Estimation of the Share of Physical
Capital in Output
NELSON RAMÍREZ-RONDÁN
Banco Central de Reserva del Perú
JUAN CARLOS AQUINO
Pontificia Universidad Católica del Perú
WITSON PEÑA
Macroconsult S.A.
(June, 2005)∗
______________________________________________________________________
ABSTRACT
The importance of calculating factor shares, specifically physical capital’s share, is to know the
elasticities of per capita income respect to saving rate and population growth rate. A our study’s
characteristic is the emphasis on level analysis more than on growth rates of real output that
removes all the long-run information in the data, since the first difference operator eliminates
low frequencies, and thus emphasizes short-term fluctuations in the data. Nevertheless due to
the problem of low availability of long samples, we have exploited the cross-section dimension
(between countries). In this sense this paper aims to estimate physical capital’s share in output
using a Cobb-Douglas production function; for this we have assumed two models: one with
constants returns to scale and the second without such a specification. Using the fully Modified
Ordinary Least Squares (FMOLS) methodology, developed by Pedroni (2001) in a panel
cointegration framework for fourteen countries in Latin America during 1960-2002 period. We
find capital’s share in output is 0.39 in the first model, and 0.41 in the second one. The results
are similar to the measures found by Bernanke and Gurkaynak (2001) and Elias (1992).
Furthermore our results differ a bit of Senhadji’s work (2000).
JEL Clasification: C23, E23, 047
Keywords: Production function, physical capital, Cointegration, panel data
______________________________________________________________________
∗
E-mail address: [email protected] (N. Ramírez-Rondán), [email protected] (Juan Carlos
Aquino), [email protected] (Wilson Peña).
1. INTRODUCTION
An important issue in economic growth literature is the estimation of capital’s share in
product; generally past studies had calculated such a parameter from national accounts,
or from a production function regression, expressed in growth rates.
The principal problem of estimating such a production function in growth rates is the
elimination of stochastic trends in series; in this way, the potentiality of the nonstationarity of the series is not being exploited, since the first difference operator
removes all the long-run information in the data (the first difference operator eliminates
low frequencies, and thus emphasizes short-term fluctuations in the data). One
important insight from the cointegration literature is that we know much more about the
long-run than the short-run relationship between macroeconomic variables, hence the
cointegration literature has clearly demonstrated the superiority of level equation versus
first-difference equations when series are nonstationary. In this context there are very
few studies that have tried to estimate a production function in levels, (Senhadji, 2000),
trying to find a long run relationship involving common stochastic trends
(cointegration). Moreover the critical technology parameter -the share of physical
capital in output- is econometrically estimated and the usual assumption of identical
technology across regions is relaxed, that is, the production function is assumed to be
identical across countries within the same region but different among countries across
regions.
A first problem in the long run estimation is the potential endogeneity of the
explanatory variables, in the case of capital and labor, which does not take account of
the non-stationary time series estimation methods (Two Step OLS), developed by Engle
and Granger (1987), this is an argument often made in the literature against the
estimation of production functions for determining the share of physical capital (the key
parameter in the accounting exercise); such problems are corrected by the Maximum
Likelihood methodology developed by Johansen (1988), and the Fully Modified OLS
and Dynamic OLS, which additionally correct the possible autocorrelation between
error terms, developed by Phillips and Hansen (1990) and Hansen (1992).
A second problem is the low availability of long samples, especially in developing
countries, such the case of Latin America, which makes the estimates very sensible to
changes in the sample and where there is a huge economic sector that is informal.
However, the cross-section dimension (between countries) in the data can be exploited.
In this sense, pioneer works such Levin et al. (2002) and Im et al. (2003) developed
tests in order to check for the presence of non-stationarity of the series in a panel data
framework; additionally works such Kao (1999), McCosKey et al. (1998) and Pedroni
(1997) developed tests to detect a possible cointegration among the series in a panel
data framework. This work indeed employs the Fully Modified OLS (FMOLS)
methodology, developed by Pedroni (2001) in a context of panel cointegration, for a
period 1960-2002 and fourteen countries, using for this a Cobb-Douglas specification.
The importance of calculated factor shares, specifically physical capital’s share, is to
know the elasticities of per capita income respect to saving rate and population growth
rate. In the methodological context, for the standard growth accounting (national
accounts approach), factor shares are used to decompose growth over time in a single
country into part explained by growth in factor inputs and an unexplained part (the
2
Solow residual) which is usually attributed to technological change. In the same way, in
the cross-country approach (from a production function regression, expressed in
growth), factor shares are used to decompose variation in income across countries into a
part explained by variation in saving and population growth rates and an unexplained
part which could attributed to international differences in the level of technology.
The remaining of this work is organized as follows: in the second section we reviewed
methodological studies of the share of factors (physical capital and labor) in output for
Latin America finding three important studies that give us main tools to estimate the
share of factors in output. These works were made by Bernanke and Gürkaynak (2001),
Senhadji (2000) and Elias (1992). In the third section the production function to be
estimated and the data set is described, in the fourth section we make a discussion of the
estimation methodology, in the fifth section we present the results of estimation and in
the sixth section some final concluding comments are made.
2. METHODOLOGICAL STUDIES FOR LATIN AMERICA
2.1. BERNANKE AND GÜRKAYNAK’S METHODOLOGICAL STUDY
In their classic study Bernanke and Gürkaynak (2001) estimate labor’s share assuming
that all the economies in their sample lie on a balanced growth path. First, economies
are buffeted by a variety of major and minor shocks, as well as changes in institutions
and policies; hence, even if their models are precisely correct, some component of
observed economic growth must be accounted for by transition dynamics. Second, they
cannot take literally the prediction of many endogenous growth models that country
growth rates may differ permanently, as that would imply counterfactually that the
cross-sectional variance of real GDP per worker grows without bound. Although
government policies and private-sector decisions may have highly persistent effects on
growth, ultimately there must be forces (such as technology transfer from leaders to
followers) that dampen the tendency toward divergence. A more direct way they found
to study the determinants of long-run growth, without having to take a stand on whether
the world’s economies are currently on a balanced growth path (or whether some are
and some aren’t), is to obtain country-by-country estimates of the growth of TFP. As is
well known, if production is Cobb-Douglas and factor markets are competitive, then
TFP growth rates can be found by standard growth accounting methods, using factor
shares to estimate the elasticities of output with respect to capital and labor.
