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Journal of Business & Economic Statistics
ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20
A New Approach to Estimating Production
Function Parameters: The Elusive Capital–Labor
Substitution Elasticity
Robert S. Chirinko, Steven M. Fazzari & Andrew P. Meyer
To cite this article: Robert S. Chirinko, Steven M. Fazzari & Andrew P. Meyer (2011) A
New Approach to Estimating Production Function Parameters: The Elusive Capital–Labor
Substitution Elasticity, Journal of Business & Economic Statistics, 29:4, 587-594, DOI: 10.1198/
jbes.2011.08119
To link to this article: http://dx.doi.org/10.1198/jbes.2011.08119
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A New Approach to Estimating Production
Function Parameters: The Elusive Capital–Labor
Substitution Elasticity
Robert S. C HIRINKO
Department of Finance, University of Illinois at Chicago, Chicago, IL 60607-7121 ([email protected])
Steven M. FAZZARI
Department of Economics, Washington University, St. Louis, MO 63130-4899 ([email protected])
Andrew P. M EYER
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Federal Reserve Bank of St. Louis, St. Louis, MO 63166-0442 ([email protected])
Parameters of taste and technology are central to a wide variety of economic models and issues. This
article proposes a simple method for estimating production function parameters from panel data, with a
particular focus on the elasticity of substitution between capital and labor. Elasticity estimates have varied
widely, and a consensus estimate remains elusive. Our estimation strategy exploits long-run variation
and thus avoids several pitfalls, including difficult-to-specify dynamics, transitory time-series variation,
and positively sloped supply schedules, that can bias the estimated elasticity. Our results are based on
an extensive panel comprising 1860 firms. Our approach generates a precisely estimated elasticity of
0.40. Although existing estimates range widely, we document a remarkable convergence of results from
two related approaches applied to a common dataset. The method developed here may prove useful in
estimating other structural parameters from panel datasets.
KEY WORDS: Capital formation; Panel data; Production function parameters.
1. INTRODUCTION
Parameters of taste and technology are central to a wide variety of economic models and issues. Production function parameters loom large in real business cycle models, the monetary
transmission mechanism, exogenous and endogenous growth
models, and tax policy evaluations. This article reports a simple method for estimating production function parameters from
long-run variation in panel data, with a particular focus on the
elasticity of substitution between capital and labor, σ . Although
the value of σ has been the subject of an enormous number of
empirical studies, a consensus remains elusive; for example, in
the Joint Committee on Taxation’s (1997, T.6) study of nine
models, σ ’s range from 0.20 to 1.00.
This wide range of estimated elasticities may be attributed to
a common source. Most econometric studies rely on relatively
high-frequency quarterly or annual time series variation in investment data to identify the elasticity. Three biases may result
that weigh more or less heavily in different studies. First, the
specification of an investment equation requires assumptions
about dynamics. Although economic theory is highly informative about the demand for the stock of capital, it is relatively
silent regarding the demand for the flow of investment. Misspecified dynamics can bias estimates of elasticity (Summers
1988). Of particular importance are the nature of adjustment
costs (Hamermesh and Pfann 1996; Caballero 1999) and the
role of financing constraints (Fazzari, Hubbard, and Petersen
1988; Hubbard 1998). Second, coefficient estimates from investment regressions may be biased if the time series variation
of investment spending largely reflects adjustments to transitory
shocks and firms respond less to transitory variation than to permanent variation. An elasticity estimated with time series data
at quarterly or annual frequencies will tend to be lower than the
“true” long-run elasticity; compare our results reported herein
with those of Chirinko, Fazzari, and Meyer (1999). Third, if the
supply curve of investment is upward sloping, as is more likely
in the short to medium-run, then studies that incorrectly maintain a perfectly elastic supply schedule will tend to understate
demand elasticities (Goolsbee 1998). Although transitory timeseries variation and positively sloped supply schedules bias the
estimated elasticity toward 0, misspecified dynamics has an indeterminate effect, and thus the bias resulting from these three
factors cannot be signed.
These potential problems all stem from the use of relatively
high-frequency data for estimation. Theory is most informative
about the long-run economic relationships among the relevant
variables, however. In the case of the first-order condition for
capital (the focus of this article), problems arise from the use of
investment data as the measure of capital formation. We avoid
these problems with an approach that estimates unobserved
long-run variables by exploiting the information available from
panel data. We apply our new estimation method to the firstorder condition relating the long-run desired capital stock, K ∗ ,
to the long-run desired values of output, Y ∗ , and the user cost
of capital, C∗ ,
587
K ∗ = G[Y ∗ , C∗ ].
(1)
© 2011 American Statistical Association
Journal of Business & Economic Statistics
October 2011, Vol. 29, No. 4
DOI: 10.1198/jbes.2011.08119
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588
Journal of Business & Economic Statistics, October 2011
This specification underlies virtually all investment studies
since Jorgenson’s path-breaking work on the neoclassical
model of capital (Hall and Jorgenson 1967; Chirinko 1993).
The difficulty in estimating (1) stems from the fact that the longrun desired values are not readily observable. Our method proxies these unobservable variables as time averages within firms.
Of course, this approach would not leave adequate degrees of
freedom in pure time series data or in short panels; however,
a large panel provides sufficient degrees of freedom for estimation in the cross-section dimension. With empirical counterparts to K ∗ , Y ∗ , and C∗ defined as time averages, estimating
G[·] from cross-sectional variation is straightforward. This intuitive and computationally straightforward approach for estimating technology parameters can be interpreted in terms of a
low-pass filter placing relatively more weight on low-frequency
movements than the traditional approach of first-differencing
the capital stock widely used in the estimation of investment
models.
Our study proceeds as follows. Section 2 introduces the estimation strategy for a representative firm’s capital accumulation decision. The econometric equation is derived from firm
profit maximization, and long-run values of the variables entering this equation are measured as time averages. Our estimation strategy accounts for a variety of productivity shocks,
omitted variables, and firm fixed effects. Section 3 discusses
the panel dataset containing 1860 firms for the period 1972–
1991. Section 4 presents our ordinary least squares (OLS) and
instrumental variables (IV) results. Both techniques yield similar estimates of the substitution elasticity, approximately 0.40.
Section 4 also examines the sensitivity of the estimates across
various subsets of the sample. Section 5 compares the method
developed in this article with two alternative approaches and
computes estimates from the three models based on a common
dataset. Section 6 offers a summary and conclusions.
2.
ESTIMATION STRATEGY
2.1 Derivation
Our econometric model follows directly from the behavior of
a firm that maximizes its discounted flow of profits over an infinite horizon. We analyze the firm’s choices in long-run equilibrium, thereby eliminating the need to model adjustment costs,
delivery lags, vintage effects, and expectations. Under these assumptions, the firm always produces its long-run desired level
of output with its long-run desired mix of inputs. The critical
consequence of this approach is that the firm’s dynamic optimization problem is transformed into a static problem. To determine the firm’s demand for capital, we need only calculate
the marginal product of capital evaluated at the long-run levels
of inputs and output.
We assume that production possibilities are described by the
following CES technology:
∗[(σ −1)/σ ]
∗[(σ −1)/σ ] [ησ/(σ −1)]
+ (1 − θ ) Xf ,t
Uf ,t ,
Yf∗,t = θ Kf ,t
(2)
Yf∗,t ,
Kf∗,t ,
Xf∗,t
where
and
are the long-run desired values of real
output, the real capital stock, and all other factors of production, respectively, for firm f at time t, and Uf ,t is a stochastic
productivity shock. An attractive feature of the CES technology is that it depends on only three parameters characterizing
returns to scale η, the distribution of factor returns θ , and—
the focus of this study—substitution possibilities between the
factors of production, σ . These parameters are the same for all
firms, although the productivity shock allows the level of the
production function to vary across firms. The CES function is
strongly separable and can be expanded to include many additional factors of production (e.g., intangible capital) without
affecting the econometric equation derived later. This feature
gives the CES specification an important advantage relative to
other technologies that allow for a more general pattern of substitution possibilities (e.g., the translog, minflex-Laurent). Our
approach does not require price and quantity data on the other
factors of production to recover the key parameter of interest.
Differentiating (2) with respect to capital, we obtain the following relation for the marginal product of capital (∂Yf∗,t /∂Kf∗,t ):
∗[1+(1−σ )/σ η]
∂Yf∗,t /∂Kf∗,t = (ηθ )Yf ,t
∗−[1/σ ]
Kf ,t
[(σ −1)/σ η]
Uf ,t
.
(3)
Profit maximization implies that this marginal product of capital equals the Jorgensonian user cost of capital (Cf∗,t ), which
combines interest, depreciation, and tax rates with relative
prices. (The details of the user cost specification are given in
Appendix A; all appendices are available at our Web sites.) Setting ∂Yf∗,t /∂Kf∗,t equal to Cf∗,t and rearranging (3), we obtain the
following expression for the long-run desired capital stock,
∗[β]
[ζ ]
Kf∗,t = Cf∗[α]
,t Yf ,t Uf ,t ,
(4)
(ηθ )σ , α
= −σ, β = (σ η + 1 − σ )/η, and ζ =
where =
(σ − 1)/η.
The central difficulty in estimating (4) is the fact that the
long-run values are not observed. Most previous research addresses this problem by differencing the log of (4) to obtain
an equation for investment. However, as discussed in Section 1,
this approach necessarily confronts the modeling challenges associated with investment data, and thus may generate biased
estimates. To avoid these potential problems, we focus on the
capital stock rather than on investment and develop an econometric method to directly estimate the long-run desired levels
of capital, output, and user cost as time averages. We refer to
the years over which an average is computed as an interval and
divide the sample into three intervals indexed by a subscript τ
(τ = 0, 1, 2). The intervals are 1972–1977 (τ = 0), 1978–1984
(τ = 1), and 1985–1991 (τ = 2). We measure Kf∗,t by Kf ,τ ,
where the latter is the mean of Kf ,t over a τ interval. (Symmetric definitions apply to Yf∗,t and Cf∗,t .) The τ = 1 and τ = 2
intervals are used for parameter estimation; the τ = 0 interval
is used only to form instruments and define classifications that
split the sample.
With the variables in (4) defined by averages computed for
the τ = 1 and τ = 2 intervals, we take logarithms and obtain
the following equation:
kf ,τ = αcf ,τ + βyf ,τ + ψ − uf ,τ ,
τ = 1, 2,
ln[Kf∗,t ], cf ,τ
(5)
ln[Cf∗,t ],
≡ ln[Cf ,τ ] =
where kf ,τ ≡ ln[Kf ,τ ] =
∗
yf ,τ ≡ ln[Yf ,τ ] = ln[Yf ,t ], ψ ≡ ln[], and uf ,τ is an error term
that follows directly from the technology and represents productivity shocks. We model productivity shocks as
uf ,τ = ζ [vf + vi + wτ + wi,τ + wf ,τ ].
