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CONSTRUCTION OF INPUT-OUTPUT
COEFFICIENTS WITH FLEXIBLE
FUNCTIONAL FORMS
P.R. Israilevich
Working Paper Series
Regional Economic Issues
Research Department
Federal Reserve Bank of Chicago
January, 1990 (WP-90-1)
Construction of Input-Output Coefficients with Flexible
Functional Forms
P.R. Israilevich *
December 6, 1989
1
In trod u ction
Input-output (1-0) models with fixed coefficients are widely recognized as the most restric
tive set of production functions. An attempt to estimate 1-0 coefficients using neoclassical
production functions with flexible functional form has been made by Hudson and Jorgen
son [1974] (H-J), and later Nakamura [1984].*1 Input shares derived within the neoclassical
framework were identified as 1-0 coefficients. However, we observe that shares of primary
resources (value-added components) in the neoclassical framework are defined differently
from the 1-0 framework, an incompatability that was neglected in the previous work.
It is shown that this incompatability can be eliminated via the use of the double de
flated value-added measure. More importantly, a methodology of imposing traditional 1-0
constraints (on the sums of rows and columns of the 1-0 table) on the neoclassical model is
*The author isan economist at the Federal Reserve Bank ofChicago and the Associate Director of the
Regional Economics Application Laboratory ofthe University ofIllinois, Urbana.
1Nakamura’swork represents clarificationofthe H-J assumptions and an extension ofthe originalmodel.
The scope ofthispaper does not cover numerous innovative topics addresssed in Nakamura. Therefore most
ofthe references are drawn from the H-J work.
1
2
developed that makes it possible to perform the task of RAS (biproportional adjustments
of 1-0 tables) with an econometric model.2 These two aspects are presented in Section 2 .
In Section 3, the RAS and H-J methods of projecting 1-0 coefficients are combined. For
this purpose, a modified RAS procedure is developed and termed Extended RAS or ERAS.
To make a consistent presentation the paper begins with the brief description of 1-0 model,
Subsection 2 . 1 , and description of the translog cost function, Subsections
2
2 .2
and 2.3.
T im e-series d ata as an aid to th e con stru ction o f inputou tp u t coefficients
2.1
1 -0 ta b le as a set o f in p u t shares
An 1-0 table is computed from transaction flows, which are dollar values of purchases of
goods and services by industries. If l 7’ is a matrix of transaction flows, then the 1-0 table
is represented as A = F X ” 1, where X is a diagonal matrix with output values for each
industry on its principal diagonal [(?) denotes, throughout the text, a diagonal matrix with
vector (.) on its principal diagonal].
The 1-0 table A represents the set of shares of intermediate input values. The sum of
intermediate shares for each industry (single column’s sum of
1 -0
coefficients), combined
with the corresponding value added per unit of output, are equal to unity for each industry.
The A matrix is assumed to be fixed in a traditional 1-0 analysis. 1-0 table A is
observed in time and can be deflated with respect to input and output prices (see Section
2.3). Therefore, via deflation, 1-0 tables can be made consistent over time, allowing changes
in the A matrix to be due to input substitution. However, 1-0 analysis does not offer an
explanation for such substitution. Instead, this phenomenon (under certain assumptions)3*
3The RAS method of projection of 1-0 tables is well known among 1-0 experts. For definitions see
Section 3.
3
must be explained with a neoclassical model, as discussed in the following Subsection.
2.2
N e o cla ssica l r ep resen ta tio n o f in p u t shares
H-J presented a methodology that allows for the use of time-series data for the estimation
of all input shares, which are interpreted as 1-0 coefficients. Earlier, we indicated that such
shares are equivalent to the 1-0 coefficients in value terms. In this Subsection, the H-J
methodology is described. This methodology is based on a neoclassical micro framework,
which is not necessarily consistent with the 1-0 framework. Differences between the two
frameworks are discussed and, based on this discussion, a modification to the H-J framework
is proposed.
Following Jorgenson [1986], price of unit of output (denote, q ) can be expressed as a
function of input prices and time:
q = Q ( p ,t )
(1)
where p and t are the price and time vectors, respectively. According with the neoclassi
cal considerations the unit price function has to be positive, homogenous, monotone, and
concave with respect to input prices.
The necessary condition for equilibrium is to have equality between shares of inputs in
the value of output and the elasticities of respective input prices:
_ Pi&i _ dinQ
q
d In pi
where s = [s{] is a vector of shares. Assuming constant returns to scales the sum of shares
adds up to unity:
l 's i = i .
(3)
Since the unit price function is monotone in positive input prices and the sum of shares
equals one, each share must be nonnegative and at least one share must be positive.
4
Next, the rate of technical change can be expressed as a negative of the change in price
of output over time, holding input prices fixed:
d in Q
(4)
dt
Shares and technical change are derived by the first order derivative of the unit price
function with respect to input prices and time. Second-order derivatives provide measures
of input substitutions, biases of shares with respect to technical changes, and changes in the
rate of technical change. The above notions can be determined by the first derivatives of
S{ and Vt , since they, in turn, are the first derivatives of Q . The matrix of share elasticities
(denote U pp) is:
ds
= UtVV'
d ln p
(5)
Bias of shares with respect to technical change (denote u pt as a technical bias) is:
da
dvt
dt
d ln p
upt.
(6)
Finally, the speed of the rate of change (denote v tt), i.e. deceleration or acceleration of
the technical changes is:
dvt
dt
= vti.
