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FRS
Chicago
#1996/2
REGIONAL ECONOMIC ISSUES
Working Paper Series
Forecasting Structural C hange with a R egional
Econom etric Input-O utput M odel
Philip R. Israilevich, Geoffrey J.D. Hewings,
Michael Sonis and Graham R. Schindler
FEDERAL RESERVE BANK
OF CHICAGO
WP- 1996/2
Forecasting Structural Change
Econometric Input-Output Model1
with
a
Regional
Philip R. Israilevich, Geoffrey J.D. Hewings, Michael Sonis and
GrahamR. Schindler
Abstract
The sophistication of regional economic models has been demonstrated in
several ways, most recently in the form of linking several modeling systems or
in the expansion in the number of equations that can be manipulated
successfully to produce impact analyses or forecasts. In this paper, an
alternative perspective is employed. What do regional macro-level forecasts
indicate about the process of structural change? A new methodology is
illustrated that enables analysts to make forecasts of detailed structural change
in the interindustry relations in an economy. Using a regional econometricinput-output model developed for the Chicago Metropolitan region, derived
input-output tables are extracted for the period 1975-2016. These tables are
then analyzed to determine the forecasted direction of structural changes for the
region. The innovation illustrated here is based on a model that exploits the
general equilibrium spirit of computable general equilibrium models through
the adjustment of input coefficients to clear markets.
Introduction
Prior work (Israilevich and Mahidhara, 1991) has demonstrated that the
Chicago metropolitan economy underwent some significant structural changes
over the period 1970-1990. Perhaps the most significant change was the
transformation of the employment profile from one dominated by
manufacturing to one dominated by services. However, this perspective offers
little insight into the changes that might have occurred in the structure of the
economy in terms of the interdependence between sectors. Did the Chicago
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economy exhibit evidence of the hollowing out process that Okazaki (1989)
found in Japan, in which interdependence declined as Japanese producers
sought lower cost production sites in other countries? Can the procedures that
were used to estimate the changes in interdependence in the past be used to
generate forecasts of structural changes? This paper provides a description of
the methodology that is used to extract input-output tables from an
econometric-input-output model for the period 1970-2018 for the Chicago
region. The system in which the input-output tables are embedded provides an
example of the changing nature of regional model construction in the last two
decades (see Hewings and Jensen, 1988 for more extended commentary along
these lines), with greater focus on integrating input-output tables in more
extensive or economy-wide models and with explicit focus on the problems of
error and sensitivity analysis (see, for example, Jackson and West, 1989; Sonis
and Hewings, 1989,1992; and Israilevich et al. 1995).
The methodology presented in this paper offers an innovative approach to the
forecasts of structural change in the context of interindustry linkages through
the use of a economy-wide model that achieves balance between supply and
demand in the commodity market for all commodities through the annual
adjustment of input coefficients. In contrast, many other attempts to estimate
changes in input-output coefficients have often operated over short time
horizons, focused on only a small set of sectors or used rather more mechanical
procedures for which economic interpretation of the resulting changes has
proven to be difficult
The analysis presented here examines input-output in a broader system-wide
context by exploiting the potential for introducing endogenously-generated
coefficient change within a macro-economic forecasting environment The
methodology is illustrated by reference to a model for the six-county Chicago
Metropolitan Region. In the next section, a review of input-output based
approaches to structural change are reviewed. Section 3 provides a discussion
of the modeling system and the procedure for the extraction of the annual
input-output tables from the model. In section 4, attention is focused on some
of the major changes; while a few specific sectors will be highlighted, the
majority of the attention will be directed to a 3-sector version of the findings.
The final section will reflect upon the exercise and provide some sense of the
opportunities for further analysis in this direction.
The Legacy
Input-output analysts have long been aware of the problems that are associated
with the use of a methodology that is based upon an assumption of constant
production relationships over time, especially if this time horizon stretches over
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a period of more than a decade. At the regional level, the issues are further
complicated by the potential for change in trading relationships and problems
that may arise if input-output components are nested inside larger modeling
systems (such as computable general equilibrium models) or linked or
integrated with other models (such as demo-economic models). In this section,
a brief review will be provided of some of the issues and approaches that have
been proposed to handle the problem of structural change as manifested in
changes in input coefficients.
There have been many different approaches to the problem of updating inputoutput tables; for the most part, the techniques that have been used have
handled changes in input coefficients separately from other changes in the
economy. One may point to the work of Stone (1961), Bacharach (1970),
Lecomber (1969) and others who have grappled with the problem of finding the
most efficient ways of updating the U.K. models; attention was devoted to the
problem of forecasting the values of r and s in the RAS procedure of the
following kind:
A (t + \) = r A ( t) s
(1)
Here, r and s may be considered as multipliers that implicitly transform a
prior matrix to new one and ensure that the row and column sums of the new
matrix accord with their observed or forecast values. The economic
interpretation of the r and s has proven to be contentious and analysts have
tended to frown on mechanical methods for adjustment - while, at the same
time, unable to offer something more attractive.
Fisher (197S) and Tiebout (1970) took different approaches from the macro
adjustment procedures implied by the RAS technique. Fisher surveyed experts
in different sectors of the economy to develop what he referred to as ex ante
input coefficients; these ex ante coefficients were then incorporated into new
input-output tables. Adjustments would have to be made and this process
depended very heavily on the weight attached to the various expert opinions. In
addition, there was no guarantee that each expert was basing his/her judgment
on a consistent view of the prospects for the macro economy. Almon (1967)
has provided one of the earliest, most comprehensive surveys of methods that
have been used to forecast changes in input coefficients. He has summarized
these as (i) technical projections, (ii) judicious extrapolation and (iii) product
mix adjustments. All three were used by Almon in making his forecasts of the
United States economy.