(6)
Chirinko, Fazzari, and Meyer: The Elusive Capital–Labor Substitution Elasticity
The productivity shock is decomposed into firm-specific (vf )
and industry-specific (vi ) components, as well as components
that vary over the τ intervals, wτ , wi,τ , and wf ,τ . With this error
structure, estimates can be obtained by differencing (5) and (6)
between the τ intervals,
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kf = −σ cf + β yf − γ − λi − wf ,
(7)
where kf ≡ kf ,τ , cf ≡ cf ,τ , yf ≡ yf ,τ , σ ≡ −α,
γ ≡ ζ wτ , λi ≡ ζ wi,τ , and wf ≡ ζ wf ,τ . Because there
are only two intervals, first-differencing eliminates the temporal dimension of the model, and τ subscripts have been omitted
in (7). Consequently, the parameters are estimated in a crosssectional regression. Fixed firm and industry effects are eliminated by differencing, and fixed interval effects (notably, biased
technical change) are captured by the constant γ . Industry effects that vary across intervals are captured by industry dummies, λi .
2.2 Discussion and Interpretation
Equation (7) is the basis for estimation; it relates long-run
growth in the capital stock to long-run growth in the user cost
and output across firms. Cross-sectional variation in the data is
key to identification and allows us to estimate the σ parameter, which is of central interest in this study. In addition, we can
recover the returns to scale elasticity, η, as a nonlinear combination of the estimated σ and β parameters.
In contrast to many previous studies using investment data,
our estimation method emphasizes low-frequency variation. We
measure the long-run values of the regression variables by taking averages over several years (intervals) and then differencing the averaged data to control for a variety of fixed effects.
This new interval-difference estimator (IDE) stands in contrast
to the standard investment regression derived by differencing
untransformed annual data; we refer to the latter approach as
the year-difference estimator (YDE). Because it averages over
several years, the IDE places substantially more weight on the
lower frequencies reflecting long-run variation in the data. (See
Appendix B for a frequency domain interpretation of these estimators using spectral methods; that analysis suggests that the
IDE defined for an interval length of 7 years seems reasonably
successful in emphasizing long-run variation, especially relative to the YDE.)
Apart from the frequency of the data entering the regression model, satisfactory OLS parameter estimates depend on
the relationship between the stochastic element, wf , and the
regressors, especially yf . This correlation is not likely to be
a problem for two reasons. First, wf is that part of the productivity shock that remains after accounting for a variety of
fixed effects. Major technological changes (e.g., telecommunications, computing, the Internet) are likely to have their greatest
effects on all firms (captured by γ ) or on all firms in an industry
(captured by λi ) with only a small residual impact that is firmspecific. Second, only part of the productivity shock enters the
error term. As noted by Shapiro (1986), including output in a
factor demand equation can completely absorb the productivity shock. When the elasticity of substitution is unity, ζ equals
0, and uf τ vanishes [cf. (4) and (6)]. When σ deviates from
unity, the impact of the productivity shock is nonetheless diminished by ζ , provided that returns to scale are not decreasing
589
too sharply. Despite the arguments that OLS estimates will not
be appreciably distorted by wf , we present two alternative estimates that are robust to simultaneity. First, in Section 4.1 we
impose constant returns to scale (η = 1 → β = 1). Thus yf ,
the variable most likely to be correlated with wf , no longer
appears as a regressor. Second, in Section 4.2 we present IV
estimates and Hausman tests using the variables in the τ = 0
interval as instruments.
The IDE is robust to four potentially important distortions.
First, parameter estimates are not sensitive to trending variables. (See Appendix C for formal consideration of this issue
and for the role played by differencing in eliminating firmspecific trends.) Second, the estimates are unlikely to be influenced by additional factors that might affect the specification of the production function. For example, the specification
of the econometric equation does not change with additional
factors of production. Markups that vary across firms are captured by a firm-specific fixed effect that is eliminated by differencing. Moreover, the information processing revolution might
have led to biased technical change. In the CES technology (2),
biased technical change is represented by temporal variation
in θ and, like wτ , will be reflected in the constant (4). Third,
studies implementing the Jorgensonian framework often have
been criticized for failing to distinguish between desired output and actual output (e.g., Coen 1969; Hall 1995). The time
averages in the econometric equation recognize this important
distinction. Fourth, the estimates are unlikely to be affected
by measurement error in the capital stock. Classical measurement error will be part of the error term, and hence innocuous. A plausible situation in which measurement error may be
systematic arises when an increase in the pace of technological change effectively increases the depreciation of fixed capital through obsolescence, an effect not captured in our fixed
depreciation rate assumption. However, an increase in depreciation rates would lead to a systematic overstatement of capital
in τ = 2 and would be captured by the constant. If omitted variables or measurement error are both firm-specific and intervalvarying, then consistent estimation might become an issue. In
this case, the IV estimates and measurement error analysis discussed in Section 4.2 provide useful checks on the parameter
estimates.
In summary, the simple, intuitive interval-difference estimation strategy developed here collapses the time dimension
of firm panel data by defining three intervals and then timeaverages the data within an interval. The first interval is used to
form instruments or sort variables into contrasting classes, and
the second and third intervals are used for estimation. Identification is provided by the cross-sectional variation in the growth
rates of capital, output, and the user cost. A variety of productivity shocks, omitted variables, and fixed effects are accounted for by estimated parameters or differencing. Thus production function parameters are estimated in a cross-section of
time-averaged, differenced firm data. This econometric model
does not solve the estimation problems inherent with investment models—difficult-to-specify dynamics, transitory time series variation, and positively sloped supply schedules—that can
bias estimates of the elasticity. Instead, our approach avoids
these problems by exploiting panel data with a method that
emphasizes low-frequency variation and directly estimates the
first-order condition for capital.
590
Journal of Business & Economic Statistics, October 2011
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3. THE PANEL DATASET
To construct the variables needed to estimate (7), we link
firm-level financial data from Compustat with industry-level
user cost data from Data Resources, Inc. (DRI). The user cost,
C, is the opportunity cost of a dollar investment in fixed capital (the interest rate plus the depreciation rate) adjusted for
taxes and multiplied by the relative price of capital. The user
cost is defined in detail in Appendix A. Measurement of the
capital stock, K, is important for our study. Accounting book
values of net plant and equipment likely understate current replacement values in periods of inflation. In addition, accounting
depreciation rules might not accurately reflect economic depreciation. We measure the replacement value of capital with an
iterative algorithm and adjust for changes due to acquisitions
and divestitures, as described in Appendix A. Output, Y, is annual gross sales. Sales will differ from output by the change
in finished goods inventories. Although this difference may be
nontrivial in the short run, it will have very little impact on the
long-run averages used in our estimation. Nominal sales figures
from Compustat are deflated by industry-specific output price
indexes from DRI.
To protect against results driven by a small number of extreme observations, we exclude observations in the 1% upper
and lower tails of the firm-specific variables. We checked the
robustness of our results to this procedure by deleting both the
0.5% and 2% tails and found a negligible effect on the results.
Because the user cost is computed from stable industry and aggregate data, we did not delete data in the tails of the user cost
variable distribution.
Our final data set contains 1860 firms from all sectors of the
economy. The data for estimating the parameters with the IDE
consist of the growth rates in model variables between τ = 1
and τ = 2 intervals that enter directly into the model. Whereas
(7) is estimated with a cross-section of firms, the value of each
observation is based on temporal variation between intervals.
The coefficient of variation for the growth rates in capital, output, and the user cost are 2.3, 2.4, and 2.7, respectively, indicating substantial differences across firms in the regression variables. (Summary statistics are provided in Appendix A.) It is
not surprising that capital and output growth rates differ across
firms, but the firm heterogeneity in the user cost growth rate is
remarkable relative to most previous studies. We rely on this
heterogeneity to identify σ with the IDE.
4. EMPIRICAL RESULTS WITH THE
INTERVAL–DIFFERENCE ESTIMATOR
4.1 Ordinary Least Squares Estimates
OLS estimates of the structural parameters from (7) appear in
Table 1. The focus of our study is on σ . In column 1, our benchmark estimate of σ is 0.367 with a standard error of 0.068. The
null hypothesis that the elasticity is 0 can be strongly rejected at
any conventional level of significance. It is also clear, however,
that we can reject the hypothesis that σ is 1, the value implied
by the Cobb–Douglas production function and often assumed
in applied work.
The estimated returns to scale elasticity, η, is a function of
the regression coefficients on the growth in both output, β, and
user cost, σ (see Table 1). The OLS estimate of η is 1.135, also
with a small standard error. It is interesting to note that the effect
of output is much stronger here than in typical panel data studies using investment data. Estimates from investment models
of coefficients that correspond to β in our study are often well
below 0.5, which suggests an implausible degree of increasing
returns. We believe that the reason for our more reasonable result is that, unlike investment equations, our estimation method
captures long-run, permanent changes in output and is not affected by the transitory variation that might unduly influence
investment regressions with annual or quarterly data.
The second column of Table 1 presents results from the inclusion of two-digit industry dummies in the benchmark regression [the λi terms in (7)]. These dummies control for industrylevel productivity shocks between intervals τ = 1 and τ = 2 or,
more generally, any industry-specific effects that vary across
the intervals. The structural parameter estimates do not change
much when the dummies are included. The σ estimate rises
Table 1. OLS and IV estimates based on the IDE
OLS
Unconstrained
Parameter
σ
η
β
γ
R2
IV
Constrained
Unconstrained
Constrained
Benchmark
With SIC dummies
η=1
η = 1, σ = 1
Benchmark
With SIC dummies
η=1
η = 1, σ = 1
0.367
(0.068)
1.135
(0.076)
0.925
(0.038)
0.084
(0.012)
0.564
0.440
(0.247)
1.152
(0.115)
0.926
(0.037)
−0.055
(0.060)
0.593
0.372
(0.072)
1.000
1.000
0.434
(0.098)
1.000
1.000
1.000
1.000
1.000
1.000
0.063
(0.013)
0.560
0.020
(0.013)
0.540
0.373
(0.363)
0.633
(0.184)
1.364
(0.166)
−0.324
(0.101)
—
1.000
1.000
0.390
(0.119)
0.603
(0.086)
1.402
(0.117)
−0.049
(0.034)
—
0.059
(0.013)
—
0.020
(0.013)
—
NOTE: Estimates of the IDE [eq. (7)] with firm-level panel data as described in Sections 2 and 3. The instrument list for the estimates in columns 5–8 includes the user cost (Ci,τ =0 ),
capital stock (Ki,τ =0 ), output-to-capital ratio [(Y/K)i,τ =0 ], and cash flow-to-capital ratio [(CF/K)i,τ =0 ]. In addition, we include the annualized growth rates of capital, output, cash
flow, accounts receivable, and cash and cash equivalents defined over the τ = 0 interval. For the estimates reported in column 6, it was not useful to include industry dummies in the
model as both regressors and instruments, because of collinearity. In this column, the instrument set for output and user cost growth variables is the same as for the other IV regressions;
the industry dummies are instrumented by themselves. Heteroscedastic-consistent standard errors are given in parentheses. The parameters are σ (the capital–labor substitution or user
cost elasticity), η (the returns to scale elasticity), β (the regression coefficient on output growth), and γ (the intercept). The returns to scale elasticity are recovered from the estimated
coefficients with the formula η = (1 − σ )/(β − σ ), when β > σ . The variance of η is computed by the delta method.