Gollin (1998) presents evidence against the conventional finding; his key insight is that
published series on “employee compensation” may significantly understate total labor
compensation, particularly in developing economies, because of the large share of
income flowing to workers who are self-employed or employed outside the corporate
sector. To try to capture the income of the latter group of workers, Gollin employs data
from the United Nations System of National Accounts and shows the UN’s method of
breaking down the cost components of GDP.
3
FIGURE 1.- COST COMPONENTS OF GDP
Indirect taxes, net
Indirect taxes
Less: Subsidies
Consumption of fixed capital
Compensation of employees by resident
producers
Resident households
Nonresidents
Operating surplus
Corporate and quasi-corporate enterprises
Private unincorporated enterprises
General government
Statistical discrepancy
Equals Gross Domestic Product
Source: Bernanke and Gurkaynak (2001).
Income received by the selfemployed and non-corporate employees is a component of
the category Operating Surplus, Private Unincorporated Enterprises (OSPUE). Gollin
considers two measures of labor’s share which use data on OSPUE. For the first
measure, he attributes all of OSPUE to labor earnings, so that labor’s share becomes
(corporate) employee compensation plus OSPUE, divided by GDP net of indirect taxes.
For his second measure, he assumes that the share of labor income in OSPUE is the
same as its share in the corporate sector. Specifically, this measure of the share of labor
income can be written:
(1)
Share Labor = Corporate employee compensation __
GDP-taxes indirect-OSPUE
They view this second measure, which allows for the existence of non-corporate capital
income, as more reasonable; they refer to it as the OSPUE measure. Gollin also
considers a third measure of labor’s share, which uses data on the ratio of corporate
employees to the total labor force less unemployed, available in various issues of the
International Labor Organization’s Yearbook of Labor Statistics. Specifically, he
assumes that corporate and non-corporate workers receive the same average
compensation, so that aggregate labor income can be calculated by scaling up corporate
employee compensation by the ratio of the total labor force to the number of corporate
employees. This measure, which we will refer to as the labor force correction, is defined
by:
(2)
Share Labor =
Corporate employee compensation
Corp. share of labor force * (GDP- indirect taxes)
They have replicated and updated Gollin’s calculations for the OSPUE measure and the
labor force correction for their sample of countries. One problem that they noted in
doing so is that OSPUE is reported for only about 20 countries; the majority of
countries report only the total operating surplus of corporate enterprises and private
unincorporated enterprises, that is, they have only the sum of OSPUE and corporate
capital income. To expand the number of countries for which labor shares could be
4
calculated, they constructed an alternative measure of labor share that combines
information about the corporate share of the labor force and the aggregate operating
surplus. To do so, they assume that the corporate share of total private-sector income
(both capital income and labor income) is the same as the share of the labor force
employed in the corporate sector. Total private-sector income is calculated as the sum of
the operating surplus and corporate employee compensation. They then compute
“imputed OSPUE” as the share of noncorporate employees in the labor force times
private-sector income. Using the imputed value of OSPUE we then estimate labor’s
share using equation (1), with imputed OSPUE in place of actual OSPUE.
FIGURE 2.- ALTERNATIVE MEASURES OF LABOR’S SHARE
Country
Bolivia
Chile
Colombia
Costa Rica
Ecuador
El Salvador
Jamaica
México
Panama
Paraguay
Peru
Trinidad & Tobago
Uruguay
Venezuela
Mean
Employee/LF
Naïve
0.55
0.68
0.68
0.72
0.56
0.60
0.60
0.59
0.65
0.62
0.53
0.77
0.74
0.68
0.64
0.37
0.42
0.45
0.44
0.25
0.35
0.53
0.34
0.50
0.32
0.31
0.55
0.43
0.38
0.41
Imputed
OSPUE
0.59
0.73
0.55
0.73
0.49
0.56
0.69
0.58
0.53
0.61
LS
0.67
0.62
0.65
0.74
0.45
0.58
0.59
0.76
0.52
0.59
0.71
0.59
0.55
0.62
Source: Bernanke and Gurkaynak (2001).
It reports a variety of data for the countries in our sample for which either 1) OSPUE is
available or 2) the share of corporate employees in the labor force is at least half, or
both. They impose the second requirement because they found that, for countries with
very low corporate employment shares (for some, this share is below 0.10), the
calculated labor shares are often unreasonable (e.g., they may exceed one). This result is
not unexpected, for two reasons: First, countries with large informal sectors are likely to
have relatively poor economic statistics, all else equal. Second, our estimates which use
the labor force correction scale up corporate employee compensation by the inverse of
the corporate employee share of the labor force. When the corporate employee share is
both small and measured with error, estimates based on the inverse of the share will be
highly unreliable. We found, on the other hand, that when the corporate employee share
exceeds 0.5 or 0.6, the estimated labor shares that result are both reasonable in
magnitude and tend to agree closely with alternative measures. All of the analyses
reported below use both 0.5 as the cutoff for the corporate employee share of the labor
force; results for samples based on a 0.6 cutoff are essentially identical. In Figure 2 the
second column gives the share of the country’s labor force employed in the corporate
sector. Columns 3 through 5 give three alternative measures of labor’s share for each
country. Column 3, the “naïve” calculation, is corporate employee compensation
divided by GDP net of indirect taxes. As emphasized by Gollin, this estimate is likely to
5
be too low, because it ignores the income of noncorporate employees. They include it
for reference and comparison to other measures. Columns 4-5 give their three primary
measures of labor’s share. Column 4 shows Gollin’s imputed OSPUE measure, and
Column 5 the measure based solely on the labor force correction. Columns 2-5 are
based on averaged data for the period 1980-1995, or for a period as close to 1980-1995
as possible. They also calculated country-by-country time series for the labor share (not
shown in their paper).
2.2. SENHADJI’S METHODOLOGICAL STUDY
Senhadji (2000) examines the source of cross-country differences in total factor
productivity (TFP) levels with a growth accounting exercise conducted for 88 countries
for 1960–94. Two differences distinguish this analysis from that of the related literature.
First, the critical technology parameter -the share of physical capital in output- is
econometrically estimated and the usual assumption of identical technology across
regions is relaxed. Second, while the few studies on the determinants of cross-country
differences in TFP have focused on growth rates of real output this analysis is on levels.