Tiebout (1970) adopted an equally novel approach; he assumed that regional
economies might reasonably be expected to evolve in similar trajectories. To
this end, he sought guidance for forecasting the future of the Puget Sound
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region's (Seattle) economy through a careful analysis of the San Francisco
economy for which an input-output table was available. The idea here was to
use the input coefficients from San Francisco at time period t as the basis for
change in the structure of the Puget Sound region between t and t+n;
furthermore, Almon's (1967) national-level forecasts were used to guide overall
structural change.
Miemyk's (1970) West Virginia model forecasts were based on a derivative of
Tiebout's idea; instead of using input coefficients from another region, he
proposed identifying best practice technology within the region. Borrowing
from the work of Carter (1970), Miernyk assumed (i) that coefficient change
was a moving average process and (ii) that the technology embodied in the best
practice firms would become the prevailing average technology ten or fifteen
years hence. The approaches of Fisher, Tiebout, Almon and Miernyk all
examine individual coefficients but with different degrees of dependence on the
macro economy for purposes of ensuring consistency.
A third major initiative in this venture of forecasting input coefficients was
initiated by Hudson and Jorgenson (1974) and subsequently, more extensively,
by Nakamura (19S4). Using a neoclassical general equilibrium approach, they
developed a macro model of the economy (in which an input-output system was
embedded) and then ran the model to examine how changes in relative prices
would effect the growth and development of the economy. All input
coefficients were endogenously determined, changes being made in response to
changes in relative prices. The translog production functions employed worked
well when the model contained just a few sectors but became more intractable
when more detailed sectoring schemes were employed, requiring consideration
of assumptions of strong and weak separability in the production function
specifications.
As Israilevich (1991) has shown, the procedure adopted here might be
considered as a columns-only adjustment process, since the only restriction was
that column sum to unity (value added was included in the estimation
procedure). The procedure was not monoproportional (in the sense of
Bacharach’s definition) since the effects of relative prices changes would likely
produce a different adjustment to each column entry. However, no effort was
made to balance across rows.
Wrigley (1970) has provided one of the few thorough analyses of the time
trends of input-output coefficients, drawing on some detailed work for the
United Kingdom; one of the important innovations in this paper is his attempt
to relate the findings to an input-output based production function. The present
paper provides a description of data that may provide opportunities for testing
some of the ideas in Wrigley’s paper. Barker (1985) developed a methodology
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that, in spirit, is not unlike the one used for Chicago; his method essentially
conjoins the detail of an input-output tables with projections of macro variables
(such as outputs, final demands, imports and exports). His findings, using the
Cambridge model of the British economy, pointed to rather small changes in
the input coefficients; however, the methodology does admit the possibility for
the inclusion of expert opinion or for partial information on changes in subsets
of coefficients.
Finally, there have been sets of approaches to the problem that have attempted
to identify analytically important coefficients (West 1981; Jensen, 1980;
Hewings, 1984; Israilevich, 1991; Bullard and Sebald, 1977, 1988 and Kpp
Jansen, 1994); however, the tests adopted here have been varied and there has
not been a concerted effort to apply these techniques in a long-run forecasting
context The notion that perhaps only a small set of coefficients might be
important sources of change is intuitively appealing; however, all the analysis
to date has been of an ex post variety and, as Hewings (1984) has shown, what
is analytically important may change over time and with the specification of the
framework in which the input-output accounts are placed.
In the next section, the approach adopted for the Chicago forecasts will be
illustrated; first, a brief description of the regional econometric-input-output
model will be provided. Thereafter, the specific methodology used to forecast
and extract the input coefficients will be revealed.
The Chicago Model and the Input-Output Extraction Method
BriefDescription ofthe Chicago Model
General Structure
The Chicago Region Econometric Input-output Model [CREIM] generates
forecasts of the Chicago economy on an annual basis, with the forecast horizon
extending up to 25 years. The model is comprised of two major components, an
input-output module and an econometric module. The modeling system is one
designed and implemented for the state of Washington by Conway (1990,
1991); the reader is referred to Conway’s papers for more complete
descriptions of the model. Figure 1 summarizes the main structure of the
model and Table 1 describes the main variables. The model is a system of
linear and nonlinear equations formulated to predict the behavior of 151
endogenous variables, and consists of 123 behavioral equations, 28 accounting
identities, and 68 exogenous variables. CREIM identifies 36 industries and
three government sectors. For each industry, there are projections of output,
employment, and earnings. Thus, out of 150 equations, only 36 relate to the
linear input-output components. Many of the non input-output equations are
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Figure 1
Structure of the Chicago Econometric Input-Output Model
P R O D U C T IO N
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Tabic 1
Description of the Major Components of the Chicago Model
Projection Horizon
1-25 years
Model Size
156 endogenous variables
55 exogenous variables
143 behavioral equations
13 identities
Industry Detail
36 industries with projections of
output
employment
earnings
Other Selected Endogenous Variables
Gross Regional Product
personal consumption expenditures
housing construction
nonresidential investment
state and local government expenditures
exports (including federal government
expenditures)
imports
labor force
unemployment rate
personal income
per capita income
net migration
population by age and sex
consumer price index
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nonlinear and estimated in a recursive fashion (usually, incorporating
autoregressive lags of order one or two). As a result, the relationships of one
sector to another include the formal input-output link as well as a set of
complex linkages through a chain of actions and reactions that could
potentially involve the whole economy. However, the output of one industry
can be related to the output of another industry, in CREIM, this is specified
through first derivatives. It would be very difficult to derive these derivatives
analytically due to the nonlinearity of many of the equations and their
incorporation of autoregressive components; in the solution to the model, these
derivatives are calculated numerically. Then, the whole system is tested to
ensure that these numerical derivatives are stable with respect to the shocks that
were used in the process of estimating the derivatives.