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Chirinko, Fazzari, and Meyer: The Elusive Capital–Labor Substitution Elasticity
from 0.367 to 0.440, and η is virtually identical. The standard
error of σ also rises, however. The σ estimate is much less precise with industry dummies in the model because a substantial
amount of the variation in the user cost growth variable is at
the industry level. For this reason, and given the modest change
in σ , the remaining OLS regressions in Table 1 exclude the industry dummies and thus do not control for industry effects that
vary across intervals.
As discussed in Section 2, the most likely source of correlation between the error term and the independent variables in
these OLS regressions comes from firm-specific productivity
shocks (embedded in the error term) and firm output growth.
This potential simultaneity problem can be avoided by imposing constant returns to scale (η = 1, implying β = 1), an assumption that removes output growth as a regressor. The third
column of Table 1 presents a regression with the output growth
coefficient constrained to unity. The σ estimate changes only
trivially (from 0.367 to 0.372) when constant returns to scale
are imposed. This result supports our contention that any bias
in the OLS estimate of sigma is small in our framework.
4.2 IV Estimates
The OLS estimates of (7) are consistent under the assumption
that the error term is independent of both output and user cost
growth. As discussed in Section 2.2 and suggested by the results
with the constant returns model presented in Section 4.1, these
are reasonable assumptions with our estimation method. Measurement error can adversely affect the OLS results, however
(Goolsbee 2000). Our analysis of the effects of measurement error affecting the capital stock and the independent variables and
arising from short-run frictions (e.g., irreversibility constraints,
asymmetric adjustment costs) and other sources (Appendix D)
shows that classical measurement error apparently has no quantitatively important effect on our estimates of σ . Despite these
results, we present IV estimates in Table 1 to explore the robustness of our OLS results in the presence of nonclassical measurement error. The instruments, listed in Table 1, are constructed
from data in the τ = 0 interval.
The benchmark IV estimate of σ in the fifth column of Table 1, 0.390, is almost identical to the benchmark OLS estimate
of 0.367 in column 1. Not surprisingly, the standard error increases with IV, but we can still strongly reject the hypotheses
that σ equals 0 or equals unity. Unfortunately, the IV estimates
of η are not as reasonable as their OLS counterparts. Because of
the large coefficient on output growth, β, the point estimate of
the returns to scale elasticity, η, is 0.603. The standard error of η
is larger with IV than with OLS, but the IV estimate still rejects
constant returns to scale in favor of decreasing returns. We do
not consider this result reliable, however, because of our inability to find good instruments for output growth. The partial R2
statistic developed by Shea (1997) confirms this interpretation.
This statistic measures the relevance of instruments for each estimated coefficient after removing the explanatory power used
for instrumenting other regressors. The partial R2 for yf (the
regressor multiplying β) is 0.040, dramatically lower than the
partial R2 of 0.515 for cf (the regressor multiplying σ ).
To pursue this issue one step further, we reestimate the model
with IV imposing constant returns to scale (η = 1). Under this
591
assumption, β = 1, and we no longer need to instrument output
growth. This IV estimate of σ , reported in the seventh column
of Table 1, rises to 0.434 from 0.390, a change much smaller
than one standard error. This result demonstrates that even if
the IV estimate of returns to scale is unreliable due to the lack
of relevant instruments for output growth, this difficulty does
not “contaminate” conclusions about σ , which is the primary
focus of our study.
The sixth column of Table 1 presents IV estimates with twodigit industry dummies. This specification accounts for industry
effects that vary across intervals [cf. (7)]. The point estimate of
σ changes little from the benchmark value (0.373 vs. 0.390).
As was the case for the OLS estimates with industry dummies,
the standard error of σ rises sharply.
As a final test of the validity of the OLS estimates, we performed Hausman tests on σ . The Hausman statistics are asymptotically distributed χ 2 (1) under the null hypothesis that
the OLS estimate is consistent. For the benchmark model, the
test statistic is 0.07, and for the constant returns to scale model
it is 0.92. (The Hausman test is not defined for the model that
includes industry dummies because, as sometimes occurs in finite samples, the standard error of the IV estimate is slightly
smaller than the standard error of the OLS estimate.) Both test
statistics are far below the 90% critical value for the χ 2 (1) distribution of 2.71. These tests support the validity of the OLS
estimates of σ .
Taken together, the unconstrained OLS and IV estimates
strongly suggest that σ is approximately 0.40.
4.3 Split-Sample Estimates
Table 2 assesses the stability of structural parameter estimates across three different subsamples chosen to address issues that have arisen with empirical investment models. First,
many empirical studies interpret significant cash flow in investment regressions as consistent with the presence of financial
constraints. In the long run, however, we expect financial constraints to be less important. To test this hypothesis, we estimate
the benchmark model for subsamples split according to the presample (τ = 0) median cash flow-to-capital ratio. Second, we
split the sample by the median average capital stock in the presampling period. The technologies used by firms may vary systematically by size, and the technology parameters estimated
here may change accordingly. Third, we split the data at the
median value of the Brainard–Tobin Q variable to address the
sensitivity of our estimation strategy to investment dynamics
and shocks. The numerator of Q is the market value of equity
plus the book value of debt less the book value of inventories;
the denominator is our measure of the replacement value of the
capital stock. Firms with high values of Q are presumably farther from their long-run equilibrium capital stock due to, among
other reasons, favorable demand or productivity shocks. Therefore, if our estimation method did not adequately account for
investment dynamics or shocks, then we might expect a difference in the estimated σ ’s across the high-Q and low-Q subsamples.
Table 2 presents OLS estimates from the benchmark model
and IV estimates from a constrained model in which η = 1 (in
accordance with the discussion in Section 4.2). The σ estimates
0.086
(0.021)
0.023
(0.016)
—
0.562
0.631
—
−0.024
(0.212)
—
1.000
0.024
(0.156)
0.107
(0.024)
Low
capital
0.329
(0.187)
1.000
0.126
(0.023)
0.002
(0.017)
0.556
—
1.000
1.000
—
0.575
−0.129
(0.130)
0.541
R2
γ
β
η
Parameter
σ
Low
CF/K
0.278
(0.081)
1.308
(0.137)
0.830
(0.058)
0.040
(0.014)
OLS
High
CF/K
0.407
(0.101)
1.019
(0.080)
0.989
(0.046)
0.125
(0.018)
IV
0.006
(0.201)
High
CF/K
0.358
(0.168)
1.000
Low
CF/K
0.364
(0.112)
1.000
Low
capital
0.435
(0.121)
1.042
(0.104)
0.977
(0.055)
0.105
(0.020)
OLS
0.141
(0.142)
0.582
High
capital
0.294
(0.076)
1.284
(0.113)
0.844
(0.046)
0.064
(0.013)
1.000
IV
High
capital
0.353
(0.100)
1.000
Low
Q
0.334
(0.083)
1.268
(0.094)
0.859
(0.036)
0.054
(0.014)
−0.003
(0.144)
0.122
(0.187)
—
1.000
1.000
IV
OLS
Split by Tobin–Brainard Q
Split by capital stock size
Split by cash flow-to-capital ratio
Table 2. OLS and IV estimates based on the IDE: Various sample splits
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NOTE: Estimates of the IDE with firm-level panel data as described in Sections 2 and 3. See the note to Table 1 for additional information. IV regressions are estimated with η and β constrained to unity because of instrument relevance concerns, as
discussed in Section 4.2. Sample splits are based on the median value of the classifying variable in the τ = 0 interval (1972–1977). The coefficient measuring the difference between the σ ’s for the contrasting classes is , which is distributed asymptotic
t under the null hypothesis of equality. The null hypothesis, = 0 = σ L − σ H , is evaluated with the following auxiliary equation based on (7): kf = −σ H cf − cf ∗ If + β L yf ∗ If + β H yf ∗ (1 − If ) − γ L ∗ If − γ H ∗ (1 − If ) − wf , where If
is an indicator variable equal to 1 for the low subsample and 0 for the high subsample. In the IV regressions, each individual instrument, zf , appears twice in the instrument list as zf ∗ If and zf ∗ (1 − If ).
High
Q
0.339
(0.148)
1.000
Low
Q
0.461
(0.115)
1.000
Journal of Business & Economic Statistics, October 2011
High
Q
0.337
(0.117)
1.029
(0.096)
0.982
(0.061)
0.107
(0.022)
592
are similar across the three subsample splits. The test statistics,
, for the equality of the σ ’s across subsamples cannot reject
the null hypothesis of equal σ ’s at any standard significance
level. These results (as well as IV results from the benchmark
model in which η is estimated freely) provide additional support
for the way in which our estimation method addresses problems
with complicated investment dynamics, avoiding these difficult
specification issues by focusing directly on the long-run growth
of the capital stock.
5. COMPARISONS AMONG ESTIMATORS
This section compares the IDE with two other estimators that
also focus on long-run relationships among variables defining
the first-order condition for capital: the dynamic OLS (DOLS)
and static OLS (SOLS) estimators. These three methods address the problem of unobservable long-run variables differently. As discussed in Section 2, the IDE solves this problem
by constructing time averages. Caballero (1994) developed the
innovative idea that the unobservability problem can be solved
if the model variables are I(1) and cointegrated. In Caballero’s
DOLS model, the long-run values of the capital/output ratio and
user cost equal the actual values (see the note to Table 3 for an
explicit equation). Estimates of σ can be biased downward in finite samples, and Caballero uses the Stock–Watson (SW 1993)
correction, which effectively accounts for short-run adjustment
dynamics to long-run values. The SOLS model is a special case
of DOLS without the SW correction, a procedure that is appropriate provided that the capital-to-output ratio and user cost
are reasonably close to their long-run values (Caballero 1994,
p. 54).