(7)
The matrix of the second-order logarithmic derivatives of the price function Q must be
symmetric. The matrix of the second-order derivatives of Q must be nonpositive definite
to satisfy concavity of the price function. This implies that the matrix Upp — 3 + s s f must
be nonpositive definite.
In most econometric studies of production processes, inputs (and input prices) are pre
sented in a very aggregated form. To consider the disaggregated input prices, economists
5
assume independence between disaggregated input prices of one product and aggregated
prices of the rest of inputs. The problem of input price disaggregation can be resolved with
the separability assumptions. Price function Q can be presented as:
Q = Q {^{Pi ? •••*Pra)j Pm+i >•••>Pn)
where P is an aggregated price as a function of individual prices
(8)
and p i the
price of good i.
Price function Q is homothetically separable if P is linearly homogeneous in its disaggre
gated components p XJ
and independent of the rest of the input prices
Under the homothetic separability, P has properties identical to those of Q. This allows
one to formulate two- (or greater) stage optimization of the production process. The first
stage represents the determination of shares as the logarithmic differentiation of Q with
respect to aggregated P , such as equation ( 2 ). In the second stage the set of “subshares”
of the share derived in equation ( 2 ) can be computed as ( d h x P / d l n p i ) .
H-J, for example, have utilized a two-stage procedure. In the first stage, each industry
was considered to be a production process with four inputs — capital, labor, energy, and
materials. In the second stage, some of these inputs were disaggregated. For example,
energy prices were represented by coal, gas, oil, and other energy product prices. Nakamura
presented a three-stage process, where the first two were similar to the H-J model. In the
third stage, input prices of the second stage were disagregated, based on a function that
consisted of domestic and import prices.
2,3
E co n o m etric m o d el o f in p u t shares
To determine the functional forms for the unit price of output, shares, and technical change
as functions of input prices and time, an econometric approach can be utilized. In this
approach, the matrix of elasticities of substitution, the vector of technical biases, and the
6
rate of technical change are considered to be unknown.
The model of cost and production can be generated by integration of the following set
of second-order partial differential equations:
Bpp = Uppi
fipt =
fitt = ^ tt
(9)
where B pp is a matrix of elasticities of shares with respect to input prices; f3pt is a vector
of technical biases, which is expressed as the elasticity of a share with respect to time or as
the elasticity of technical change with respect to input prices; and /3tt is a constant equal
to the elasticity of technical change with respect to time.
By integrating system (9), one obtains a system of first-order partial differential equa
tions:
ap+
Bpp
In p + (3ptt
(1 0 )
a q + /3'pt In p + P ttt
(1 1 )
where vector s and v t are vectors of shares and technical change, respectively. The intercept
of equation (10) is a constant of integration. If prices are normalized to unity and the level
of technology is normalized to zero, then the intercept, a p, is equal to s. In this case, <xp
represents the share of an input expenditure per unit of output for the base period, i.e.,
the period when all prices are set to unity, and technology is represented by time t = o.
Similarly, a constant of integration ctq for the base year establishes the level of technology,
which is equal to v t for t — o.
Integrating the system of partial differential equations (10) and (11), one can derive an
expression for the unit price of output in translog form:
lnp
a 0
+ a plnp
+ OLtt +
o.slnp'JSpplnp + Inp'(3ptt +
0.5/3«(t)2,
(12)
7
where a Q is a constant of integration and p is unitary price of output.
Traditionally, linear homogeneity in prices is imposed on the set of equations (10) (12). This implies the following adding-up restrictions on parameters:
1
otp —
1
Ppt —
and
(13)
Bnn
pp
B*p p 9, l'Bpp
where 1 and 0 are vectors of ones and zeroes.
The described system of equations can be used to model an individual industry. Naka
mura used the unit price function to model a given industry. This approach is adopted here.
Collecting equations (10) - ( 1 2 ) into one system, the H-J specification can be delineated as:
+ Bpp ln p +
/3ptt
S
—
OLp
vt
=
otq + /3'pt In p + (3ut
In p
=
€xc + t t pl np +
0 .5
(14)
In j / 2 ?pp In p
+\np'pptt + 0.5/3u(ty
subject to a set of constraints (13). Constant returns to scale are imposed on the system
(14), due to the adding up restrictions. For the system of equations (14), input prices p are
exogenous.34
The estimation of shares s is the primary objective of this paper. The price equation in
(14) adds efficiency to the estimation process. Estimates of the output prices are beyond
the scope of this paper. To simplify notations, input and output prices are assumed to be
the same (denoted by p )A
3In the H-J system these pricesare projected with the growth model. Therefore, by separating equations
(14) from the rest ofthe H-J system the assumptions related to (14) are not violated.
4Under the assumption oflinear relations between input and output prices, the simultaneity for price
8
2.4
I n p u t-o u tp u t and th e N e o cla ssica l approach
The H-J procedure models optimal behavior over the historical period, and shares are
derived as a result of the assumed optimization. To compare the derived shares in time,
the shares must be expressed in real terms, i.e. prices must have a common deflator for
all periods. However, the observed shares are reported by the 1-0 tables in current values.
Therefore, 1-0 tables must be transformed in time with the deflator common to the H-J
model.
There are two ways to model real shares. First, estimate equation (10) with current
values and then deflate shares. This is the method used by H-J and Nakamura.
Our
alternative is to use real shares and prices on both sides of equation (10). These two
methods are illustrated as follows.
Assume s* is input i real share of its respective output, and p j is the real price of input
j . Then P j = Py P), where superscripts c and b stand for the current and base periods.
