unsatisfactory.
Starting from the initial extreme prop-
osition that no economic content can be read into the auxiliary regression
equation parameters, we have come a long way.
A related measure is shown in Table 2-II.
The average absolute dif-
ference between B-weight and nroportion divided by the average corresponding
B-weight or proportion
is a sensible normalization for comparative pur-
The steel industry as before shows the smallest relative difference
poses.
while Retail is the worst.
Petroleum is fair for both coefficients while
Machinery does poorly on sales and adequately on assets.
Order of mag-
nitude zero is good and order of magnitude unity or larger is bad according
to the basic hypothesis.
*
These averages are identical since each series sums to unity,
Table
I.
2
Regression of B-Weight on Proportion
Sales
Slope
Intercept
Slope
Intercept
1.05 75
(31.8271)
-0.0025
(-0.9767)
0.8356
(31.3322)
0.0071
(3.0041)
1.0391
(37.9362)
forced through
origin
0.8747
(32.1531)
forced through
origin
-0.3114
(-1.1032)
0.0690
0.6174
(2.0663)
0.0201
(0.8785)
0.1421
(0.5339)
forced through
origin
0.7974
(3.6911)
forced through
origin
-0.1220
(-0.7194)
0.0510
(4.2623)
1.4166
(15.4912)
-0.0189
(-3.1438)
0.3439
(1.9665)
forced through
origin
1.2182
(15.4298)
forced through
origin
T-stat
0.3136
(1.6463)
0.0167
(2.1568)
1.0240
(11.0235)
-0.0006
(-0.1426)
Coeff.
T-stat
0.5596
(3.5102)
forced through
origin
1.0168
(13.2754)
forced through
origin
Industry
Steel
Coeff.
T-stat
Coeff.
T-stat
Retail
Coeff.
T-stat
Coeff.
T-stat
Petroleum
Coeff.
T-stat
Coeff.
T-stat
Machinery
II.
Assets
Coeff.
(2.7318)
Average Absolute Difference Between B-Weight and Proportion Divided
by Average B-Weight or Average Proportion
Industry
Sales
Assets
Steel
0.,158
0.191
Retail
0..889
0.984
Petroleum
0..419
0.369
Machinery
0.,793
0.478
-21-
Table
3
presents detailed information bearing on another basic assump-
tion, which is that corresponding B-weights are large relative to noncorres-
ponding B-weights.
For the Steel industry, the assumption held up for all
23 asset auxiliary regressions and for 19 out of 23 sales auxiliary regres-
The Machinery industry had eight and nine failures to conform, or
sions.
success on the order of 80%.
A similar average success rate holds up in
Petroleum, while Retail had under 50% correspondence to this particular
*
assumption.
A comparison of relative statistical significance between corres-
ponding and non-corresponding B-weights in Table
sion obtained from comparing only magnitudes.
4
reinforces the impres-
Retail again excepted,
corresponding B-weights have the greater statistical significance most of
the time, overwhelmingly so for Steel and Petroleum with somewhat less,
though clearly evident superiority in Machinery.
Table
4
Number of Times Absolute Value of t-Statistic for Corresponding B-Weight
Exceeds Absolute Value of t-Statistic for Non-Corresponding B-Weight
No.
of Firms
Sales
Assets
Steel
23
18
23
Retail
19
10
13
Petroleum
22
19
18
Machinery
41
35
29
*
Table 2-A (Appendix) reports the auxiliary regressions in full detail.
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-22-
Once again, there appears to be a definite relation to collinearity
The quality of the results is inverse to the extent of linear dependence:
when there is not excessive correlation among the explanatory variables,
the assumptions of the statistical aggregation model are more nearly
validated.
Table 5
Industry
Sales-Asset Simple Correlation
Steel
,8378
Retail
.9978
Petroleum
.9902
Machinery
.9696
The individual t-statistic on the difference between corresponding B's
and proportions has been calculated using the B weight's standard error
and treating the proportion as a known parameter.
This understates the
true variability so that the outcome is relatively unfavorable to a finding of no significant difference.