The DOLS model of Caballero, Engel, and Haltiwanger
(CEH 1995) is conceptually closest to our IDE model. (See
Chirinko 2008 for a list of DOLS and other studies estimating σ .) Both studies use panel data, and both emphasize longrun variation. CEH obtained a range of elasticities across twodigit industries for equipment capital from 0.01 to 2.00, with an
unweighted average of approximately unity. This corresponds
to an elasticity for total capital (comparable to those presented
herein) of approximately 0.70 (see Table 3). Notwithstanding
the similarities in the two studies, there is a marked gap between
σDOLS = 0.70 and σIDE = 0.40. This gap might be due to several differences, most notably data construction (documented in
Berndt 1976) and the sample period.
To control for these factors, we use a common dataset (described in Appendix E) to generate estimates of σ from the
three models. The DOLS model of CEH, which does not include time fixed effects and is estimated on plant-level panel
data, generates estimates of σ = 0.70. (The order of integration
of the model variables is examined in Appendix F.) Due to a
concern about degrees of freedom, CEH used only lags (as opposed to leads and lags) when implementing the SW correction.
When we follow this procedure with a maximum of three lags
and remove firm fixed effects through mean differencing, our
DOLS point estimate of σ = 0.720 (0.026) matches the result
of CEH. (CEH’s reported estimates are based on five lags, but
they noted that that their results are “fairly robust to the number of lags” (fn. 19); we confirmed the stability to lag length in
results not reported here.) When we include both leads and lags
Chirinko, Fazzari, and Meyer: The Elusive Capital–Labor Substitution Elasticity
Table 3. OLS and IV estimates of σ : DOLS, SOLS, and IDE
Fixed effects
Model
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DOLS (Caballero, Engel, and
Haltiwanger 1995)
Firm: Yes
Time: No
Firm: Yes
Time: Yes
0.700@
—
DOLS(3, 0)
Mean-differenced
0.720
(0.026)
0.754
(0.054)
DOLS(3, 3)
Mean-differenced
0.666
(0.036)
0.540
(0.059)
SOLS
Mean-differenced
0.645
(0.023)
0.520
(0.041)
DOLS(3, 3)
First-differenced
0.913
(0.089)
0.582
(0.095)
IDE
OLS, mean-differenced = first-differenced
0.372
(0.072)
IDE
IV, mean-differenced = first-differenced
0.434
(0.098)
NOTE: Estimates of σ are from various models with firm-level panel data as described
in Section 5. DOLS(M, N) is based on the following equation:
kyf ,t = −σ cf ,t +
LAG
M
m=0
LAG c
θm
f ,t−m +
LEAD
N
θnLEAD cf ,t+n + ξf ,t ,
n=1
where kyf ,t is the logarithm of the capital/output ratio, cf ,t is the log of the user cost
of capital, and M LAG and N LEAD indicate the number of lags and leads, respectively;
LAG = θ LEAD = 0 for all m and
SOLS is a special case of DOLS with no leads or lags (θm
n
n); and IDE [eq. (7)]. A common dataset containing 12,617 observations is used for all
estimates reported in this table; see Appendix E for details. The models are either firstdifferenced or mean-differenced to remove firm fixed effects. The models in column 2 also
contain time fixed effects. The instruments for IDE are discussed in the note to Table 1.
Heteroscedastic-consistent standard errors are given in parentheses. The @ indicates that
the estimate of 0.70 equals 1.0 (the estimate taken from the cited article) multiplied by 0.70.
This latter adjustment factor converts an equipment elasticity to an equipment + structures
elasticity and is based on the assumptions that the structures elasticity is one-third as large
as that for equipment (Cummins and Hassett 1992; Pindyck and Rotemberg 1983), and that
equipment constitutes 0.55 of total capital. The estimate from the cited article is an average
of estimated coefficients across two-digit industries for equipment capital; we do not have
sufficient information to compute the standard error associated with the estimate presented
in the table. For the IDE models presented in the last two rows, mean-differencing and
first-differencing are equivalent because there are only two intervals.
to account for the short-run adjustment dynamics, the DOLS
estimate of σ = 0.666 (0.036) is a bit lower. In summary, the
dataset used in the present study does not lead to different estimates than those of CEH.
The SOLS model excludes the SW correction and thus allows us to assess the impact of accounting for short-run adjustment dynamics. Following Caballero (1994), we expect
σSOLS < σDOLS . The SOLS model without time fixed effects
yields σ = 0.645 (0.023), close to the DOLS estimates. This
indicates that the SW correction, intended to account for deviations from long-run values, has little impact on the estimated
elasticity in the DOLS model.
Insofar as short-run adjustment dynamics are largely driven
by aggregate factors, time fixed effects can be interpreted as a
second correction factor that should affect σ . Estimates of the
DOLS and SOLS models with time fixed effects are presented
in column 2 of Table 3. For the SOLS model with time fixed effects, σSOLS = 0.520 (0.041), which is 0.125 lower than the estimate without time fixed effects. The estimate from the DOLS
model with three lags and leads, σDOLS = 0.540 (0.059), is sub-
593
ject to a similar decline. Unlike the SW correction, time fixed
effects matter for estimates of σ .
The IDE also emphasizes long-run variation but, rather than
accounting for short-run adjustment dynamics with the SW correction and fixed time effects, avoids this issue by inserting estimates of long-run values into the first-order condition (4). Differencing the data in the two disjoint intervals accounts for the
impact of both firm and time fixed effects. The OLS and IV estimates based on the IDE (Table 1 with η = 1) are σ = 0.372
(0.072) and σ = 0.434 (0.098), respectively; our preferred estimate is σIDE = 0.40.
The use of the common dataset reduces the gap between elasticity estimates based on the IDE and DOLS. The most important factor in lowering the initial DOLS estimate of 0.70
is the inclusion of time fixed effects. The remaining gap is
0.14 = 0.54 − 0.40 = σDOLS − σIDE .
6. SUMMARY AND CONCLUSIONS
The substitution elasticity between capital and labor has been
the focus of much research over the past 40 years. Almost all
previous work based on the first-order condition for capital has
relied on time series variation in investment data at the aggregate, industry, or firm level to estimate this elasticity. This
article offers a simple, intuitive, and computationally straightforward approach for estimating production function parameters that focuses on long-run relations. The estimation strategy
developed here groups annual data into longer time intervals
and then averages the firm-level panel data within each interval to estimate unobservable long-run variables. The data are
differenced across intervals, and production function parameters are estimated in a cross-section of time-averaged, differenced firm data. Our approach accounts for a variety of productivity shocks, omitted variables, and firm effects. This econometric model does not solve the estimation problems, including
difficult-to-specify dynamics, transitory time series variation,
and positively sloped supply schedules, inherent in regression
models using relatively high-frequency investment data that
may bias estimates of the substitution/user cost elasticity. Instead, our approach avoids these problems by exploiting panel
data with a simple approach that emphasizes low-frequency
variation and directly estimates the first-order condition for capital.
We find that the elasticity can be consistently and precisely
estimated with IDE and is approximately 0.40. Extant estimates
of σ range widely. However, when two other models that emphasize long-run variation—DOLS and SOLS—are applied to
a common dataset, we find a remarkable convergence of results. Although DOLS will be more efficient than IDE for large
T, Monte Carlo simulations suggest that IDE is preferred over
DOLS for T < 30 for a particular specification of the datagenerating process calibrated to our data set (see Chirinko and
Fazzari 2011).
Econometric issues aside, however, our results offer a strikingly negative answer to the central question of whether the
Cobb–Douglas assumption is valid. This robust finding raises
questions about the frequent use of the Cobb–Douglas production function in theoretical and empirical models.
594
Journal of Business & Economic Statistics, October 2011
Apart from our immediate objective of estimating production
function parameters, our method may prove useful in estimating
other structural parameters from panel datasets. Our intervalaverage approach could be applied to other problems in which
short-run dynamics may obscure long-run structural relations.
There are likely to be a number of applications in public economics, labor economics, and industrial organization, where the
availability of panel datasets and interest in long-run structural
parameters may make this method feasible and informative.
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ACKNOWLEDGMENTS
We thank Gib Bassett, Stephen Bond, Hashem Dezhbakhsh,
Uwe Hassler, Harry Huizinga, Daniel Levy, Doug Meade,
James Morley, Werner Roeger, Julio Rotemberg, Lawrence
Summers, Giovanni Urga, Philip Vermuellen, Kenneth West,
an anonymous referee, an anonymous associate editor, the editors, and seminar participants at the International Conference
on Panel Data, Center for Economic Studies, the Centro de Investigacion y Docencia Economicas (Mexico City), the Eastern
Economics Association meetings, Emory University, the European Central Bank, the European Commission, the University
of Illinois at Chicago, and the University of Missouri for comments and suggestions. An earlier draft circulated under the title “That Elusive Elasticity: A Cross-Section/Long-Panel Approach to Estimating the Sensitivity of Business Capital to Its
User Cost.” The views expressed here do not necessarily reflect
those of the Federal Reserve Banks of San Francisco and St.
Louis or the Federal Reserve System. All errors and omissions
remain the sole responsibility of the authors, and the conclusions do not necessarily reflect the views of the organizations
with which they are associated.
[Received April 2008. Revised September 2010.]
