Nº 507
ISSN 0104-8910
Testing production functions used in empirical growth studies
Pedro Cavalcanti Ferreira
João Victor Issler
Samuel de Abreu Pessôa
Outubro de 2003
Testing Production Functions Used in Empirical Growth Studies∗
Pedro Cavalcanti Ferreira†
João Victor Issler
Samuel de Abreu Pessôa
Graduate School of Economics - EPGE
Getulio Vargas Foundation
March 2003
Abstract
We estimate and test two alternative functional forms, which have been used in the growth literature,
representing the aggregate production function for a panel of countries: the model of Mankiw, Romer
and Weil (Quarterly Journal of Economics, 1992), and a mincerian formulation of schooling-returns to
skills. Estimation is performed using instrumental-variable techniques, and both functional forms are
confronted using a Box-Cox test, since human capital inputs enter in levels in the mincerian speciÞcation
and in logs in the extended neoclassical growth model.
1
Introduction
In this paper we estimate and test two alternative functional forms, that have been used in the growth
literature, representing the aggregate production function for a panel of countries. The Þrst model has a
long tradition in this literature and was proposed by Mankiw Romer and Weil (1992), among others. The
second is a mincerian formulation of schooling-returns to skills, traditionally used in the labor-economics
literature, e.g., Mincer (1974), but recently incorporated into the growth literature as well; see Klenow,
Rodriguez-Clare (1997), Hall and Jones (1999), and Bils and Klenow (2000)1 . Islam(1995) recognizes that
human capital is important in the growth process, but claims that the unresolved question is: “in what
exact way”? Since the basic difference between these two competing models is the way in which human
capital affects output — in the mincerian model human capital enters the production function exponentially
∗ We gratefully acknowledge the comments and suggestions of an anonymous referee, Costas Azariadis, Gary Hansen, Marcos
B. Lisboa, Carlos Martins-Filho, Naércio Menezes, Alberto Trejos, Farshid Vahid, Martin Uribe, and several seminar participants
around the world. All remaining errors are ours. Rafael Martins de Souza provided excellent research assistance. The authors
also acknowledge the Þnancial support of CNPq-Brazil and PRONEX.
† Corresponding author. Praia de Botafogo 190, 1125, Rio de Janeiro, RJ, 22253-900, Brazil. Email addresses of the authors
are, respectively, [email protected], [email protected], [email protected]
1 Strictly speaking, both models are “neoclassical,” in the sense that both have marginal diminishing returns to input
factors and that we have imposed constant returns to scale in both production functions. Despite that, we have labelled them
“augmented neoclassical growth model” and “mincerian model” respectively, for the sake of tradition.
1
while in the Mankiw, Romer and Weil model it enters the production function in levels — we propose here to
distinguish between them by means of an econometric test based on the Box and Cox(1962) transformation.
Contrary to the bulk of the previous empirical growth literature, we estimate directly the logarithmic
version of the production function, which makes performing a Box-Cox test a direct exercise. Panel-data
estimation is performed using instrumental-variable techniques, after a careful search for appropriate instruments. Also, a variety of speciÞcation tests are performed to validate the Þnal choice of model. The data
used in our study are extracted from Summers and Heston (1991, mark 5.6) and Barro and Lee (1996), and
includes 95 countries in different stages of economic development, with observations ranging from 1960 to
1985.
2
Model SpeciÞcation
The Mankiw, Romer and Weil model (MRW),2 has the following speciÞcation when constant returns of scale
are not imposed:
α φ
Hit (Lit exp (g · t))β ,
(1)
Yit = Ait Kit
where Yit , Kit , Hit , Lit , and Ait are respectively output, physical capital, human capital, raw labor inputs,
and productivity for country i in period t, where i = 1, · · · , N, and t = 1, · · · , T and g is the exogenous
technological progress, which is the same across countries. In per-worker terms, the equation above reduces
to:
(2)
ln yit = ln Ai + α ln kit + φ ln hit + (α + β + φ − 1) ln Lit + βg · t + ηit ,
where lower-case variables are in per-worker terms, and Ait is decomposed into a time-invariant component
Ai and an error component that varies across i and t, η it .
In the mincerian speciÞcation (Mincer (1974)) there is only one type of labor in the economy, which has
skill-level determined by educational attainment. It is assumed that the skill-level of a worker with h years of
schooling is exp (φh) greater than that of a worker with no education at all, leading to the following function:
α
(exp (φhit ) Lit exp (g · t))β .
