WPS3808
Network Effects of the Productivity of Infrastructure
in Developing Countries*
Christophe Hurlin **
Abstract
Using panel data models we examine the threshold effects of the productivity of infrastructure
investment in developing countries. We consider various specifications of an augmented
production function that allow for endogenous thresholds. More precisely, these specifications are
tested in a panel threshold regression. Our main robust result is the presence of strong threshold
effects in the relationship between output and private and public inputs. Whatever the transition
mechanism used, the testing procedures lead to strong rejection of the linearity of this
relationship. In particular, the productivity of infrastructure investment generally exhibits some
network effects. When the available stock of infrastructure is very low, investment in this sector
has the same productivity as non-infrastructure investment. On the contrary, when a minimum
network is available, the marginal productivity of infrastructure investment is generally largely
greater than the productivity of other investments. Finally, when the main network is achieved, its
marginal productivity becomes similar to the productivity of other investment.
Keywords : Infrastructure, Threshold Panel Regression Models.
J.E.L Classi.cation : C82, E22, E62.
World Bank Policy Research Working Paper 3808, January 2006
The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the
exchange of ideas about development issues. An objective of the series is to get the findings out quickly,
even if the presentations are less than fully polished. The papers carry the names of the authors and should
be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely
those of the authors. They do not necessarily represent the view of the World Bank, its Executive Directors,
or the countries they represent. Policy Research Working Papers are available online at
http://econ.worldbank.org.
*I would like to thank Santiago Herrera for his support and his comments on a previous version of this work. I
also thank Mohamed Belkir for his comments.
**LEO, University of Orléans. Rue de Blois. BP 6739. 45067 Orléans Cedex 2. France. Email address:
christophe.hurlin@univ-orleans.fr.
1 Introduction
Most of the public investments, and particularly public investments in infrastructure,
are generally devoted to the construction of networks. It is particularly clear when
we consider public investments in roads, railways, telecommunications, electricity, wa-
ter and sewer systems, but it is also true when we consider some residential public
investments devoted for instance to educational buildings. In this case, as noted by
Romp and De Haan (2005), the internal composition of the stock matters, since the
marginal productivity of one link depends on the capacity and con...guration of all the
links in the network. Using measures of the total stock, hence, may allow estimation
of the average marginal product of roads in the past. However these estimates may
not be appropriate for evaluating the marginal product of additional roads today. This
argument was in particular used by Fernald (1999) to asses the link between public
capital and productivity in the road sector in the United States. To quote the title of
his paper, his evaluation based on industry data does not support the idea that public
investment o¤ers a continuing and neglected route to prosperity. More precisely, his
estimations do not allow rejecting the hypothesis that roads now o¤er a normal (or
even zero) rate of return. He concludes that "the data seem consistent with a story in
which the massive road-building of the 1950's and 1960's o¤ered a one-time boost to the
level of productivity, rather than a path to continuing rapid growth in productivity. This
conclusion - that roads were exceptionally productive before 1973 but not exceptionally
productive at the margin - is consistent with simple network argument. In particular,
building an interstate network might be very productive; building a second network may
not"(Fernald, 1999, page 621). This network character is likely to be generalized to
the main components of public investments not only in the United States, but also in
most of developing countries.
What is the main implication of this network dimension for the evaluation of the
productivity of infrastructure? It clearly implies that the relationship between the
output and the level of public capital stock (or infrastructure stocks) is strongly non
linear. More precisely, this relationship may depend on the level of infrastructure
actually available and may be represented as a threshold model. For instance, let us
admit, as Fernald (1999) does, that the construction of the network boosts substantially
the total factor productivity and the output, but when the construction of the network
is completed, the infrastructure investments may be not exceptionally productive at
the margin. Thus, the marginal productivity of these investments is not identical
given the level of stock actually available in the country: a low level of infrastructure
stock (relatively to the population or to the workforce for instance) indicates that the
construction of the network is not completed and implies a high productivity, and a
high level of stocks implies, if the network is built, a null or low marginal productivity.
These properties clearly correspond to the de...nition of a threshold regression model:
"threshold regression models specify that individual observations can be divided into
classes based on the value of an observed variable"(Hansen, 1999, page 346).
2
In this paper, we propose to use these threshold regression models in order to
take into account this original dimension in the estimation of the rates of return on
public capital stocks, i.e. the potential presence of threshold e¤ects. Applying the
so-called production function approach introduced by Aschauer (1989), we consider
various speci...cations of public capital stock augmented production functions. At this
stage, a remark has to be made. So far, we indi¤erently used the terms of public capital
and infrastructure. However, the notion of infrastructure does not fully correspond to
the concept of public sector investment expenditure. This distinction is particularly
important when it comes to interpret the threshold e¤ects. Indeed, it would be naïve to
directly interpret the threshold e¤ects in the productivity of public capital as network
e¤ects: the public investments are so heterogeneous, that the level of public investments
or the public capital stocks can not be used to reveal information on the completion of
the main infrastructure networks. For this reason, in this paper, we only consider the
threshold e¤ects of the productivity of the infrastructure investments.
More precisely, we propose to re-evaluate the marginal productivity of infrastructure
stocks using threshold regression models, in which the threshold variable may be de...ned
as the level of the infrastructure stock actually available. The main concern with these
models is that they require an important number of observations to be estimated since
they depend on an important number of parameters. This concern is particularly
relevant in our context, since the time dimension of the series of infrastructure stocks is
generally not su¢ cient to estimate such models, even if we consider only two regimes1.
One solution is to use panel data as it is generally done in the literature devoted to
linear representations of the productivity of infrastructure. In a seminal paper, Hansen
(1999) proposed the ...rst procedure to estimate and to test the threshold e¤ects in non-
dynamic panels. His Panel Threshold Regression (PTR) model allows for dividing the
individual observations into classes according to an observable variable. In this case,
the time series and cross sections are used in order to identify the regimes. Based on
his procedure, it is then possible to test and to estimate the threshold e¤ects in the
marginal productivity of infrastructures in a panel of countries, without assuming the
homogeneity of the aggregated production function. This model assumes a transition
from one regime to another based on the value of a threshold variable, the infrastructure
stock for instance: in a model with two regimes, if the threshold variable is below a
certain value, called the threshold parameter, the productivity is de...ned by one model,
and it is de...ned by another model if the threshold variable exceeds the threshold
parameter.
It is important to note that the existence of threshold e¤ects in the productivity
of infrastructure is compatible with the three major criticisms addressed to the linear
speci...cations of the production function in the huge literature devoted to this topic (see
Gramlich, 1994; Sturm, 1998 or Romp and De Haan, 2005 for a survey). Indeed, when
it comes to estimate the rates of return on infrastructure (or public capital) with an
augmented production function in a panel, three major problems are generally raised.
The ...rst one is the potential reverse causation. If public investments depend on income,
1This is the main reason why these models have not been used even in the case of OECD countries.
To the best of our knowledge no study devoted to the productivity of public capital and based on
threshold models has yet been proposed.