REFERENCES
Berndt, E. R. (1976), “Reconciling Alternative Estimates of the Elasticity of
Substitution,” The Review of Economics and Statistics, 63, 59–68. [592]
Caballero, R. J. (1994), “Small Sample Bias and Adjustment Costs,” Review of
Economics and Statistics, 76, 52–58. [592,593]
(1999), “Aggregate Investment,” in Handbook of Macroeconomics,
Vol. 1B, eds. J. B. Taylor and M. Woodford, Amsterdam: Elsevier NorthHolland, pp. 813–862. [587]
Caballero, R. J., Engel, E. M. R. A., and Haltiwanger, J. C. (1995), “PlantLevel Adjustment and Aggregate Investment Dynamics,” Brookings Papers
on Economic Activity, 2, 1–54. [592,593]
Chirinko, R. S. (1993), “Business Fixed Investment: A Critical Survey of Modeling Strategies, Empirical Results, and Policy Implications,” Journal of
Economic Literature, 31, 1875–1911. [588]
(2008), “σ : The Long and Short of It,” in The CES Production Function in the Theory and Empirics of Economic Growth, eds. R. Klump and
C. Papageorgiou; conference proceedings published in a special issue of the
Journal of Macroeconomics, 30, 671–686. [592]
Chirinko, R. S., and Fazzari, S. M. (2011), “A Monte Carlo Comparison of
Estimators Emphasizing Long-Run Frequencies,” manuscript, Washington
University and the University of Illinois at Chicago. [593]
Chirinko, R. S., Fazzari, S. M., and Meyer, A. P. (1999), “How Responsive Is
Business Capital Formation to Its User Cost?: An Exploration With Micro
Data,” Journal of Public Economics, 74, 53–80. [587]
Coen, R. M. (1969), “Tax Policy and Investment Behavior: Comment,” American Economic Review, 59, 370–379. [589]
Cummins, J. G., and Hassett, K. A. (1992), “The Effects of Taxation on Investment: New Evidence From Firm Level Panel Data,” National Tax Journal,
45, 243–252. [593]
Fazzari, S. M., Hubbard, R. G., and Petersen, B. C. (1988),
ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20
A New Approach to Estimating Production
Function Parameters: The Elusive Capital–Labor
Substitution Elasticity
Robert S. Chirinko, Steven M. Fazzari & Andrew P. Meyer
To cite this article: Robert S. Chirinko, Steven M. Fazzari & Andrew P. Meyer (2011) A
New Approach to Estimating Production Function Parameters: The Elusive Capital–Labor
Substitution Elasticity, Journal of Business & Economic Statistics, 29:4, 587-594, DOI: 10.1198/
jbes.2011.08119
To link to this article: http://dx.doi.org/10.1198/jbes.2011.08119
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Date: 11 January 2016, At: 23:21
A New Approach to Estimating Production
Function Parameters: The Elusive Capital–Labor
Substitution Elasticity
Robert S. C HIRINKO
Department of Finance, University of Illinois at Chicago, Chicago, IL 60607-7121 ([email protected])
Steven M. FAZZARI
Department of Economics, Washington University, St. Louis, MO 63130-4899 ([email protected])
Andrew P. M EYER
Downloaded by [Universitas Maritim Raja Ali Haji] at 23:21 11 January 2016
Federal Reserve Bank of St. Louis, St. Louis, MO 63166-0442 ([email protected])
Parameters of taste and technology are central to a wide variety of economic models and issues. This
article proposes a simple method for estimating production function parameters from panel data, with a
particular focus on the elasticity of substitution between capital and labor. Elasticity estimates have varied
widely, and a consensus estimate remains elusive. Our estimation strategy exploits long-run variation
and thus avoids several pitfalls, including difficult-to-specify dynamics, transitory time-series variation,
and positively sloped supply schedules, that can bias the estimated elasticity. Our results are based on
an extensive panel comprising 1860 firms. Our approach generates a precisely estimated elasticity of
0.40. Although existing estimates range widely, we document a remarkable convergence of results from
two related approaches applied to a common dataset. The method developed here may prove useful in
estimating other structural parameters from panel datasets.
KEY WORDS: Capital formation; Panel data; Production function parameters.
1. INTRODUCTION
Parameters of taste and technology are central to a wide variety of economic models and issues. Production function parameters loom large in real business cycle models, the monetary
transmission mechanism, exogenous and endogenous growth
models, and tax policy evaluations. This article reports a simple method for estimating production function parameters from
long-run variation in panel data, with a particular focus on the
elasticity of substitution between capital and labor, σ . Although
the value of σ has been the subject of an enormous number of
empirical studies, a consensus remains elusive; for example, in
the Joint Committee on Taxation’s (1997, T.6) study of nine
models, σ ’s range from 0.20 to 1.00.
This wide range of estimated elasticities may be attributed to
a common source. Most econometric studies rely on relatively
high-frequency quarterly or annual time series variation in investment data to identify the elasticity. Three biases may result
that weigh more or less heavily in different studies. First, the
specification of an investment equation requires assumptions
about dynamics. Although economic theory is highly informative about the demand for the stock of capital, it is relatively
silent regarding the demand for the flow of investment. Misspecified dynamics can bias estimates of elasticity (Summers
1988). Of particular importance are the nature of adjustment
costs (Hamermesh and Pfann 1996; Caballero 1999) and the
role of financing constraints (Fazzari, Hubbard, and Petersen
1988; Hubbard 1998). Second, coefficient estimates from investment regressions may be biased if the time series variation
of investment spending largely reflects adjustments to transitory
shocks and firms respond less to transitory variation than to permanent variation. An elasticity estimated with time series data
at quarterly or annual frequencies will tend to be lower than the
“true” long-run elasticity; compare our results reported herein
with those of Chirinko, Fazzari, and Meyer (1999). Third, if the
supply curve of investment is upward sloping, as is more likely
in the short to medium-run, then studies that incorrectly maintain a perfectly elastic supply schedule will tend to understate
demand elasticities (Goolsbee 1998). Although transitory timeseries variation and positively sloped supply schedules bias the
estimated elasticity toward 0, misspecified dynamics has an indeterminate effect, and thus the bias resulting from these three
factors cannot be signed.
These potential problems all stem from the use of relatively
high-frequency data for estimation. Theory is most informative
about the long-run economic relationships among the relevant
variables, however. In the case of the first-order condition for
capital (the focus of this article), problems arise from the use of
investment data as the measure of capital formation. We avoid
these problems with an approach that estimates unobserved
long-run variables by exploiting the information available from
panel data. We apply our new estimation method to the firstorder condition relating the long-run desired capital stock, K ∗ ,
to the long-run desired values of output, Y ∗ , and the user cost
of capital, C∗ ,
587
K ∗ = G[Y ∗ , C∗ ].
(1)
© 2011 American Statistical Association
Journal of Business & Economic Statistics
October 2011, Vol. 29, No. 4
DOI: 10.1198/jbes.2011.08119
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588
Journal of Business & Economic Statistics, October 2011
This specification underlies virtually all investment studies
since Jorgenson’s path-breaking work on the neoclassical
model of capital (Hall and Jorgenson 1967; Chirinko 1993).
The difficulty in estimating (1) stems from the fact that the longrun desired values are not readily observable. Our method proxies these unobservable variables as time averages within firms.
Of course, this approach would not leave adequate degrees of
freedom in pure time series data or in short panels; however,
a large panel provides sufficient degrees of freedom for estimation in the cross-section dimension. With empirical counterparts to K ∗ , Y ∗ , and C∗ defined as time averages, estimating
G[·] from cross-sectional variation is straightforward. This intuitive and computationally straightforward approach for estimating technology parameters can be interpreted in terms of a
low-pass filter placing relatively more weight on low-frequency
movements than the traditional approach of first-differencing
the capital stock widely used in the estimation of investment
models.
Our study proceeds as follows. Section 2 introduces the estimation strategy for a representative firm’s capital accumulation decision. The econometric equation is derived from firm
profit maximization, and long-run values of the variables entering this equation are measured as time averages. Our estimation strategy accounts for a variety of productivity shocks,
omitted variables, and firm fixed effects. Section 3 discusses
the panel dataset containing 1860 firms for the period 1972–
1991. Section 4 presents our ordinary least squares (OLS) and
instrumental variables (IV) results. Both techniques yield similar estimates of the substitution elasticity, approximately 0.40.
Section 4 also examines the sensitivity of the estimates across
various subsets of the sample. Section 5 compares the method
developed in this article with two alternative approaches and
computes estimates from the three models based on a common
dataset. Section 6 offers a summary and conclusions.
2.
ESTIMATION STRATEGY
2.1 Derivation
Our econometric model follows directly from the behavior of
a firm that maximizes its discounted flow of profits over an infinite horizon. We analyze the firm’s choices in long-run equilibrium, thereby eliminating the need to model adjustment costs,
delivery lags, vintage effects, and expectations. Under these assumptions, the firm always produces its long-run desired level
of output with its long-run desired mix of inputs. The critical
consequence of this approach is that the firm’s dynamic optimization problem is transformed into a static problem. To determine the firm’s demand for capital, we need only calculate
the marginal product of capital evaluated at the long-run levels
of inputs and output.
We assume that production possibilities are described by the
following CES technology:
∗[(σ −1)/σ ]
∗[(σ −1)/σ ] [ησ/(σ −1)]
+ (1 − θ ) Xf ,t
Uf ,t ,
Yf∗,t = θ Kf ,t
(2)
Yf∗,t ,
Kf∗,t ,
Xf∗,t
where
and
are the long-run desired values of real
output, the real capital stock, and all other factors of production, respectively, for firm f at time t, and Uf ,t is a stochastic
productivity shock. An attractive feature of the CES technology is that it depends on only three parameters characterizing
returns to scale η, the distribution of factor returns θ , and—
the focus of this study—substitution possibilities between the
factors of production, σ . These parameters are the same for all
firms, although the productivity shock allows the level of the
production function to vary across firms. The CES function is
strongly separable and can be expanded to include many additional factors of production (e.g., intangible capital) without
affecting the econometric equation derived later. This feature
gives the CES specification an important advantage relative to
other technologies that allow for a more general pattern of substitution possibilities (e.g., the translog, minflex-Laurent). Our
approach does not require price and quantity data on the other
factors of production to recover the key parameter of interest.
Differentiating (2) with respect to capital, we obtain the following relation for the marginal product of capital (∂Yf∗,t /∂Kf∗,t ):
∗[1+(1−σ )/σ η]
∂Yf∗,t /∂Kf∗,t = (ηθ )Yf ,t
∗−[1/σ ]
Kf ,t
[(σ −1)/σ η]
Uf ,t
.
(3)
Profit maximization implies that this marginal product of capital equals the Jorgensonian user cost of capital (Cf∗,t ), which
combines interest, depreciation, and tax rates with relative
prices. (The details of the user cost specification are given in
Appendix A; all appendices are available at our Web sites.) Setting ∂Yf∗,t /∂Kf∗,t equal to Cf∗,t and rearranging (3), we obtain the
following expression for the long-run desired capital stock,
∗[β]
[ζ ]
Kf∗,t = Cf∗[α]
,t Yf ,t Uf ,t ,
(4)
(ηθ )σ , α
= −σ, β = (σ η + 1 − σ )/η, and ζ =
where =
(σ − 1)/η.