The results of the growth accounting exercise therefore depend on the specification of
the production function. The bulk of the literature has adopted the Cobb-Douglas
production function, which typically sets its parameter, the share of the remuneration of
physical capital in aggregate output, to a benchmark value of one-third as suggested by
the national income accounts of some industrial countries.
For the growth accounting exercise in Senhadji’s paper, the assumption of identical
technologies across regions is relaxed. The 88 countries in the sample are divided into
six regions. The production function is assumed to be identical across countries within
the same region but different among countries across regions. The estimates of the
production function for each region are obtained by averaging individual country
estimates belonging to each region. An argument often made in the literature against the
estimation of production functions for determining the share of physical capital (the key
parameter in the accounting exercise) is the problem of potential endogeneity of the
explanatory variables, namely capital and labor inputs. The Fully Modified estimator,
which is used to estimate the production function of each country in this paper, corrects
for this potential problem as well as for the likely autocorrelation of the error term.
The production function is estimated in levels, since the first difference operator
removes all the long-run information in the data (the first difference operator eliminates
low frequencies, and thus emphasizes short-term fluctuations in the data). One
important insight from the cointegration literature is that we know much more about the
long-run than the short-run relationship between macroeconomic variables.
Consequently, by differencing, we disregard the most valuable part of information in
the data. Few studies have attempted to explain cross-country differences in TFP. A
notable study show a significant share of the cross-country variation in TFP level can be
explained by “social infrastructure” is the work of Hall and Jones (1999).
Three factors explain why levels matter more than growth rates. First, growth rates are
important only to the extent that they are a determining factor of levels. Second, recent
contributions to the growth literature focus on levels instead of growth rates. For
example, Easterly and others (1993) show that growth rates over decades are only
6
weakly correlated, suggesting that cross-country differences in growth rates may
essentially be transitory. Moreover, several recent models of technology transfer across
countries imply convergence in growth rates as technology transfers prevent countries
from drifting away from each other indefinitely. In these models, long-run differences in
levels are the pertinent subject of analysis. And, third, the cointegration literature has
clearly demonstrated the superiority of level equation versus first-difference equations
when series are nonstationary. Formal unit-root tests show indeed that these variables
cannot reject the unit-root hypothesis.
This paper uses the Fully-Modified (FM) estimator developed by Phillips and Hansen
(1990) and Hansen (1992) to estimate the production function. The FM estimator is an
optimal single-equation method based on the use of OLS with semiparametric
corrections for serial correlation and potential endogeneity of the right-hand variables.
The FM estimator has the same asymptotic behavior as the full systems maximum
likelihood estimators. The correction for potential endogeneity of the explanatory
variables is an attractive property of the FM estimator since physical capital per capita is
likely to be endogenous. The production function was estimated for 66 countries, 46 of
which are developing countries. This paper provides estimates of α (the share of
physical capital in aggregate output) in both levels and first differences for comparison.
FIGURE 3.- COBB-DOUGLAS PRODUCTION FUNCTION
ESTIMATES FOR LATIN AMERICA COUNTRIES
Argentina
Bolivia
Colombia
Costa Rica
Ecuador
Guatemala
Honduras
Jamaica
Mexico
Panama
Paraguay
Trinidad & Tobago
Uruguay
Venezuela
Mean
Median
Standard Deviation
Min
Max
α in level
0.70
0.72
0.61
0.32
0.36
0.75
0.69
0.81
0.38
0.45
0.39
0.53
0.24
0.64
0.52
0.48
0.18
0.24
0.81
α in first difference
0.76
0.63
0.11
0.88
0.32
0.73
0.86
0.81
0.96
0.58
0.49
0.80
0.24
0.74
0.62
0.68
0.25
0.11
0.96
Note: This figure shows the Fully Modified (FM) estimates of the share of physical capital (α) for the
following Cobb-Douglas production function: Yt = At K tα ( Lt H t )1−α , where At is total factor productivity,
K t is the stock of physical capital, Lt is the active population, y H t is an index of human capital.
Source: Senhadji (2000).
7
For the equations in levels, it remains to be verified whether coefficient estimates
provide a meaningful economic relationship that is not the result of a spurious
regression. This amounts to testing whether output per capita and capital per capita are
cointegrated. The cointegration tests used are the Phillips-Ouliaris (P-O) test, which has
non-cointegration as the null hypothesis and the Shin (SH) test, which has cointegration
as the null. While P-O rejects the null of noncointegration for only 26 countries (which
is likely the result of the test’s low power in small samples), the SH test fails to reject
the null of cointegration for all 66 countries. Thus, the combined evidence from both
tests favors the hypothesis of cointegration.
3.3. ELIAS’S METHODOLOGICAL STUDY
On his study of seven Latin American economies, Elias (1992) discusses some of the
characteristics of production function used to adjust to the aggregative data for Latin
American economies. These characteristics which have important implications for the
method he has followed in his accounting estimates of growth. First the degree of return
to scale is difficult to capture in an aggregative function because it is basically a concept
for use at the firm, making its meaning difficult to interpret in an aggregative level.
However, one possibility he find is to interpret the return to scale as the measurement of
the effect of increased in market size, which produces benefits through labor
specialization (the so-called Adam Smith effect). Second the constancy of output–input
elasticities, which depend on the elasticity of substitution. The traditional Cobb-Douglas
production function implies constant output–input elasticity and, consequently, constant
inputs weights in the sources of growth equation. Third the kind of technology enters
into production function in different forms. The simple form is in a Hicks neutral way,
as a variable multiplying the function production. Other way is considered technology
as any other input variable.
FIGURE 4.- ESTIMATIONS OF COBB-DOUGLAS PRODUCTION FUNCTION
OLS Estimations of Cobb-Douglas Production Function, with the form
LnYt = a + bt + α ln( K / L)t + dLnLt + ut
t-test (absolute values)
α estimate
Argentina
0.40
1.33
Brazil
0.24
4.71
Chile
0.34
2.94
Colombia
-0.28
1.71
Mexico
-0.13
1.56
Peru
-0.38
4.45
Venezuela
0.06
0.20
OLS Estimations of Cobb-Douglas Production Function, Pooling Time Series with
Cross Country Data, with the form
LnYt = a + bt + α ln( K / L)t + dLnLt + dummies + ut
Latin America
0.39
9.89
Note: Ln K t is the log of stock of physical capital; Ln Lt is the log active population.
Source: Elias (1992).
8
His work has as a main objective is to complement the sources-of-growth approach
methodology providing an initial econometric approximation to the production function
approach.