The arithmetic mean of the
t
statistics
tends to be small, amounting to less than unity for all sales coefficients
(and nearly zero in two of the four industries) and about unity for two
industry asset coefficients and magnitude one-half for the other two.
Calculated
t
statistics rank ordered from largest positive to most negative
appear in Appendix Table 3-A. Casual inspection is enough to indicate that
there is too much density in the tails for these tabulations to represent
an identically distributed random drawing from a
zero.
t
distribution with mean
Retail for both coefficients, and Steel for the sales coefficient
are comparatively tightly bunched about zero, while the others are much
less so.
One problem with these inferences is apparent from examination
-23-
of Appendix Table 2-A showing proportions and auxiliary regressions.
Good-
ness of fit is nearly always very high and the corresponding coefficient
standard errors are very small.
As a consequence, even though a visual
scan reveals that the two series are close to each other most of the time
(with several glaring exceptions in every industry, to be sure) "significant"
differences arise with substantial frequency.
The upshot of this section will now be summarized.
Qualified sup-
port has been found for the assumptions relating corresponding B-weights to
proportions, and the magnitudes of corresponding and non-corresponding 3weights.
When there is excessive collinearity among the explanatory macro
variables, however, the assumptions tend to break down.
Yet under this
very circumstance it is difficult or impossible to obtain sensible macro
estimates anyway.
In short, when estimation feasibility breaks down most,
so do the aggregation assumptions.
Finally, the comprehensive tests sum-
marizing information about all pairs of proportions and B-weights stand up
somewhat better than pair-wise tests based on the t-statistic.
B.
Model Predictions; Micro Variances, Macro Variances and Aggregation Gain
This theory's main innovation is to predict how aggregation gain can
be expected to occur, and its magnitude.
That there is any aggregation gain
wliatsoever until recently would have been a puzzling result of itself.
Table 6 reports the major empirical results most revealing on this central
feature.
Aggregation gain is measured by the reduction of the estimated regression coefficient variances.
In the macro relative to micro estimates,
the theoretical gain can be calculated in two ways.
Assuming, first, that
proportions are good approximations to the corresponding B-weights, their
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-25-
squared sum designated as H (for reasons given in the next section) shows
the expected reduction in the macro variance as a consequence of aggregation.
Its reciprocal is therefore termed the aggregation gain.
Second,
and in parallel fashion, the sum of squared corresponding B-x<?eights and
its reciprocal provides another
tion gain.
Since the postulated
relevant estimate of potential aggrega-
empirical relation between the two
measures has been considered already at length, it is noteworthy that the
two sums of squares turn out to be highly similar figures.
Retail bombed out, as in every other aspect it least complied with
the theoretical assumptions of the basic model.
I
The remaining industries:
1.
All show positive aggregation gain;
2.
All show gains of rather similar magnitude to the theoretical prediction:
consider these findings to be the most sigrificant quantitative result
in the paper.
C.
Microproportion Stability
Some proportions grow rapidly, others decline swiftly, while some
barely change at all.
Since the stability of proportions was assumed at
the outset, some tests were made to evaluate an assumption we knew would
be violated in actuality.
A straightf onward test is presented in Appendix
Table 4-A from which Table
7
has been extracted.
The two quartiles and
median relative growth rate (defined as the linear trend slope of the proportion
divided by the ratio of variable means, dP/dt 7 x/X) summary statistics are presented here.
The top quartile rate of growth is about
2
1/2 - 4 1/2%
per annum in most industries while the median trend rate of change is in
the neighborhood of 1% - 2 1/2%, so that for most firms - not all - rel-
ative shares did not radically change over the twenty year sample period.
Table
7
Quartiles for Relative Rate of Growth of Proportions
First Quartile
Median
Third Quartile
Sales
0.026
0.011
0.00^
Asset s
0.030
0.014
0.008
Retail
First Quartile
Median
Third Quartile
0.031
0.011
0.005
0.039
0.025
0.007
Petroleum
First Quartile
Median
Third Quartile
0.03A
0.024
0.008
0.021
0.013
0.003
Machinery
First Quartile
Median
Third Quartile
0.045
0.025
0.014
0.045
0.031
0.013
Steel
-27-
5.