The central difficulty in estimating (4) is the fact that the
long-run values are not observed. Most previous research addresses this problem by differencing the log of (4) to obtain
an equation for investment. However, as discussed in Section 1,
this approach necessarily confronts the modeling challenges associated with investment data, and thus may generate biased
estimates. To avoid these potential problems, we focus on the
capital stock rather than on investment and develop an econometric method to directly estimate the long-run desired levels
of capital, output, and user cost as time averages. We refer to
the years over which an average is computed as an interval and
divide the sample into three intervals indexed by a subscript τ
(τ = 0, 1, 2). The intervals are 1972–1977 (τ = 0), 1978–1984
(τ = 1), and 1985–1991 (τ = 2). We measure Kf∗,t by Kf ,τ ,
where the latter is the mean of Kf ,t over a τ interval. (Symmetric definitions apply to Yf∗,t and Cf∗,t .) The τ = 1 and τ = 2
intervals are used for parameter estimation; the τ = 0 interval
is used only to form instruments and define classifications that
split the sample.
With the variables in (4) defined by averages computed for
the τ = 1 and τ = 2 intervals, we take logarithms and obtain
the following equation:
kf ,τ = αcf ,τ + βyf ,τ + ψ − uf ,τ ,
τ = 1, 2,
ln[Kf∗,t ], cf ,τ
(5)
ln[Cf∗,t ],
≡ ln[Cf ,τ ] =
where kf ,τ ≡ ln[Kf ,τ ] =
∗
yf ,τ ≡ ln[Yf ,τ ] = ln[Yf ,t ], ψ ≡ ln[], and uf ,τ is an error term
that follows directly from the technology and represents productivity shocks. We model productivity shocks as
uf ,τ = ζ [vf + vi + wτ + wi,τ + wf ,τ ].
(6)
Chirinko, Fazzari, and Meyer: The Elusive Capital–Labor Substitution Elasticity
The productivity shock is decomposed into firm-specific (vf )
and industry-specific (vi ) components, as well as components
that vary over the τ intervals, wτ , wi,τ , and wf ,τ . With this error
structure, estimates can be obtained by differencing (5) and (6)
between the τ intervals,
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kf = −σ cf + β yf − γ − λi − wf ,
(7)
where kf ≡ kf ,τ , cf ≡ cf ,τ , yf ≡ yf ,τ , σ ≡ −α,
γ ≡ ζ wτ , λi ≡ ζ wi,τ , and wf ≡ ζ wf ,τ . Because there
are only two intervals, first-differencing eliminates the temporal dimension of the model, and τ subscripts have been omitted
in (7). Consequently, the parameters are estimated in a crosssectional regression. Fixed firm and industry effects are eliminated by differencing, and fixed interval effects (notably, biased
technical change) are captured by the constant γ . Industry effects that vary across intervals are captured by industry dummies, λi .
2.2 Discussion and Interpretation
Equation (7) is the basis for estimation; it relates long-run
growth in the capital stock to long-run growth in the user cost
and output across firms. Cross-sectional variation in the data is
key to identification and allows us to estimate the σ parameter, which is of central interest in this study. In addition, we can
recover the returns to scale elasticity, η, as a nonlinear combination of the estimated σ and β parameters.
In contrast to many previous studies using investment data,
our estimation method emphasizes low-frequency variation. We
measure the long-run values of the regression variables by taking averages over several years (intervals) and then differencing the averaged data to control for a variety of fixed effects.
This new interval-difference estimator (IDE) stands in contrast
to the standard investment regression derived by differencing
untransformed annual data; we refer to the latter approach as
the year-difference estimator (YDE). Because it averages over
several years, the IDE places substantially more weight on the
lower frequencies reflecting long-run variation in the data. (See
Appendix B for a frequency domain interpretation of these estimators using spectral methods; that analysis suggests that the
IDE defined for an interval length of 7 years seems reasonably
successful in emphasizing long-run variation, especially relative to the YDE.)
Apart from the frequency of the data entering the regression model, satisfactory OLS parameter estimates depend on
the relationship between the stochastic element, wf , and the
regressors, especially yf . This correlation is not likely to be
a problem for two reasons. First, wf is that part of the productivity shock that remains after accounting for a variety of
fixed effects. Major technological changes (e.g., telecommunications, computing, the Internet) are likely to have their greatest
effects on all firms (captured by γ ) or on all firms in an industry
(captured by λi ) with only a small residual impact that is firmspecific. Second, only part of the productivity shock enters the
error term. As noted by Shapiro (1986), including output in a
factor demand equation can completely absorb the productivity shock. When the elasticity of substitution is unity, ζ equals
0, and uf τ vanishes [cf. (4) and (6)]. When σ deviates from
unity, the impact of the productivity shock is nonetheless diminished by ζ , provided that returns to scale are not decreasing
589
too sharply. Despite the arguments that OLS estimates will not
be appreciably distorted by wf , we present two alternative estimates that are robust to simultaneity. First, in Section 4.1 we
impose constant returns to scale (η = 1 → β = 1). Thus yf ,
the variable most likely to be correlated with wf , no longer
appears as a regressor. Second, in Section 4.2 we present IV
estimates and Hausman tests using the variables in the τ = 0
interval as instruments.
The IDE is robust to four potentially important distortions.
First, parameter estimates are not sensitive to trending variables. (See Appendix C for formal consideration of this issue
and for the role played by differencing in eliminating firmspecific trends.) Second, the estimates are unlikely to be influenced by additional factors that might affect the specification of the production function. For example, the specification
of the econometric equation does not change with additional
factors of production. Markups that vary across firms are captured by a firm-specific fixed effect that is eliminated by differencing. Moreover, the information processing revolution might
have led to biased technical change. In the CES technology (2),
biased technical change is represented by temporal variation
in θ and, like wτ , will be reflected in the constant (4). Third,
studies implementing the Jorgensonian framework often have
been criticized for failing to distinguish between desired output and actual output (e.g., Coen 1969; Hall 1995). The time
averages in the econometric equation recognize this important
distinction. Fourth, the estimates are unlikely to be affected
by measurement error in the capital stock. Classical measurement error will be part of the error term, and hence innocuous. A plausible situation in which measurement error may be
systematic arises when an increase in the pace of technological change effectively increases the depreciation of fixed capital through obsolescence, an effect not captured in our fixed
depreciation rate assumption. However, an increase in depreciation rates would lead to a systematic overstatement of capital
in τ = 2 and would be captured by the constant. If omitted variables or measurement error are both firm-specific and intervalvarying, then consistent estimation might become an issue. In
this case, the IV estimates and measurement error analysis discussed in Section 4.2 provide useful checks on the parameter
estimates.
In summary, the simple, intuitive interval-difference estimation strategy developed here collapses the time dimension
of firm panel data by defining three intervals and then timeaverages the data within an interval. The first interval is used to
form instruments or sort variables into contrasting classes, and
the second and third intervals are used for estimation. Identification is provided by the cross-sectional variation in the growth
rates of capital, output, and the user cost. A variety of productivity shocks, omitted variables, and fixed effects are accounted for by estimated parameters or differencing. Thus production function parameters are estimated in a cross-section of
time-averaged, differenced firm data. This econometric model
does not solve the estimation problems inherent with investment models—difficult-to-specify dynamics, transitory time series variation, and positively sloped supply schedules—that can
bias estimates of the elasticity. Instead, our approach avoids
these problems by exploiting panel data with a method that
emphasizes low-frequency variation and directly estimates the
first-order condition for capital.
590
Journal of Business & Economic Statistics, October 2011
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3. THE PANEL DATASET
To construct the variables needed to estimate (7), we link
firm-level financial data from Compustat with industry-level
user cost data from Data Resources, Inc. (DRI). The user cost,
C, is the opportunity cost of a dollar investment in fixed capital (the interest rate plus the depreciation rate) adjusted for
taxes and multiplied by the relative price of capital. The user
cost is defined in detail in Appendix A. Measurement of the
capital stock, K, is important for our study. Accounting book
values of net plant and equipment likely understate current replacement values in periods of inflation. In addition, accounting
depreciation rules might not accurately reflect economic depreciation. We measure the replacement value of capital with an
iterative algorithm and adjust for changes due to acquisitions
and divestitures, as described in Appendix A. Output, Y, is annual gross sales. Sales will differ from output by the change
in finished goods inventories. Although this difference may be
nontrivial in the short run, it will have very little impact on the
long-run averages used in our estimation. Nominal sales figures
from Compustat are deflated by industry-specific output price
indexes from DRI.
To protect against results driven by a small number of extreme observations, we exclude observations in the 1% upper
and lower tails of the firm-specific variables. We checked the
robustness of our results to this procedure by deleting both the
0.5% and 2% tails and found a negligible effect on the results.
Because the user cost is computed from stable industry and aggregate data, we did not delete data in the tails of the user cost
variable distribution.
Our final data set contains 1860 firms from all sectors of the
economy. The data for estimating the parameters with the IDE
consist of the growth rates in model variables between τ = 1
and τ = 2 intervals that enter directly into the model. Whereas
(7) is estimated with a cross-section of firms, the value of each
observation is based on temporal variation between intervals.
The coefficient of variation for the growth rates in capital, output, and the user cost are 2.3, 2.4, and 2.7, respectively, indicating substantial differences across firms in the regression variables. (Summary statistics are provided in Appendix A.) It is
not surprising that capital and output growth rates differ across
firms, but the firm heterogeneity in the user cost growth rate is
remarkable relative to most previous studies. We rely on this
heterogeneity to identify σ with the IDE.
4. EMPIRICAL RESULTS WITH THE
INTERVAL–DIFFERENCE ESTIMATOR
4.1 Ordinary Least Squares Estimates
OLS estimates of the structural parameters from (7) appear in
Table 1. The focus of our study is on σ . In column 1, our benchmark estimate of σ is 0.367 with a standard error of 0.068. The
null hypothesis that the elasticity is 0 can be strongly rejected at
any conventional level of significance. It is also clear, however,
that we can reject the hypothesis that σ is 1, the value implied
by the Cobb–Douglas production function and often assumed
in applied work.
The estimated returns to scale elasticity, η, is a function of
the regression coefficients on the growth in both output, β, and
user cost, σ (see Table 1). The OLS estimate of η is 1.135, also
with a small standard error. It is interesting to note that the effect
of output is much stronger here than in typical panel data studies using investment data. Estimates from investment models
of coefficients that correspond to β in our study are often well
below 0.5, which suggests an implausible degree of increasing
returns. We believe that the reason for our more reasonable result is that, unlike investment equations, our estimation method
captures long-run, permanent changes in output and is not affected by the transitory variation that might unduly influence
investment regressions with annual or quarterly data.
