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REGIONAL ECONOMIC ISSU ES
W o r k in g P a p e r S e r ie s
A N e w A p p r o a c h to R e g io n a l
C a p ita l
S t o c k E s tim a tio n : M e a s u r e m e n t a n d P e r fo r m a n c e
Alenka S Giese and Robert H Schnorbus
.
.
FEDERAL RESERVE B A N K
O F C H IC A G O
W P - 1 9 8 9 /6
A
N e w
A p p r o a c h
E s tim a tio n :
to
R e g io n a l
M e a su re m e n t
a n d
C a p ita l
S to c k
P e r fo r m a n c e
Alenka S. Giese and Robert H. Schnorbus*
Regional productivity analysis has become a major topic of interest among
regional economists and economic developers. The construction of a model
to analyze regional productivity has, however, been stymied by the lack of
regional capital stock data, which are a crucial input to productivity mod
els.1 In order to facilitate the analysis of regional productivity, we have
formulated a method to estimate regional net capital stock (in constant
dollars). Although we are not pioneers in estimating regional capital stock,
we have introduced a practical modification to the traditional perpetual
inventory approach, the most commonly used, that can successfully esti
mate capital stock from a limited time series of available regional data.
The perpetual inventory approach estimates net capital stock as being a
function o f previous gross investment net of depreciation and converted
from historical to constant dollars. Because this standard technique re
quires a regional gross investment series that extends back to the early
1900s, it is hampered by a lack of regional data. Another disadvantage of
this technique is that any estimation errors in the depreciation pattern
and/or deflators used will be magnified over a relatively long time period.
In order to mitigate these estimation problems, our model begins with 1982
regional net capital stock estimates and applies the perpetual inventory
equation in reverse for the years 1955-1981 and forward for the years 1983
and 1984. Our net capital stock estimates cover the manufacturing sector
in aggregate for each o f the nine Census divisions.2
The purpose o f this working paper is to provide a detailed explanation of
our methodology and to compare it to the traditional perpetual inventory
method. The paper is divided into five sections: methodology, data, de
preciation, price deflators, and diagnostic checks of accuracy. The first
section begins by describing the standard perpetual inventory method and
its advantages and disadvantages over other methods. Next, we present our
reverse perpetual inventory model. The second section describes the data
used in our calculations. The third and fourth sections discuss two crucial
*The views expressed by the authors do not necessarily reflect the views of the Federal Reserve
Bank of Chicago or the Federal Reserve System Alenka S. Giese is now a consultant for First
.
Boston Co. Robert H. Schnorbus is an economist with the Regional Section of the Federal
Reserve Bank of Chicago’s Research Department. The authors wish to thank Philip R.
Israilevich for his comments on drafts of this paper. The authors accept sole responsibility for
any errors.
FRB CH ICAGO W orking Paper
June 1989 , W P -1989-6
1
used in our calculations. The third and fourth sections discuss two crucial
components o f the model, the depreciation rates and the deflators, and the
debates surrounding them. The last section contains diagnostic checks o f
the accuracy o f our net capital stock estimates by comparing the sum of
our series across all regions to national totals compiled by the Office of
Business Analysis (OBA). Comparisons are also made between our re
gional capital stock estimates and those calculated by Hulten and Schwab
(1984).
I.
M e th o d o lo g y
Perpetual Inventory Me th od
The most common method to estimate net capital stock is the perpetual
inventory method. The Bureau o f Labor Statistics (BLS) and O B A used
it to construct their 1947-1982 gross capital stock and net capital stock se
ries and the Bureau o f Economic Analysis (BEA) currently uses it to cal
culate its fixed nonresidential private capital stock series (except for the
capital stock series for autos). Nongovernmental organizations have ap
plied it as well. For example, Jack Faucett Associates, Inc. uses a similar
technique that is applied extensively by academicians (Faucett 1973 and
1975) (Garafalo and Malhotra 1987). More recently, Charles Hulten and
Robert Schwab used it to construct a regional net capital stock series for
their productivity model (1984).
Basically, the perpetual inventory method as it applies to the construction
of net capital stock series involves the cumulative summing of past gross
investment less depreciation. The sum is then adjusted by a price deflator,
which converts historical dollars (i.e., capital goods valued at acquisition
cost) to constant dollars. Constant dollar values are used because they
most accurately reflect the actual change in net capital stock. The net
capital stock for any one year is equal to the cumulative value o f past real
gross investment less cumulative depreciation. In mathematical form, the
standard formula for real net capital stock (R N K ) is:
RNKt = £
{(H G I -
D )!P G I )t
/=i
It comprises three components: firstly, a historical dollar gross investment
time-series (HGI); secondly, annual depreciation (D); and thirdly, a price
deflator which is a ratio o f historical to constant dollars (PGI). Each
component will be discussed in detail in their respective section below.
Although the perpetual inventory method proves to be the most feasible
choice, it is hampered by four problems which are, fortunately, surmount
able. In fact, the first two problems are avoided by our reverse perpetual
FRB CH ICAGO W orking Paper
lune 1989 , W P -1989-6
2
inventory methodology and by the level of aggregation. The first problem
is that the perpetual inventory method assumes that no capital goods are
purchased prior to the first year of investment used. T o mitigate this
problem, BLS, O BA , and B E A extend their gross investment series as far
back as possible.
