metadata: authors: 0: author: Hurlin,Christophe
count: World
docna: 0: docna: Network effects of the productivity of
            infrastructure in developing countries
docty: Policy Research Working Paper
subtopic: Telecommunications Infrastructure,Railways Transport,Energy Policies & Economics,Hydrology,Transport Services
teratopic: Macroeconomics and Economic Growth,Industry,Public Sector Development,Finance and Financial Sector Development
abstracts: cdata!: Using panel data models, the author
            examines the threshold effects of the productivity of
            infrastructure investment in developing countries. He
            considers various specifications of an augmented production
            function that allow for endogenous thresholds. More
            precisely, these specifications are tested in a panel
            threshold regression model. The author's 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
            noninfrastructure 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 investment. Finally, when the main
            network is achieved, its marginal productivity becomes
            similar to the productivity of other investment.
repnme: repnme: Network effects of the productivity of
            infrastructure in developing countries
display_title: Network effects of the productivity
            of infrastructure in developing countries
listing_relative_url: /research/2006/01/6523128/network-effects-productivity-infrastructure-developing-countries
topicv3: Water m1326522 137,Transport m1326661 135,Power and Electricity Sector m1327094 515,Telecommunications and Broadband Access
            m1327340 286,Railways m1327400 166
publishtoextweb_dt: 2010-07-01T00:00:00Z
historic_topic: Macroeconomics and Economic Growth,Industry,Public Sector Development,Finance and Financial Sector Development
no_of_pages: 26
author_keys: 0: author_key: 775680
model_extractions: 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, water 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 regimes<sup>1</sup> . 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 nondynamic 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,

> 1 This 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 regression 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 equation 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), HoltzEakin (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 in‡uence of the infrastructures on the output in the ...rst period and the in‡uence 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 in‡uence 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 capital) 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 countries<sup>2</sup> . Canning (1999) or Canning and Bennathan (2000) investigate 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 insignificant, 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 in‡uence of the income heterogeneity on the production technology. A last section concludes.

# 2 The Productivity of Infrastructure: Toward a Threshold 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: 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

> 2 With 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: 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 productivity. 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) correspond 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 interpret 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 capital stocks. However, as noted by Calderon and Serven (2004), because infrastructure stocks typically account for relatively small portions of overall capital stock, the di¤erence 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 adopting a Panel Threshold Regression (PTR) model similar to that proposed by Hansen (1999): where qit denotes a threshold variable and � denotes a threshold parameter. This model can be rewritten as: 0 0 where �j = ��j �j �j� for j = 1; 2 and Wit = (kit hit xit) , and where I(:) is the indicator function. The error "it is assumed to be independent and identically distributed with mean zero and ...nite variance �<sup>2</sup> : 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: 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 �2 can be estimated by OLS. Let us denote<sup>b</sup> �1 (�) and<sup>b</sup> �2 (�) ; the corresponding estimates. Thus, conditionally to a value of � it is possible to compute the sum of squared errors, denoted S1 (�) : The threshold parameter � is then estimated by minimizing the sum of squared S1 (�).

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 �;<sup>b</sup> it is then possible to get the estimates of the elasticities in the regimes, i.e.<sup>b</sup> �1 �b and<sup>b</sup> �2 �b ; and the estimates of individual e¤ects bai: As � � � � it was stressed by Hansen (1999), it is undesirable for a threshold �<sup>b</sup> to be selected 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 the observations lie 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 �:<sup>b</sup> In our 7
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 statically 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: b where �<sup>2</sup> denotes a convergent estimate of �<sup>2</sup> : The main problem is that under the null the threshold parameter � is not identi...ed. Consequently, the asymptotic distribution 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 simulations<sup>3</sup> 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 where �<sup>b</sup> 1 and �<sup>b</sup> 2 denote the threshold estimates of the model with three regimes (equation 5), and S2 �b1; b�2 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 thresholds (three regimes) against the alternative of three thresholds (four regimes). The corresponding likelihood ratio statistic, denoted F3:; is de...ned as:

> 3 All the Matlab codes are available upon request.

8
where S3 �b1; b�2; b�3 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 regimes<sup>4</sup> .

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: Another solution for the threshold variable would consists in using the lagged level of GDP per worker, i.e. qit = yi;t�1: It is necessary to use the lagged value of yit in order 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:

> 4 The 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 Bennathan (2000). For output we use purchasing power parity<sup>5</sup> GDP per worker (chain index). In both studies, physical capital stocks are constructed using a perpetual inventory 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 capitaloutput 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 above<sup>6</sup> , 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, kilowatts 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 re‡ect the quality di¤erences in infrastructure 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

> 5 It is important to note that in Penn World Tables, ”real” means ”PPP converted” instead of ”in constant price”.

> 6 Variable 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 zero<sup>7</sup> . 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.

> 7 We 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 re‡ect 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 32 909 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¤erent 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 32 909 kilowatts per worker and 62 536 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 62 536 and 595 710, 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 measure<sup>8</sup> of the elasticity of infrastructures re‡ects 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 595 710 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 appendix 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 exhibits 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 intermediate 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

> 8 Which 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 productivity 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 telephones by worker. When the network is not important, the infrastructure investments in the telecommunication sector have the same productivity as the other forms of capital. 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 telecommunication 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;t�1 (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 re‡ect 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 in‡uence 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 productivity of infrastructures and public capital stocks in developing countries. Our assessment is based on the estimation of various threshold panel speci...cations of infrastructure 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 transition 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 linear. 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 available 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 productive 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 specialization of the public investments in infrastructure in developing countries. This specialization 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 completely 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