The second column of Table 1 presents results from the inclusion of two-digit industry dummies in the benchmark regression [the λi terms in (7)]. These dummies control for industrylevel productivity shocks between intervals τ = 1 and τ = 2 or,
more generally, any industry-specific effects that vary across
the intervals. The structural parameter estimates do not change
much when the dummies are included. The σ estimate rises
Table 1. OLS and IV estimates based on the IDE
OLS
Unconstrained
Parameter
σ
η
β
γ
R2
IV
Constrained
Unconstrained
Constrained
Benchmark
With SIC dummies
η=1
η = 1, σ = 1
Benchmark
With SIC dummies
η=1
η = 1, σ = 1
0.367
(0.068)
1.135
(0.076)
0.925
(0.038)
0.084
(0.012)
0.564
0.440
(0.247)
1.152
(0.115)
0.926
(0.037)
−0.055
(0.060)
0.593
0.372
(0.072)
1.000
1.000
0.434
(0.098)
1.000
1.000
1.000
1.000
1.000
1.000
0.063
(0.013)
0.560
0.020
(0.013)
0.540
0.373
(0.363)
0.633
(0.184)
1.364
(0.166)
−0.324
(0.101)
—
1.000
1.000
0.390
(0.119)
0.603
(0.086)
1.402
(0.117)
−0.049
(0.034)
—
0.059
(0.013)
—
0.020
(0.013)
—
NOTE: Estimates of the IDE [eq. (7)] with firm-level panel data as described in Sections 2 and 3. The instrument list for the estimates in columns 5–8 includes the user cost (Ci,τ =0 ),
capital stock (Ki,τ =0 ), output-to-capital ratio [(Y/K)i,τ =0 ], and cash flow-to-capital ratio [(CF/K)i,τ =0 ]. In addition, we include the annualized growth rates of capital, output, cash
flow, accounts receivable, and cash and cash equivalents defined over the τ = 0 interval. For the estimates reported in column 6, it was not useful to include industry dummies in the
model as both regressors and instruments, because of collinearity. In this column, the instrument set for output and user cost growth variables is the same as for the other IV regressions;
the industry dummies are instrumented by themselves. Heteroscedastic-consistent standard errors are given in parentheses. The parameters are σ (the capital–labor substitution or user
cost elasticity), η (the returns to scale elasticity), β (the regression coefficient on output growth), and γ (the intercept). The returns to scale elasticity are recovered from the estimated
coefficients with the formula η = (1 − σ )/(β − σ ), when β > σ . The variance of η is computed by the delta method.
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Chirinko, Fazzari, and Meyer: The Elusive Capital–Labor Substitution Elasticity
from 0.367 to 0.440, and η is virtually identical. The standard
error of σ also rises, however. The σ estimate is much less precise with industry dummies in the model because a substantial
amount of the variation in the user cost growth variable is at
the industry level. For this reason, and given the modest change
in σ , the remaining OLS regressions in Table 1 exclude the industry dummies and thus do not control for industry effects that
vary across intervals.
As discussed in Section 2, the most likely source of correlation between the error term and the independent variables in
these OLS regressions comes from firm-specific productivity
shocks (embedded in the error term) and firm output growth.
This potential simultaneity problem can be avoided by imposing constant returns to scale (η = 1, implying β = 1), an assumption that removes output growth as a regressor. The third
column of Table 1 presents a regression with the output growth
coefficient constrained to unity. The σ estimate changes only
trivially (from 0.367 to 0.372) when constant returns to scale
are imposed. This result supports our contention that any bias
in the OLS estimate of sigma is small in our framework.
4.2 IV Estimates
The OLS estimates of (7) are consistent under the assumption
that the error term is independent of both output and user cost
growth. As discussed in Section 2.2 and suggested by the results
with the constant returns model presented in Section 4.1, these
are reasonable assumptions with our estimation method. Measurement error can adversely affect the OLS results, however
(Goolsbee 2000). Our analysis of the effects of measurement error affecting the capital stock and the independent variables and
arising from short-run frictions (e.g., irreversibility constraints,
asymmetric adjustment costs) and other sources (Appendix D)
shows that classical measurement error apparently has no quantitatively important effect on our estimates of σ . Despite these
results, we present IV estimates in Table 1 to explore the robustness of our OLS results in the presence of nonclassical measurement error. The instruments, listed in Table 1, are constructed
from data in the τ = 0 interval.
The benchmark IV estimate of σ in the fifth column of Table 1, 0.390, is almost identical to the benchmark OLS estimate
of 0.367 in column 1. Not surprisingly, the standard error increases with IV, but we can still strongly reject the hypotheses
that σ equals 0 or equals unity. Unfortunately, the IV estimates
of η are not as reasonable as their OLS counterparts. Because of
the large coefficient on output growth, β, the point estimate of
the returns to scale elasticity, η, is 0.603. The standard error of η
is larger with IV than with OLS, but the IV estimate still rejects
constant returns to scale in favor of decreasing returns. We do
not consider this result reliable, however, because of our inability to find good instruments for output growth. The partial R2
statistic developed by Shea (1997) confirms this interpretation.
This statistic measures the relevance of instruments for each estimated coefficient after removing the explanatory power used
for instrumenting other regressors. The partial R2 for yf (the
regressor multiplying β) is 0.040, dramatically lower than the
partial R2 of 0.515 for cf (the regressor multiplying σ ).
To pursue this issue one step further, we reestimate the model
with IV imposing constant returns to scale (η = 1). Under this
591
assumption, β = 1, and we no longer need to instrument output
growth. This IV estimate of σ , reported in the seventh column
of Table 1, rises to 0.434 from 0.390, a change much smaller
than one standard error. This result demonstrates that even if
the IV estimate of returns to scale is unreliable due to the lack
of relevant instruments for output growth, this difficulty does
not “contaminate” conclusions about σ , which is the primary
focus of our study.
The sixth column of Table 1 presents IV estimates with twodigit industry dummies. This specification accounts for industry
effects that vary across intervals [cf. (7)]. The point estimate of
σ changes little from the benchmark value (0.373 vs. 0.390).
As was the case for the OLS estimates with industry dummies,
the standard error of σ rises sharply.
As a final test of the validity of the OLS estimates, we performed Hausman tests on σ . The Hausman statistics are asymptotically distributed χ 2 (1) under the null hypothesis that
the OLS estimate is consistent. For the benchmark model, the
test statistic is 0.07, and for the constant returns to scale model
it is 0.92. (The Hausman test is not defined for the model that
includes industry dummies because, as sometimes occurs in finite samples, the standard error of the IV estimate is slightly
smaller than the standard error of the OLS estimate.) Both test
statistics are far below the 90% critical value for the χ 2 (1) distribution of 2.71. These tests support the validity of the OLS
estimates of σ .
Taken together, the unconstrained OLS and IV estimates
strongly suggest that σ is approximately 0.40.
4.3 Split-Sample Estimates
Table 2 assesses the stability of structural parameter estimates across three different subsamples chosen to address issues that have arisen with empirical investment models. First,
many empirical studies interpret significant cash flow in investment regressions as consistent with the presence of financial
constraints. In the long run, however, we expect financial constraints to be less important. To test this hypothesis, we estimate
the benchmark model for subsamples split according to the presample (τ = 0) median cash flow-to-capital ratio. Second, we
split the sample by the median average capital stock in the presampling period. The technologies used by firms may vary systematically by size, and the technology parameters estimated
here may change accordingly. Third, we split the data at the
median value of the Brainard–Tobin Q variable to address the
sensitivity of our estimation strategy to investment dynamics
and shocks. The numerator of Q is the market value of equity
plus the book value of debt less the book value of inventories;
the denominator is our measure of the replacement value of the
capital stock. Firms with high values of Q are presumably farther from their long-run equilibrium capital stock due to, among
other reasons, favorable demand or productivity shocks. Therefore, if our estimation method did not adequately account for
investment dynamics or shocks, then we might expect a difference in the estimated σ ’s across the high-Q and low-Q subsamples.
Table 2 presents OLS estimates from the benchmark model
and IV estimates from a constrained model in which η = 1 (in
accordance with the discussion in Section 4.2). The σ estimates
0.086
(0.021)
0.023
(0.016)
—
0.562
0.631
—
−0.024
(0.212)
—
1.000
0.024
(0.156)
0.107
(0.024)
Low
capital
0.329
(0.187)
1.000
0.126
(0.023)
0.002
(0.017)
0.556
—
1.000
1.000
—
0.575
−0.129
(0.130)
0.541
R2
γ
β
η
Parameter
σ
Low
CF/K
0.278
(0.081)
1.308
(0.137)
0.830
(0.058)
0.040
(0.014)
OLS
High
CF/K
0.407
(0.101)
1.019
(0.080)
0.989
(0.046)
0.125
(0.018)
IV
0.006
(0.201)
High
CF/K
0.358
(0.168)
1.000
Low
CF/K
0.364
(0.112)
1.000
Low
capital
0.435
(0.121)
1.042
(0.104)
0.977
(0.055)
0.105
(0.020)
OLS
0.141
(0.142)
0.582
High
capital
0.294
(0.076)
1.284
(0.113)
0.844
(0.046)
0.064
(0.013)
1.000
IV
High
capital
0.353
(0.100)
1.000
Low
Q
0.334
(0.083)
1.268
(0.094)
0.859
(0.036)
0.054
(0.014)
−0.003
(0.144)
0.122
(0.187)
—
1.000
1.000
IV
OLS
Split by Tobin–Brainard Q
Split by capital stock size
Split by cash flow-to-capital ratio
Table 2. OLS and IV estimates based on the IDE: Various sample splits
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NOTE: Estimates of the IDE with firm-level panel data as described in Sections 2 and 3. See the note to Table 1 for additional information. IV regressions are estimated with η and β constrained to unity because of instrument relevance concerns, as
discussed in Section 4.2. Sample splits are based on the median value of the classifying variable in the τ = 0 interval (1972–1977). The coefficient measuring the difference between the σ ’s for the contrasting classes is , which is distributed asymptotic
t under the null hypothesis of equality. The null hypothesis, = 0 = σ L − σ H , is evaluated with the following auxiliary equation based on (7): kf = −σ H cf − cf ∗ If + β L yf ∗ If + β H yf ∗ (1 − If ) − γ L ∗ If − γ H ∗ (1 − If ) − wf , where If
is an indicator variable equal to 1 for the low subsample and 0 for the high subsample. In the IV regressions, each individual instrument, zf , appears twice in the instrument list as zf ∗ If and zf ∗ (1 − If ).
High
Q
0.339
(0.148)
1.000
Low
Q
0.461
(0.115)
1.000
Journal of Business & Economic Statistics, October 2011
High
Q
0.337
(0.117)
1.029
(0.096)
0.982
(0.061)
0.107
(0.022)
592
are similar across the three subsample splits. The test statistics,
, for the equality of the σ ’s across subsamples cannot reject
the null hypothesis of equal σ ’s at any standard significance
level. These results (as well as IV results from the benchmark
model in which η is estimated freely) provide additional support
for the way in which our estimation method addresses problems
with complicated investment dynamics, avoiding these difficult
specification issues by focusing directly on the long-run growth
of the capital stock.