Yit = Ait Kit
(3)
In per-worker terms, the equation above reduces to:
ln yit = ln Ai + α ln kit + βφhit + (α + β − 1) ln Lit + βg · t + η it .
(4)
Econometrically, the basic difference between equations (2) and (4) is whether human capital enters
the (logarithmic version of the) production function in levels or in logs. We therefore estimate a general
regression model:
µ θ
¶
hit − 1
(5)
ln yit = ln Ai + λ1 ln kit + λ2
+ λ3 · t + λ4 ln Lit + ηit ,
θ
which nests (2) and (4) even when constant returns to scale do not hold.
2 Recently
revisited by Bernanke and Gürkynak (2001).
2
3
Econometric Estimation, Testing, and Results
Consider the Box and Cox (1962) transformation for a given regressor:
xθit − 1
.
θ
xθ −1
(6)
xθ −1
Since lim itθ = ln (xit ), and lim itθ = xit − 1, for the logarithmic transformation to be valid we must
θ→0
θ→1
have θ = 0, and for the series xit to enter the regression in levels we must have θ = 1. The hypotheses
that human capital enters the production function in levels or in logs can thus be tested using a Box-Cox
transformation for the human-capital stock by means of a Wald test, with H0 : θ = 1 and H0 : θ = 0
respectively.
In estimating either (2) or (4) we must take into account that ln kit , ln hit , and hit are correlated with
ηit , which requires using instrumental-variable techniques. We propose using the following instruments for
ln kit , ln hit , and hit , respectively:
1 X
ln kjt−1 ,
Ni
i
(7)
j∈{N }
1 X
ln hjt−1 ,
Ni
i
(8)
j∈{N }
1 X
hjt−1 ,
Ni
i
(9)
j∈{N }
where N i represents the number of additional countries in the same continent that country i is in, and
© iª
N represents the set of countries in that continent that are not country i, i.e., (7), (8), and (9) represent,
respectively, rest-of-the-continent average lagged (log of the) capital stock, (log of the) human capital stock,
and level of the human capital stock. Hence, instruments are country speciÞc.
Rest-of-continent instruments such as (7), (8), and (9) are promising, since one could expect a priori that
countries in the same continent are similar in several dimensions and are relatively more integrated, making
instruments correlated with regressors. The fact that country i is excluded from computing rest-of-continent
averages makes the latter more likely to be uncorrelated with country’s i error: ηit , which can be formally
tested for over-identiÞed models; a basic references is Sargan (1958).
With appropriate instruments we estimate (5)3 . We assume that the variance of η it does not change
across t, although it is allowed to change across i. Due to this structure, our estimation method weights
data in each equation by the reciprocals of the standard deviation of country-speciÞc errors, similar to the
procedure in weighted two-stage least squares. The only difference here to weighted two-stage least squares
is that we use equation-speciÞc instruments and not the whole set of instruments. The constant term in each
regression equation uses country-speciÞc dummies, similar to Þxed-effect estimation in a completely linear
setup.
Based on the results of Wald tests with H0 : θ = 1 and H0 : θ = 0, we could either choose one of the
models (2) or (4), or reject (not reject) both. If one is chosen we then perform estimation of the chosen
3 In
(5) we also need instruments for ln Lit . We have used ln Lit−1 .
3
one using the same procedure described above: weighted instrumental-variable technique. Orthogonality
tests between errors and instruments using the procedure in Sargan(1958) is performed. Since we have
country-speciÞc instruments we perform Sargan tests equation-by-equation.
The panel data set used ranges from 1960 to 1985, and combines macroeconomic data for 95 countries in
the mark 5.6 of the Summers and Heston (SH, from now on) data set with human-capital measures extracted
from Barro and Lee (1996). We decided to interpolate human-capital measures to Þt annual frequency.4 The
time span was restricted from 1960 to 1985. The speciÞc series used are the following: yit is the ratio of
real GDP (at 1985 international prices) and the number of workers in the labor force, extracted from SH;
kit is the physical capital series per worker. The physical capital series is constructed using real investment
data from SH (at 1985 international prices) and the Perpetual Inventory Method;5 hit is Barro and Lee’s
(1996) series of average years of completed education of the labor force, interpolated (in levels) to Þt annual
frequency.
3.1
Model Estimation Results
Estimates of parameters in (5) are presented in Table 1 below.