3
it implies a feedback from income to the capital stocks. Consequently, the linear regres-
sion of the output on the public and private factors does not allow to directly identify
the parameters of the production function. Several solutions have been proposed in
the literature in order to take into account this problem of reverse causation. Canning
(1999) and Canning and Bennathan (2000) argue that the use of panel estimates under
various assumptions allows to identify the long run production function relationship.
Another solution consists in estimating a system of simultaneous equations: one equa-
tion for the production function and another equation for the relationship between the
public capital stock and the production (Demetriades and Mamuneas, 2000). Finally,
the reverse causation problem can be tackled with an instrumental variable approach
or a generalized method of moments. It is for instance the case in Finn (1993), Holtz-
Eakin (1994), Baltagi and Pinnoi (1995), Ai and Cassou (1995), Otto and Voss (1998)
and more recently in Calderon and Serven (2004). But the reverse causation issue can
also be interpreted as the consequence of threshold e¤ects. For instance, let us assume
that the true data generating process of the marginal productivity is a threshold model
as suggested by Fernald (1999). The infrastructure investments enhance the output
until the completion of the main networks, and once they are ...nished it is mainly the
output which causes the changes in the infrastructure capital stocks. In this case, the
use of a linear speci...cation leads to an estimated correlation which takes into account
both the inuence of the infrastructures on the output in the ...rst period and the inu-
ence of the output on the infrastructure stock in the second period. On the contrary,
the use of a threshold model based on the value of the capital stock per capita for
instance, makes it possible to identify the inuence of the infrastructures on the total
factor productivity in the ...rst period.
The second major problem raised in the literature is the non stationarity and the
non cointegration of the data used in the augmented production function (Tatom, 1991;
Sturm and Haan, 1995; Crowder and Himarios, 1997). In time series, it is generally
recognized that unit root tests based on linear speci...cations (ADF, KPSS etc.) are
likely to conclude to the non stationarity of series when the true data generating model
is a threshold model (SETAR, STAR etc.). Even if similar studies have not been done
with panel unit root tests, it is obvious that the ...rst generation tests, as the Im, Pesaran
and Shin's test (2003) based on an average of individual ADF statistics, are likely to
yield the same kind of results.
The last major problem, which is speci...c to the panel estimates, is the potential
heterogeneity of the parameters of the production function and more particularly the
heterogeneity of the elasticities of output with respect to infrastructure (or public cap-
ital) stocks. It is common, both in cross section and in panel data studies, to assume
that the parameters are common across countries. However, this assumption may raise
important problems when the countries of the panel are very di¤erent, as it is the case
in our sample of developing countries. Consequently, the studies based on a production
approach are generally based on speci...cations with ...xed or random individual e¤ects
(Evans and Karras, 1994; Holtz-Eakin; 1994). But, to the best of our knowledge, few
studies in the literature allow the other parameter of the augmented production function
4
to vary across countries2. Canning (1999) or Canning and Bennathan (2000) investi-
gate the possible heterogeneity of the production function by splitting their sample into
two groups of countries based on their levels of income per worker in a baseline year.
Their results based on a Cobb-Douglas production function show that the coe¢ cients
on the infrastructure terms in poorer countries are very small, and statistically insignif-
icant, but they remain large, and signi...cant, in richer countries. They conclude that
"infrastructure in the poorer countries appears to have the same e¤ectiveness in raising
output as other types of physical capital while having a greater e¤ectiveness than other
types of capital in richer countries"(Canning and Bennathan, 2000, page 13). In this
case, the cross-country heterogeneity of the production technology can be interpreted
as the consequence of threshold e¤ects. The idea is very simple: at each date in the
threshold model, the countries are divided into a small number of classes with the same
elasticities according to an observable variable. This threshold variable can be de...ned
for instance as the level of income per capita according to the decomposition used by
Canning and Bennathan (2000). But, the main di¤erence is that the heterogeneity of
the production technology is then endogenously determined by the threshold model and
not speci...ed ex-ante by splitting the sample into two or three groups of countries. To
sum it up, the existence of a nonlinearity and more speci...cally of threshold e¤ects in
the productivity of public capital and infrastructure stocks is largely compatible with
the main empirical observations done in the literature.
The paper is organized as follows. In a second section, we explore the threshold
e¤ects in the productivity of infrastructures in road, electricity, telephone and railways
sectors. For that we consider various panel threshold speci...cations of an infrastructure
augmented production function. In a third section the data are presented and we report
the results obtained in linear speci...cations in order to get a benchmark. In sections
four and ...ve, we test the hypothesis of network e¤ects and the inuence of the income
heterogeneity on the production technology. A last section concludes.
2 The Productivity of Infrastructure: Toward a Thresh-
old Speci...cation
The basis of our empirical approach is exactly the same as that used by many authors
since the seminal paper of Aschauer (1989), and more recently by Canning (1999),
Canning and Bennathan (2000) or Calderon and Serven (2004) for developing countries.
It consists in estimating the parameters of an infrastructure augmented production
function. We follow the literature in adopting a Cobb-Douglas speci...cation of the
production function. If we consider a country i = 1;::;N at a time t = 1;:;T; we
assume that:
Yit = AiKitHitXitL1it Vit (1)
where Yit is the aggregate added value, Kit is the physical capital, Hit is human capital,
Xit is infrastructure and Vit is an error term. As usual in this kind of literature, we
2With random coe¢ cient models for instance (Swamy, 1970).
5
assume that the infrastructure services are proportional to the infrastructure capital
stock. Moreover, we assume constant returns to scale, so that the sum of exponents is
one. Dividing through by Lit and taking logs, we have the following expression:
yit = ai + kit + hit + xit + vit (2)
where vit = log (Vit) and capital stocks and output are in log per worker terms. The
individual ...xed e¤ects ai capture all the timeless components of the total factor pro-
ductivity. It is also possible to include in this linear speci...cation, some time e¤ects
to capture the common factors in the total factor productivity. The equation (2) cor-
respond exactly to the model estimated by Canning (1999), Canning and Bennathan
(2000) or Calderon and Serven (2004). As noted by these authors it is di¢ cult to inter-
pret directly the parameters of equation (2), since infrastructure capital appears twice,
once its own but also as a part of aggregate capital Kit. Consequently, the parameter
cannot be interpreted as the infrastructure elasticity. More precisely, Canning (1999)
shows "that in this case the parameter captures the extent to which the productivity
of infrastructure exceeds (if > 0) or falls short (if < 0) the productivity of non
infrastructure capital" (Calderon and Serven, 2004, page 98). Thus, the elasticity of
output with respect to infrastructure is not constant and depends on the ratio of cap-
ital stocks. However, as noted by Calderon and Serven (2004), because infrastructure
stocks typically account for relatively small portions of overall capital stock, the di¤er-
ence between the genuine elasticity evaluated around the sample mean and the naïve
estimate should be fairly modest in practice.