5. COMPARISONS AMONG ESTIMATORS
This section compares the IDE with two other estimators that
also focus on long-run relationships among variables defining
the first-order condition for capital: the dynamic OLS (DOLS)
and static OLS (SOLS) estimators. These three methods address the problem of unobservable long-run variables differently. As discussed in Section 2, the IDE solves this problem
by constructing time averages. Caballero (1994) developed the
innovative idea that the unobservability problem can be solved
if the model variables are I(1) and cointegrated. In Caballero’s
DOLS model, the long-run values of the capital/output ratio and
user cost equal the actual values (see the note to Table 3 for an
explicit equation). Estimates of σ can be biased downward in finite samples, and Caballero uses the Stock–Watson (SW 1993)
correction, which effectively accounts for short-run adjustment
dynamics to long-run values. The SOLS model is a special case
of DOLS without the SW correction, a procedure that is appropriate provided that the capital-to-output ratio and user cost
are reasonably close to their long-run values (Caballero 1994,
p. 54).
The DOLS model of Caballero, Engel, and Haltiwanger
(CEH 1995) is conceptually closest to our IDE model. (See
Chirinko 2008 for a list of DOLS and other studies estimating σ .) Both studies use panel data, and both emphasize longrun variation. CEH obtained a range of elasticities across twodigit industries for equipment capital from 0.01 to 2.00, with an
unweighted average of approximately unity. This corresponds
to an elasticity for total capital (comparable to those presented
herein) of approximately 0.70 (see Table 3). Notwithstanding
the similarities in the two studies, there is a marked gap between
σDOLS = 0.70 and σIDE = 0.40. This gap might be due to several differences, most notably data construction (documented in
Berndt 1976) and the sample period.
To control for these factors, we use a common dataset (described in Appendix E) to generate estimates of σ from the
three models. The DOLS model of CEH, which does not include time fixed effects and is estimated on plant-level panel
data, generates estimates of σ = 0.70. (The order of integration
of the model variables is examined in Appendix F.) Due to a
concern about degrees of freedom, CEH used only lags (as opposed to leads and lags) when implementing the SW correction.
When we follow this procedure with a maximum of three lags
and remove firm fixed effects through mean differencing, our
DOLS point estimate of σ = 0.720 (0.026) matches the result
of CEH. (CEH’s reported estimates are based on five lags, but
they noted that that their results are “fairly robust to the number of lags” (fn. 19); we confirmed the stability to lag length in
results not reported here.) When we include both leads and lags
Chirinko, Fazzari, and Meyer: The Elusive Capital–Labor Substitution Elasticity
Table 3. OLS and IV estimates of σ : DOLS, SOLS, and IDE
Fixed effects
Model
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DOLS (Caballero, Engel, and
Haltiwanger 1995)
Firm: Yes
Time: No
Firm: Yes
Time: Yes
0.700@
—
DOLS(3, 0)
Mean-differenced
0.720
(0.026)
0.754
(0.054)
DOLS(3, 3)
Mean-differenced
0.666
(0.036)
0.540
(0.059)
SOLS
Mean-differenced
0.645
(0.023)
0.520
(0.041)
DOLS(3, 3)
First-differenced
0.913
(0.089)
0.582
(0.095)
IDE
OLS, mean-differenced = first-differenced
0.372
(0.072)
IDE
IV, mean-differenced = first-differenced
0.434
(0.098)
NOTE: Estimates of σ are from various models with firm-level panel data as described
in Section 5. DOLS(M, N) is based on the following equation:
kyf ,t = −σ cf ,t +
LAG
M
m=0
LAG c
θm
f ,t−m +
LEAD
N
θnLEAD cf ,t+n + ξf ,t ,
n=1
where kyf ,t is the logarithm of the capital/output ratio, cf ,t is the log of the user cost
of capital, and M LAG and N LEAD indicate the number of lags and leads, respectively;
LAG = θ LEAD = 0 for all m and
SOLS is a special case of DOLS with no leads or lags (θm
n
n); and IDE [eq. (7)]. A common dataset containing 12,617 observations is used for all
estimates reported in this table; see Appendix E for details. The models are either firstdifferenced or mean-differenced to remove firm fixed effects. The models in column 2 also
contain time fixed effects. The instruments for IDE are discussed in the note to Table 1.
Heteroscedastic-consistent standard errors are given in parentheses. The @ indicates that
the estimate of 0.70 equals 1.0 (the estimate taken from the cited article) multiplied by 0.70.
This latter adjustment factor converts an equipment elasticity to an equipment + structures
elasticity and is based on the assumptions that the structures elasticity is one-third as large
as that for equipment (Cummins and Hassett 1992; Pindyck and Rotemberg 1983), and that
equipment constitutes 0.55 of total capital. The estimate from the cited article is an average
of estimated coefficients across two-digit industries for equipment capital; we do not have
sufficient information to compute the standard error associated with the estimate presented
in the table. For the IDE models presented in the last two rows, mean-differencing and
first-differencing are equivalent because there are only two intervals.
to account for the short-run adjustment dynamics, the DOLS
estimate of σ = 0.666 (0.036) is a bit lower. In summary, the
dataset used in the present study does not lead to different estimates than those of CEH.
The SOLS model excludes the SW correction and thus allows us to assess the impact of accounting for short-run adjustment dynamics. Following Caballero (1994), we expect
σSOLS < σDOLS . The SOLS model without time fixed effects
yields σ = 0.645 (0.023), close to the DOLS estimates. This
indicates that the SW correction, intended to account for deviations from long-run values, has little impact on the estimated
elasticity in the DOLS model.
Insofar as short-run adjustment dynamics are largely driven
by aggregate factors, time fixed effects can be interpreted as a
second correction factor that should affect σ . Estimates of the
DOLS and SOLS models with time fixed effects are presented
in column 2 of Table 3. For the SOLS model with time fixed effects, σSOLS = 0.520 (0.041), which is 0.125 lower than the estimate without time fixed effects. The estimate from the DOLS
model with three lags and leads, σDOLS = 0.540 (0.059), is sub-
593
ject to a similar decline. Unlike the SW correction, time fixed
effects matter for estimates of σ .
The IDE also emphasizes long-run variation but, rather than
accounting for short-run adjustment dynamics with the SW correction and fixed time effects, avoids this issue by inserting estimates of long-run values into the first-order condition (4). Differencing the data in the two disjoint intervals accounts for the
impact of both firm and time fixed effects. The OLS and IV estimates based on the IDE (Table 1 with η = 1) are σ = 0.372
(0.072) and σ = 0.434 (0.098), respectively; our preferred estimate is σIDE = 0.40.
The use of the common dataset reduces the gap between elasticity estimates based on the IDE and DOLS. The most important factor in lowering the initial DOLS estimate of 0.70
is the inclusion of time fixed effects. The remaining gap is
0.14 = 0.54 − 0.40 = σDOLS − σIDE .
6. SUMMARY AND CONCLUSIONS
The substitution elasticity between capital and labor has been
the focus of much research over the past 40 years. Almost all
previous work based on the first-order condition for capital has
relied on time series variation in investment data at the aggregate, industry, or firm level to estimate this elasticity. This
article offers a simple, intuitive, and computationally straightforward approach for estimating production function parameters that focuses on long-run relations. The estimation strategy
developed here groups annual data into longer time intervals
and then averages the firm-level panel data within each interval to estimate unobservable long-run variables. The data are
differenced across intervals, and production function parameters are estimated in a cross-section of time-averaged, differenced firm data. Our approach accounts for a variety of productivity shocks, omitted variables, and firm effects. This econometric model does not solve the estimation problems, including
difficult-to-specify dynamics, transitory time series variation,
and positively sloped supply schedules, inherent in regression
models using relatively high-frequency investment data that
may bias estimates of the substitution/user cost elasticity. Instead, our approach avoids these problems by exploiting panel
data with a simple approach that emphasizes low-frequency
variation and directly estimates the first-order condition for capital.
We find that the elasticity can be consistently and precisely
estimated with IDE and is approximately 0.40. Extant estimates
of σ range widely. However, when two other models that emphasize long-run variation—DOLS and SOLS—are applied to
a common dataset, we find a remarkable convergence of results. Although DOLS will be more efficient than IDE for large
T, Monte Carlo simulations suggest that IDE is preferred over
DOLS for T < 30 for a particular specification of the datagenerating process calibrated to our data set (see Chirinko and
Fazzari 2011).
Econometric issues aside, however, our results offer a strikingly negative answer to the central question of whether the
Cobb–Douglas assumption is valid. This robust finding raises
questions about the frequent use of the Cobb–Douglas production function in theoretical and empirical models.
594
Journal of Business & Economic Statistics, October 2011
Apart from our immediate objective of estimating production
function parameters, our method may prove useful in estimating
other structural parameters from panel datasets. Our intervalaverage approach could be applied to other problems in which
short-run dynamics may obscure long-run structural relations.
There are likely to be a number of applications in public economics, labor economics, and industrial organization, where the
availability of panel datasets and interest in long-run structural
parameters may make this method feasible and informative.
Downloaded by [Universitas Maritim Raja Ali Haji] at 23:21 11 January 2016
ACKNOWLEDGMENTS
We thank Gib Bassett, Stephen Bond, Hashem Dezhbakhsh,
Uwe Hassler, Harry Huizinga, Daniel Levy, Doug Meade,
James Morley, Werner Roeger, Julio Rotemberg, Lawrence
Summers, Giovanni Urga, Philip Vermuellen, Kenneth West,
an anonymous referee, an anonymous associate editor, the editors, and seminar participants at the International Conference
on Panel Data, Center for Economic Studies, the Centro de Investigacion y Docencia Economicas (Mexico City), the Eastern
Economics Association meetings, Emory University, the European Central Bank, the European Commission, the University
of Illinois at Chicago, and the University of Missouri for comments and suggestions. An earlier draft circulated under the title “That Elusive Elasticity: A Cross-Section/Long-Panel Approach to Estimating the Sensitivity of Business Capital to Its
User Cost.” The views expressed here do not necessarily reflect
those of the Federal Reserve Banks of San Francisco and St.
Louis or the Federal Reserve System. All errors and omissions
remain the sole responsibility of the authors, and the conclusions do not necessarily reflect the views of the organizations
with which they are associated.
[Received April 2008. Revised September 2010.]
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