< include Table 1 here>
First, the estimate of θ is 0.86, closer to unity, favoring the mincerian speciÞcation for the production
function. Indeed, Wald test results for θ = 1 and θ = 0 did not reject the mincerian model, although strongly
rejected the MRW model. Hence our Þnal conclusion is in favor of the mincerian model against the MRW
model. This is a relevant result, since a group of authors in the growth literature have recently made the
case for the use of the mincerian form of the production function; see Hall and Jones (1999), Klenow and
Rodriguez-Clare (1997) and Bils and Klenow (1999), among others. Because their argument is entirely based
on microeconomic evidence — e.g., Psacharopoulos (1994) — the evidence presented here using the Box-Cox
transformation conÞrms the appropriateness of their approach from a formal econometric point of view,
using a macroeconomic model. At the same time, our evidence points toward the rejection of a competing
alternative model also used extensively in the growth literature.
Second, despite the fact that Hall and Jones, Klenow and Rodriguez-Clare, and Bils and Klenow impose
constant returns in production, a formal test rejected this hypothesis, since λ4 is statistically signiÞcant6 . To
check whether or not our model choice depends on ruling out constant returns, we have performed Box-Cox
testing assuming constant returns beforehand. The Þnal choice of model is unaltered: we still do not reject
the mincerian model although we do reject the MRW model.
4 Although
this induces measurement error in human capital, the problem is relatively small, since human capital changes
with a highly predictable pattern and the estimation technique used allows for regressors that are measured with error. To
check the robustness of estimation results, we compared those using data with Þve-year intervals and those using yearly data.
They were very close indeed.
5 As for the initial capital stock, we followed Hall and Jones (1999) among many and approximated it by K = I / (g + δ),
0
0
I
where K0 is the initial capital stock, I0 is the initial investment expenditure, gI is the growth rate of investment, and δ is the
depreciation rate of the capital stock. The latter was set equal to 9%, but results barely changed when we used different values
for δ.
6 See also the evidence agaisnt constant returns in Duffy and Papageorgiou (2000).
4
In Table 2 we present the estimates of parameters of the mincerian model, i.e., equation (4).
< include Table 2 here>
The reported estimates for α, φ, β, and g in Table 2 are close to what could be expected a priori: several
calibrated studies use a capital elasticity α = 1/3 (see Cooley and Prescott (1995)). Estimates in Gollin
(2002) are close to 0.43 for a variety of countries. As discussed above, φ can be interpreted as a measure
of the percentage increase in income of an additional year of schooling. Mincerian regressions usually Þnd
b ' 10% (Mincer (1974)). Moreover, Psacharopolos (1994), who surveys schooling-return evidence using
φ
a large set of countries, Þnds an average of 6.8% for OECD countries and 10.1% for the world as a whole
— very close to our estimate. An average growth rate of productivity of about 2.2% a year is in line with
the evidence on long-run growth presented by Maddison (1995). Last, we perform a Wald test for ln (Ai ) =
ln (A), ∀i, when estimating (4). Results overwhelmingly reject these restrictions (a p-value of 0.000), showing
that productivity indeed varies across countries.
Because we want to check the speciÞcation of our structural model, we performed a series of Sargan (1958)
tests (equation-by-equation). If we take the signiÞcance level to be 5%, from a total of 95 country-regressions,
15 countries rejected the null in this test. This is about 16% of the sample of countries, a relatively low
number. Although in terms of number of countries these rejections are relatively small, since the data for each
country are weighted by the reciprocal of the standard deviation of its error term in computing estimates,
it could happen that including these countries makes a big difference in terms of parameter estimates. To
check if this was a potential problem, we ran mincerian regressions excluding from our sample of countries
those for which we rejected orthogonality at the 5% level in the Sargan test. The results of this exercise
showed overwhelmingly that estimates did not changed very much when these countries were excluded. For
b = 0.39, φ
b = 0.089,
the restricted sample of countries, parameter estimates are the following: α
b = 0.42, β
and gb = 0.029.
4
Conclusion
In this paper, we examined several features of the production functions used in the growth literature. The
most relevant result is related to the form human capital should enter in the production function. Since
the basic difference between the two most popular models used in the Þeld — the mincerian model and the
Mankiw, Romer and Weil model — is whether human capital affects output exponentially or in levels, we
propose here to distinguish between them by means of an econometric test based on the Box and Cox(1962)
transformation. Contrary to the bulk of the previous empirical growth literature, we estimate directly the
logarithmic version of the production function, which makes performing a Box-Cox test a direct exercise.