Among the other variables depicted by the model are gross regional product,
personal consumption expenditures, investment, state and local government
expenditures, exports, labor force, unemployment rate, personal income, net
migration, population, and the consumer price index.
The Input-Output Module
This module was constructed from establishment-level data obtained from the
U.S. Bureau of the Census. Two models have been developed, one based on
1982 and one on 1987 data; the possibility for updating these models with 1992
data will exist when the various censuses are made available in late 1995.
Since survey-based systems are prohibitively expensive, researchers developing
regional input-output models have relied on a variety of adjustments of national
level data. There are many problems with this approach; first, for many years,
the latest available U.S. national table was for 1982, and this table only
appeared in mid-19912. While updates have been made annually, the reliability
of these updates is not known. Secondly, the adjustment process in developing
regional from national tables relies on a large number of assumptions; the most
critical being the one that assumes that the technology at the regional and
national levels is identical. Since there has been little survey work done to test
this assumption, it often reverts to an assertion.3 Preliminary analysis with the
Census data suggests that differences between national and regional
technologies may be significant
REAL'S approach to table construction avoids many of these problems, since
survey data is used to build the manufacturing portions of the tables. Since the
data have already been collected by the Bureau of the Census, the tables are
constructed at a fraction of the time and expense usually associated with
survey-based methods, once constructed, the input-output table reveals the
linkages that exist between the sectors in the region. Thirty six sectors were
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Table 2
Sectoring Scheme
Sector Description
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
Livestock and Other Agricultural Products
Forestry and Fishery, Agricultural Services
Mining
Construction
Food and Kindred Products
Tobacco Manufactures
Textiles and Apparel
Lumber and Wood Products
Furniture and Fixtures
Paper and Allied Products
Printing and Publishing
Chemicals and Allied Products
Petroleum Refining and Related Industries
Rubber and Miscellaneous Plastics Products
Leather and Leather Products
Stone, Clay, Glass, and Concrete Products
Primary Metal Industries
Fabricated Metal Products
Machinery, Except Electrical
Electrical and Electronic Machinery
Transportation Equipment
Scientific Instruments; Photographic and Medical Goods
Miscellaneous Manufacturing Industries
Transportation and Warehousing
Communication
Electric, Gas and Sanitary Services
Wholesale and Retail Trade
Finance and Insurance
Real Estate and Rental
Hotels; Personal and Business Services
Eating and Drinking Places
Automobile Repair and Services
Amusement and Recreation Services
Health, Educational and Nonprofit Organizations
Federal Government Enterprises
State and Local Government Enterprises
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SIC Codes
01,02
07-09
10-14
15-17
20
21
22,23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40-42,44-47
48
49
50-57, 59
60-64,67
65,66
70-73, 76, 81, 89
58
75
78,79
80, 82-84, 86
9
identified for Chicago - essentially, the two-digit SIC manufacturing sectors
and somewhat more aggregated sectors for non-manufacturing. Table 2
describes the sectoring scheme used. While data are available at the individual
establishment level, Federal Disclosure Rules preclude the publication of data
that would reveal the transactions of individual firms or would enable
reasonable estimation from information presented.
In addition to the transactions between sectors, the table also records the
purchases made from labor (wages and salaries), capital (profits and
undistributed dividends) and imports from outside the state. Complementing
the sales made to other sectors are sales to households (consumers),
government, investment and exports outside of Chicago. With this table one
has, in essence, an economic photograph of the state of Chicago, captured at
one point in time. Adding the econometric component enables the analyst to
extend this photograph back in time to test the reliability of the system in
tracking the changes that have been observed in the economy and to redevelop
this photograph each year for the next twenty to twenty-five years producing
the annual forecasts.
Solving the M odel
The model is solved in a number of ways; in this example, assume that US
exports increase as a result of a stimulus generated by increased demand in
Eastern Europe or the Former Soviet Union. In Stage I, file model first allocates
a share of these exports to the Chicago region and these provide the first
stimulus to an increase in local production. In Figure 1, the stimulus would be
shown as entering the system through the US economy model (in this case
DRI's model) to generate an increase in Chicago's exports.
In Stage n, production of local exports generates a set of internal demands i.e., the regional interindustry demands. The individual output equations
capture these internal demands using the input-output relationships. Unlike
many other models that use national input-output coefficients, CREIM uses
Chicago-specific input-output transactions. In addition, input-output
coefficients are adjusted for changing supply-demand relationships, thus
creating the possibilities for changes in interindustry dependencies on an
annual basis. This equilibrium adjustment process - that includes a complex
system of interacting equations - avoids one of the major criticisms of the inputoutput models, namely their static nature.