For plants, data begin in the 1800’s and for equipment, in the early
1900’s.3 Faucett applied the perpetual inventory method using investment
flows back to 1890 (Faucett 1973 and 1975). The problem of underesti
mating capital stock is, however, attenuated for later years because the
portion of unaccounted for capital decreases as the capital goods depreciate
and are finally discarded. Thus, for the post-1950 years, the capital stock
series are more accurate than they are for the pre-1950 years. We have
circumvented this problem by estimating 1982 regional net capital stock
values and then building the time-series backwards and forwards using the
perpetual inventory method.
The second problem is that the allocation of capital goods to specific
manufacturing industries is permanent. Thus, no account is taken for
transfers of capital goods from one industry to another or for the reclassi
fication of an establishment to a different industry. Because our capital
stock series are not disaggregated by industry, they are not distorted by this
problem.
The third problem is that the depreciation pattern, a crucial component of
the equation, is hard to specify. As a result, many different methods have
been applied and are discussed in S ection III. The primary source o f the
problem is that there is a wide variation in the service lives and depreciation
pattern among types of capital goods and a paucity of data to determine
these variations. The problem o f estimating service lives is mitigated in part
by disaggregating the capital stock series into different categories of asset
types as BLS and B E A do. Because o f the lack of regional data, we could
only separate the capital goods into plant and equipment and have used
averages of B E A service lives.
The fourth problem is the uncertain magnitude of the “values” of the net
capital stock series in constant dollars and their intertemporal comparabil
ity. Problems arise from the two assumptions that are made when historical
dollar values are converted into constant dollars. The first assumption
called into question is that old and new capital goods are materially the
same. Because of the potential differences in old and new capital goods,
the accuracy of the use of “value” as a proxy for quantity is uncertain. The
second assumption called into question is that any change in technology
and productive capability is reflected only in the change in real costs. Thus,
costless improvements in capital goods are excluded. If these improvements
FRB CH ICAGO W orking Paper
June 1989, W P -1989 - 6
3
are significant, the perpetual inventory method would underestimate the
“value” o f the capital stock in later years. Thus, setting a benchmark
against which constant dollar capital stock can be compared is made diffi
cult. These problems, however, can and have been mitigated by adjusting
price deflators for quality changes.4 The details of the debate regarding
price deflators are discussed in Section IV.
The Perpetual Inventory Method Compared to Other Methods
Other methods to construct capital stock series have been proposed, but
because there are significant problems with these methods, they are less
preferable than the perpetual inventory method.
Among these other
methods are the book-value technique and the combined bookvalue/perpetual inventory technique.5 These two techniques use IRS and
Annual Survey o f Manufactures and Census of Manufactures (ASM /CM )
data on the gross book-value of depreciable assets.
There are two advantages in estimating capital stock values from bookvalue data as opposed to gross investment data and the perpetual inventory
method. Firstly, book-value data are relatively more accurate at the de
tailed industry level because they account for the transfer of capital goods
from one industry to another. Secondly, the use of book-value data avoids
measurement error resulting from estimates on asset service lives. These
advantages, however, are counteracted by disadvantages.
Both the book-value and combined techniques are mined with problems
stemming from the availability and questionable accuracy and compar
ability across years of the book-value data. The IRS data have two limi
tations: they exclude noncorporate assets and do not take into account
physical depreciation. Using A S M /C M data avoids the first problem but
results in a shorter time series because separate plant and equipment data
are available only from 1967 on. More of a problem is that both IRS and
A S M /C M data are in historical dollars. Estimation o f a constant dollar
series is extremely difficult, if not impossible, because it requires data on the
age distribution of the capital which are not available. The second alter
native, the combined technique, tries to overcome the historical-constant
dollar conversion problem by using the relationship between historical and
constant dollar capital stock series derived from the perpetual inventory
technique. Although the combined technique may overcome the deflation
problem (although this is uncertain), it still is faced with book-value data
problems.
In contrast to book-value data, gross investment data—the key input o f the
perpetual inventory method—are relatively reliable and consistent in their
definition. In addition, in terms of availability at the regional level and
FRB CH ICAGO W orking Paper
lune 1989 , W P -7 9 8 9 -6
4
length o f time series, gross investment data have an advantage over
A S M /C M book-value data.
A Reverse Perpetual Inventory Me th od
Our approach to estimate regional net capital stock differs from the tradi
tional perpetual inventory method applied by BEA, BLS, and others (e.g.,
Hulten and Schwab). Instead o f building up the capital stock estimates
from a gross investment series, we begin by estimating 1982 regional net
capital stock and then use the perpetual inventory method in reverse to
calculate the 1955-1981 estimates and forward to calculate the 1983 and
1984 values. The reason that we took a slightly different approach is that
it overcomes some o f the estimation problems o f the standard perpetual
inventory method. Firstly, our approach does not require a regional gross
investment series that extends back to the early 1900s and thus is not
hampered by regional data limitations. Unlike Hulten and Schwab who
had to use two different data sources for regional gross investment
(A SM /C M data for the years 1951-1978 and their own estimates for
1920-1950), we were able to use one consistent regional gross investment
series. Secondly, any estimation errors in our methodology will accumulate
over a relatively shorter period o f time and will be compounded for earlier
years. In contrast, with the perpetual inventory technique, any estimation
errors will be magnified over a relatively long time period and will be
compounded for most recent years.