As it was previously mentioned, in this study we propose to consider exactly the
same framework as that studied in this literature, except the fact that we introduce a
non-linearity in order to test and to match the network dimension of infrastructure. In
order to take into account this speci...city of infrastructure, a solution consists in adopt-
ing a Panel Threshold Regression (PTR) model similar to that proposed by Hansen
(1999):
ai + 1 it
k +
1hit + 1xit + "it if qit
yit = ( (3)
ai + 2 it
k +
2hit + 2xit + "it if qit >
where qit denotes a threshold variable and denotes a threshold parameter. This
model can be rewritten as:
yit = ai + 01WitI(qit ) + 02WitI(qit> + "it
) (4)
where j= j j j 0for j = 1;2 and Wit = (kit hit xit)0, and where I(:) is the indicator
function. The error "it is assumed to be independent and identically distributed with
mean zero and ...nite variance 2: Two remarks must be made here. In this model the
observations are divided into two regimes depending on whether the threshold variable
qit is smaller or larger than the threshold parameter . No constraint is imposed on the
choice of the threshold variable except the fact that it cannot be the contemporaneous
endogenous variable and it cannot be time independent. We will discuss further the
choice of this threshold variable. The second remark is that in Hansen (1999) the
individual e¤ects ai are not di¤erent in the two regimes. It would be possible to consider
6
a constant or N individual constant speci...c to each regime but it would largely increase
the number of parameters of the model. Thus, the regimes are distinguished by di¤ering
elasticities 1 and 2. The elasticities of output with respect to the three inputs (total
capital, infrastructure and human capital) are assumed to be regime dependent.
Naturally, a general speci...cation can be studied with more than two regimes. The
procedure of estimation proposed by Hansen allows considering a model with K regimes,
however in this study we limit our investigation to models with at the most four regimes
(three threshold parameters). For example, threshold models with respectively three
and four regimes (respectively two and three threshold parameters) take the form:
yit = ai + 01Wit I(qit + + + "it (5)
1) 02Wit I( 12)
yit = ai + 01Wit I(qit +
1) 02 Wit I( 1 3)
where the threshold parameters j are sorted, 1 < :: < K : In all the cases, the
parameters j; j = 1;;::K are estimated according to the same simple least square
sequential procedure as that used for the standard STAR or SETAR models for times
series. If we consider the single threshold model (equation 4), for a given value of the
threshold parameter , the slope coe¢ cients 1 and 2can be estimated by OLS. Let us
denote
of it is possible to compute the sum of squared errors, denoted S1 ( ):
b1 ( ) and b2 ( ) ; the corresponding estimates. Thus, conditionally to a value
XXb
N T
S1 ( ) = "2it ( ) (7)
i=1 t=1
The threshold parameter is then estimated by minimizing the sum of squared S1 ( ).
b = ArgMinS1 ( ) (8)
2
Since this sum of squared residuals depends on only through the indicator function,
it is a step function with at most NT steps, with the steps occurring at distinct values
of qit in the sample. Thus, the minimization problem can be reduced to searching
over values of equalling at the most distinct values of qit in the sample. Given the
value of the estimate ; it is then possible to get the estimates of the elasticities in
the regimes, i.e.
it was stressed bybHansen (1999), it is undesirable for a threshold
1 b
b and a
which sorts too few observations into one or other regime. That is why we consider an
optimization domain which assures that a minimal percentage of thebobservations lie
b2 b ; and the estimates of individual e¤ects bi: As
to be selected
in each regime. More precisely, the minimization program (8) can be solved by a direct
search over the values of equalling the at most NT distinct values of the threshold
variable qit in the sample. If we sort the NT distinct values of the observations on
qit, the previous constraint implies to eliminate the smallest and largest values. The
remaining values constitutes the set of values of which can be searched for : In our
7 b
application, smallest and largest 5% values are eliminated. The same kind of procedure
is used for models with three or four regimes. Thus, for each model, at least 5% of the
total of the NT observations is available to estimate the elasticities in each regime.
In this threshold model, there are two main problems. The ...rst one consists of
testing the number of regimes or equivalently in testing the threshold speci...cation.
The second issue consists of choosing the threshold variable. Let us assume that the
threshold variable is known. If one comes to test whether the threshold e¤ect is sta-
tically signi...cant in the model with two regimes (equation 4), the null hypothesis is
H0 : 1 = 2: This null hypothesis corresponds to the hypothesis of no threshold e¤ect.
Under H0 the model is then equivalent to a linear model (equation 2). This hypothesis
could be tested by a standard test. If we note S0 the sum of squared of the linear
model, the approximate likelihood ratio test of H0 is based on:
S0 S1
F1 = b (9)
where b2 denotes a convergent estimate of 2
the threshold parameter is not identi...ed. Consequently, the asymptotic distribution
b2
: The main problem is that under the null
of F1 is not standard and in particular does not correspond to a chi-squared distribution.
This issue has been largely studied in the literature devoted to the threshold models,
notably since the seminal paper of Hansen (1996). One solution consists in simulating
by Bootstrap the asymptotic distribution of the statistic F1. Hansen (1996) shows that
a bootstrap procedure attains the asymptotic distribution, so p-values constructed from
the bootstrap are asymptotically valid. As proposed by Hansen (1999) in the context
of panel models, we use bootstrap simulations3 to compute the critical values of the
distribution of the statistics of tests on the number of thresholds.
The same kind of procedure can be applied in general models (equations 5 or 6) in
order to determining the number of thresholds. If the p-value associated to F1 leads to
rejects the linear hypothesis, we then discriminate between one and two thresholds. A
likelihood ratio test of one threshold versus two thresholds is based on the statistic
S1
F2 = b S2
b1;b2 b2 b1;b2 (10)
where
tion 5),b1and Sb2
and 2denote the threshold estimates of the model with three regimes (equa-
denotes the corresponding residual sum of squares. The hy-
pothesis of one threshold is rejected in favor of two thresholds if F2 is larger than the
critical value of the non simulated distribution. The corresponding asymptotic p-value
can be approximated by bootstrap simulations (Hansen, 1999). If the model with two
thresholds (three regimes) is accepted, we propose to test the hypothesis of two thresh-
olds (three regimes) against the alternative of three thresholds (four regimes). The
corresponding likelihood ratio statistic, denoted F3:; is de...ned as:
S2
F3 = b1;b2 S3
(11)
3All the Matlab codes are available upon request.b2 b1;b2;b3
8
where S3 b1;b2;b3 denotes the residual sum of squares of the model with four regimes
and three threshold parameters. In models with m + 1 > 2 regimes, the estimates of
the m threshold parameters can be done sequentially by using the estimated threshold
parameters obtained in the model with m regimes as suggested by Bai and Perron (1998)
in the context of multiple changepoint models. In this paper, we limit our analysis to
a model with 4 regimes at the maximum given the computational cost of such models.
Thus, a sequential procedure based on F1, F2 and F3 allows to determining the number
of regimes4.