The tests conducted here show unequivocally that the mincerian model is more appropriate than the
augmented neoclassical growth model when we consider the way in which the production function should
be written. This is a relevant result, since a group of authors in the growth literature have recently made
the case for the use of the mincerian form of the production function; e.g., Hall and Jones (1999), Klenow,
Rodriguez-Clare (1997) and Bils and Klenow (1999), among others. Because their argument is entirely based
5
on microeconomic evidence, the results presented here validate their previous choice of production function
from a formal econometric point of view. Moreover, our estimated coefficients are consistent with previous
microeconomic evidence.
References
[1] Barro, R. and J. W. Lee, 1996. “International Measures of Schooling Years and Schooling Quality,”
American Economic Review, 86(2): 218-223.
[2] Bernanke, Ben S. and Gürkaynak, Refet S. 2001. “Is Growth Exogenous? Taking Mankiw, Romer
and Weil Seriously,” NBER Macroeconomics Annual 2001, Ben S. Bernanke and Julio J. Rotemberg,
Eds.Cambridge, MA: The MIT Press, 11-57.
[3] Bils, M. and P. Klenow 2000. “Does Schooling Causes Growth?,” American Economic Review,
90(5): 1160-1183.
[4] Box, G. E. P. and Cox, D. R. 1962. “An Analysis of Transformations,” Journal of the Royal
Statistical Society, Series B: 211-243.
[5] Cooley, T. and Prescott, E., 1995. “Economic Growth and Business Cycles,” in Cooley, T. and
Prescott, E. (eds.), Frontiers of Business Cycle Research, Princeton, Princeton University Press.
[6] Duffy, J. and C. Papageorgiou, 2000. “A Cross-Country Empiriacal Investigation of the Aggregate
Production Function SpeciÞcation,” Journal of Economic Growth, 5(1): 87-120.
[7] Gollin, D., 2002.“Getting Income Shares Right: Self Employment, Unincorporated Enterprise, and
the Cobb-Douglas Hypothesis,” Journal of Political Economy, 110(2): 458-474.
[8] Hall, R.E. and C. Jones, 1999. “Why do some countries produce so much more output per worker
than others?” Quarterly Journal of Economics, 114(February): 83-116.
[9] Hausman, Jerry A., 1978. “SpeciÞcation Tests in Econometrics,” Econometrica, 46: 1251-1272.
[10] Islam, N., 1995. “Growth Empirics: A Panel Data Approach,” Quarterly Journal of Economics, 110:
1127-1170.
[11] Klenow, P. J. and A. Rodríguez-Clare, 1997. “The neoclassical revival in growth economics: has
it gone too far?” NBER Macroeconomics Annual 1997, Ben S. Bernanke and Julio J. Rotemberg, Eds.
Cambridge, MA: The MIT Press, 73-103.
[12] Mankiw, N. G., Romer, D. and D.Weil. 1992. “A Contribution To The Empirics of Economic
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[13] Maddison, A., 1995. “Monitoring the World Economy: 1820-1992,” Paris: OECD.
6
[14] Mincer, J., 1974. Schooling, Experience, and Earning, National Bureau of Economic Research, distributed by Columbia University Press.
[15] Psacharopoulos, G., 1994. “Returns to Investment in Education: A Global Update,” World Development, 22(9): 1325-1343.
[16] Sargan, J. D., 1958. “The Estimation of Economic Relationships using Instrumental Variables,”
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[17] Summers, R. and A. Heston, 1991. “The Penn World Table: An Expanded Set of International
Comparisons,1950-1988,” Quarterly Journal of Economics, 106(May): 327-368.
7
Table 1: Estimates of the Models
³ θ in ´Box-Cox Form
Equation: ln yit = ln Ai + λ1 ln kit + λ2 hitθ−1 + λ3 · t + λ4 ln Lit + ηit
λ1
Estimated Parameters/(Std. Errors)
λ2
λ3
θ
λ4
0.4285
(0.0059)
0.0537
(0.0102)
0.0098
(0.0004)
0.8622
(0.0994)
-0.1216
(0.0193)
Box-Cox Wald Test
θ = 1 (p-value) θ = 0 (p-value)
0.1657
0.0000
Table 2: Estimates of the Mincerian Growth Model ( Log-Level Model)
Equation: ln yit = ln Ai + α ln kit + βφhit + βg · t + (α + β − 1) ln Lit + ηit ,
Estimated Parameters/(Std. Errors)
α
φ
g
β
0.4306
(0.0059)
0.0909
(0.0090)
0.0221
(0.0016)
0.4501
(0.0200)
8
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