Our reversed version of the perpetual inventory formula is presented in
Exhibit 1. Although the most significant difference between our formula
and others is that we use the perpetual inventory method in reverse, there
are some additional differences that exist in terms of the data, depreciation
pattern, and price deflators used. These items are discussed in their re
spective section.
Exh ib it 1
Th e M odel—Regional Real Net C ap ital S to c k
Notes:
1. Net capital stock series are calculated separately for plant and
equipment using the same equations and are then summed.
2. All “estimated” variables are described in detail in the appropriate
sections that follow.
3. Real denotes constant 1972 dollars.
FRB CH ICAGO W orking Paper
June 1989, W P -1989-6
5
4.
Regional values are aggregates o f state values. The regions mirror
the nine Census divisions.
Formula:
R N K h%1- = real net capital stock in i in region r for the year 1982-t.
t
For years 1955 to 19826 :
R N K [ ,2-t = R N K t 82
“I
t=i
R B V i ,2_X ]
For years 1983 and beyond:
R N K [ ,2_ t = R N K i S2+
E I R G I i n - - ^ RBV\M _ t_ x) ^
t= -i
where
i = type of capital good (e = equipment and p = plant),
r = region.
t = - 2 (1984) to 27 (1955),
0.
R N K l S = 1982 estimated net capital stock of i in region r.
2
R G If = H G l\ jP G I l real gross investment of i in r (for years 82-t + 1
or 82-t).
H G I\ = A S M /C M historical dollar gross investment of i in r.
PGI\ — estimated regional gross investment price deflator.
4
= estimated depreciation rate for i.
R B V J = estimated real gross book value of i in r (for years 82-t or
82-t-l).
To begin, we needed to estimate 1982 regional net capital stock for plant
and equipment (i.e., R N K t; see Exhibit 2 for the estimation procedure).
Because no government data on real net capital stock are available at the
regional level, we needed to devise a method by which we could allocate
real national net capital stock data across regions. To calculate estimates
of regional shares of national net capital stock, we used regional and na
tional A S M /C M data on gross book value of depreciable assets. The im
plicit assumption is that the regional share of national gross book value and
net capital stock are commensurate. In support of this assumption’s cred-
FRB CH ICAGO W orking Paper
June 1989, W P -1989-6
6
ibility is the fact that net capital stock is a derivative o f gross capital stock
which, when valued in historical dollars, is the same as gross book value.
In addition, the depreciation rate used to calculate the difference between
gross and net capital stock should not vary greatly across regions.
Ex h ib it 2
T h e Fo rm u las For E stim atin g R N K In 1982 And
T h e R B V T im e S e rie s
R N K f c = R N K % 2 * ( R B V y R B V % 2)
(1)
(2)
(3)
39
(1) R N K f l2 =
Z [w,r82 *
or 1982 real net capital stock o f i in the
7=20
U.S. adjusted for region r’s industry mix,
where h£8 = { E M P J j ^ l E M P Q U E M P ^ E M P g ) .
2
j = two-digit SIC code industry (20-39).
E M P j % — 1982 BLS production worker employment in industry j (in r
2
or in the U.S.).
E M P % = 1982 BLS total production worker employment (in r or in the
1
U.S.).
=
1982 O B A real national net capital stock o f i in industry j.
(2) R B V 'm = 1982 estimated real gross book value of i in region r.
Years
1957+ : R B V { sl+t = R B V y -, + m % N E T D , 51+l * R G I [ S1+,)']
t=1
1955-56: N o separate plant and equipment gross book value data available.
In order to allocate total gross book value for the years
1955 and 1956, plant and equipment shares o f total
gross investment were used.
R B V l 57+ = R B V i51 t
FRB CH ICAG O W orking Paper
June 7989, W P -7 9 8 9 -6
i l ( % N E T D , S7+l+l + R G I ^ +l+l) l ;
t=1
7
where t = -2 (1955) to 27 (1984), t+ 0.
R B V \ 51 = H B V y / PBV\j51
H B V r51 = a v e [ H G I rj H G I r'] * H B V r1, or estimated 1957 historical gross
i
i
5
book value for i in r.
ave = average ratio between 1958-1960.
H G I r = A S M /C M historical dollar gross investment o f i in r.
i
A S M /C M total historical dollar gross investment in r.
H G Ir =
H B V r1 = 1957 C M total historical dollar book value in r.
5
P B V t 57 = 1957 estimated regional book value price deflator.
% N E T D t — { R G I f — R D I S C f ) I R G I m9 or average percent of
real national gross investment in i remaining after
discards (for years 57 + 1 or 57 + 1 + 1).
R G I f = O B A real national gross investment in i.
R D IS C f =
O B A real value of discards in i.
R G I f = H G I sup r sub i / P G I j or real gross investment in i in r
(for years 57 + 1 and 57 + 1 + 1).
H G I r = A S M /C M historical dollar gross investment in i in r.
i
P G I r = estimated regional gross investment price deflator for i.
i
39
(3)
R B V f a = Z [Mfo * R B V * 8 ], or 1982 estimated real book value
2
y- 20
o f i in the U.S. adjusted for r’s industry mix.
where w£8 = same as above.
2
R B V f M = R B V % 51 + 2[ ( % N E T D , 57+l * R G I ij57+,)~\,
i
t1
=
or 1982 estimated real national gross book value o f i in
industry j.