The second issue consists of determining the threshold variable. Few technical
constraints are imposed to the choice of the threshold variable. The only constraint is
that the threshold variable can not be time invariant. Therefore, this choice is mainly
an economic issue. If we want to assess the idea that the infrastructure investments have
a network character, it implies that the threshold variable should be an indicator of the
completion of the network. In this perspective, a natural candidate for the threshold
variable is the existing level of available infrastructure. According to Fernald, a low level
of capital (relatively to the population or to the workforce for instance) indicates that
the construction of the network is not completed and may imply a high a productivity
of stocks, and a high level of stocks may imply, if the network is built, a null or low
marginal productivity. Two choices are then possible: the threshold variable can be
de...ned as the level of infrastructure stock or as the level infrastructure stock per worker
(or per capita). The second variable allow to avoid the scale e¤ects in the network.
Thus, our ...rst speci...cation is based on the following threshold variable:
Model A: qit = xit (12)
Another solution for the threshold variable would consists in using the lagged level
of GDP per worker, i.e. qit = yi;t : It is necessary to use the lagged value of yit in order
1
to avoid a simultaneity since this variable is the endogenous variable of our regressions.
This speci...cation corresponds to a heterogeneity between "rich"and "poor "countries
as suggested by Canning and Bennathan (2000). However, contrary to Canning and
Bennathan, our heterogeneity is endogenous in the sense that it is the the threshold
variable yi;t 1 (and the threshold parameters j) which determine the list of countries
included in the di¤erent regimes of productivity. Besides, in our speci...cation a country
with low productivity (which belongs to regime 1 for instance) in the beginning of the
period can have a medium or high productivity at the end of the sample period (i.e.
belongs to the regime 2 or 3). Therefore, we consider a second model with:
Model B: qit = yi;t 1 (13)
4The codes for the estimation and inference procedures are done with Matlab. All codes are available
upon request.
9
3 Data and Linear Models of Productivity
We consider the same data as those used by Canning (1999) and Canning and Ben-
nathan (2000). For output we use purchasing power parity5 GDP per worker (chain
index). In both studies, physical capital stocks are constructed using a perpetual in-
ventory method. The initial stock is obtained by assuming a capital-output ratio of
3% in the base year. The ows of investments are taken from the Penn World Tables
6.00. As Canning, we assume a constant depreciation rate of 7% for the private capital.
Canning shows that his results are robust to variations in the initial choice of capital-
output ratio and the depreciation rate. Finally, human capital per worker is taken to
be the average years of schooling of the workforce. Given the data availability, the
average years of schooling of the workforce is approximated here by the average years
of schooling of the total population aged 15 and above6, from Barro and Lee (2000).
Since these human capital data are available only every ...ve years, Canning proposes
to interpolate to give annual data.
We use four infrastructure capital stock variables: the number of telephones, kilo-
watts of electricity generating capacity and the length of paved roads and the railways
line length, both expressed in kilometers. The infrastructure ...gures are the processed
data from Canning (1998). The corresponding maximal sample is 1950-1995. As stated
by Canning, these physical measures do not reect the quality di¤erences in infrastruc-
ture across countries and over time. The e¤ectiveness of infrastructure may depend
on its quality both initially and in terms of maintenance. But as noted by Canning, a
simple ...xed e¤ect speci...cation allows to capture a part of these cross country quality
di¤erences, more precisely the part which is constant over time. Consequently, in all
our models, individual ...xed e¤ects are introduced.
The estimation of a threshold model requires the use of a balanced panel. For each
speci...cation, i.e. for each type of infrastructure, we select countries (among a total
of 146 countries) for which we have a complete data set on the relevant variables over
the longest period. The period of estimation is chosen in order to maximize the total
number of observations. That is why, as in previous studies, the composition of our
panel varies with the type infrastructure considered. For each speci...cation, the sample
used and the number of included countries are reported in Table 1. The countries can
be divided in four groups: low income countries (LIC), lower middle income countries
(LMI), upper middle income countries (UMI) and high income countries (HIC). These
groups are based on World Bank de...nitions.
In order to asses the comparability of our data sets to the data sets used in previous
studies, we ...rst estimate the augmented production function in linear panel models.
As in Calderon and Serven (2004), we propose simple estimators of the equation (2).
The results are reported in Table 2. First, we consider a model in which the four
infrastructures are simultaneously introduced in the production function. It can be seen
5It is important to note that in Penn World Tables, "real"means "PPP converted"instead of "in
constant price".
6Variable code: TYR15, Barro and Lee (2000), "International Data on Educational Attainment:
Updates and Implications", Harvard University, February 2000.
10
that all regressors carry positive coe¢ cients, all signi...cantly di¤erent from zero7. The
same qualitative results are obtained when the infrastructure variables are introduced
separately (column 2 to 4). Our results are not directly comparable to that of Canning
(1999), since (i) he considers a speci...cation with lead and lagged explanatory variables
in order to obtain consistent estimates of the long-run parameters (Kao and Chiang,
2000), (ii) he considers only three types of infrastructures and (iii) his panel is not
balanced and covers more countries than our data set. However, we can observe that
the values of the various estimated parameters are roughly similar. In our sample,
the estimated elasticity of physical capital is slightly greater than those obtained by
Canning (0.37) with a panel of 57 countries in a speci...cation with 2 lags and 1 lead. But,
it is exactly the same value as those obtained by Canning in a model without leads
and lags, and with only physical and human capital stocks as explanatory variables
(table 32, page 11, Canning, 1999). Moreover, the estimated parameters when the
infrastructures are introduced one by one are similar to that reported by Canning and
Bennathan (2000). For instance, the estimated parameter associated to the paved
roads is 0.07 in our balanced sample of 50 countries whereas it is equal to 0.083 in
the dynamic speci...cation (2 lags, 1 lead) estimated by Canning and Bennathan in a
panel of 67 countries. For electricity, our estimated parameter is 0.052 whereas their
estimated parameter is equal to 0.085 and to 0.057 when electricity generating capacity
is introduced simultaneously with the kilometers of paved roads.
Thus, our results of linear models based on the four balanced panels are quite similar
to those generally obtained in the literature. Consequently, our selection procedure of
countries can be considered as robust in the sense that it does not distort the estimates
of productivity previously reported. Given these balanced panels, we now consider the
estimation of the productivity of infrastructures when threshold e¤ects are introduced.
4 Threshold Models and Network E¤ects
We now consider models in which the threshold variable is de...ned as the level of
available infrastructure. The aim of this speci...cation is to capture the network e¤ects
of the infrastructure in order to estimate robust rates of returns. As it was previously
mentioned, the ...rst step consists in determining the number of regimes, or equivalently
in testing the threshold. For that, we consider the sequential procedure proposed by
Hansen (1999). The model is estimated, allowing for sequentially zero, one, two and
three thresholds. For each speci...cation, the tests statistics F1, F2 and F3 along with
their bootstrap p-values are reported. The results of these tests for the model A,
where the threshold variable qit is de...ned as the infrastructure stock per worker xit;
are reported in Table 3.
7We do not report here t-ratios based on the long-run auto-covariance matrix as in Canning (1999).
Since, in the threshold models such corrected t-ratios are not computed, we choose to report the same
t-statistics in linear and non linear models. Moreover, the t-ratios based on estimated long run variances
with a Bartlett kernel and a common lag truncation parameter of 3 (as in Canning, 1999) give the same
qualitative results, even the values of the t-ratios are smaller than those reported in Table 2.