Elias have also reported results of the share of capital income in GDP during 1940-1985
period for seven Latin America economies (for detail see appendix 4).
3. THE MODEL AND DATA
3.1. THE MODEL
First, we consider a Cobb-Douglas production function which depends of physic capital
( K ), labor ( L ) and the level of total factor productivity – the level of technology ( A )
as shown in equation (3), where we assume constant returns to scale and perfect
competition in factors market.
(3)
Y = A( K )α ( L)1−α
Rewriting equation (3) we can get the production function in terms of product per
worker and physical capital per worker.
(3’)
Y / L = A( K / L)α
Taking logarithms to both sides of equation (3’), we arrive to the following expression:
(3’’)
y / l = B + α (k / l )
Where y/l is the logarithm of Y/L, k/l is the logarithm of K/L, B is the logarithm of A and
α is the parameter to be estimated.
Let assume a production function where there are not necessarily constant returns to
scale, in the following form.
(4)
Y = A( K )α ( L) β
In the same way, we take logarithms to both sides of equation (4), getting the following
expression.
(4’)
y = B +αk + βl
Where y is the logatirhm of Y, k is the logatirhm of K, l is the logatirhm of L, B is the
logatirhm of A, and α and β are the parameters to be estimated.
9
3.2. CONSTRUCTION OF PHYSICAL CAPITAL
The stock of physical capital series were constructed following Nehru y Dareshwar
(1993), who calculate the stock of physical capital using the perpetual inventory
method, which is based in the following capital accumulation equation
t −i
(5)
K t = (1 − d )t K (0) + ∑ I t −i (1 − d )i
i =0
Where K t is the stock of capital in period t, K (0) is the initial stock of capital (in period
0), I t −i is the gross domestic investment t-i, and d is the rate of depreciation. Nehru and
Dareshwar (1993) estimate K (0) using a modification of the technique proposed by
Harberger (1978).
The proceeding is based in the assumption that in steady state, the output growth rate
(g) is equal to the capital growth rate. Rewriting equation (5):
(6)
( K t − K t −1 ) / K t −1 = − d + I t / K t −1
Which implies:
(7)
K t −1 = I t /( g + d )
Then in period 0, the stock of capital can be calculated as:
(8)
K (0) = I1 /( g + d )
The rate of depreciation is assumed to be 4% and g is derives of real Gross Domestic
Product in market prices; in this way, the remaining of the series is calculated from
equation (5)
In order to construct the stock of capital series, we have used the Gross Domestic
Investment from World Bank’s World Development Indicators (2004).
3.3. DATA
The variables employed in this paper are taken from The World Development Indicators
(2004), Gross Domestic Investment are in constant 1995 U.S. dollars, and consists of
outlays on additions to the fixed assets of economy plus net changes in the level of
inventories. Fixed assets include land improvements (fences, ditches, drains, and so on);
plant, machinery, and equipment purchases; and the construction of roads, railways, and
the like, including schools, offices, hospitals, private residential dwellings, and
commercial and industrial buildings. Inventories are stocks of goods held by firms to
meet temporary or unexpected fluctuations in production or sales.
Also, Gross Domestic Product at purchaser's prices are in constant U.S. dollars, and the
variable Total labor Force comprises people who meet the International Labor
Organization definition of the economically active population.
10
The panel includes the following countries: Argentina, Bolivia, Brazil, Chile, Colombia,
Ecuador, Guatemala, Honduras, Mexico, Paraguay, Peru, Dominic Republic, Uruguay
and Venezuela.
4. METHOD OF ESTIMATION
The fact of temporal dimension (T) growing to infinite in macro panel data generates
two important ideas: The first one rejects the homogeneity of the parameters of the
regression in the use of a pooled regression in favor of heterogeneous regressions, that
is, one for each individual. The second one is the application of the proceedings of time
series in panel data. The addition of transversal dimension to the temporal dimension
gives an advantage in the tests of non-stationarity and cointegration.
The technique of cointegration analysis is a powerful tool in the combination of nonstationary time series. However, it has encountered statistical limitations to its
application in a context of low availability if homogeneous statistics and sufficiently
large. Due to this, it has been recently proposed the use panel cointegration techniques.
In this way, it has been taken advantage of transversal availability.
Is in this way that in the last decade it has been developed an interesting literature about
the possible stationarity of series in a panel data framework, and the possible
cointegration that could exist between non-stationary variables. A general overview of
such literature can be found in Banerjee (1999), Phillips and Moon (2000), and Baltagi
and Kao (2000).
4.1. UNIT ROOTS IN PANEL DATA
Quah’s work (1994) appears as the pioneer one in developing a unit root test, assuming
a simple model of panel data with random disturbances, independent and identically
distributed in both dimensions, and shows the asymptotic normality of the Dickey Fuller
test in presence of a unit root when temporal and transversal dimension growth to
infinity at the same rate, that is, with N/T constant.
Levin and Lin constitute one of the first unit root tests in panel data that originally
appeared in working papers in 1992 and 1993, and was published together with Chu in
2002; Levin and Lin extend the work of Quah, using an augmented version of the
contrast DF, in order to test the common presence of a unit root in opposition to the
alternative hypothesis of stationarity among the individuals in the panel, letting the two
dimensions to grow in an independent way. Additionally, it permits a greater transversal
heterogeneity in the random disturbances behavior and let a greater flexibility with
respect to the appearance of deterministic terms in the data, generating process assumed
for the distinct individuals in the panel.
Im, Pesaran and Shin (2003) complement the results of Levin and Lin, developing new
unit root tests in panel data based on group measures of the Lagrange Multiplier (LM)
type. The principal characteristic of this test consists in the flexible formulation of the
11
hypothesis that does not restrict to the stationarity of all the individuals of the panel
under the alternative hypothesis.
A third orientation was proponed by Choi (1999) and Madala and Wu (1999), who
propose tests of the Fisher type, in order to average the values of the significance levels
associated to the unit root tests obtained in each cross-section panel. Such test has some
desirable properties like the application possibility to panels with different time
observations for each individual and a relax of the cross-section independence
hypothesis.