FRB C H IC AG O W orking Paper
June 1989, W P -1989-6
8
RBV%S = H B V fa IP B V fc ,
7
H BVl
57
= 1957 estimated historical dollar national gross book value
for i in industry j.
P B V f # = estimated national book-value price deflator for i in
industry j.
% N E T D i 51+t = same as above.
+t
=
OB A real national gross investment of i in industry j.
An elaboration of the methodology used to estimate the regional gross
book value time series (R B V r is useful. The regional R B V r time series was
)
derived using the perpetual inventory method, beginning with 1957 gross
book values and building up with gross investment net discards data to ar
rive at the 1982 values. The reason that A S M /C M 1982 regional gross
book values were not used is that they are in historical dollars and con
verting them to real dollars would not have been as accurate as the method
we have chosen to use. We felt that more accurate deflators could be esti
mated for gross investment than for gross book value.7 With our method,
we have to estimate only one book value deflator, 1957, whose importance
to the 1982 estimate is very small.
The regional book values calculated from this formula were summed for
each year and the totals were compared to O B A ’s real national gross capital
stock time series. Theoretically, the aggregate of our book values should
be commensurate with national gross capital stock values because the latter
are equal to cumulative gross investment net discards. For our diagnostic
checks, we calculated the correlation between the two series and ran simple
regressions (regressing our book values on O B A ’s gross capital stocks). For
both plants and equipment, the correlation was nearly one (.999) while the
regression coefficients were .95 for plants and 1.02 for equipment.
II.
D a ta
In our model, we have used two key sources of data (excluding BLS em
ployment data).
For national data except book values, we have used
O B A ’s capital stock database.8 The O B A variables used were: national
gross investment, discards, gross capital stock, and net capital stock. Price
deflators were calculated from the historical and constant 1972 dollar series
of these variables (see price deflator section). For regional data and book
values, we have used A S M /C M data, which include data on expenditures
FRB CH ICAGO W orking Paper
June 1989, W P -1989-6
9
for plant and equipment by state and gross book values o f depreciable as
sets.
The use o f both O B A and A S M /C M data did not present any inconsistency
or comparability problems because O B A ’s 1949-1982 investment time series
was supplied by the Bureau of the Census. The O BA , however, could not
use the data directly and had to make two adjustments. Firstly, changes
in industry definitions had to be considered. Secondly, some three and
four-digit SIC code level detail were missing for earlier years and had to
be approximated.
As a sidenote, data differences do exist between
O B A /B LS’s and B E A ’s capital stock time series.9
The time series for the above variables were taken as given without any
adjustments. However, data for some variables were not available for cer
tain years. Firstly, as noted above, 1957 regional gross book value for plant
and equipment were not available separately and had to be estimated from
C M total gross book value. Secondly, A S M /C M gross investment data by
state were missing for the years 1979, 1980, and 1981. In order to estimate
these values, we used the Longitudinal Research Data file (LRD), which
was developed by the Bureau of the Census and contains all the data in the
1972 and 1977 C M and the 1973-1976 and 1978-1981 A S M (see Exhibit 3).
E x h ib it 3
T h e F o r m u la T o E s tim a te M is s in g D a ta :
1 9 7 9 -1 9 8 1
H G I\ tX = H G I L R D \ tX * lb r a c k e t(H G I“s H G I L R D ? sX l I N D E X ]
J
t)
where i = plant or equipment,
tl = 1979-1981.
H G I L R D it = L R D historical gross investment in i (in r or the U.S.)
H G I it = A S M /C M historical gross investment in i (in r or the U.S.)
1978
IN D E X = [ Z
[ (H G I?s l H G I L R D f s ) * { H G I r2 /
a
a
t
12=1972
H G I L R D rtl)]] / 7, or seven year (1972-1978) average value
i
of the ratio of A S M /C M to L R D data
t2 = 1972 to 1978
FRB CH ICAGO W orking Paper
June 1989, W P -1989-6
10
Once all the necessary data for book values and gross investment were
compiled, the next step was to estimate depreciation rates to derive net
capital stock and to estimate price deflators to convert historical dollars to
constant dollars.
III.
D e p r e c ia tio n
The depreciation pattern, which accounts for capital consumption, is a
crucial element in calculating net capital stock estimates using the perpetual
inventory method. In its most accurate form, called efficiency depreciation,
it reflects the assets decline in productivity, which may or may not be
commensurate with economic depreciation.10 Because o f the difficulty in
measuring efficiency depreciation, economic depreciation is most com
monly used.11 It accounts for the decline in efficiency due to physical dete
rioration and the decline in remaining potential output that is affected by
declining service life and obsolescence. Although economic depreciation is
easier to measure, there is no consensus as to which depreciation pattern is
the most accurate. Thus, the patterns used have varied widely and have
triggered much debate. (See Exhibit 4 for a listing o f alternative methods).
E x h ib it 4
C o m m o n D e p r e c ia t io n P a t t e r n s
1. Straight-line depreciation (used by BEA, commonly used in company
annual reports), which assumes that the physical deterioration o f the capital
good is linear, that is, equal dollar depreciation across the asset’s service
life. “Expected” obsolescence is accounted for in B E A ’s estimates o f the
asset service life.12 “Unexpected” obsolescence is written off in the final year
2. N IP A depreciation (used in the National Income Product Account),
which is similar to straight-line but allocates depreciation over the asset’s
service life in proportion to its estimated service in each period and charges
obsolescence when the asset is retired.