Within CREIM, there are two types of output; we refer to these as a c tu a l and
output. An industry's actual output is the historical or forecasted value of
shipments (vector X ). Predicted output of a given industry is the output that is
p re d ic te d
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calculated by the Chicago input-output table (vector 2). In Israilevich e t al. 1995,
it is shown that Conway’s (1990, 1991) specification of actual and expected
outputs can be expressed as one equation:
Xt
= f i t A X t + P ,f t
where A is the Chicago input-output matrix, / is the final demand vector, t time
A
subscript, and is a diagonal matrix with elements defined as follows:
Pu =
exp
a 0+ a t
In w ‘W s & . r
lxu-J
J
(2)
where a and /? are estimated parameters from the regression that relates z t to
x ., and g t are exogenous variables (for detailed description see ibid).
Equation (1) transforms the static input-output equation into a dynamically
determined relationship between the intermediate demand and final consumption
matrix. Notice, that /? are determined by outputs of the previous period;
therefore, if a shock is introduced to the system, will 'transmit' this shock to the
current period. To measure the effect of a shock, we would have to accumulate the
shock effects over several periods. The derivative of an output related to a shock
would have to be a function of outputs affected by the shocks over time. In other
words, if we collect terms in (1) we can derive a reduced form:
If we assume that shock is expressed as/ and introduced in t and (t-1) and we want
to determine the effect of two shocks on X , then:
(4)
&
These derivatives would have to be accumulated and averaged over time. (4) was
simplified for the exposition purposes by assuming that/ is a vector, and that f t is
independent of f t_x. However, in CREIM / is matrix and only one column
(export) can be shocked; thus, the actual reduced form is more complicated than
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(4). In addition, final demands are related over time. We will show that other
blocks of the model complicate even further the relationship between the shocks
and output; therefore, analytic derivative derivation is impractical. As a result, we
are utilize a numeric derivative.
In Stage m, forecasts of output (obtained using national data and exports) are
combined with forecasts of labor productivity and wage rates to predict
employment and earnings by industry. These projections are further combined
with projections of the labor force participation rate, the unemployment rate
and natval population changes to obtain population forecasts. Meanwhile, total
earnings are obtained by predictions of property income, transfer payments,
residence adjustments and personal contributions to social insurance.
The productivity relationship equation determines employment (N). This
equation explains the linkage between an industries’ total shipments and total
employment by modeling productivity changes through time. This is an
econometrically estimated specification. Typically, we model productivity
changes using changes of employment-related variables such as hours worked,
unemployment, earnings, or aggregate output indices. As with the output
correction equations, the estimated equation is normalized to isolate
employment on the left-hand-side:
Here N t is total employment within industry /, and 8 is a white noise.
Notice, that employment is a function of output
Total earnings are then combined with population forecasts to obtain estimates
of personal income in Stage IV. The final equation in the industry block is the
wage equation. This equation describes the relationship between an industry's
employment and income through changes in average wages and salaries per
employee. This is an econometrically-specified relationship where the change
in average wage is modeled as dependent on the changes of variables impacting
employees’ income such as compensation rates, hours worked, total production,
relative unemployment rates, and economic growth. Again the estimated
equation is normalized to isolate income on the left-hand-side:
(6)
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Here w t is total income within industry /, and $ is a white noise. Again
income is a function of output via employment Typically, the structural
econometric specifications for the equations for each industry will vary.
This completes the path of the first set of demands - those originating initially
outside the system (Chicago Region). Personal income and population now
expand internal demands, the final demand sectors, comprised of consumption,
investment and government (Stage V). Very briefly, four types of consumption
expenditures and three types of investment expenditures are considered, along
with one type of state and local government expenditure.
Within CREIM, there are ten final demand variables that are modeled: gross
regional product four types of consumption, three types of investment and two
types of government spending. These ten series are then combined to calculate
net exports through an accounting identity. For each final demand series, there
is an econometrically-specified equation that links it to the 36 production
sectors of the economy. A typical final demand equation is estimated with the
dependent variable measured in per capita terms or as a proportion of personal
income. The econometric specification is then normalized so that final demand
is isolated on the left-hand-side providing:
F D ,= f ( P O P ,Y ( Y ,) p )
(7)
where p is a set of exogenous variables, Y is income and POP is population.
Again final demand is a function of output via income.
The growth of gross regional product, which equals the total value added
produced in the economy, is econometrically-related to total personal income
that includes all wages and salaries along with net income from non-work
related activities and transfer payments. For each of the four types of
consumption, the change in the proportion of personal income spent on each
category of consumption is related to similar changes for the nation as well as
other factors such as population growth and interest rates. Growth of the
investment series is then related to changes in value added, cost of capital,
savings, and population. Finally, the growth of government expenditures is
determined as a function of changes in personal income, and population.
The population block within CREIM links the demographic characteristics of a
region with their underlying economic determinants. This block has been
designed to forecast total population along with eight age and sex cohort groups
along with birth, death, and net-migration totals. Only total population is
directly related to economic variables. Total population is directly determined
through a combination of employment, labor force participation, and
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unemployment information. Changes in total population due to changes in
employment demand or other employment series will be met by an adjustment
of migration flows. Regional changes of employment characteristics will not
affect local birth or death rates, and hence cohort groups, although, at the
national level in the long term, this constraint does not hold. The age- and sexcohorts are estimated as shares of total population whose change over time is
then related to similar changes for the nation. Birth and death rates are
similarly related to female and total population respectively; these are then
related to corresponding national characteristics.