The Im, Pesaran and Shin (IPS) and Fisher tests combine information based in the
individual unit root tests. However, the Fisher test has the advantage on IPS test in not
requiring the panel to be balanced. Additionally, the Fisher test can use different lag
lengths in the ADF individual regression and can be applied for other unit root test. The
disadvantage is that the significance levels have to be derived from a Monte Carlo
simulation. Choi (1999) finds similar advantages for the Fisher tests: a) the crosssection dimension, N, can be finite or infinite, b) each group can have different types of
stochastic or non stochastic components, c) the time series dimension, T, can be
different for each individual in the panel and d) the alternative hypothesis can show
some group having unit root while other may not.
Finally, Hadri (2000) propose a residual version of unit root tests base on the Lagrange
Multiplier (LM) test for the null hypothesis that time series for each individual in the
panel are stationary around a deterministic trend in contrast to the alternative
hypothesis, in an analogous way to the KPSS (Kwiatowski et. al., 1992) in a time series
framework.
Others studies on unit roots in panel data are developed by Sarno y Taylor (1998); in
their test, a simple autoregressive parameter is estimated on a panel, using the Zellner’s
SUR estimators for N equations, corresponding to the N individuals in the panel,
Nyblom y Harvey (2000) develop a series of tests for common stochastic trends, proof
the validity of a specific value of the range for the variance-covariance matrix of the
residual using a multivariate random walk, which is indeed equal to the number of
common trends in series set.
4.2. PANEL COINTEGRATION TEST
A first panel cointegration tests was proponed by Kao (1999), who’s proposed residual
tests of no cointegration and critical values in an analogous way to the residual bietapic
test of Engel and Granger (1987) for time series. This way, the estimation of the
autorregresive parameter of the residual equation using OLS, obtaining asymptotic
normal distributions (0,1) for four tests.
McCoskey and Kao (1998) presented an adaptation to panel data case of the LM
(Lagrange Multiplier) and LBUI (Locally Best Unbiased Invariant) tests for an unit root
in a MA process of Harris and Inder (1994) in a time series framework. In these residual
tests, they use the null of cointegration ahead to the model proponed by Kao (1999), so,
the tests distribution in the limit of derived tests does not depend on asymptotic
properties of the spurious regression estimated; however the tests properties are highly
12
sensible to estimation quality of cointegration vector (efficiency and consistency). So,
they propose the use of alternative estimators to the OLS estimator, like FMOLS (Fully
Modified Ordinary Least Squares) based from the previous developments of Phillips
and Hansen (1990) or DOLS (Dynamic Ordinary Least Squares) in a similar way to the
time series case, developed by Saikkonen (1991) and Stock and Watson (1993).
Pedroni (1997) proponed a first four-test set, three of them include the use
nonparametric corrections similar to the Phillips and Perron (1988), while the fourth is a
parametric test ADF-like; this first group includes a statistics averages form of Phillips
and Ouliaris (1990). A second-test set proposed, consider construction in pieces of
average tests, so that limiting distributions of the same will be derived from limits
combination of each pieces of tests’s numerator and denominator; because its test is
based on average of numerator and denominator terms respectively, and not in the
statistics average as a whole.
Pedroni (1999) derives asymptotic distributions and critical values for several tests,
based on residuals of no cointegration mull in panels, where there are multiple
regressors. The model includes regressions with fixed individual effects and temporal
tendencies.
Finally, Larsson, Lyhagen and Löthgren (1998) present an extension of time series
proceedings, developed by Johansen (1988) to homogeneous panel case, based on
Likelihood Ratios tests, obtained as averages of trace statistics for each individual in the
panel. However, they find that the proposed test requires long series in time, including
the case in which panel has a large individuals number in the transversal dimension, test
length could be seriously distorted.
4.3. FMOLS AND DOLS ESTIMATORS IN COINTEGRATED PANELS
The method employed is the Fully Modified Least Squares (FMOLS) developed
originally by Phillips and Hansen (1990), and later by Phillips (1995) in a time series
context, to provide a general framework about asymptotic behavior of FMOLS
estimators in models with regressors I(1), I(1) and I(0), and I(0).
Pedroni (1996, 2001) was the first-one researchers who applied estimators mentioned
above in a “panel cointegration context”, the section is based in this author. An
important point is to build estimators in a way that does not restrict transitional
dynamics to be similar between different countries from panel. This is indeed a subject
in panel fully modified OLS test that was developed by Pedroni (2001). Pedroni studies
three estimators’s versions, Residual-FM, Adjustment-FM and Group-FM, he shows
that the last estimator exhibits a low large of distortion in small samples.
Panel FMOLS estimators have two dimensions: between-dimension and withindimension. An important advantage of between-dimension estimators, in the way data is
pooled, allow a greater flexibility in heterogeneity presence of cointegrating vectors.
Another estimators’s advantage is point estimates have a more useful interpretation in
the event that true cointegrating vectors are heterogeneous. Specifically, point estimates
for between-dimension estimator can be interpreted as a mean value for cointegrating
vector. This is not true for the within-dimension estimator.
13
Consider the regression of the equation (9):
(9)
y it = α i + β i * x it + µ it
Where yit y xit are cointegrated with slopes βi , which may or may not be
homogeneous across countries.
The expression for between-dimension, group mean panel FMOLS estimator is given as
(10)
βˆ *
GFM
−1
⎛ T
⎞ ⎛ T
⎞
= N ∑ ⎜ ∑ ( xit − xi ) 2 ⎟ * ⎜ ∑ ( xit − xi ) yit* − Tγˆi ⎟
i =1 ⎝ t =1
⎠ ⎝ t =1
⎠
−1
N
Where:
ˆ
Ω
21i
∆xit
ˆ
Ω 22i
(11)
yit* = ( yit − yi ) −
(12)
ˆo −
γˆi ≡ Γˆ 21i + Ω
21i
ˆ
Ω
21i ˆ
ˆo )
(Γ22i + Ω
22 i
ˆ
Ω 22i
Let Ω i be the long-run covariance matrix which in typically estimated using any one of
a number of HAC estimators, such as the Newey and West estimator. It can be
decomposed as Ω i = Ω io + Γi + Γi' , where Ω io is the contemporaneous covariance and Γi
is a weighted sum of autocovariances.