11
We ...nd that the test for a single threshold F1 is highly signi...cant with a bootstrap
p-value smaller than 0.001 for all the infrastructure variables considered. The lower
value of the F1 statistic is obtained for the telecommunication sector, but even in this
case the value of the test statistic is largely below the critical values at standard levels.
It implies that there is strong evidence that the relationship between output and the
considered inputs is non-linear. As it was raised by Fernald (1999) or Gramlich (1994),
the network aspect of infrastructure investments induces a strong non linearity in the
productivity of these equipment and structures. The test for a double threshold F2
is also strongly signi...cant with a bootstrap p-value smaller than 0.001. The last test
for a third threshold F3 is also signi...cant, even if in this case the value of the F-
statistic is largely less important than those reported for the model with one or two
thresholds. According to the Hansen's procedure, it would be necessary to pursue
and to test for four thresholds, ...ve thresholds etc. until the corresponding F-test is
statistically not signi...cant. However, as it was previously mentioned, we limit our
analysis in this section to a model with at most three threshold parameters and four
regimes. This choice is ...rstly justi...ed by the computational cost of the estimation
and inference procedures for panel models with more than 4 regimes. Secondly, we
show that, given the sequential estimation procedure proposed by Hansen, when a
supplementary regime is introduced, it does not a¤ect (or slightly) the estimates of the
other threshold parameters and the estimates of the slope parameters in the existing
regimes. For instance, if we compare the estimated elasticities (and particularly that
of infrastructure) obtained in a model with three regimes (see appendix A1) to the
estimated elasticities obtained in a model with four regimes, we get similar results in
the three existing regimes. For all these reasons, we limit our analysis to a model with
three threshold parameters and four regimes in the relationships between output and
capital stocks.
The estimates of the parameters of the PTR models with four regimes and the
corresponding t-statistics based on corrected standard errors for heteroskedasticity are
reported in Table 4. We use as threshold variable the infrastructure stock per worker
(model A). In the electricity sector, the results clearly reect the network dimension
of the investments devoted to the electricity generating capacity. Indeed, we observe
that when the electricity generating capacity per worker is very low (less than 32909
kilowatts per worker), the investments in this sector have the same productivity as the
other private investments since the estimated parameter is not signi...cantly di¤er-
ent from zero. On the contrary, when a minimum network is available, the marginal
productivity of the electricity generating capacity is signi...cantly and largely greater
than to the productivity of the other investments. When the electricity generating
capacity ranges from 32909 kilowatts per worker and 62536 kilowatts per worker, the
estimated parameter is equal to 0.295. This value is largely greater than the values
generally obtained in linear speci...cations in panel models. Recall that in our sample,
the estimated parameter obtained in a linear speci...cation with individual e¤ects and
time e¤ects was only equal to 0:05 (Table 2). It implies that if we do not take into
account the non linearity of the relationship between infrastructure and output, the
productive e¤ect of these investments can be undervalued during the period of building
of the network. When the network is near to be completed, i.e. when the capacity by
12
worker ranges from 62536 and 595710, the marginal productivity of infrastructure is
always signi...cantly greater than the productivity of other investments, but the value of
the parameter is smaller than in the previous regime. The estimated parameters falls
from 0:29 to 0:12, which is a value similar to that generally obtained in linear models
(Canning, 1999; Calderon and Serven, 2004). This decrease of the naïve measure8 of
the elasticity of infrastructures reects the progressive maturity of the network and not
only a standard argument of decreasing returns, which only implies a simple constraint
on the position of this elasticity with respect to the unity. Finally, when the electricity
generating capacity per worker exceeds 595710 kilowatts per worker, the productivity
of investments in this sector becomes similar to the productivity of other investments.
The estimated parameter is slightly signi...cantly di¤erent from zero, but relatively
small compared to the value obtained in linear models. The t-statistic associated to
the test of the nullity of is in this case largely smaller than the value obtained in the
regimes 2 or 3. The productivity of infrastructure investment in the electricity sector is
then slightly positive (or even null if we consider not corrected t-statistics) in the last
regime. In other words, when the network is completed, the use of a linear speci...cation
may lead to overvalue the productivity of infrastructure when the network is achieved.
It is exactly the idea pointed out by Fernald (1999) for the road network in the United
States: the construction of the network booms substantially the productivity and the
output, but when the construction of the network is completed, the public capital is not
exceptionally productive at the margin. As it was previously mentioned, these results
are robust when we consider a model with three regimes and two threshold parameters.
The corresponding results for the model A (qit = xit) are reported in Table A1 in ap-
pendix A1. We obtain roughly the same estimated elasticities as reported in a model
with four regimes. The only di¤erence is that the two intermediate regimes with the
highest productivity are now replaced by only one regime with an estimated parameter
equal to 0.127
As far as the road sector is concerned, the productivity of infrastructures also ex-
hibits strong network e¤ects when the threshold variable is de...ned as the number of
kilometers of paved road per worker (Table 4). As for in previous case, we observe
a pro...le low - high - low for the productivity given the regimes. When the paved
road length actually available is very low, smaller than 0:36 kilometers per worker, the
parameter is signi...cantly positive and greater than the value obtained in a linear
speci...cation (Table 2). But, when the network is more important (regime 3), the value
of this parameter is greater and increases from 0:13 to 0:15. On the contrary, when
the network is completed or near to be completed, the value of the naïve elasticity
decreases to 0:12, but is still signi...cantly di¤erent from zero, contrary to the electricity
sector. It is highly probable that in a model with more than four regimes, an extreme
regime would appear with a smaller value for in which some industrialized countries
would appear. Finally, the last di¤erence with the electricity sector is that an interme-
diate regime appears in the model with four regimes in which the estimated parameter
is negative and signi...cantly di¤erent from zero. As noted by Canning (1999) the
correct interpretation of the coe¢ cient on the infrastructure variable represents the
8Which is probably near to the true elasticity value, as pointed out by Calderon and Serven (2004).
13
output e¤ect of increasing infrastructure capital while holding overall physical capital
constant. "That is, we measure the e¤ect as an increase in infrastructure assuming an
equal (measured of terms of the cost of investment) o¤setting decrease in other forms of
capital"(Canning, 1999, page 12). When this o¤set has a negative impact on the GDP
per worker, it implies that the marginal productivity of infrastructure capital is less
important than the marginal productivity of other capital. Thus, in the intermediate
regime 2, the marginal productivity of road infrastructure is smaller than the produc-
tivity of other capital. However, this result is not robust when we consider a model
with three regimes (Table A1, appendix A1). In this case, we still observe the low -
high - low productivity pro...le, but the intermediate regime with a negative value of
disappears.