Then according to equation (10):
Si = (<xPt - 2 2 bP,Pj ln P j) + bP , P j P jc + Pp,tt-
(15)
3
Since p \ • is constant, the components in parentheses in equation (15) represent an intercept.
To derive current shares, expression J2j bPiPj lnp^ is set equal to zero.
Therefore, the
transformation from real to current prices is derived via the change of the intercept in the
translog formulation.
In contrast H-J compute Si in current values (expression
bPiPj In p h- is set to zero),
then Si is multiplied by p bk/ p \ where subscript k denotes output. Obviously, projection of
real shares by equation (15) or by H-J approach would yield different results. As indicated
equations in (14) can be considered. For the case of n industries, n endogenous prices and (n2— n ) share
equations can be considered as endogenous. Thus, only primary resource prices would be exogenous.
9
in the next Subsection, equations estimated for real shares provide the necessary framework
for projection of 1 - 0 tables.
As indicated, to express shares in real values H-J pre- and post-multiply shares by the
base year reciprocal of input prices and the base year prices of output. This methodology
was applied to both intermediate inputs and the value-added components. Although this
transformation of the shares from current values to real values is consistent with the
1 -0
methodology [this formula is derived in equation (18)] for the intermediate inputs, it is not
consistent for the components of value added. This inconsistency can be illustrated with
the dual
1 -0
approach.
1-0 analysis assumes value added for each sector as an exogenous variable. For the given
vector of value added, V , the 1-0 matrix A = [at-j] determines the vector of prices, P
P
-
[ I - A T]~XV .
For example, if the vector of value added, V , is such that l —
:5
(16)
a ij = v jf f°r anY JS
then according to equation (16) P = l .6 This example corresponds to the base year for
which the price deflator is set, i.e., all real prices are equal to unity for this year. Matrix
A with P =
1
is an 1-0 table for a given year [textbooks report these tables as 1-0 tables,
see footnote (5)]. Given matrix A of the current year, correspondence between real prices
for any current year to real value added, is defined below. If equation (16) considers the
current year to be the base year, then:
1 = [ J - A t ] - 1V
this equation can be rewritten as,
P
XP = [I - A T] - XP - XP V
5For details see Miller Sz Blair [1985].
6This follows from [ I — A T ] 1 = V .
(17)
10
or,
P = [I - P A TP ~ 1]~1P V .
(18)
Elements of the matrix P A T P _1 are shares of current expenditures expressed in constant
dollars P . Vector P represents real values of one unit of output and vector P V represents
value added necessary to produce the output value P . Equation (18) represents the current
year. If this year is selected as a base year, then equation (18) transforms into equation
(17).
Equation (18) presents a convenient way to deflate the A matrix. However, this equation
does not define share of value added in real terms. Since equation (18) relates output in
real terms to
1 -0
coefficients in real terms, the share of value added can be computed as a
residual. This is the double deflation method, which can be illustrated as follows:
[I - P A
tP
~ 1]1 = V d
(19)
where Vd is double deflated value added. It is important to note that only in the base year
does Vd — V .
In transforming shares from current to real values, H-J do not use double deflation.
As a result, shares may not sum up to unity. This problem is eliminated with the shares
expressed in real terms using the translog formulation. In this formulation the double
deflated value-added figure is necessary. Value added in real terms, V^, can be projected
with the share equations (10), as the neoclassical model would suggest. Yet, for projection
purporses value added in 1-0 analysis is treated as exogenous variable. In that case, sum of
shares for the intermediate consumption are constrained( i.e. sum of shares is less or equal
to unity) as opposed to the neoclassical constraints on total sum of shares (with the strict
equality to unity). This case is addressed in the next Subsection.
11
2.5
E co n o m etrica lly e stim a te d 1 -0 as a s u b s titu te for th e R A S p ro ced u re
In this subsection the econometric 1-0 link is considered for the period when prices, GNP,
shipments, and value added are reported. This information is sufficient to estimate margins
for the
1 -0
model.7
There is an extensive literature on the use of margins to update 1-0 tables. The most
recent
1 -0
table available is called the base table (not to be confused with the base year for
the deflator). The predicted 1-0 is called the target table. The target table is, obviously,
predicted for the year when margins are available.
The prediction is based on the choice of the objective function that is minimized subject
to marginal constraints in order to find the shortest distance between the base and the target
tables.
The well-known objective functions in the literature are linear, quadratic, and minimum
information objective functions. Only the last of the three has a theoretic justification.
However, the theoretic justification for the minimum information principle comes from
information theory and has no reflection on the economic nature of the problem .8 In the
economic literature the minimum information principle for the 1-0 update is called the RAS
procedure, which is discussed in detail in Section 3.
In this Subsection an alternative to the above mentioned techniques is provided. The
proposed method is based on the H-J approach, applied to the historic period with known
margins but the unknown 1-0 table. For example, in 1989 the latest available data for
margins are for 1987. The most recent update of the U.S. 1-0 table is for 1983.9 In this
7One setofmarginsforthe1-0 analysisarethecolumn totals,determined asaresidualbetween shipments
and value added. Another set ofmargins are row totals as a residual between shipments and finaldemand.
GNP can be considered as a proxy for the finaldemand.
8For the most fundamental reference on this subject see M. Bacharach [1970], or footnote (5).
9The annualestimates of1-0 tablesareproduced, among others, by Bureau ofEconomic Analysis. These
tables are provided in the electronic forms.