Empirical Evidence
—
All Manufacturing H Indexes
According to the previously developed theory, expected gains from
aggregation depend on the number of elements in the aggregate and their
size distribution.
the same size.
Maximum aggregation gain arises when all elements are
This is evident, since the variances of proportions will
be zero when all shares are identical and the correlative fact that the
sum of squared proportions (or variances) is nearly one for very skewed
shares.
The sum of squares for equal shares is 1/N.
Data on the sum of squared shares exist for manufacturing industries
because this market parameter has begun to acquire descriptive and analytical significance in the study of industrial organization.
It
has been
named the H concentration measure or index after the two originators who
independently promoted it, O.C. Herflndahl and
A. P.
Hirschmann.
The most
interesting interpretation is that of M.A. Adelman [1], who shows that the
H Index can be viewed as the weighted average slone of the cumulative con-
centration curve.
In a second interpretation, suppose that there were N
equal sized firms in an industry; then N turns out to equal 1/H.
Adel-
man suggests that the reciprocal of H can properly be viewed as a "num-
bers equivalent".
Since its introduction into industrial organization literature by
Herflndahl [8], the H index has been calculated for various firm attributes
for four (and some five) digit industries in a 1963 volume on concentration
in manufficturing by Ralph L. Nelson
is stringent but useful.
[14].
Resort to four digit industries
It is stringent in the sense that such a fine
subdivision has comparatively few firms in it, useful because one might
-28-
reasonably expect that substantial homogeneity of markets and production
methods will prevail at this classification level, in contrast to the more
common econometric category of two-digit aggregates.
The exhaustive
tabulations of Nelson yield several impressions on which we briefly comment.
First, concentration and H measures by value added, by shipments,
by production manhours and by employees yield highly similar results, so
that estimates of aggregation gain can be equally well derived from any
of them.
A second impression is that these various measures are quite
stable over time.
These census groupings are much more realistic than the
arbitrarily formed industries selected out of COMPUSTAT, which, for the
most part, is composed of large New York Stock Exchange listed firms.
Several quantitative tabulations bearing aggregation properties in manu-
facturing deserve explicit presentation.
The first is a tabulation of
ranked H indexes for 1947 shown in the first column of Table
8.
While
these data are now twenty-two years old, they are more complete, but still
similar to the latest census information available to Nelson for 1954, plus
annual survey information for 1955, 1956 and 1957.
Later tabulations in-
dicate that these measures are stable according to visual impressions, so
that broad inferences drawn from 1947 will be reliable, even though some
industries now have widely different H and concentration indexes than they
had in 1947.
An experiment with these data turned out to be a partial success.
The purpose of the exercise was to determine whether a simple one para-
meter function like the exponential can be used to derive H indexes from
It is generally true that establishment based indexes show considerably less concentration than company data.
Thus, subsequent evaluation
of these tabulations are restricted to the series listed only.
-29-
usually more readily available data.
More concretely, if the concentration
density curve could be adequately represented by an exponential density
function with exponential parameter designated
it is a simple matter
jc
*
to estimate
to use concentration indexes
c^.
Furthermore, the sum of
squares of the density function will then be cjl.
If the initial hypothesis
of equivalency is correct, ojl will then be equal to the industry H index
and in addition, estimates of
c^
for the top four, eight and twenty firms
in each industry will turn out to be approximately the same.
While the
implicit estimates of the squared sum of the hypothesised exponential were
indeed similar in magnitude to the actual H index, the second implication
of the exponential form was not supported by the data.
four digit industries,
c^(4)
>
c^(8)
>
c^(20)
where
c^
For all but a few
has been estimated for
each of the four, eight and twenty largest firms in Table
Suppose
8.