Within each industry group, the structural equations, when normalized, are all
interrelated. That is, predicted output is dependent on actual output of all
industries including its own, actual output is dependent on predicted output,
employment is dependent on actual output, and income is dependent on
employment. Each industry grouping has output, employment, and income
linked in a similar manner. These linkages are incorporated with econometric
specifications of final demand, and thus directly influences sectoral predicted
output. It can also appear on the right-hand-side of the sectoral econometric
equations, and the demographic block, which influences many of the final
demand and other sectoral equations. The complete system of equations is then
solved simultaneously to forecast all the left-hand-side variables.
This unique structure of REAL'S models overcomes one of the primary
problems of most input-output based forecasting models, namely, the static
input-output relationships. In fact, the model allows for the extraction and
forecasting of input-output tables on an annual basis. The interindustry
relationships detailed within these input-output tables have been shown to have
changed significantly over the past twenty years and are forecast to continue to
do so in the future (for prior analysis, refer to Israilevich, et al. 1994, and
Schindler et al. 1994).
Until now, the entire stimulus to the Chicago economy has come from external
demand (in this case, exports). Tracing through the effects, one arrived at
increases in personal income, the expenditures of these increases in personal
income give rise to the second set of demands that drive the model. These are
the internal demands and, in some cases, these can account for the more
significant part of the total changes in the regional economy.
In the final stage, Stage VI, the model is brought to closure as the internallygenerated final demands are feed into the input-output sectors; the system then
produces a set of outputs that are derived from the local stimulus. The increased
demand so generated works its way through the input-output module in exactly
the same way as the export stimulus did - resulting in another chain of
increases in output, employment, earnings, population, income, and again, final
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demand. This is a slice of the ripple or multiplier effect; it will continue to
work its way through the system but each time around the impact will become
smaller and smaller until the effects are negligible. The model mimics this
process through a series of iterations until convergence is obtained for each
year the model is run; the more complex the economy, the greater the number
of iterations required before convergence is achieved.
The original formulation of the WPSM system (Conway, 1990, 1991)
equilibrates outputs demanded by both intermediate and final sectors with the
supply of output Hence, to draw on Takayama (1985), the dynamic output
adjustment equation for this system of markets can be presented as:
? = *[£>(?)-*?(?)]
(8)
where both demand, D(q) and supply S(q) are expressed as functions of output
and k is the speed of adjustment of the market. The process described above
may be referred to as an adjustment along the lines of a Marshallian output
adjustment process (see Takayama, 1985 p. 295fi).4 Takayama (1985) noted
that there has been some confusion about the differences between Walrasian
and Marshallian adjustment processes; he notes that
....the Marshallian adjustment is better suited (for the case in
which) the adjustment of output is explicitly considered. It is
important to note that the Marshallian output adjustment
process is.... perfectly relevant for a competitive market
In CREIM, there is an underlying (though not observed) price adjustment
process but the operation of the model focuses on the market clearing quantity
adjustment mechanisms. Hence, the system shares more of a Marshallian
character in the terms defined by Takayama (1985). Since regional price
differentials for goods and services are generally unavailable, Marshallian
equilibrium adjustment is easier to model than a Walrasian process.
Extracting the Input-Output Coefficients
However, CREIM offers a different path for calculating economic
interdependence over time. Recall that individual time series regressions that
include input-output relations, make up only 36 of the 150 equations in the
system; many of the remaining equations are highly nonlinear and recursive,
involving AR lags of 1 or 2. As a result, the output of one sector is related to
the output of other sectors through a complex, multiple chain process in which
the input-output relationship is only one component of the chain. Nevertheless,
one can relate the output of one industry to the output of another industry
within CREIM; this is accomplished through the use of first derivatives. It
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would be a difficult procedure to derive such derivatives analytically, as a result
of the nonlinearities and AR components that are present in many equations.
Therefore, these derivatives are calculated numerically; then, tests are
conducted to examine the stability of these derivatives.
Essentially, CKEIM has an expanded input-output structure - the model is
closed in a far more complex fashion than ones that involve making households
endogenous - and in many ways retains the appearance and character of a
general equilibrium formulation. However, CREIM is solved through
adjustment of quantities rather than prices, but market clearing assumptions do
hold. Shocks are introduced through the final demand components; given a
8X
change in final demand of 4 fj> the model is solved to calculate — -. For
each Af j , n partial derivatives can be obtained; if all A/"; j = 1,...n (where
n is the number of sectors) are considered, then w2 partial derivatives will
result However, these derivatives represent the elements of Leontief inverse
matrix. One vexing question that arose concerned the degree to which the
8X
magnitude of the change, A/y . might affect the value of the derivative, — -.
5f j
If the value is sensitive to the size of the initial shock, then this procedure
would be somewhat limited; fortunately it was demonstrated that significant
variations in 4 f j implied the same derivative. This is shown in Figure 2; the
sector illustrated in the oil refining sector but all sectors exhibit the same
stability properties.
The inverse of the whole matrix of such derivatives then becomes the matrix of
direct input coefficients. However, note that this matrix is not derived from the
traditional process of using the direct coefficient matrix derived from observed
technologies, or derived through the application of Shephard's Lemma
(neoclassical approach). The process described here is a reverse process. While
these input coefficients can be interpreted as traditional coefficients, the world
from which these coefficients are derived is created by CREIM. As a result,
these input coefficients will be sensitive to changes in some of the macro
economic indicators (such as a change in national GNP, bond rates, or steel
exports). The degree of sensitivity has yet to be explored together with an
appreciation of the varying degrees of relationships that selected coefficients
might have with different external changes. In this regard, the procedure opens
up a whole new vision for the definition of error and sensitivity analysis within
the context of extended input-output systems (see Sonis and Hewings, 1991).