For the telecommunication sector, the results are similar and also reveal strong
threshold e¤ects. In this case, the threshold variable is de...ned as the number of tele-
phones by worker. When the network is not important, the infrastructure investments
in the telecommunication sector have the same productivity as the other forms of cap-
ital. When the network is more important their productivity increase. In the second
regime, the estimated parameter is then greater than 0:20, whereas it is only equal
to 0:10 in a linear speci...cation (see Table 2). In the third regime, the value of falls
to 0:12 as it is suggested in the Fernald's analysis. However, when the telecommuni-
cation network is largely developed, with more than 126 telephones per worker, this
kind of investments has another regime of high productivity. This particular regime
is undoubtedly linked to the importance of the telecommunications networks in the
post-industrialized countries included in our sample.
Finally, the results are less clear when we consider the railways sector. In this case,
the regime of high productivity is found when the network is largely incomplete with
less than 0:24 kilometers of railways per worker. When the network is more important,
the estimated parameter becomes null or even negative. This particular evolution
of the productivity of railways can be explained by the substitution of this mean of
transport by other means of transport in the developing countries. After the building
of the main lines between the main urban areas of the countries, the productivity of
railways investments falls (comparatively to other means of transport) when ones comes
to build the secondary network between the less important urban areas. Consequently,
compared to the road sector for instance, the decline of the marginal productivity due
to the network completion is more rapidly achieved.
5 Threshold Models and Income Heterogeneity
As it was previously mentioned, the threshold e¤ects in the productivity can also be
induced by the heterogeneity of the production structure between the countries of
our sample. The idea is that the marginal productivity of the same infrastructure
may not be equivalent for rich countries and poor countries: this heterogeneity may
be the consequence of various network e¤ects (which depend on the completion of
the main infrastructure networks, and not only on the completion on the considered
14
infrastructure network), but also of many other reasons (the heterogeneity of the level
and the quality of private capital, the heterogeneity in the economic systems etc.). So,
we propose to consider a threshold speci...cation of the production function in which the
threshold variable is de...ned by the lagged level of real GDP per worker, i.e. qit = yi;t1
(Model B).
It is important to note that the speci...cation of a threshold production function,
where the threshold variable is de...ned as the level of lagged GDP per worker can
be considered as a technical solution to circumvent the reverse causation problem.
Indeed, in linear speci...cation the correlation between infrastructure stocks and the
productivity of private capital may reect causation from the changes in the stock
to changes in productivity. In other words, the infrastructure stocks are likely to be
endogenous. This point has been largely document in the literature devoted to the
measure of the productivity public capital (Gramlich, 1994). It generally leads to the
use of instrumental variable methods (GMM, simultaneous equations etc.). In our
threshold speci...cation, the correlation between GDP per worker and infrastructure
stocks is conditional to the level of the GDP per worker observed in the past period.
Thus, if the infrastructure level is linearly linked to the GDP, the reverse causation
would imply a number of regimes identical to the number of possible values of the
GDP. If the number of regime is restricted, the inuence of the reverse causation in
each regime would be less important than in a linear representation and may be ignored
if the number of regimes is su¢ ciently large. A similar observation can be done for the
other speci...cations as soon as the threshold variable is correlated with the level of the
lagged GDP.
As in the previous case, the ...rst step of the Hansen estimation procedure consists
in determining the number of regimes which must be adopted in the estimation of the
production function. For that, the model is estimated, allowing for sequentially zero,
one, two and three thresholds. For each speci...cation, the statistics F1, F2 and F3 along
with their bootstrap p-values are reported. The results of these tests for the model B
are reported in Table 5. First, we can observe that whatever the hypothesis tested, the
values of the F-tests are largely greater than those obtained in the case of the model
A (see Table 2). It clearly indicates that the threshold e¤ects (and the rejection of
the linearity assumption) are largely more important when we consider the per worker
income heterogeneity than when we consider the network e¤ects of infrastructures.
Secondly and consequently, in all speci...cations the F-tests lead to the conclusion that
there are more than 4 regimes in the production function given the level of the lagged
value of the real GDP per capita. However, for the same reasons as exposed in the case
of the model A, we limit our analysis to a model with at most four regimes.
The estimates of the parameters of the PTR models with four regimes and the
corresponding t-statistics based on corrected standard errors for heteroskedasticity are
reported in Table 6. The three threshold estimates (de...ned on the GDP per worker)
identify four production functions and four separate groups of countries. The four
regimes can be interpreted as four separated groups of countries as in the World Bank
classi...cation: "lower income", "lower middle income", "upper middle income" and
"high income"countries. However, the distribution of the countries in these groups is
15
endogenous and not time invariant since it may evolve given the level of the GDP per
worker. Globally, in all sectors we can observe that the estimated parameters are
generally positive and signi...cant in each group of countries. One exception is the case
of the road sector in the last regime (high income countries), in which this parameter
becomes not signi...cant at 5% level. This result is compatible with the conclusions
of Fernald (1999) for the road sector in the United States, if we admit that when a
country is in this last regime, its road network is generally achieved. More surprisingly,
the values of the parameters are quite similar in all the regimes, compared to the
values obtained when the threshold variable was de...ned as the amount of infrastructure
available per worker. It is particularly true for the telecommunication sector, where the
parameter ranges from 0:13 to 0:18 in the four groups of countries and for the road
sector in the ...rst three groups of countries. This result implies that the heterogeneity
of the productivity of infrastructure is less important when we regroup the countries
according to their GDP per worker than we consider groups according to the amount
of infrastructure per worker actually available. In other words, the heterogeneity seems
to be more related to the network e¤ect than to the level of per capita income, even
if this last measure is sometimes (but not always) a proxy of the completion of the
network. The network e¤ects seem to be more important to explain the non linearity
of the marginal productivity of infrastructures than the threshold e¤ects based on the
income heterogeneity.
6 Conclusion
In this paper we provide an empirical evaluation of the threshold e¤ects of the pro-
ductivity of infrastructures and public capital stocks in developing countries. Our
assessment is based on the estimation of various threshold panel speci...cations of in-
frastructure or public capital augmented production functions. More precisely, we
consider various speci...cations of a Panel Threshold Regression (PTR) model (Hansen,
1999) in which the individual parameters of the augmented production functions are
divided into a small number of groups according to an observable variable. The tran-
sition mechanism between the di¤erent regimes of productivity is then determined by
the level of a threshold variable. If the threshold variable exceeds a certain value, the
production technology switches from one regime to another. Our main results can be
summarized in two main points.
First, the relationship between the output and the infrastructure stocks is non lin-
ear. More precisely, strong threshold e¤ects can be identi...ed in these relationships.
This conclusion is robust to changes in the panel model used, in the testing procedure
applied and to changes in the composition of the panel sample. These threshold e¤ects
are clearly identi...ed when physical measures of the infrastructure stocks actually avail-
able in the country per worker are used as threshold variable. Second, the productivity
of infrastructure (road, electricity, telephones and railways) exhibits strong threshold
e¤ects which could be interpreted as network e¤ects. In a ...rst step, when the stock of
infrastructure in a sector actually available per worker is very low, the infrastructure
investments in this sector have the same productivity as the other investments. On the
16
contrary, when the network is su¢ ciently developed but not achieved, the infrastructure
investments have a productivity which is generally largely higher than the productivity
of other investments. Finally, when the level of physical infrastructure stock per worker
exceeds a certain value indicating that the main network is achieved, the productivity
rapidly decreases and the infrastructure investments may be not exceptionally produc-
tive at the margin. In other words, the highest marginal productivity of investments is
reached when a network is su¢ ciently developed, but not completely achieved.