12
case the base 1-0 table is for 1983 and the target table is for 1987.
Earlier the importance of expressing 1-0 in real prices was emphasized. Thus, equations
(14) should represent a time-series of 1-0 coefficients and prices in real terms [computed
according to equation (19)]. The difficulty with projecting shares for the target year is
in generating estimates for the double deflated value added, because the
1 -0
table is not
available for that year.
This problem can be resolved by using the target period as a base period fo r the price
deflator because, for the base year, Vd = V . Furthermore, all real prices in the target period
are equal to unity, under this consideration. In addition prices for all other periods will
have to be deflated with the prices of the target period.
By designating the target period as the base year, share equations [in the system of
equations (14)] are greatly simplified for the target period.
Consider industry j , since
prices for the target year are equal to unity, the shares for this year are computed as:
s3 =
+ Pit* + H p°
where s J is a vector of shares for industry j , which is equivalent to the j column of the
(2°)
1 -0
table for the target year. Superscript j denotes industry-specific vector of coefficients. As
a result, an estimated share for the target year represents the constant of integration plus
the technical bias.
To simplify equation (20) further, the time variable can be presented in such a way that
t = o for the target year and t is negative (in ascending order) for all the previous years.
Then, new equations for the shares in the target year are:
=
a j + ^ 0 + 6^0.
(21)
To estimate the target 1-0 matrix, constraints on rows and columns should be introduced
13
for the H-J system. Since the sum of shares of the intermediate consumption is determined
by each column’s margins, constraints on the shares of expenditures for the intermediate
consumption have to be imposed. For example, for industry j this constraint is
si =
a j , where D - denotes the set of subscripts of value added components. According to
equation (2 1 ), these constraints on shares can be expressed as:
(2 2 )
^ 2 **Pi ~~
i£D
Another set of constraints, which is not required for the neoclassical models, is for the rows
of the 1-0 table. It is required that ]Cj=1 s i ^
j
^ /*.? f°r * £ -D, where
X j
is output in real
terms for the target year and /*. is total sales of product i to the intermediate consumers.
Again, utilizing equation (21), constraints on the intermediate sales can be expressed as
= /*'.>
where
i g D.
(23)
3= i
Since X j is known, equation (23) is linear in the unknown parameters <xJp..
In the presented model for the 1-0 table, only estimation constraints on intercepts
[equations (22) and (23)] are required. Therefore constraints on B pp and /3pt in (13) can be
relaxed, making the estimated model more general than the H-J model.
The method demonstrated here allows one to estimate the target 1-0 table for the year
that ha-s the observed variables necessary to calculate margins. The same method can be
utilized to estimate the
1 -0
table for any year with predicted margins.
To summarize, the advantages of the developed method relative to the H-J model and
1-0 model are listed below. Advantages relative to the H-J model include:
• To achieve consistency between the H-J and 1-0 models, prices in both models should
14
be in real terms.
This means that the value added should be double deflated . 10
However double deflation is not possible for the target year, unless the target year is
chosen as a base year for the price deflators. The proposed method equates double
deflated value added with the real value added for the target year. This insures
consistency between the H-J estimation of the input shares and the traditional 1-0
constraints for the target period.
• The proposed method allows one to introduce constraints on the intermediate con
sumption [equations (22)] for the target period. The H-J model allows constraints on
the sum of all shares only.
• The H-J model considers the technological structure for an individual industry. How
ever there are no constraints on the intermediate sales. The proposed method imposes
such constraints (23).
• Constraints (23) and (22) are less restrictive than the H-J constraints (13) making
the presented model more general.
The 1-0 model benefits from the proposed method as follows:
• In the traditional approach, 1-0 coefficients represent a fixed technology. In the pro
posed approach
1 -0
coefficients are based on the neoclassical behavioral assumptions.
This allows one to determine the effect of input prices on 1-0 coefficients according
to the estimated elasticities of substitutions.
• The traditional methods of projection of 1-0 tables use only one observation on 1-0
coefficients. The proposed method utilizes a time-series of 1-0 tables. This allows one
to derive the statistical properties of the estimated
1 -0
coefficients.
10Components of the double deflated value added should represent the dependent variables in the H-J
model.
15
3
E xten d ed R A S as a m eth od o f d isaggregation o f shares
estim a ted by th e H -J m ethod
3.1
H -J m e th o d o f d isa g g reg a tio n
Due to the low degrees of freedom, H-J’s analysis has to be applied to a very aggregated
1-0 table. For example, H-J considers a 9 sector table with 4 input components. Nakamura
considers 12 sectors with 5 input components.
Both works estimate aggregated 1-0 tables as a first stage of estimation. In the second
stage, under the assumption of homothetic separability between aggregate input prices, they
disaggregated individual 1-0 coefficients (shares). The sum of these disaggregated shares is
constrained to be equal to the corresponding aggregated share. Aggregated shares corre
spond to the aggregated prices P and disaggregated shares correspond to prices p 15
in equations (14).
Due to the limited degrees of freedom, only a small number of input prices can be
considered for the estimation of the disaggregated shares (H-J and Nakamura used on
average four prices for the shares’ regressions). The sum of these disaggregated shares are
not constrained across rows. As a result, intermediate consumption can exceed the supply
of input, i.e. violate equation (23). Finally, prices are not available on the time-series bases
for some inputs. This is common for different types of services (medical, business etc.). At
the same time, detailed service outlays axe reported in the
1 -0
table and, therefore, can be
projected with the RAS or other methods of 1-0 projection.