This result
""C i
gives the i'th firm's proportion of
total industry shipments. Then the proportion of total shipments of
the M largest firms is given by the cumulated density function.
s(M)
Tims, if
to give
=
f(i)
ce
f(i)di
=
s(H)
1
M
=
is known,
1
- e
"^^
the last expression may be solved for
c
1
1
^
1
- s(M)
**This table is based on all census industries in Nelson's Table A:l,
Company Concentration Indexes, Industries, 1947, 1954, 1955, and 1956,
with the exception of the following which could not be used because sufficient data were not available, or were not reported to avoid disclosing
figures for individual companies:
2224, 2233, 2651, 2824, 3211, 3331, 3332, 3334,
3553, 3555, 3616, 3821.
3429,
3461, 3497, 3519,
-30suggests that typically industry is less concentrated than exponential
curves derived in this fashion.
Yet it is still possible to use concen-
tration indexes to obtain an approximate notion of potential aggregation
gain when, as usual, these data are available but H indexes are not.
For
industries with high to medium concentration c^(4)/2 provides the best estimate of H (actual H >.07); £(8)/2 is closest to H for medium to low con-
centration (.02
<
actual H
<
.07)
and c\20) 12) is closest to H for the
least concentrated industries (actual H
<
.02).
While close estimates of
H vjere obtained for medium and low concentration industries, the most
highly concentrated industries usually provided the poorest information
about H from the concentration indexes, as an examination of the top 15 20 industries in Table 8 indicates.
H indexes in manufacturing four digit industries have a median value
of .09 according to Table
8.
Translated into the terms of our approach to
aggregation, the gain in terms of variance reduction from aggregation is
twelve fold.
Even at the first quartile mark,
industries, a seven fold
above which lie 25% of the most concentrated
instead of micro information
gain will occur from using aggregated data
directly.
about .04 implies that a
At the third quartile, an H index of
twenty-eight fold aggregation gain may be expected.
of concentration in AmerTo what extent does the existing degree
ican industry reduce estimation efficiency?
To recall the earlier state-
distribution of elements in an
ment of the problem, a highly skewed size
of collapsing
aggregate reduces the beneficial aggregation effects
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-31-
parameter variances.
Subtracting H's minimum theoretical value 1/N
(which occurs when all elements are equal) from actual H offers a simple
measure of this loss.
This difference, H - 1/N, also has significance
for the analysis of industrial organization.
Table 9.
The results are arrayed in
The median within manufacturing is about .07, which is almost
the same as the value of H itself.
While the H - 1/N distribution is
somewhat more concentrated than the H distribution and some rearrangement
in the rank orderings do occur,
each other.
the two distributions strongly resemble
Since for all but a handful of industries 1/N tends to be
negligible, we may conclude that it is not fewness of firms but the skewness of their size distribution that limits potential aggregation gain.
The stability of concentration measures over time depends on which
measure has been selected according to Table 10.
H indexes show greater
sensitivity than the more standard concentration index.
Noticable insta-
bility is apparant for one fifth of all manufacturing industry total shipments between 1947 and 1956 for the H index, while a mere three out of 113
industries appeared to deviate significantly from the base share held by
the four largest firms over the same period.
In studying aggregation,
pertinent magnitude.
the more sensitive H index is the most
While far from ideal, the overall picture for sta-
bility is quite satisfactory.
A batting average of .800 over a decade
(what happened in between we cannot say) is good enough to warrant broad
confidence to a basic presumption of stability.
While these data are more
inclusive than the micro data previously analyzed, they are not as powerful,
A quick subjective examination reveals that the industries 2031, 2043,
2045, 2073, 2085, 2111, 2825, 3021, 3261, 3272, 3275, 3312, 3331, 3493, 3562,
3576, 3613, 3631, 3717, 3722, and 3861 had marked changes in their H index
between 1947 and 1956, while industries 2031, 2045 and 3613 had striking
changes in the concentration index for the four largest firms.
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-32since the stability in an industry aggregate does not preclude shift in
the underlying individual distributions.
-33-
6.
Heterogeneous Parameters amd Research Strategy
A complementary statistical aggregation approach starts with the
assumption of heterogeneity, but concludes that structural simplification
through clustering of similar objects can lead to increased understanding
and manageability, although at some cost in precision.