3. Accelerated depreciation (e.g., double-declining balance, primarily used
in income tax returns), which allocates the largest portion o f the depreci
ation in the beginning years and accounts for obsolescence by assuming
that it occurs at a constant percentage rate.
4. Discounted value (advocated by Faucett), which discounts the value of
the asset’s future services and results in a pattern opposite o f the accelerated
FRB CH ICAG O W orking Paper
June 198% W P -1989-6
11
depreciation pattern, that i, l s depreciation in earlier years and more in
s es
later years.
5 Beta-decay function (used by BLS, see Bulletin 2034), which allows the
.
pattern of efficiency depreciation to vary depending on asset type.
W e have chosen to use the straight-line depreciation pattern and are s t s
ai
fied with our choice for several reasons. Although i i a simple pattern,
t s
studies have shown that overall straight-line depreciation i an accurate es
s
timate of the actual depreciation pattern. B E A regards i as a close ap
t
proximation and Hulten and Schwab (1984) were satisfied with i s use in
t
their net capital stock estimates. Young and Musgrave (1980), in their
overview of the empirical results of studies of depreciation patterns con
clude that “for broad aggregates, straight-line depreciation comes reason
ably close to the measure called for by either definition [.. the NIPA
ie,
definition and the discounted value definition].”1 In response to Faucett’
3
s
argument in favor of the discounted value model, Young and Musgrave
wrote that Faucett’ depreciation pattern did not differ significantly from
s
the straight-line pattern.
The reason that straight-line depreciation i comparable to the discounted
s
value approach i that there are two offsetting errors. F r t y for certain
s
isl,
types of equipment, straight-line may provide too slow of a depreciation
pattern. In contrast, for certain types of plants, i may overstate depreci
t
ation during the early years. Thus, when plant and equipment net capital
stock are aggregated, the estimation errors counteract each other. Sec
ondly, i straight-line depreciation i too low in early years and thus too
f
s
high in later years (as some believe), these errors may be counteracted by
straight-line depreciation’ treatment of obsolescence. Our diagnostic
s
checks compared our aggregated estimates of national net capital stock
with O B A ’. The findings showed that our use of the straight-line method
s
did not cause any significant deviations in our s r e from O B A ’ s r e that
eis
s eis
are based on a more complex depreciation pattern.
The next step, after choosing a depreciation pattern, was to determine the
asset service l v s to be used. Similar to depreciation patterns, asset service
ie
l v s vary by source. Lack of regional data on type of assets precluded us
ie
from disaggregating beyond the two broad categories of plant and equip
ment. W e chose our service l v s based on B E A estimates. For equipment,
ie
we chose an average service l f of 18 years (a depreciation rate of . 5 )
ie
08,
and for plants, we used an average service l f of 30 years (a depreciation
ie
rate of . 3
0 3). These estimations were compared to O B A ’ depreciation
s
rates that we approximated from O B A ’ gross and net capital stock s r e
s
eis
FRB CHICAGO Working Paper
lune 1989, WP-1989-6
12
summed across a l industries. Our rates were close to our estimates of
l
O B A ’ rates (1956-1982 averages, .062 for equipment and .026 for plants).
s
Our rate for plants i close to that developed by Hulten and Schwab (1984)
s
of .0361, but our rate for equipment was l s than half of theirs of .1464.
es
Their equipment r t , which r f e t an asset service l f of 6.83 years, ap
ae
elcs
ie
pears to be underestimated given B E A service l v s estimates.
ie
IV. Price Deflators
As highlighted in the methodology section, the selection of accurate price
deflators poses some d f i u t e . The major problem that plagues national
ifclis
deflators i how to account for new assets and significant quality changes
s
in existing a s t . Our task of overcoming these problems was simplified
ses
because we used O B A ’ historical and constant dollar s r e which have
s
eis
already been adjusted for price differences among asset types and to the
extent possible, quality changes. W e did, however, have to make adjust
ments for regional differences in industry mix. I i important to account
t s
for varying regional industry mixes because deflators vary significantly
across industries ( . . substantially greater increases have occurred in the
eg,
costs of the food and kindred product industry than in the costs of the
elec r c l machinery industry). See Exhibit 5 for estimation procedures.
tia
E x h ib it 5
T h e F o r m u la U s e d T o E s tim a te T h e D e f la t o r s
Gross Investment:
For years 1955 to 1982:
39
P G Irn -t = Z [yhi-t * ^ ^ 782- ] or estimated regional gross
i
/>
7=20
investment price deflator for i
;
where t = -2 (1984) to 27 (1955), t=f=0.
yj ,&2 -t ~ E A fP gi _//E M P %2 —
t
j = same as above.
E M P j^ -t = BLS production worker employment in industry j in r
.
E M P r1 = BLS total production worker employment in r
%_t
.
FRB CHICAGO Working Paper
lune 1989; WP-7989-6
13
P G l}fa _ t = H G l f a J R G l f a . ,
H G I ^ 2__ = O B A historical dollar national gross investment of i in
t
industry j
.
_ = O B A real national gross investment of i in industry j
t
.