According to both works the shares estimated at the first stage of the H-J procedure
present good proxy for a very aggregated 1-0 tables. However, due to the outlined short
comings of the disaggregation (second and further stages of the H-J procedure), the number
of estimated shares may not be adequate, as was found by Nakamura.
The procedure that combines H-J disaggregated shares with the RAS method is devel-
16
oped in the following Subsection.
3.2
R A S m e th o d and th e n o tio n o f a sso cia tio n s
Methods of projection of an 1-0 table from the base to the target period are known as
nonsurvey procedures. These procedures are based on the principal that the estimate of an
1 -0
table should be as “close” as possible to the latest observed
m in
1 -0
table, which means:
d ( A ° , A p)
s.t.
aPij =
ai
(24)
I Zai
j = °'i
i
where the objective function represents the distance between base 1-0 table A 0 and target
(unknown) table Ap, and a*. and a m
j are margins for rows and columns. This system consists
of (2n —1 ) equations, since one of the equations is linearly dependent on the rest. To derive
the RAS procedure, the objective function in (24) should be expressed as an information
measure of distance (for details see Bacharach). Then, system (24) is rewritten as:
m in
s.t.
J2 za
^ 2 z fj = f i .
(25)
3
Zi3 = f>3
i
where transaction flows Z{j are substituted for the direct coefficients a*j. It will be shown
in the following that this substitution does not alter the results of the RAS procedure.
Applying the Lagrange multipliers method, one derives:
( Z i j l Z ^3 ) /
"b ln(z2-j ! z ij) )
\
or
In z fj = A,- + In
+ (fij — i) ,
17
then
Zij = eXiZ i j e \
This expression can be written in matrix form as:
(26)
Z p = R Z °S
where 1% = e Xi and Sj = e ^ -1 . Matrix Z p is computed via a series of iterations: the base
matrix is multiplied by the ratio of row sums to row margins, then the resultant matrix is
multiplied by the ratio of column sums to column margins, and so on until sums of rows
and columns of the resultant matrix are identical to the corresponding margins. The RAS
procedure is convergent and has a unique solution (for the proof see Bacharach).
Next, the notion of associations between the elements of a matrix is introduced. For any
matrix Z , the association among any four elements, which are chosen at the intersection of
two given rows (i and k ) and two given columns (j and m) is a number:
jjkm _ ZijZkm
3
zkjzim
where j ^ m and k ^ i.
It can be seen that the assocations for matrix Z p projected by RAS are the same as for
the base matrix Z ° since:
P*km _
^ij ~
z i j z km
z k j z im
_
( T i z ij& j){?*kz k m S m )
(r k Z k j S i ) ( r i z i m &m )
_
z i j z km
(27)
z kj z im
Also, Df™ is the same whether we consider the transaction flows matrix Z or the direct
coefficients matrix A . Since Z{j = a ^ X j, the association D*™ can be written as:
jjkm
3
z i j z km
z kj z im
i P ' i j X j X^Arra-^m)
( a k j X j ) ( a im X m
)
Q' i j Q' km
&kj d im
RAS procedure preserves associations [equation (27)] and provides unique projection.
Since matrices A and Z have the same associations the RAS will provide identical projection
18
wether the base matrix is A or Z.
Similarly, it can be shown that, whether the base
11
matrix is in current or real values, the RAS projection will be the same:
jyk m
^
_
Z i j Z km
_
(P iQ ij)(P k Q k m
%kj Z im
)
(P kQ h j ){p iQ im
)
Q ijQ k m
Q k jQ im
where q is a physical measure of input. The price units of the projected matrix are deter
mined by the margins and not by the base matrix.
3 .3
E R A S m e th o d as a d isa g g reg a tio n to o l for th e H -J e stim a te s
The system (24) was formulated for ( 2 n —1 ) equations and, correspondingly, the RAS
procedure would require (2n — 1 ) margins. In traditional 1-0 analysis, data on margins
are considered as the only information available for the target period. The H-J procedure
estimates aggregated coefficients that can be considered as totals of subcolumns of the
1 -0
table. This can be expressed as (( 2 n —1 ) + &), where A; is a number of subcolumns for which
total sums are estimated in the form of the share. Consider one such subcolumn l with two
elements, the constraint for the subcolumn is: Zki + z ^ i = z/. This extra constraint for
system (25) can be expressed as:
min
s.t.
^2 *ij
/zij)
y 1 zij — fi.
(28)
3
53 z *j ~ f-j for j
i
Z il + *m l ~ z l
i
11Itiscommon tosuggesttobegin the RAS procedure with theA matrix toderive Z matrix, sincemargins
are given asflow values (/) see footnote (5). This isunnecessary, since the whole process can begin from Z .
19
Table 1 : Base matrix for the RAS procedure.
1
2
3
4
1
2
X
X
X
X
X
X
X
X
4
X
X
X
X
3
X
X
X
X
Table 2: Modification of the base matrix when a single subcolumn constraint is given. This
constraint is zl'5 + z^5 = z 5. (•) denotes elements replaced by zero; (X) denotes elements
which are not replaced.
1
2
3
4
1
2
3
X
X
X
X
X
X
X
X
0
0
X
X
X z*> =
j^k,m
4
X
X
X
X
5
•
•
0
0
- Zl
Applying the Lagrangian method, in a manner similar to system (25), we derive z fj —
ViZijSj. With the traditional set of margins, such as system (25), multipliers R and S are
of dimension n. However, in the above example of system (28), S is of dimension (n + i )
because of the one additional constraint in the form of the new column. The number of
rows remains the same, making R of dimension n.