Walter Fisher
[4]
provides operational solutions to this problem through statistical decision
theory.
The present treatment of aggregation uses a quadratic cost function
i.e.
least squares, on the aggregate relation whose implications were long
ago illuminated by Theil.
While
I
have nothing to add to Theil's theory
under conditions where micro paramaters differ, the interpretation given
here to the R-weights has a definite bearing on this question.
When the B-weights for corresponding parameters resemble proportions,
and ncncorresponding parameter aggregation weights are negligible term-byterm, ordinary least squares parameter estimates of macro relations will
have desirable properties, even when the micro parameters are different.
In this exposition, my debt to Theil should be evident, even though it is
my inclination to emphasize the economic substance of B-weights and hence
the feasibility of using macro relations effectively to predict and study
behavior.
Assume the following conditions hold:
1.
Corresponding B.j weights are proportions i.e.
_ \Si _
-34-
2.
Noncorresponding B weights are zero.
N
6^ = .1^ P.^6,^
Then
K
Y=.Z 6„Xj
A
The macro predictions will then be
^
^= /_n
p
^1
„
,
"^11
••••x^
(3^^ii+-----^)^V
[^
ihI il il
„
.
.
^1 +••••+
.""iK „
^K
^l^^iK^'---^^W \
.
.
1=1 iK iK
In these circumstances the micro predictions and macro predictions are
the same.
As Theil characterizes this sort of outcome, there is no contra-
diction between the micro relations and the macro relation.
The forecast-
ing benefits from relying on macro equations are evident, even when these
conditions hold only approximately.
Much interest thus attaches on the
stability of shares through time for members of economic population.
The
law of proportional growth analyzed for instance by Simon and iionmi [17]
suggests that stability in shares is not implausible among economic populations.
liere
The matter is fundamentally an empirical one that has been examined
for several subsets of manufacturing data.
Furthermore, the optimal properties of least squares estimation apply
to the auxiliary equation parameter estimates as an estimator of propor-
tions under the conditions stated above.
When shares are changing, ordinary
least squares provides an efficient estimate of the average proportion over
the sample period.
Efficient predictions can be obtained when shares are stable and
parameters differ or when parameters are similar and shares are not.
Of-
ten one, the other or both sets of conditions will hold - this explains
much of the modest success that highly aggregative macro models have had.
-35Many failures have their origins in the failure of one or both conditions.
The ideal situation, of course, is one where underlying behavior is rel-
atively homogeneous so that compositional aspects of the data universe become secondary.
But macro model building strategy can and should proceed
on several fronts simultaneously, relying on aggregates
(and hence share
stability) more heavily in the short term while searching for significant
behavioral homogeneities in the longer run.
This observation actually
characterizes much econometric model building activity of the past twenty
years, beginning with the simplest three to nine equation Keynesian models.
Research activity then expanded in several directions to disaggregate be-
havior equations that obviously contained disparate elements.
Thus aggre-
gate investment was separated into inventory, plant, equipment and resi-
dential construction.
Total consumption has by now come to be divided
into at least three components - durables, nondurables and services - in an
effort to isolate significant differences in underlying behavior.
Dif-
ferential industry behavior has led to the proliferation of economic sectors.
When the disaggregated data can be obtained, it becomes possible
to weigh aggregation gains arising from grouping similar behavior units
against the losses from subsuming too much disparate behavior in one
relationship, and the perils or benefits inherent in shifting or stable
shares and, more generally, aggregation weights.
Material in Section
is fragmentary.
5
The empirical propositions
held up well in numerous situations, poorly in others.
While
clear
advan^-eges accrue from proper aggregation, one noticable instance that
failed especially badly was
be extrei^e collinearity.
Retail Trade.
The main reason seemed to
In this case no aggregation
gain occured: in
fact, if the calculations are to be believed, there were actual aggrega-
tion losses.
In instances where collinearity is pervasive, we may
-36-
speculate that the only possibility for successful estimation is to rely
more heavily on disaggregated data.
ficient to assure success.
This is neither necessary not suf-
It is a conjecture based partly on this study
and other econometric work with micro data.