For years 1983 and beyond:
P G I\%_t = ave{P G IrIP G I t ) * a v e (P G P " ID P G r *)* D P G I$ 2 _t;
1
e
where ave = 3 year average of 1980-1982 r t o .
ais
PG Ir$2 -t = same as above.
i
PGIj$2 -t = HGI%2-tlRGIu*2 -t same as above.
D P G lfa -t = B E A national industry implicit price deflator for nonresi-
dential fixed investment of i
.
Gross Book Value
39
P B V r = j [ ;57*Pj
i51
v
BF^5 ] or 1957 estimated regional gross book value
7,
j=20
price deflator for i
;
where vj57 = same as above.
PBVff57 = H G K f 51! R G K f 51,or 1957 national book value price deflator
of i in industry j
.
H G ^ f 57 = 1957 O B A historical dollar national gross capital stock in i
( . . cumulative investment net discards).
ie,
rgk
%51 = 1957 O B A real national gross capital stock in i
.
W e do not account for regional cost variations for two reasons. F r t y the
isl,
market for equipment i national. Thus, the deflator should be similar
s
across a l regions. Secondly, while the market for plants may be regional
l
and re l c differing construction costs, the s z of our regions i probably
fet
ie
s
large enough to make these differences negligible.
FRB CHICAGO Working Paper
June 1989, WP-1989-6
14
Y. Diagnostic Checks
To gauge the accuracy of our regional capital stock (RNK) s r e , we per
eis
formed several diagnostic checks. F r t y we compared the aggregate of
isl,
our R N K s r e with O B A ’ national capital stock s r e . Secondly, we
eis
s
eis
made comparisons between our R N K s r e and that of Hulten and Schwab
eis
(1984). The diagnostic tools that we used included graphs (absolute and
indexed l v l ) and correlations.
ees
A priori, we would expect the aggregate of our s r e to be close to O B A ’
eis
s
s r e because our estimates are partially based on O B A estimates for real
eis
national gross investment, value of discards, and net capital stock. The
results from the correlation check reveal a strong positive relationship be
tween the two s r e for both plant and equipment (see Table 1 . The
eis
)
strong correlation i demonstrated graphically in terms of absolute l v
s
e els
(see Graph 1 . There i a s
)
s
light divergency between the two s r e at the
eis
beginning of the period from 1956 to 1965. The probable reason for t i
hs
gap i that any errors in our methodology are compounded for earlier years
s
because we use the perpetual inventory equation in reverse ( . . beginning
ie ,
with 1982 RNK). This divergency translates into a gradually increasing
gap between the trend in our national aggregate and O B A ’, which can best
s
be seen by indexing both s r e to the base year of 1956 (see Graph 2 )
eis
.
T a b le 1
C o r r e la t io n B e t w e e n T h r e e M e a s u r e s o f N a t io n a l R e a l
N e t C a p it a l S to c k
RNK Series3
OBA
OBA
(n=28)
Correlations*3
FRB
H&S
10
.
FRB
(n=30)
.999
H&S
(n=24)
.999
10
.
.998
10
.
OBA = Office of Business Analysis (U.S. Department of Commerce)
FRB = Federal Reserve Bank of Chicago
H&S = Hulten and Schwab
^All correlations are significant at the 1 percent level or less.
FRB CHICAGO Working Paper
June 1989, WP-1989-6
15
Graph 1
C o m p a r is o n O f A b s o lu te L e v e ls O f R e a l N e t C a p ita l S to c K ( R N K ) N a t io n a l A g g r e g a te s
million dollars
G ra p h 2
C o m p a r is o n O f In d e x e d L e v e ls O f R N K - N a t io n a l A g g r e g a te s
FRB CHICAGO Working Paper
June 1989, WP-1989-6
76
Before comparing our regional R N K ser e to that of Hulten and Schwab,
is
we examined the aggregate of the Hulten and Schwab s r e . Similar to our
eis
s r e , their aggregated ser e was highly correlated with O B A ’ (see Table
eis
is
s
1 . In terms of absolute l v l , however, Hulten and Schwab’ ser e dif
)
ees
s is
fered from O B A ’. Hulten and Schwab’ national aggregate ranged from
s
s
12 percent (1956) to 18 percent (1973) lower than O B A ’ (see Graph 1 .
s
)
As the gap widened, however, Hulten and Schwab’ national aggregate
s
showed a weaker upward trend than O B A ’ (see Graph 2 . In f c , the in
s
)
at
dexed values of Hulten and Schwab’ s r e matched more closely our pat
s eis
tern of growth than they did O B A ’.
s
At the regional l v l there were no startling divergencies between our s r e
ee,
eis
and Hulten and Schwab’. The region by region inter-series correlations
s
were 0.99 for a l but one region. The one exception was the East North
l
Central region for which the correlation was 0.95. 14
A comparative examination of regional shares of national R N K revealed
a slightly different regional distribution of R N K in the Hulten and Schwab
s r e . In our s r e , the industrial heartland ( . . the New England, Mideis
eis
ie,
Atlantic, and East North Central regions) holds a greater share of national
R N K (about 55 percent in 1978) than they do in the Hulten and Schwab’
s
s r e (about 50 percent). The converse i true for the peripheral regions,
eis
s
that i, those regions whose manufacturing sector has developed relatively
s
recently ( . . East South Central, West South Central, and Mountain).