Because the expression that follows the summation over the i is identical to the
conventional time series FMOLS, the between-dimension estimator can be constructed
N
*
*
ˆ*
simply as βˆGFM
= N −1 ∑i =1 βˆ FM
,i , where β FM ,i is the conventional FMOLS estimator,
applied to ith panel member. Likewise, the associated t-statistics for the betweendimension estimator can be constructed as:
N
(13)
(14)
t βˆ *
= N −1/ 2 ∑ t βˆ *
t βˆ *
= ( βˆ *
GFM
FM , i
i =1
FM ,i
FM ,i
⎛ ˆ −1 T
2⎞
− β 0 )⎜ Ω
11i ∑ ( x it − x i ) ⎟
t =1
⎝
⎠
1/ 2
In similar way, a between-dimension, group-mean panel DOLS estimator can be
constructed as follows. First, augmenting the cointegrating regression with lead and
lagged differences of the regressor to control for the endogenous feedback effect. Then,
the DOLS regression becomes:
14
(13)
yit = α i + β i * xit +
Ki
∑γ
k =− K
ik
∆pit − k + µ it*
From this regression, Pedroni constructs the group-mean panel DOLS estimator as:
(14)
−1
⎡ −1 N ⎛ T
⎞ ⎛ T ~ ⎞⎤
*
ˆ
β GD = ⎢ N ∑ ⎜ ∑ zit zit′ ⎟ ⎜ ∑ zit sit ⎟⎥
i =1 ⎝ t =1
⎠ ⎝ t =1
⎠⎥⎦1
⎢⎣
Where:
zit is 2( K + 1) ×1 vector of regressors zit = ( xit − xi , ∆xit − K ,..., ∆xit + K ) , ~
sit = sit − si , and
the subscript 1 outside brackets indicates that it is taking only the first vector element to
obtain the pooled slope coefficient.
Because the expression that follows the summation over the i is identical to the
conventional time series DOLS estimators, the between-dimension estimator can be
N
*
constructed simply as βˆGD
= N −1 ∑ i =1 βˆ D* ,i , where βˆ D* ,i is the conventional DOLS
estimator, applied to ith panel member. Likewise, the associated t-statistics for the
between-dimension estimator can be constructed as:
N
(15)
t βˆ * = N −1/ 2 ∑ t βˆ *
GD
i =1
D ,i
1/ 2
(16)
T
⎛
⎞
t βˆ * = ( βˆ D* ,i − β 0 ) ⎜ σˆ i−2 ∑ ( xit − xi ) 2 ⎟
D ,i
t =1
⎝
⎠
5. ESTIMATION RESULTS
The estimator used was the Fully Modified OLS (FMOLS) between-dimension, because
it lets a greater flexibility in heterogeneity presence in cointegration vector; another
advantage of between-dimension estimator is that the estimated value has an easier
interpretation in the case that cointegration vectors are heterogeneous. Specifically,
estimating a value for between-dimension estimator can be interpreted as the average
value of cointegration vector.
Estimations are presented in Figures 5 and 6: assuming a model with constant returns to
scale in physic capital’s share, the coefficient is 0.39, while assuming a model without
such specification the coefficient calculated is 0.41, and labor’s share in output is 0.49,
an important aspect is that the sum of two coefficients is 0.9, indicating diminishing
returns to scale.
15
FIGURE 5.- ESTIMATION OF THE EQUATION (1’’), ASSUMPTION CONSTANT RETURNS TO
SCALE
α
t-statistic
0.39
13.72*
14 countries, 1960-2002
FMOLS
Group-Mean Panel
Between-dimension
* Indicate 1% rejections levels for Ho: α =0
FIGURE 6.- ESTIMATION OF THE EQUATION (2’), WITHOUT ASSUMPTION CONSTANT
RETURNS TO SCALE
14 countries, 1960-2002
FMOLS
Group-Mean Panel
Between-dimension
α
t-statistic
β
t-statistic
0.41
10.56*
0.49
5.19*
* Indicate 1% rejections levels for Ho: α =0
The estimation of labor’s share in production function with constant returns to scale is
0.61, similar to three different measures found by Bernanke y Gurkaynak (2001), which
are in average 0.64, 0.61 and 0.62 for several Latin American countries; however, the
share in the second model where constant returns to scale are not assumed is 0.49, lower
than three measures mentioned (see appendix 4). It’s useful to indicate that such authors
have calculated values for several countries of Latin America that are not equal to the
sample used in this work.
6. CONCLUDING REMARKS
This work aimed to estimate the capital’s share on output for fifteen countries of LatinAmerica in 1960-2002 period, where the Fully Modified OLS (FMOLS) estimator
(developed by Pedroni, 2001) was used in a panel cointegration framework.
The results show that the share of capital in output is 0.39, assuming a Cobb-Douglas
production function with constant returns to scale, while assuming a model without
specification about the returns to scale; it is found a share of 0.41. The results are
similar to the measures found by Bernanke and Gurkaynak (2001) where estimation of
physical capital’s share is calculated through 1-share of labor and on Elias’s work
(1992) in his study of seven Latin American economies, where he finds a share of
physical capital of 0.39, using OLS estimator in a panel data framework. Furthermore
our results differ a bit of Senhadji’s work (2000) where the share of physical capital for
Latin America found is of 0.52 in level analysis and 0.62 in first difference analysis.
16
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19
APPENDIX 1.- STOCK OF PHYSICAL CAPITAL PER WORKER (K/L) AND OUTPUT PER WORKER (Y/L)
Argentina
Bolivia
K/L
Y/L
Brasil
K/L
Y/L
Y/L
1999
2002
2002
1999
1996
1993
1990
1987
1984
2002
1999
1996
1993
1990
1987
1984
1981
K/L
Y/L
20
Y/L
2002
1999
3,1
1996
3,3
3,15
1993
2002
1999
1996
1993
1990
1987
1984
1981
1978
3,35
1990
3,4
3,5
3,2
1987
3,6
3,25
1984
3,45
1981
3,5
3,7
3,3
1975
3,55
1972
3,8
3,35
1969
3,6
1966
3,9
1975
Y/L
3,4
1963
3,65
1960
3,85
3,8
3,75
3,7
3,65
3,6
3,55
3,5
3,45
3,4
3,7
1972
3,8
3,75
3,7
3,65
3,6
3,55
3,5
3,45
3,4
3,35
Honduras
Guatemala
1969
Y/L
K/L
4
1966
1978
1975
2002
1999
1996
1993
1990
3,4
1978
K/L
1987
1984
1981
1978
1975
1972
1969
1966
1963
1960
3,45
1972
3,5
1969
3,55
1966
3,6
1963
3,65
1960
3,7
1963
3,5
Ecuador
4,28
4,26
4,24
4,22
4,2
4,18
4,16
4,14
4,12
4,1
3,75
1960
1981
K/L
3,8
K/L
1978
3,6
Colombia
3,4
1996
3,7
1975
3,5
3,8
1972
2002
1999
1996
1993
1990
1987
1984
1981
1978
1975
1972
1969
1966
1963
3,6
3,9
1969
3,7
4
1966
3,8
4,1
1963
3,9
4,2
1960
4
4,2
4,15
4,1
4,05
4
3,95
3,9
3,85
3,8
3,75
Y/L
Chile
4,55
4,5
4,45
4,4
4,35
4,3
4,25
4,2
4,15
4,1
4,05
4,1
1960
4,6
4,5
4,4
4,3
4,2
4,1
4
3,9
3,8
3,7
3,6
1993
2002
1999
1996
1993
1990
1987
1984
1981
1978
1975
1972
1969
1966
1963
K/L
1990
3,2
1987
3,25
3,6
4
1984
3,65
4,05
1981
3,3
4,1
1978
3,35
3,7
4,15
1975
3,4
3,75
4,2
1972
3,45
3,8
1969
3,85
1966
3,5
4,3
1963
3,9
1960
4,35
4,25
1960
4,85
4,8
4,75
4,7
4,65
4,6
4,55
4,5
4,45
4,4
3,05
APPENDIX 1.- STOCK OF PHYSICAL CAPITAL PER WORKER (K/L) AND OUTPUT PER WORKER (Y/L)
(CONT.)