Therefore, these threshold e¤ects are an argument in favour of the sectorial special-
ization of the public investments in infrastructure in developing countries. This spe-
cialization may be less important in the industrialized countries in which infrastructure
networks are already available (water, sewer, road, rail, electricity, telecommunications
etc.). On the contrary, this specialization of infrastructure investments may be essential
in developing countries where none of these main networks is near to be operational.
In this case, one or two particular infrastructure networks have to be considered as a
priority compared to all other infrastructure networks, and the majority of investments
must be devoted to these priority sectors. In the poorest developing countries, it could
be the case for the water distribution and sanitation systems, for instance as suggested
by some many recent programs lead by the international institutions. This idea of a
specialization of infrastructure investments may be compared to the idea of a universal
primary service, as for instance the universal primary education service supported by
the Education for All (EFA) international commitment (1990, 2000). In this context,
the goal is to ensure a primary education for all children in developing countries before
investing in the secondary education and more evidently in the university system which
would bene...t only to a minority of the population. The mechanism is exactly the same
in our network perspective: the highest marginal productivity of investments is reached
when a network (primary education buildings) is su¢ ciently developed, but not com-
pletely achieved. During this period it would be inappropriate to begin investments in
another network (university facilities) for which the marginal productivity may be very
low.
A Appendix
A.1 Threshold Models with Three Regimes
In order to asses the sensitivity of our results to the choice of the number of regimes,
we report on the Tables A1 and A2 the results of the estimation of the Models A and
B with three regimes (two threshold parameters).
17
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18
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19
Table 1. Sample Properties
Sample N T Obs. LIC LMI UMI HIC
Paved Roads 1965-1990 50 26 1300 13 12 9 16
Electricity Capacity 1961-1995 76 35 2660 17 22 11 26
Number of Telephones 1961-1990 56 30 1680 15 18 12 11
Railways 1960-1992 57 33 1881 9 18 8 22
All Types of Infrastructure 1965-1990 24 26 624 5 9 5 5
Notes: The value of Obs. denotes the total number of observations in the balanced panel. The
values of LIC, LMI, UMI and HIC correspond respectively to the number of Low Income Coun-
tries, Lower Middle Income countries, Upper Middle Income countries and High Income Countries
included in the sample.
Table 2. Infrastructure Augmented Production Function
Linear Panel Models with Fixed E¤ects and Year Dummies
Within Estimates Model 1 Model 2 Model 3 Model 4 Model 5
Physical Capital 0:473 0:616 0:529 0:460 0:576
(12:4) (44:4) (49:0) (31:32) (47:7)
Human Capital 0:143 0:131 0:104 0:109 0:207
(4:55) (7:06) (7:42) (6:28) (10:37)
Roads 0:078 0:070 -- -- --
(3:72) (7:20)
Electricity 0:096 -- 0:052 -- --
(4:89) (6:09)
Telephones 0:041 -- -- 0:104 --
(1:377) (7:42)
Railways 0:063 -- -- -- 0:192
(2:41) (11:19)
RSS 5:88 13:6 41:0 22:1 24:2
Number of observations 624 1 300 2 660 1 680 1 881
Number of countries 24 50 76 56 57
Notes: Dependent variable is log GDP per worker. All variables are measured per worker
and expressed in logs. The t-statistics are in parenthesis.
20
Table 3. Tests for Threshold E¤ects: Model A, qit = xit
Roads Electricity Telephones Railways
Test for single threshold
F1 90.2 112.1 59.5 128.4
P-value 0.00 0.00 0.00 0.00
1% Critical Values 13.7 14.3 14.3 14.9
5% Critical Values 15.3 15:5 16.8 16.5
10% Critical Values 19.3 20.6 22.0 21.3
Test for double threshold
F2 84.7 74.3 82.6 122.6
P-value 0.00 0.00 0.00 0.00
1% Critical Values 50.4 19.8 17.2 30.7
5% Critical Values 55.2 22.7 19.2 36.9
10% Critical Values 68.2 26.2 25.1 42.2
Test for triple threshold
F3 41.7 43.8 55.7 85.9
P-value 0.00 0.00 0.00 0.00
1% Critical Values 13.6 13.6 13.6 13.2
5% Critical Values 15.2 15.6 15.7 15.5
10% Critical Values 18.0 21.2 19.2 18.4
Notes: P-values and critical values are computed from 300 simulations. F1 de-
notes the Fisher type statistic associated to the test of the null of no threshold against
one threshold. F2 corresponds to the test one threshold against two thresholds and
F3 corresponds to the test of two thresholds against three thresholds.
21
Table 4. Four Regimes Panel Models. Model A: q it = xit
Roads Electricity Telephones Railways
Regime 1: qit 1
Physical Capital per Worker 0:593 0:520 0:417 0:612
(38:4) (30:1) (24:6) (49:1)
Human Capital per Worker 0:046 0:165 0:062 0:564
(2:44) (10:2) (2:43) (21:5)
Infrastructure per Worker 0:131 0:001 0:030 0:183
(5:51) (0:08) ( 0:73) (7:51)
Regime 2: 1 < qit 2
Physical Capital per Worker 0:496 0:629 0:384 0:600
(24:5) (29:0) (22:4) (50:0)
Human Capital per Worker 0:194 0:179 0:120 0:398
(6:52) (6:60) (6:94) (16:3)
Infrastructure per Worker 0:676 0:295 0:210 0:070
( 5:44) (6:56) (13:8) ( 1:64)
Regime 3: 2 < qit 3
Physical Capital per Worker 0:580 0:591 0:453 0:624
(39:8) (41:9) (17:7) (51:7)
Human Capital per Worker 0:219 0:071 0:127 0:230
(12:8) (4:46) ( 3:42) (11:8)
Infrastructure per Worker 0:155 0:126 0:129 0:158
(7:42) (11:7) (2:81) ( 6:42)
Regime 4: qit > 3
Physical Capital per Worker 0:590 0:556 0:365 0:524
(36:6) (39:3) (17:1) (32:7)
Human Capital per Worker 0:099 0:256 0:116 0:717
(3:62) (9:91) (3:22) (15:4)
Infrastructure per Worker 0:122 0:035 0:226 0:014
(4:58) (2:41) (12:25) ( 0:36)
Threshold Estimates
First Threshold 1:021 3:414 1:445 1:427
Second Threshold 0:642 2:772 4:314 0:861
Third Threshold b1b2 1:249 0:518 4:839 0:975
Residual Sum of Squares
b3 12:4 42:4 23:3 26:6
Notes: The t-statistics in parenthesis are computed with an estimator of the covariance
matrix which allows for heteroskedasticity. The con...dence intervals for the threshold pa-
rameters are not reported. See Appendix A1, for the con...dence intervals in a model with
three regimes.