eg,
Differences between our s ries and Hulten and Schwab’ were found, how
e
s
ever, when regional levels of R N K were examined. As was visi l in the
be
aggregate comparisons, our s r e results in greater absolute l v
eis
e els than
Hulten and Schwab’ s r e . On a region by region basis, the absolute l v l
s eis
ees
of our s r e exceed that of Hulten and Schwab for s x out of the nine re
eis
i
gions. The three regions for which t i i not the case are the same three
hs s
peripheral regions mentioned above. Graph 3 displays examples of the two
contrasting cases: East North Central (our s r e i greater) and West
eis s
South Central (Hulten and Schwab’ i greater).
s s
When transformed into indexed l vels (1955 = 100), our s r e shows
e
eis
stronger growth than Hulten and Schwab’ for a l but two regions (East
s
l
North Central and West North Central). The greatest divergency in trends
across the two s r e occurred for three regions: East North Central, West
eis
South Central, and Mountain. For the East North Central region, Hulten
and Schwab’ s r e shows a stronger trend. For the la t r two regions,
s eis
te
Hulten and Schwab’ s r e lags ours. The probable explanation for the
s eis
FRB CHICAGO Working Paper
June 1989, WP-1989-6
17
Graph 3
C o m p a r is o n O f A b s o lu te L e v e ls O f R N K - E N C A n d W S C R e g io n s
E a s t N o r t h C e n t r a l R e g io n
million dollars
Note: ENC = East North Central (II, IN, Ml, OH, and Wl) and WSC = West South Central
(AR, LA, OK, and TX).
G ra p h 4
C o m p a r is o n O f T r e n d s In C a p ita l S to c k A n d In v e s t m e n t E a s t N o r t h C e n t r a l R e g io n
1956 ’58 ’60 ’62 ’64 ’66 ’68 7 0
72
Note: East North Central Region is IL, IN, Ml, OH, Wl.
FRB CHICAGO Working Paper
June 7989, WP-1989-6
74
76
78
’80
’82
’84
18
divergency existing for the l t e two peripheral regions i that the base
atr
s
value (1955) of these regions’ R N K in our s r e i relatively small. As a
eis s
re u t when transformed into an index, our s r e shows a stronger growth
sl,
eis
trend ( . . steeper slope).
ie,
In trying to interprete the differing inter-series patterns, the trend in re
gional real gross investment (RGI) was plotted along with the trends in
RNK. For a l but the two western regions (Mountain and Pacific), our
l
s r e tended to track the trend in RGI better than Hulten and Schwab’
eis
s
s i s (a trendline was added because the gross investment s r e i relatively
er e
eis s
c c i a ) For example, Graph 4 displays the trend of Hulten and Schwab’
ylcl.
s
RNK, our RNK, and RGI for the East North Central region, plus RGI’
s
trendline. For the period 1955-1964, our s r e shows slightly stronger
eis
R N K growth than Hulten and Schwab’ and the RGI trendline. For the
s
period 1964-1978, however, the trend in our s
eries r f ects closely that in
el
RGI.
VI. Conclusion
Our belief that we have found a feasible and accurate way to estimate re
gional capital stock (post-World War I ) has been reinforced by our diag
I
nostic checks. Our primary intent was to capture the pattern of capital
stock growth with our reverse perpetual inventory model. To that extent,
as evidenced by our comparisons with O B A ’ s r e , we have been very
s eis
successful in terms of both absolute and indexed l v l . At the regional
ees
l v l we also seem to be capturing regional patterns of growth that are
ee,
consistent with those displayed by the Hulten and Schwab’ s r e , even
s eis
though the levels tend to d f e . An additional advantage to our approach
ifr
i that i can easily be applied to any region and/or industry for which a
s
t
gross investment time-series and gross book-value data (only one year
needed) are available. In other words, our reverse inventory model can be
generalized to provide a real net capital stock s r e required of most pro
eis
ductivity models.
FRB CHICAGO Working Paper
June 1989,, WP-1989-6
19
Footnotes
1 The model referred to allocates the growth of real value added by region across
the growth rates of regional capital stock, regional labor, and regional total factor
productivity. Fo r an example see Charles Hulten and Robert Schwab, “ Regional
Productivity Growth in U .S. Manufacturing: 1951-1978,” T h e A m e r i c a n E c o
n o m i c R e v i e w , Vol. 74, No. 1, 1984, pps. 152-162.
2 The nine Census divisions are:
New England = Connecticut, Maine, Massachusetts, New Hampshire,
Rhode Island, and Vermont.
Middle Atlantic = New Jersey, New Yo rk, and Pennsylvania.
South Atlantic = Delaware, Florida, Georgia, Maryland, North Carolina,
South Carolina, Virginia, and West Virginia.
East North Central = Illinois, Indiana, Michigan, Ohio, and Wisconsin.
West North Central = Iowa, Kansas, Minnesota, Missouri, Nebraska,
North Dakota, and South Dakota.
East South Central = Alabama, Kentucky, Mississippi, and Tennessee.
West South Central = Arkansas, Louisiana, Oklahoma, and Texas.
Mountain = Arizona, Colorado, Idaho, Montana, Nevada, New Mexico,
Utah, and Wyoming.
Pacific = Alaska, California, Hawaii, Oregon, and Washington.
3 B L S ’s gross investment series begins in 1921 for equipment and in 1890 for
plants. The beginning of B E A ’s gross investment series varies within plant and
equipment groupings. Most of the equipment gross investment data begin around
1900 while the plant data begin around 1830.