K/L
4,05
4
3,95
3,9
3,85
3,8
3,75
3,7
3,65
3,6
3,55
4,4
3,8
4,2
3,7
4
3,6
3,8
3,5
3,6
3,4
3,4
K/L
Uruguay
4,7
4,5
4,15
4,65
4,6
4,05
4,55
Y/L
K/L
1999
1996
1993
1990
2002
2002
1999
1996
1993
2002
1999
1996
1993
1990
1987
1984
1981
1975
1972
1969
2002
1999
1996
1993
1990
1987
4,35
1984
4,4
3,85
1981
3,9
4,25
1978
4,3
1966
4,45
3,95
1975
1990
4,5
4
4,35
1963
4,4
4,1
1960
4,45
1972
Y/L
Venezuela
4,2
1969
1987
3.1
Y/L
4,55
1966
1987
3.2
1960
2002
1999
1996
1993
1990
3,6
3.3
1978
K/L
1987
1984
1981
1978
1975
1972
1969
1966
1963
3,65
1984
4,2
3.4
1981
3,7
3.5
1978
3,75
1975
4,25
3.6
1972
3,8
1969
4,3
3.7
1966
3,85
3.8
1963
4,35
1960
1984
4.15
4.1
4.05
4
3.95
3.9
3.85
3.8
3.75
3.7
3,9
1963
Y/L
Republica Dominicana
3,95
1960
1981
K/L
Perú
K/L
1978
Y/L
4,4
4,15
1975
1972
1969
3,2
1966
3,3
3
1963
3,2
1960
2002
1999
1996
1993
1990
Paraguay
1987
1984
1981
1978
1975
1972
1969
1966
1963
1960
México
4,6
4,55
4,5
4,45
4,4
4,35
4,3
4,25
4,2
4,15
Y/L
APPENDIX 2A.- PANEL UNIT ROOT TEST: LEVIN, LIN AND CHIU
14 countries, 1960-2002
coefficient
t-value
Variables:
Y
-0.04111
-4.242
K
-0.01176
-4.640
L
-0.00462
-4.367
Y/L
-0.04848
-4.118
K/L
-0.01393
-4.439
Note: Test was taken with a constant and two lags.
Null Hypothesis: all the time series in the panel are unit root.
21
t-star
P>t
-0.94454
-2.05636
-2.40141
-0.33056
-1.32573
0.1724
0.0199
0.0082
0.3705
0.0925
4,2
4,15
4,1
4,05
4
3,95
3,9
3,85
3,8
3,75
3,7
APPENDIX 2B.- PANEL UNIT ROOTS TEST: IM, PESARAN AND SHIN
14 countries, 1960-2002
t-bar
10%
5%
Variables:
Y
-1.157
-1.810
-1.890
K
-1.194
-1.810
-1.890
L
-1.080
-1.810
-1.890
Y/L
-1.495
-1.810
-1.890
K/L
-1.445
-1.810
-1.890
Note: Test was taken with a constant and two lags.
Null Hypothesis: all the time series in the panel are unit root.
1%
Psi[t-bar]
P-value
-2.050
-2.050
-2.050
-2.050
-2.050
1.405
1.252
1.730
-0.007
0.201
0.920
0.895
0.958
0.497
0.580
APPENDIX 2C.- PANEL UNIT ROOT TEST: HADRI
14 countries, 1960-2002
Zτ
P-value
61.453
55.440
20.475
70.734
59.492
23.050
65.101
61.721
21.310
59.260
54.252
19.711
69.890
64.639
22.832
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
P-value
Zµ
Variables:
Y (a)
95.984
0.0000
Y (b)
94.196
0.0000
Y (c)
32.342
0.0000
K (a)
98.790
0.0000
K (b)
97.468
0.0000
K (c)
33.133
0.0000
L (a)
101.469
0.0000
L (b)
101.156
0.0000
L (c)
34.140
0.0000
Y/L (a)
69.127
0.0000
Y/L (b)
60.484
0.0000
Y/L (c)
22.875
0.0000
K/L (a)
87.728
0.0000
K/L (b)
65.905
0.0000
0.0000
K/L (c)
29.017
(a) Homoskedastic disturbances across units.
(b) Heteroskedastic disturbances across units.
(c) Controlling for serial dependence in errors (lags truncation = 2).
Null Hypothesis: all 14 time series in the panel are stationary processes.
APPENDIX 3.- THE PEDRONI PANEL COINTEGRATION TESTS
Pedroni (1999) developed seven statistics. In each case, the statistics can be constructed using
the residuals of the cointegrating regression of the equation (7) in combination with various
nuisance parameter estimators which can be obtained from these.