In many price-adjusted computable general equilibrium models, the direct input
FRB CHICAGO Working Paper
January, 1996, WP-1996-2
16
coefficients are often placed at the bottom of a nested production function that
admits (via a CES specification) substitution between aggregate inputs (i.e., the
sum of intermediate inputs) and value added but usually employs a Leontief
technology for individual inputs. CREIM, in essence, allows for substitution
between inputs in response to changes in demand generated by shocks to the
system. Potentially, underlying this system, there is a dual in which some
implicit price elasticities could be calculated.
Figure 2
Multiplier Stability and Exogenous Shock: Sector 13,1999
1 .7 9 I
1l.fO
7ft ■
0)
7 fi .
dB l.fO
3
2
■
1
l . f7 7f -
Al . 7K
f 0 .
l1. /7HA .
1 .7 3
I I I I I I M M
1111111111
I I I I I I M M
Percent Shock
The input coefficients that are derived from this process may be considered to
reflect the underlying technology; even though they are derived from a much
more complex system than is usually associated with input-output analysis, they
are not amalgams of relationships that embody the full complexity of
interactions described in earlier sections.
The degree of what may be referred to as Leontief closure determines the
degree to which the input-output multipliers fully capture the interactions in the
economy. It is well known that input-output multipliers derived only from the
nxn interindustry transactions table will yield lower multipliers than those
derived from an (n+l)x(n+l) system such as would be the case in which
households were made endogenous. In the latter case, the feedback effects
derived from increases in wages and salaries and their expenditures are
accounted for in this expanded system. In our case, where the coefficients are
FRB CHICAGO Working Paper
January. 1996, WP-1996-2
17
derived, the completeness of information is judged by the A matrix not the
Leontief inverse; the derived matrix is a static (annual) representation of the
interactions that is based on all the interactions modeled in CREIM. In this
paper, the derived input coefficient matrix corresponds to the output of n
sectors of the economy while the number of variables in CREIM is about five
times the number of sectors. Hence, the derived input coefficients may be
considered to be analogous to those associated with a Type I multiplier. We
recognize that the derived system is, in this sense, incomplete but our main
interest is in the properties of the derived tables and not the precision of these
tables. The correspondence between the derived tables and observed tables is
beyond the scope of this paper.
In this paper, we derive the matrix B H, associated with the output of n sectors;
by deriving numeric derivatives with respect to other variables, we could
potentially form a B n+k matrix where (n+k) is the total number of variables in
CREIM. Hence, A n is derived (from B n ) by assuming that the derivatives
represented by B k . rows and B .M columns are all zero. However, the process
is not easily tractable and it is often unclear how to include this system of
numeric derivatives into the form of the traditional Leontief inverse. For
example, changes in demographic variables, in respect to an output shock,
would not easily place itself in the form of the traditional Leontief matrix.
To appreciate the limitation of the incompletely derived input-output system,
we will illustrate the differences between the input-output coefficients that we
have obtained with those that are based on a greater number of variables from
CREIM. Let B n+X indicate the Leontief matrix that includes an additional row
and column (such as the household sector); from B ^ x we can derive the input
coefficient matrix,
, and then exclude from this matrix the
A n+l
(n
+ 1)*
sector resulting in the matrix, A H . Obviously, A n and A n will differ, we
will show that A n < A n . Therefore, additional rows and columns of
derivatives from CREIM, while adding new rows and columns to the input
coefficient matrix will reduce the size of the remaining (interindustry) input
coefficients. Schematically, the process is as follows:
B„=>A n
^n+l
FRB CHICAGO Working Paper
January, 1996, WP-1996-2
-4i+l —^ A n
(9)
18
This implies that the information missing from CREIMin the estimation of the
input coefficients will cause underestimation of the direct relationships between
the n sectors. However, by expanding the matrix B n towards B n+k, A n will
approach the true static representation of the complete matrix that would be
found in CREIM [The proofis provided in an Appendix.]
Numeric derivatives beyond the traditional A matrix are much harder to
construct This is the first paper to address this approach and we have limited
ourselves to the derivation of the A matrix only. At this point, we are more
interested in the properties of the derived input-output matrix rather than the
precision of the derived input-output Again, if precision was an issue, then
more work should be done in expanding numeric derivatives for other variables
such as population, capital expenditures, employment etc.
Hence, this matrix is not a matrix of pure technical coefficients or regional
purchase coefficients in the sense in which these have been employed in
regional models. However, the extraction process is consistently applied over
the time period in terms of the type of coefficients that are derived, facilitating
year-to-year comparisons. There is precedence for the development of a
somewhat non-traditional input coefficient in the earlier work of Solow (1952).
In his case, a series of difference equations was developed to analyze an
expenditure-lagged input-output system:
x (t)
= A x (t - l ) +
c
(10)
where, in this case, the input coefficient is defined as:
a
XA ‘)
(11)
However, in our case, the numerator and denominator in (11) have the same
time subscript although they reflect the influence of the lag effects of prior
years’ activities.
Prior to a discussion of some of the results, it is interesting to point out that the
d e riv e d in p u t-o u tp u t c o e ffic ie n ts fluctuate during the model calibration period
but that this annual fluctuation end once the model enters the projection period.
This results from the fact that the projection of exogenous variables are entered
into CREIM in the form of trends, while during the historical period these
variables fluctuate. However, the changes in the coefficients that result in the
model are not linear, in part because the full effect of any shock does not
manifest itself immediately, due to the AR process included in CREIM.