4 B E A attempts to account for quality changes by adjusting Producer Price In
dexes (PPI). The theoretical basis of B E A ’s quality adjustments is explained in
Edward Denison’s “Theoretical Aspects of Quality Change, Capital Consumption,
and Net Capital Formation,” in Denison’s P r o b l e m s o f C a p i t a l F o r m a t i o n : C o n
c e p t s , M e a s u r e m e n t , a n d C o n t r o l l i n g F a c t o r s , Conference on Research in Income
and Wealth: Studies in Income and Wealth, Vol. 19, Princeton: Princeton U n i
versity Press for National Bureau of Economic Research, 1957. Fo r an alternative
method that attempts to measure equipment according to performance charac
teristics see Robert Gordon, T h e M e a s u r e m e n t o f D u r a b l e G o o d s P r i c e s , 1989,
forthcoming.
5 Fo r greater detail on these methods see Young and Musgrave, and Faucett’s
Comment in T h e M e a s u r e m e n t o f C a p i t a l (pps. 23-46 and p. 70, respectively); and
B L S , C a p i t a l S t o c k E s t i m a t e s f o r I n p u t - O u t p u t I n d u s t r i e s : M e t h o d s a n d D a t a , pps.
1-4.
6 Fo r the years 1955-1965, a different methodology was used for real net capital
stock in equipment ( R N I Q for the West South Central and Mountain regions
(W SC and M T N ). The reason for this was that when the methodology described
in the paper was used to calculate R N K g for these regions, their R N K g at the be-
FRB CHICAGO Working Paper
June 1989, WP-1989-6
20
ginning of the series turned negative. The probable reason is that these regions’
R N K ^ is relatively small (around 6.0 percent of the nation’s for W SC and 1.0
percent for M T N ). Thus, when the perpetual inventory method was applied in
reverse, the real gross investment subtractions eventually eroded these region’s
R N K f , beyond zero.
In order to correct for this problem, W S C ’s and M T N ’s share of national O B A
in 1966 was multiplied by 1955-1964 national O B A R N K ^ to derive an es
timate of their 1955-1964 R N K e (assuming their share remained constant during
this period).
R N K e
To gauge the impact of the above adjustments, the change in the other regions’
share of total national R N K and the change in the trend of total national R N K
were examined. Regarding the results for both, there were negligible changes.
As the last section demonstrates, the aggregate of the regional total R N K ’s tracks
well O B A ’s national R N K .
7 Because gross book-value data are valued at acquisition cost, the estimation of
gross book-value deflators is nearly impossible. It would require regional data
on the age distribution of the capital stock for each year which is unknown. In
order to estimate our 1957 gross book-value deflator, we used O B A ’s historical
and constant dollar national gross capital data (comparable to gross book-value)
to calculate the ratio between historical and constant dollars (see section IV).
8 Fo r published information on the methodology used by O B A to construct their
capital stock database, see B LS , C a p i t a l S t o c k E s t i m a t e s f o r I n p u t - O u t p u t I n d u s
trie s :
M e t h o d s a n d D a t a , Bulletin 2034, 1979. There is one important difference
between B L S and O B A calculations. B L S assumed that the pattern of assets
purchased remained fixed across all years (i.e., assets were purchased in the same
proportion) whereas O B A accounted for changes in the pattern of assets pur
chased by using N IP A data on types of assets purchased.
9 The three major differences between B E A and O B A /B LS (and thus A SM /C M )
gross investment data are that B E A totals unlike O B A totals include: first, pur
chases of government surplus assets; second, passenger automobiles owned by
households used for business purposes; and third, capitalized trade margins on
purchases of used equipment assets. In addition, as mentioned above, B E A ’s and
B L S ’s investment series begin in different years. Fo r more information see, B L S
Bulletin 2034.
1 Fo r a detailed definition of these two types of depreciation, see B L S Bulletin
0
2034.
1 There are other types of depreciation such as tax depreciation and change in
1
market value which should not be confused with efficiency depreciation.
1 12For an explanation of B E A ’s methodology, see B E A ,
2
T a n g i b l e W e a l t h i n t h e U . S . , 1 9 2 5 - 1 9 8 5 , June, 1987.
F ix e d
R e p ro d u c ib le
1 Allan Young and John Musgrave, “Estimation of Capital Stock in the U .S .,”
3
in Dan Usher, ed., T h e M e a s u r e m e n t o f C a p i t a l , Chicago: University of Chicago
Press for National Bureau of Economic Research, 1980, p. 36.
FRB CHICAGO Working Paper
June 1989, WP-1989-6
21
1 The slightly lower correlation for the East North Central (E N C ) region stems
4
from the relatively low correlation between the plant component of our series and
Hulten and Schwab’s (.85). In order to shed some light on why this occurred,
we examined the share of total R N K and book value (1957) held by the plant
component. Across all regions, Hulten and Schwab’s plant component consist
ently held a higher share of the total than our plant component whose share was
on average close to its share of total book value. Fo r example, for the E N C re
gion, plant’s share using Hulten and Schwab’s series went from 58 percent (1957)
to 47 percent (1978), whereas its share according to our series was a relatively
stable 34 percent (which was equal to its share of book value in 1957).
FRB CHICAGO Working Paper
June 1989, WP-1989-6
22
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FRB CHICAGO Working Paper
June 7989, WP-1989-6
23
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