1. Panel ν -Statistics:
T N
2. Panel ρ -Statistics: T
2
3. Panel t -Statistics:
2
3/ 2
Zνˆ N ,T ≡ T N
N Z ρˆ N ,T −1
2
3/ 2
⎛ N T ˆ−2 2 ⎞
⎜ ∑∑ L11i eˆí ,t −1 ⎟
⎝ i =1 t =1
⎠
⎛ N T −2 2 ⎞
ˆ
≡ T N ⎜ ∑∑ Lˆ11
i eí ,t −1 ⎟
⎝ i =1 t =1
⎠
N T
⎛
⎞
−2 2
ˆ
Z t N ,T ≡ ⎜ σ% N2 ,T ∑∑ Lˆ11
i eí ,t −1 ⎟
i =1 t =1
⎝
⎠
(non-parametric)
22
−1/ 2 N
−1
−1 N
T
∑∑ Lˆ
i =1 t =1
∑∑ Lˆ
i =1 t =1
T
−2
11i
−2
11i
(eˆí2,t −1∆eˆi ,t − λˆi )
(eˆí2,t −1∆eˆi ,t − λˆi )
4. Panel t -Statistics:
Z
*
t N ,T
N T
⎛
⎞
−2 2
ˆ
≡ ⎜ s%N2 ,T ∑∑ Lˆ11
i eí ,t −1 ⎟
i =1 t =1
⎝
⎠
−1/ 2 N
T
∑∑ Lˆ
−2 *
11i í ,t −1
i =1 t =1
eˆ
∆eˆi*,t
(parametric)
5. Group ρ -Statistics: TN
6. Group t -Statistics:
−1/ 2
N
⎛ T
⎞
Z% ρˆ N ,T −1 ≡ TN −1/ 2 ∑ ⎜ ∑ eˆí2,t −1 ⎟
i =1 ⎝ t =1
⎠
−1 T
∑ (eˆ
i , t −1
t =1
N
T
⎛
⎞
N −1/ 2 Z%t N ,T ≡ N −1/ 2 ∑ ⎜ σˆ i2 ∑ eˆí2,t −1 ⎟
i =1 ⎝
t =1
⎠
∆eˆi ,t − λˆí )
−1/ 2 T
∑ (eˆ
i ,t −1
t =1
∆eˆi ,t − λˆí )
(non-parametric)
7. Group t -Statistics:
N
−1/ 2
N
⎛ T
⎞
Z%t* N ,T ≡ N −1/ 2 ∑ ⎜ ∑ sˆi*eˆí2,t −1 ⎟
i =1 ⎝ t =1
⎠
−1/ 2 T
∑ eˆ
t =1
*
i ,t −1
∆eˆi*,t
(parametric)
TABLE 3A.- VARIABLES K/L AND Y/L
14 countries, 1960-2002
With heterogeneous trends
Without heterogeneous trends
Test one
Test two
Test three
Test four
-0.67341
1.26651
0.71967
-1.37700
-0.41190
1.06934
0.53329
-0.14295
Test five
Test six
Test seven
2.25129
1.56033
-1.09618
2.15201
1.46762
0.17065
Note: Under the alternative hypothesis, large positive values imply that the null of no cointegration is
rejected in the first test. For each of the other six test statistics, large negative values imply that the null of
no cointegration is rejected.
TABLE 3B.- VARIABLES Y, K AND L
14 countries, 1960-2002
With heterogeneous trends
Without heterogeneous trends
Test one
Test two
Test three
Test four
0.66787
1.35507
1.32013
-1.30112
1.63489
0.32212
-0.03876
-1.82953
Test five
Test six
Test seven
2.39971
2.15324
-1.41494
1.60787
0.85183
-1.84919
Note: Under the alternative hypothesis, large positive values imply that the null of no cointegration is
rejected in the first test. For each of the other six test statistics, large negative values imply that the null of
no cointegration is rejected.
23
APPENDIX 4.- SHARE OF CAPITAL INCOME IN GDP, 1940-1985
Year
Argentina
1940
0.580
1941
1942
1943
1944
1945
0.577
1946
1947
0.559
1948
0.522
1949
0.466
1950
0.503
1951
0.526
1952
0.502
1953
0.503
1954
0.492
1955
0.523
1956
0.547
1957
0.562
1958
0.556
1959
0.623
1960
0.620
1961
0.592
1962
0.602
1963
0.610
1964
0.611
1965
0.593
1966
0.563
1967
0.545
1968
0.556
1969
0.567
1970
0.542
1971
0.535
1972
0.573
1973
0.531
1974
0.553
1975
0.566
1976
0.721
1977
0.732
1978
0.704
1979
0.678
1980
0.629
1981
1982
1983
1984
1985
Mean
0.574
Source: Elias (1992)
Brazil
0.472
0.474
0.475
0.481
0.491
0.492
0.458
0.456
0.453
0.443
0.426
0.422
0.436
0.442
0.426
0.410
0.414
0.422
0.421
0.414
0.435
0.423
0.428
0.428
0.592
0.616
0.621
0.462
Chile
0.487
0.524
0.509
0.535
0.479
0.408
0.567
0.548
0.558
0.581
0.584
0.552
0.542
0.544
0.511
0.519
0.501
0.559
0.547
0.556
0.5440
0.553
0.568
0.656
0.528
0.520
0.538
Colombia
0.640
0.638
0.645
0.638
0.645
0.628
0.644
0.653
0.647
0.644
0.633
0.621
0.605
0.598
0.617
0.608
0.606
0.596
0.605
0.589
0.589
0.589
0.600
0.623
0.637
0.590
0.653
0.650
0.634
0.538
0.528
0.522
0.522
0.547
0.609
24
Mexico
0.749
0.733
0.724
0.738
0.757
0.739
0.724
0.714
0.721
0.687
0.721
0.697
0.689
0.675
0.680
0.670
0.666
0.679
0.670
0.662
0.657
0.660
0.655
0.655
0.653
0.659
0.607
0.609
0.585
0.557
0.589
0.610
0.683
0.688
0.684
0.676
Peru
0.638
0.692
0.706
0.707
0.707
0.690
0.685
0.675
0.679
0.669
0.676
0.663
0.661
0.634
0.664
0.653
0.657
0.606
0.621
0.606
0.580
0.583
0.612
0.565
0.604
0.616
0.651
0.682
0.672
0.659
0.649
0.641
0.675
0.718
0.653
Venezuela
0.520
0.539
0.559
0.536
0.543
0.557
0.569
0.601
0.558
0.514
0.500
0.502
0.520
0.505
0.547
0.544
0.539
0.526
0.588
0.573
0.574
0.576
0.580
0.611
0.597
0.587
0.577
0.545
0.581
0.573
0.568
0.558
0.542
0.608
0.582
0.557