Therefore, to determine full effect of the shock, the model has to be simulated
FRB CHICAGO Working Paper
January, 1996, WP-1996-2
19
for several periods. Somewhat arbitrarily, a five year period was chosen;
however, changing this to 6 or 4 years would not necessarily change the results.
After the model is shocked, output variables from this simulation are calculated
and the relative difference between the simulated variables and variables from
the basic runs of the model are determined. This difference determines the
numeric value of
This procedure has been performed for the output
values only since it corresponds to one of the traditional applications of inputoutput analysis; the analysis might be expanded to other variables such as
gross regional product change as a result of a shock, unemployment change, or
investment, in other words, second order changes
5 lX i
' S ft8 G N P '
Empirical Evaluation
General Observations
In this paper, attention will be focused on a general description of some of the
changes that have been observe; a more complete, detailed evaluation will be
provided in later papers. The empirical work is presented first for the 36-sector
version, then a 3x3 analysis and finally focus is directed to one sector. Figure 3
presents a summary of the differences between the structures of the region’s
input-output coefficients measured in 1973-80 compared to 2103-2018 (recall
the moving average-type process that was used to extract the coefficients to
accommodate the nonlinearities and the lead and lag structures in many of the
equations). The changes have been summarized into three categories - positive,
negative and no change. However, some of the changes are very small and the
appearance in the figure of dramatic changes in the economy overstates the
modest nature of many of the changes. Sectors 1 through 4 represent primary
activities and construction; sectors 3 through 23 are manufacturing and the
remaining sectors (24 through 36 are services broadly defined). Some general
features stand out: first, the changes provide little evidence for some
monotonic process operating. Secondly, the changes in the manufacturing
sector noted by Israilevich and Mahidhara (1991) for the period from the early
1970s to the late 1980s seem to be continuing although at a slower rate.
Negative changes in manufacturing seem to be larger in the manufacturing to
service flows than within manufacturing. Finally, the largest (absolute)
changes appear to be concentrated in the services’ sectors.
FRB CHICAGO Working Paper
January, 1996, WP-1996-2
20
Figure 3
Changes in Input Coefficients between 1973-80 and 2013-2018
2
3 4
5 (
7
*
9 10 11 12 13 M 15 16 17 It 19 20 21 22 2) 34 25 25 27 2t 29 30 31 32 33 34 35 34
■■111
I
FRB CHICAGO Working Paper
January, 1996, WP-I996-2
21
These findings are not dramatic in the sense of portending a significant change
in the structure of the region’s economy, a parsimonious view of the expected
changes would be ones that continue some of the prior trends albeit with a less
dramatic set of outcomes than the ones observed earlier. Part of the reason for
this comment relates to the significant growth in manufacturing productivity
that was observed over the period 1970-1987; during this period, Chicago led
the nation, and for many years, Japan in manufacturing productivity (output per
employee). The general sense is that the remaining manufacturing base is now
more nationally and internationally competitive and less likely to erode (in
employment terms) quite as rapidly in the 1990s. The growth in the non
manufacturing coefficients reflect a continuing trend that reflects overall
system-wide growth in service demand.
The 3-sector analysis
To provide some sense of the temporal nature of the changes, the projected
input-output tables were aggregated into three sectors (primary, manufacturing
and services) to enable an evaluation of the macro-level changes. Figure 4
shows the changes in the direct coefficients for each of the nine elements of this
aggregated (3x3) matrix; note that the scales are not the same for each
coefficient. They are presented in this fashion to provide a broad-brush
comparison across coefficients. Recall, also, that the forecast period for the
model runs from 1992 on. There are some potentially interesting findings:
note, for example, that intra-sectoral transactions in the primary sector are
projected to increase while the reverse will be true for purchases of primary
goods from the other two sectors. The manufacturing coefficients provide
intriguing findings; of especial interest is the shape of the intra-manufacturing
sector coefficient which first drops (extending a downward trend from the
calibration period), then holds constant before beginning a slow rise in 2006.
The manufacturing to services coefficient exhibits a slight upward trend.
Similar upward trends are observed for the services coefficients, although the
pattern of the services to manufacturing coefficient is reminiscent of the intramanufacturing curve.
FRB CHICAGO Working Paper
January, 1996, WP-1996-2
22
Figure 4
Changes in Individual Coefficients, 1975*80 to 2015-2018,3 x 3 matrix
TTTTrrrrrrrnTrrri
lllllllll
mini
lllllllil(lHllillilil
liiiiumiiiiimi
urn
A single sector: fo o d and kindred products
Finally, the changes for a single sector are highlighted. In examining the
structure of the economy at an earlier time period, the behavior of the food and
kindred products sector appeared to be of considerable interest, especially given
the $2 billion ($1982 dollars) intrasectoral flow observed in 1987. Figures 5
and 6 provide a summary of the projected changes in purchases and sales
coefficients for this sector using the complete 36-sector tables. Some striking
results may be found; for example, a rather large increase in purchases from
the stone, clay and glass sector and an increase in the intra-sectoral
transactions, perhaps reflecting increased segmentation of the production
processes. These gains are balanced by decreases in purchases from fabricated
metals, machinery and transportation equipment and utilities (adoption of
energy saving technology?). The sales coefficients reflect more systematic
trends; with the exception of the increase in intra-sectoral transactions, almost
all sales to other manufacturing sectors exhibit decreases. Sales to the leather
sector are the other exception. The largest increase is recorded in sales to
eating and drinking establishments.