The difference in methodology between RAS and ERAS is in the specification of con
straints, as was illustrated in system (28). Representation of the base matrix constitutes
another difference between the two methods. This difference is illustrated in Tables 1 and
2.
20
The ERAS method has exactly the same basis in the minimum information principle as
does RAS. The iterative procedure applied to ERAS is the same as for RAS, so the theorems
of uniqueness and existence of RAS are equivalent for ERAS. In the above example, one
subcolumn constraint was applied to two elements of one column. Obviously, system (28)
can be designed for many rows and columns. The H-J system predicts shares (in our
example it is z/). ERAS can disaggregate this share following outlined modifications for
margins and the base matrix.
The main difference between RAS and ERAS is the change in associations. The RAS
projection postulates that associations in the base and target matrices are the same. The
ERAS relaxes this assumption. This can be explained with the example based on elements
from Table 2:
33
_ (ri*n*i)(r3*33*3 = £3 ^ 3 3
(29)
(»*3*3iSi)(»,i*i3<T)
whereas D ^i =
etc, where D denotes associations derived by ERAS and D represents
associations derived by the RAS procedure. Therefore associations of the target matrix
derived by ERAS are not necessarily identical to those of the base matrix.
4
C onclusion
H-J estimated the value of input shares in current dollars using translog relations between
input prices and shares. Afterwards, the estimated shares were deflated using 1-0 relations.
This paper has pointed out that estimation of deflated shares with the deflated input prices
via translog relations would provide results different from those derived by the H-J method.
It was further stated that the H-J method does not provide an appropriate deflator for value
added (from the
1 -0
point of view).
Double deflated value added should be used for the econometric estimation of deflated
shares and prices. If the price of the target year is chosen as base price for deflation, then
21
the set of constraints that is typical for the RAS method can easily be incorporated into
the H-J model.
The H-J method can only provide estimates for the highly aggregated 1-0 table. The
ERAS method presented here allows to disaggregate the estimated 1-0 table according with
the associations of the base table.
R eferences
Bacharach, M. (1970), Biproportional M atrices & Input-O utput Change . Cambridge
University Press, London.
Hudson E., and D.W. Jorgenson (1974), “US Energy Policy and Economic Growth, 1975 2000” B ell Journal o f Econom ics and M anagem ent Science 5, 461-514.
Nakamura, S. (1984), A n Inter-Industry Translog Model o f Prices and Technical Change
fo r the W est G erm an Econom y . Springer-Verlag, Berlin.
Miller, R.E. and P.D. Blair (1985), Input-O utput Analysis: Foundations and E xtensions .
Prentice-Hall, Inc., New Jersey.
Jorgenson, D.W. (1986), “Econometric Methods for Modeling Producer Behavior” in Z.
Griliches and M.D. Intriligator, (ed.), Handbook o f Econom etrics, Volume III.
Amsterdam: Elsevier.
Federal Reserve Bank of Chicago
RESEARCH STAFF MEMORANDA, WORKING PAPERS AND STAFF STUDIES
The following lists papers developed in recent years by the Bank’s research staff. Copies of those
materials that are currently available can be obtained by contacting the Public Information Center
(312) 322-5111.
Working Paper Series—A series of research studies on regional economic issues relating to the Sev
enth Federal Reserve District, and on financial and economic topics.
Regional Economic Issues
*WP-82-l
Donna Craig Vandenbrink
“The Effects of Usury Ceilings:
the Economic Evidence,” 1982
David R. Allardice
“Small Issue Industrial Revenue Bond
Financing in the Seventh Federal
Reserve District,” 1982
WP-83-1
William A. Testa
“Natural Gas Policy and the Midwest
Region,” 1983
WP-86-1
Diane F. Siegel
William A. Testa
“Taxation of Public Utilities Sales:
State Practices and the Illinois Experience”
WP-87-1
Alenka S. Giese
William A. Testa
“Measuring Regional High Tech
Activity with Occupational Data”
WP-87-2
Robert H. Schnorbus
Philip R. Israilevich
“Alternative Approaches to Analysis of
Total Factor Productivity at the
Plant Level”
WP-87-3
Alenka S. Giese
William A. Testa
“Industrial R&D An Analysis of the
Chicago Area”
WP-89-1
William A. Testa
“Metro Area Growth from 1976 to 1985:
Theory and Evidence”
WP-89-2
William A. Testa
Natalie A. Davila
“Unemployment Insurance: A State
Economic Development Perspective”
WP-89-3
Alenka S. Giese
“A Window of Opportunity Opens for
Regional Economic Analysis: BEA Release
Gross State Product Data”
WP-89-4
Philip R. Israilevich
William A. Testa
“Determining Manufacturing Output
for States and Regions”
WP-89-5
Alenka S.Geise
“The Opening of Midwest Manufacturing
to Foreign Companies: The Influx of
Foreign Direct Investment”
WP-89-6
Alenka S. Giese
Robert H. Schnorbus
“A New Approach to Regional Capital Stock
Estimation: Measurement and
Performance”
**WP-82-2
^Limited quantity available.
**Out of print.
Working Paper Series (cont'd)
WP-89-7
William A. Testa
“Why has Illinois Manufacturing Fallen
Behind the Region?”