22
Table 5. Tests for Threshold E¤ects: Model B, qit = yi;t 1
Roads Electricity Telephones Railways
Test for single threshold F1 172.4 324.5 201.0 495.6
P-value 0.00 0.00 0.00 0.00
Test for double threshold F2 198.9 319.9 200.4 363.4
P-value 0.00 0.00 0.00 0.00
Test for triple threshold F3 181.4 324.2 204.2 165.6
P-value 0.00 0.00 0.00 0.00
Notes: P-values are computed from 300 simulations. F1 denotes the Fisher type
statistic associated to the test of the null of no threshold against one threshold. F2
corresponds to the test one threshold against two thresholds and F3 corresponds to
the test of two thresholds against three thresholds.
23
Table 6. Four Regimes Panel Models. Model B: q it= yi;t 1
Roads Electricity Telephones Railways
Regime 1: qit 1
Physical Capital per Worker 0:500 0:390 0:319 0:515
(32:9) (29:2) (19:1) (40:6)
Human Capital per Worker 0:085 0:152 0:050 0:450
(4:71) (12:4) (1:87) (21:5)
Infrastructure per Worker 0:061 0:037 0:179 0:035
(3:11) (3:96) (8:11) (1:88)
Regime 2: 1 < qit 2
Physical Capital per Worker 0:526 0:464 0:355 0:582
(35:1) (35:8) (22:6) (46:2)
Human Capital per Worker 0:063 0:023 0:053 0:172
(3:52) ( 1:31) (2:56) (6:76)
Infrastructure per Worker 0:081 0:102 0:135 0:051
(5:07) (9:81) (7:37) (3:36)
Regime 3: 2 < qit 3
Physical Capital per Worker 0:552 0:475 0:391 0:630
(37:1) (35:8) (22:8) (49:8)
Human Capital per Worker 0:073 0:033 0:068 0:027
(3:17) ( 1:30) ( 3:28) ( 1:27)
Infrastructure per Worker 0:071 0:007 0:166 0:053
(4:95) ( 0:57) (12:2) (3:96)
Regime 4: qit > 3
Physical Capital per Worker 0:591 0:503 0:428 0:653
(37:8) (39:7) (23:2) (50:8)
Human Capital per Worker 0:041 0:007 0:170 0:087
( 1:68) ( 2:76) ( 6:29) ( 3:38)
Infrastructure per Worker 0:028 0:114 0:158 0:042
(1:89) (7:80) (11:7) (2:95)
Threshold Estimates
First Threshold 7:838 8:325 7:833 8:530
Second Threshold b1 8:332 9:234 8:291 9:076
Third Threshold bb2
3 9:222 9:674 9:234 9:602
Residual Sum of Squares 8:6 30:2 17:1 17:2
Notes: The t-statistics in parenthesis are computed with an estimator of the covariance
matrix which allows for heteroskedasticity. The con...dence intervals for the threshold pa-
rameters are not reported. See Appendix A1, for the con...dence intervals in a model with
three regimes
24
Table A1. Three Regimes Panel Models. Model A: q it = xit
Roads Electricity Telephones Railways
Regime 1: qit 1
Physical Capital per Worker 0:604 0:522 0:422 0:620
(40:3) (36:7) (26:7) (52:9)
Human Capital per Worker 0:078 0:165 0:099 0:453
(4:21) (10:9) (5:95) (19:6)
Infrastructure per Worker 0:138 0:012 0:207 0:135
(5:34) (0:81) (14:9) (6:73)
Regime 2: 1 < qit 2
Physical Capital per Worker 0:591 0:584 0:493 0:629
(42:2) (41:9) (18:5) (52:9)
Human Capital per Worker 0:239 0:072 0:170 0:256
(14:1) (4:69) ( 4:46) (13:0)
Infrastructure per Worker 0:145 0:127 0:128 0:139
(8:82) (11:9) (2:53) ( 5:56)
Regime 3: qit > 3
Physical Capital per Worker 0:604 0:550 0:414 0:532
(38:4) (39:3) (20:5) (33:2)
Human Capital per Worker 0:096 0:254 0:124 0:731
(3:48) (9:84) (3:47) (15:3)
Infrastructure per Worker 0:114 0:040 0:195 0:007
(4:23) (2:75) (10:85) ( 0:17)
Threshold Estimates
First Threshold 1:105 2:773 4:808 0:863
95% Con...dence Interval
b1 [ 1:13; 0:99] [ 3:16 ; 2:73] [4:78; 4:84] [ 0:88; 0:82]
Second Threshold 1:249 0:518 4:314 0:975
95% Con...dence Interval
b2 [1:10; 1:38] [ 0:60; 0:13] [4:11; 4:35] [0:91; 0:99]
Residual Sum of Squares 12:8 43:1 24:1 27:9
Notes: The t-statistics in parenthesis are computed with an estimator of the covariance matrix which
allows for heteroskedasticity. The con...dence interval for the threshold parameters corresponds to the no
rejection region of con...dence level 95% associated to the likelihood ratio statistic for test on the values of
the threshold parameters (see Hansen, 1999). This con...dence interval can be not symetric.
25
Table A2. Three Regimes Panel Models. Model B: q it = yi;t 1
Roads Electricity Telephones Railways
Regime 1: qit 1
Physical Capital per Worker 0:553 0:451 0:401 0:575
(38:2) (33:9) (24:8) (47:7)
Human Capital per Worker 0:085 0:177 0:139 0:422
(4:56) (13:7) (7:27) (19:3)
Infrastructure per Worker 0:902 0:030 0:137 0:023
(4:41) (2:98) (8:26) (1:17)
Regime 2: 1 < qit 2
Physical Capital per Worker 0:580 0:526 0:455 0:641
(41:5) (40:8) (25:9) (5:3:6)
Human Capital per Worker 0:084 0:033 0:046 0:145
(5:74) ( 1:78) ( 2:13) (5:63)
Infrastructure per Worker 0:118 0:083 0:141 0:041
(11:3) (7:55) (9:91) (2:63)
Regime 3: qit > 3
Physical Capital per Worker 0:624 0:556 0:492 0:686
(40:7) (44:6) (26:08) (55:6)
Human Capital per Worker 0:072 0:096 0:154 0:029
( 2:976) ( 4:13) ( 5:38) ( 1:37)
Infrastructure per Worker 0:047 0:091 0:130 0:047
(3:26) (8:10) (9:21) (3:54)
Threshold Estimates
First Threshold 7.838 8.310 8.289 8.374
95% Con...dence Interval
b1 [7:79; 7:83] [8:25; 8:33] [8:28; 8:33] [8:37; 8:37]
Second Threshold 9.222 9.234 9.234 9.076
95% Con...dence Interval
b2 [9:20; 9:23] [9:20; 9:29] [9:20; 9:33] [8:99; 9:16]
Notes: The t-statistics in parenthesis are computed with an estimator of the covariance matrix which
allows for heteroskedasticity. The con...dence interval for the threshold parameters corresponds to the no
rejection region of con...dence level 95% associated to the likelihood ratio statistic for test on the values of
the threshold parameters (see Hansen, 1999). This con...dence interval can be not symetric.
26