F R B C H I C A G O W o r k in g P a p e r
J a n u a r y . 1 9 9 6 , W P - 1 9 9 6 -2
23
Figures
Changes in Input Coefficients, 1975*80 to 2015-2018: Food and Kindred
Products
•0020
-0015
-0010
0005
OOOO 0006
Clung**
0010
0015
0020
Figure 6
Changes in Output Coefficients, 1975-80 to 2015-2018: Food and Kindred
Products
F R B C H I C A G O W o r k in g P a p e r
J a n u a r y , 1 9 9 6 , W P - 1 9 9 6 -2
24
Interpretation
The results presented here are broad-brush in scope. In addition, the trends
projected here need to be compared with those being made at the national level
for similar, broad industry aggregates; this is not to suggest that Chicago will
myopically follow whatever it projected for the nation as a whole. In fact, the
Chicago region has revealed a penchant for moving in directions that are often
counter to the United States or even in anticipation of trends in the country as a
whole.
Unfortunately, there is really no completely satisfactory way of verifying these
results; a proposal has been made to present them to major actors in the
relevant industrial sectors to gauge their reaction to the trends that have been
projected. While this approach may seem to be in the same spirit as Fisher’s
(1975) ex ante method, the major difference here is that the coefficients are
derived from the model not from industry analysts. Further, the coefficients are
consistent in the holistic sense articulated by Jensen (1980); whether they
make sense from an industry specific sense remains to be seen.
Conclusions
In attempting to evaluate this exercise, it might be tempting to engage in some
of the usual exercises of critical self-immolation that usually characterize
analysis based on input-output methodology. By moving away from the
classical, but comfortable, world of Leontief to a system in which direct
coefficients are derived from an analytically-derived inverse matrix, problems
of evaluation arise. Is this system to be judged on the basis of a sensitivity
exercise within a quantity-adjusted general equilibrium format? What type of
production function is subsumed in this system? These questions, and many
others, need to be answered to place the analytical work within the framework
that will allow careful, rigorous scrutiny.
In this regard, one other important concept may be raised. The procedure
described here may also be considered a good test of the soundness of the
model. Usual forecasts of the variables may not reveal incorrect structures in
the regression equations; however, the derived input-output system is very
sensitive to incorrect specifications. For example, in the first 100 or so runs,
negative input-output coefficients were obtained. While initially it was thought
that these values reflected problems with the procedure, it turned out that these
negative coefficients were always an indicator of a mistake in the specification
of one or more equations in the model. It so happened that, initially, it was a
F R B C H I C A G O W o r k in g P a p e r
J a n u a r y , 1 9 9 6 , W P - 1 9 9 6 -2
25
serious effort to discern the exact cause of this negativity, since there were
multiple errors.
The next steps will center around detailed evaluation of the projections,
including the application of various decomposition procedures, and a careful
evaluation of the sensitivity of the derived coefficients to alternative
specifications of the exogenous variables. The challenge now is not to produce
more forecasts but rather to produce forecasts that can be interpreted and
explained.
Appendix
Proof of the relationship between Derived Input Coefficients
Using the bordering method (Faddeeva, 1959, p. 107), we can present the
Leontief inverse as:
(Al)
where
6^, . represents the first n elements of the n+1 rowvector of
represents the corresponding column vector,
; similarly,
and
a ~ ^n+ln+l ~
Ai+i.» •
Since:
(A2)
we can rewrite (Al) as:
(A3)
F R B C H I C A G O W o r k in g P a p e r
J a n u a r y , 1 9 9 6 , W P - 1 9 9 6 -2
26
Denote Z = —C R . Now we can prove that Z 'Z .O .
a
Since R and C
represent the first n elements of row and columns of
we can
present the row as R = -a w+1. and the column as C = —3 . ^ . Therefore,
C -R Z O .
Using the determinant of the bordered matrix (see Henderson and Searle, 1981,
p. 54), we can define a as follows:
det5s+1
detB „ - ^ +l..(adj5j6.^,
(A4)
then:
detBn* = b.
detBn
K+\.»{Bn )b*n+1 = a
(A5)
Since B is positive definite, a > 0, then Z i> 0. Now we can rewrite part of
(A3) as:
( l - A n) = ( l - A „ ) + Z
(A6)
or
A = A „ -z
(A7)
therefore,
(A8)
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1 Earlier versions of this paper were presented at the Regional Science
Association International British and Irish Section Meeting, Dublin, Ireland,
September 14-16,1994 and the Western Regional Science Meeting in San
Diego, California, February, 1995. The authors would like to thank the many
comments received from these presentations as well as the comments of Jan
Oosterhaven. In addition, we would like to acknowledge the contributions of
Ramamohan Mahidhara and Eduardo Martins and the computational assistance
of Jiemin Guo. Richard Conway's contribution to the creation and development
of the econometric-input-output system that is used in this application should
also be acknowledged.
2 The 1987 benchmark tables appeared in April, 1994.
3 The work of Stevens and Trainer (1980) would refute this claim (in favor of
the importance of the regional purchase coefficients) while Giarratani and
Garhart (1991) work offers support. Israilevich et al. (1995) show that the
choice of input-output table is an important consideration in undertaking
impact analyses and forecasting with econometric-input-output tables.
4 As Takayama (1985) notes, the Walrasian equilibrium system would solve
p = Jc[D(p) - 5(p)] where demand and supply are functions of prices, p.
F R B C H I C A G O W o r k in g P a p e r
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@FedFRASER