Another reason that micro data is desirable is fundamental to the
acquisition of a solid basis for using macro equations, namely, the un-
derstanding how much heterogeneity exists, where the main tool would be
some form of covariance analysis, and in what catagories it is most
prevalent.
Thus sensible research strategy requires simultaneous ex-
ploration of micro and macro data; micro data to test hypotheses and explore homogeneity, macro data for statistical efficiency and as a way to
hold expenses down to tolerable levels.
Intensive analysis of aggregation
v;as
restricted to manufacturing,
using some concocted two digit industries from COMPUSTAT for which all
possible parameters were calculated that bear on the theory and "real"
four digit cencus data using the sum of squared shares
,
or H index as
a direct indicator of the sort of aggregation gains that can be expected
from such industries
in future investigations.
With the tools developed
in Section 6 for translating concentration indexes into H indexes, per-
sonal income distribution data should be explored.
One may speculate that
H indexes are smaller for personal income - income is more evenly dis-
tributed than are firm sizes - so that potential aggregation gain is even
greater
in the study of consumption-
It should now be more possible to explore
wider areas of economic
behavior in an operational framework that will enable econometricians
more readily to adopt a research strategy by choosing aggregation levels on
the basis of systematic information rather than hunch or random availability.
-37-
7.
Related Literature and Acknowledgements
Zellner
[
was first to point out that the random coefficient model
22]
provided a natural approach to the problem of consistent aggregation.
He
demonstrated that least squares estimation in the random coefficient case,
where data had been aggregated, was unbiased.
Theil [20] has postulated a random coefficient model with identical
properties to those we have adopted.
N
In the regression equation:
N
'
iSl
i=l_^iit^Vt- ^
~N
"".t^-"'^-^
.
^t =
1=1
the
1=1
lit
-
^iKt^'iKt
^Kt
,,.
^^^
•
iKt
typical aggregate coefficient is a function of time and sample size:
N
,
jSl ^iilt^iJLt
_
_
\
iil'^iit
,
,^.
i
/
Then the covariance of macro coefficient Bn with
X,
3,
h
can be written as
follows.
^hU
^
I
N
_
N
1
^ N iil^^it" ^ht^^^Ut"
_
.,
\t^ \
(C)
\t\\
I
where
O,
„
.
is the micro covariance among micro parameters g.
The variance of
3n
.
and
can thus be written:
^^^
lii£t(l^^Jt>]/«
where
y
Cj
3. j^-
is the coefficient of variation of x.
.
during the
th
t
period.
»
-38-
Theil concludes that as the numbers in the aggregate N increase, the
variance of the estimated parameters will tend to zero.
His result is the
same as mine, but there are differences in approach and some possible
First, Theil's macro coefficient
contradictions that arise.
is not an estimated coefficient in the standard sense:
Sp
it cannot be
estimated in the absence of the micro parameters themselves.
This may be
seen from the fact that the parameters are strictly functions of time (in-
volving both
6
p
and x..
)
in ways that the more conventional macro para-
.
meters
I
have postulated are not.
Second,
I
have shown that, on admittedly
extreme assumptions, it is possible for the variance of the macro parameters
not to diminish as N increases.
The same is possible in Theil's presenta-
tion of the problem, but the precise conditions relating to coefficients
of variation for the explanatory variables do not seem to be related to
my requirements.
This possible contradiction is not readily explicable.
An advantage of my treatment, in addition to its being directly re-
lated to standard least squares estimation, is that the macro parameters
variance is closely though approximately related to important
available
incomes
and often
information on size distributions of firms, individuals, or
for instance.
The relative stability of these distributions and
their properties provide genuine insight into when, and how much,
ag-
gregation gain can arise from enlarging the population aggregate.
Coef-
ficients of variation do not have the same intuitive significance, are
less readily available, and we know less of their stability over time.
is reassuring, however, that two rather different approaches
similar assumptions, do reach the same conclusions.
using
It
-39-
Charles Revier with some help from Dan Luria and Joe Steuart provided
admirable research assistance, relying heavily on M.I.T. computational
facilities.