WP-89-8
Alenka S. Giese
William A. Testa
“Regional Specialization and Technology
in Manufacturing”
WP-89-9
Christopher Erceg
Philip R. Israilevich
Robert H. Schnorbus
“Theory and Evidence of Two Competitive
Price Mechanisms for Steel”
WP-89-10
David R. Allardice
William A. Testa
“Regional Energy Costs and Business
Siting Decisions: An Illinois Perspective”
WP-89-21
William A. Testa
“Manufacturing’s Changeover to Services
in the Great Lakes Economy”
WP-90-1
P.R. Israilevich
“Construction of Input-Output Coefficients
with Flexible Functional Forms”
Issues in Financial Regulation
WP-89-11
Douglas D. Evanoff
Philip R. Israilevich
Randall C. Merris
“Technical Change, Regulation, and Economies
of Scale for Large Commercial Banks:
An Application of a Modified Version
of Shepard’s Lemma”
WP-89-12
Douglas D. Evanoff
“Reserve Account Management Behavior:
Impact of the Reserve Accounting Scheme
and Carry Forward Provision”
WP-89-14
George G. Kaufman
“Are Some Banks too Large to Fail?
Myth and Reality”
WP-89-16
Ramon P. De Gennaro
James T. Moser
“Variability and Stationarity of Term
Premia”
WP-89-17
Thomas Mondschean
“A Model of Borrowing and Lending
with Fixed and Variable Interest Rates”
WP-89-18
Charles W. Calomiris
“Do "Vulnerable" Economies Need Deposit
Insurance?: Lessons from the U.S.
Agricultural Boom and Bust of the 1920s”
WP-89-23
George G. Kaufman
“The Savings and Loan Rescue of 1989:
Causes and Perspective”
WP-89-24
Elijah Brewer III
“The Impact of Deposit Insurance on S&L
Shareholders’ Risk/Retum Trade-offs”
*Limited quantity available.
**Out of print.
Working Paper Series (cont'd)
Macro Economic Issues
WP-89-13
David A. Aschauer
“Back of the G-7 Pack: Public Investment and
Productivity Growth in the Group of Seven”
WP-89-15
Kenneth N. Kuttner
“Monetary and Non-Monetary Sources
of Inflation: An Error Correction Analysis”
WP-89-19
Ellen R. Rissman
“Trade Policy and Union Wage Dynamics”
WP-89-20
Bruce C. Petersen
William A. Strauss
“Investment Cyclicality in Manufacturing
Industries”
WP-89-22
Prakash Loungani
Richard Rogerson
Yang-Hoon Sonn
“Labor Mobility, Unemployment and
Sectoral Shifts: Evidence from
Micro Data”
WP-90-2
Lawrence J. Christiano
Martin Eichenbaum
“Unit Roots in Real GNP: Do We Know,
and Do We Care?”
*Limited quantity available.
**Out of print.
4
Staff Memoranda—A series of research papers in draft form prepared by members of the Research
Department and distributed to the academic community for review and comment. (Series discon
tinued in December, 1988. Later works appear in working paper series).
**SM-81-2
George G. Kaufman
“Impact of Deregulation on the Mortgage
Market,” 1981
**SM-81-3
Alan K. Reichert
“An Examination of the Conceptual Issues
Involved in Developing Credit Scoring Models
in the Consumer Lending Field,” 1981
Robert D. Laurent
“A Critique of the Federal Reserve’s New
Operating Procedure,” 1981
George G. Kaufman
“Banking as a Line of Commerce: The Changing
Competitive Environment,” 1981
SM-82-1
Harvey Rosenblum
“Deposit Strategies of Minimizing the Interest
Rate Risk Exposure of S&Ls,” 1982
♦SM-82-2
George Kaufman
Larry Mote
Harvey Rosenblum
“Implications of Deregulation for Product
Lines and Geographical Markets of Financial
Instititions,” 1982
♦SM-82-3
George G. Kaufman
“The Fed’s Post-October 1979 Technical
Operating Procedures: Reduced Ability
to Control Money,” 1982
SM-83-1
John J. Di Clemente
“The Meeting of Passion and Intellect:
A History of the term ‘Bank’ in the
Bank Holding Company Act,” 1983
SM-83-2
Robert D. Laurent
“Comparing Alternative Replacements for
Lagged Reserves: Why Settle for a Poor
Third Best?” 1983
**SM-83-3
G. O. Bierwag
George G. Kaufman
“A Proposal for Federal Deposit Insurance
with Risk Sensitive Premiums,” 1983
*SM-83-4
Henry N. Goldstein
Stephen E. Haynes
“A Critical Appraisal of McKinnon’s
World Money Supply Hypothesis,” 1983
SM-83-5
George Kaufman
Larry Mote
Harvey Rosenblum
“The Future of Commercial Banks in the
Financial Services Industry,” 1983
SM-83-6
Vefa Tarhan
“Bank Reserve Adjustment Process and the
Use of Reserve Carryover Provision and
the Implications of the Proposed
Accounting Regime,” 1983
SM-83-7
John J. Di Clemente
“The Inclusion of Thrifts in Bank
Merger Analysis,” 1983
SM-84-1
Harvey Rosenblum
Christine Pavel
“Financial Services in Transition: The
Effects of Nonbank Competitors,” 1984
SM-81-4
**SM-81-5
^Limited quantity available.
**Out of print.
Staff Memoranda (cont'd)
SM-84-2
George G. Kaufman
“The Securities Activities of Commercial
Banks,” 1984
SM-84-3
George G. Kaufman
Larry Mote
Harvey Rosenblum
“Consequences of Deregulation for
Commercial Banking”
SM-84-4
George G. Kaufman
“The Role of Traditional Mortgage Lenders
in Future Mortgage Lending: Problems
and Prospects”
SM-84-5
Robert D. Laurent
“The Problems of Monetary Control Under
Quasi-Contemporaneous Reserves”
SM-85-1
Harvey Rosenblum
M. Kathleen O’Brien
John J. Di Clemente
“On Banks, Nonbanks, and Overlapping
Markets: A Reassessment of Commercial
Banking as a Line of Commerce”
SM-85-2
Thomas G. Fischer
William H. Gram
George G. Kaufman
Larry R. Mote
“The Securities Activities of Commercial
Banks: A Legal and Economic Analysis”
SM-85-3
George G. Kaufman
“Implications of Large Bank Problems and
Insolvencies for the Banking System and
Economic Policy”
SM-85-4
Elijah Brewer, III
“The Impact of Deregulation on The True
Cost of Savings Deposits: Evidence
From Illinois and Wisconsin Savings &
Loan Association”
SM-85-5
Christine Pavel
Harvey Rosenblum
“Financial Darwinism: Nonbanks—
and Banks—Are Surviving”
SM-85-6
G. D. Koppenhaver
“Variable-Rate Loan Commitments,
Deposit Withdrawal Risk, and
Anticipatory Hedging”
SM-85-7
G. D. Koppenhaver
“A Note on Managing Deposit Flows
With Cash and Futures Market
Decisions”
SM-85-8
G. D. Koppenhaver
“Regulating Financial Intermediary
Use of Futures and Option Contracts:
Policies and Issues”
SM-85-9
Douglas D. Evanoff
“The Impact of Branch Banking
on Service Accessibility”
SM-86-1
George J. Benston
George G. Kaufman
“Risks and Failures in Banking:
Overview, History, and Evaluation”
SM-86-2
David Alan Aschauer
“The Equilibrium Approach to Fiscal
Policy”
*Limited quantity available.
**Out of print.
6
Staff Memoranda (cont'd)
SM-86-3
George G. Kaufman
“Banking Risk in Historical Perspective”
SM-86-4
Elijah Brewer III
Cheng Few Lee
“The Impact of Market, Industry, and
Interest Rate Risks on Bank Stock Returns”
SM-87-1
Ellen R. Rissman
“Wage Growth and Sectoral Shifts:
New Evidence on the Stability of
the Phillips Curve”
SM-87-2
Randall C. Merris
“Testing Stock-Adjustment Specifications
and Other Restrictions on Money
Demand Equations”
SM-87-3
George G. Kaufman
“The Truth About Bank Runs”
SM-87-4
Gary D. Koppenhaver
Roger Stover
“On The Relationship Between Standby
Letters of Credit and Bank Capital”
SM-87-5
Gary D. Koppenhaver
Cheng F. Lee
“Alternative Instruments for Hedging
Inflation Risk in the Banking Industry”
SM-87-6
Gary D. Koppenhaver
“The Effects of Regulation on Bank
Participation in the Market”
SM-87-7
Vefa Tarhan
“Bank Stock Valuation: Does
Maturity Gap Matter?”
SM-87-8
David Alan Aschauer
“Finite Horizons, Intertemporal
Substitution and Fiscal Policy”
SM-87-9
Douglas D. Evanoff
Diana L. Fortier
“Reevaluation of the Structure-ConductPerformance Paradigm in Banking”
SM-87-10
David Alan Aschauer
“Net Private Investment and Public Expenditure
in the United States 1953-1984”
SM-88-1
George G. Kaufman
“Risk and Solvency Regulation of
Depository Institutions: Past Policies
and Current Options”
SM-88-2
David Aschauer
“Public Spending and the Return to Capital”
SM-88-3
David Aschauer
“Is Government Spending Stimulative?”
SM-88-4
George G. Kaufman
Larry R. Mote
“Securities Activities of Commercial Banks:
The Current Economic and Legal Environment'
SM-88-5
Elijah Brewer, III
“A Note on the Relationship Between
Bank Holding Company Risks and Nonbank
Activity”
SM-8 8 -6
G. O. Bierwag
George G. Kaufman
Cynthia M. Latta
“Duration Models: A Taxonomy”
G. O. Bierwag
George G. Kaufman
“Durations of Nondefault-Free Securities”
♦Limited quantity available.
♦♦Out of print.
7
Staff Memoranda (corn'd)
SM-88-7
David Aschauer
“Is Public Expenditure Productive?”
SM-8 8 -8
Elijah Brewer, III
Thomas H. Mondschean
“Commercial Bank Capacity to Pay
Interest on Demand Deposits:
Evidence from Large Weekly
Reporting Banks”
SM-88-9
Abhijit V. Banerjee
Kenneth N. Kuttner
“Imperfect Information and the
Permanent Income Hypothesis”
SM-88-10
David Aschauer
“Does Public Capital Crowd out
Private Capital?”
SM-88-11
Ellen Rissman
“Imports, Trade Policy, and
Union Wage Dynamics”
Staff Studies—A series of research studies dealing with various economic policy issues on a national
level.
SS-83-1
♦♦SS-83-2
Harvey Rosenblum
Diane Siegel
“Competition in Financial Services:
the Impact of Nonbank Entry,” 1983
Gillian Garcia
“Financial Deregulation: Historical
Perspective and Impact of the Garn-St
Germain Depository Institutions Act
of 1982,” 1983
♦Limited quantity available.
♦♦Out of print.
@FedFRASER