J.
Philip Cooper made useful comments.
Support by the National
Science Foundation is gratefully acknowledged.
My colleague Gordon Kaufman supplied generous assistance.
the lemma ;n a more ingenious form than
I
He devised
had contemplated as well as help-
ing to set up and prove the theorems in their full generality.
BIBLIOGRAPHY
[1]
Adelman, M.A. : "Comment on the 'H' Concentration Measure as a NumbersEquivalent", Review of Economics and Statistics , Vol.51 (1969), pp.
99-101.
[2]
Boot, J.C.G., and CM. de Wit: "Investment Demand: An Empirical
Contribution to the Aggregation Problem", International Economic
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[3]
Chenery, H.: "Overcapacity and the Acceleration Principle",
Econometrica , Vol.20 (1952), pp. 1-28.
[4]
Fisher, W.D.
Clustering and Aggregation in Economics
University, Baltimore, 1969.
[5]
Friedman, M. , and D. Meiselman: "The Relative Stability of Monetary
Velocity and the Investment Multiplier in the United States, 189 71958", in Stabilization Policies , by E.G. Bro^-m et. al.: PrenticeHall, Englewood Cliffs, N.J., 1964.
[6]
Griliches, Z. and Y. Grunfeld: "Is Aggregation Necessarily Bad?"
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[7]
Grunfeld, Y. The Determinants of Corporate Investment , unpublished
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[8]
Herfindahl, O.C: Three Studies in Mineral Economics
University, Baltimore, 1961.
[9]
Jorgenson, D.W. and CD. Siebert,: "A Comparison of Alternative
Theories of Corporate Investment Behavior", American Economic
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:
,
Johns Hopkins
:
[10] Klein, L.R.
A Textbook of Econometrics
:
.
Johns Hopkins
,
New York: Harper and Row, 1953.
"Note on Convenient Notations in Multivariate Statistical
Analysis and in the Theory of Linear Aggregation".
International
Economic Review , Vol.2 (1961), pp. 351-360.
[11] Kloeck, T.
:
Distributed Lags and Investment Analysis
Publishing Company, Amsterdam, 1954.
[12] Koyck, L.M.
:
.
North Holland
E.
Captial Stock Growth: A Micro-Econometric Approach
Holland Publishing Company, Amsterdam, 1963.
[13] Kuh
,
:
.
North
R.L.: Concentration in Manufacturing Industries of the United
States, A Midcentury Report Yale University Press, New Haven, 1963.
[14] Nelson,
.
[15] Orcutt,
CH.
,
H.W. Watts and J.B. Edwards: "Data Aggregation and
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-41-
[16]
"The Theory of Least Squares When the Parameters are
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[17] Simon, H.A.
,
[18]
Standard Statistics Company, Inc.: Compustat Information Manual , (1967)
[19]
Swamy, P.A.V.B.: "Statistical Inference in Random Coefficient Regression Models Using Panel Data". S.S.R.I., University of Wisconsin,
Systems Formulation Methodology and Policy Workshop Paper 6701,
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[22]
:
.
Amsterdam, North
Zellner, A.: "On the Aggregation Problem: A New Approach to a Troublesome Problem". Report 6628, Chicago, Center for Mathematical Studies
in Business and Economics, ( October, 1966).
APPENDIX
APPENDIX
Table 1-A
Investment Regressions - Micro and Macro
Table 2-A
Estimated Coefficients of Auxiliary Equations and Firm
Proportions in Industry Totals, Ranked by Proportions
in Industry Sales
Table 3-A
Rank Ordering of t-Statistics for (Proportion - B-weight)
Table 4-A
Time Trend Regressions for Firm Proportions, Ranked by Size
of Trend Coefficient
Table 5-A
List of Firms and their COMPUSTAT Numbers and ID Numbers
J
Note:
The computer used for calculating summary statistics in this
study did not round off, but simply truncated all figures
after the ones printed. Therefore, the means and other
summary statistics reported here cannot be duplicated exactly
by using the numbers in these tables or in Table 3.
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