metadata: authors: 0: author: Calderon, Cesar
1: author: Moral-Benito, Enrique
2: author: Serven, Luis
count: World
docna: 0: docna: Is infrastructure capital productive ? a
            dynamic heterogeneous approach
docty: Policy Research Working Paper
subtopic: Telecommunications Infrastructure,Power & Energy Conversion,Infrastructure Regulation,Infrastructure Finance,Infrastructure Economics,Energy Demand,Energy and Environment,Energy and Mining,Transport Services,Infrastructure and Law
teratopic: Transport,Public Sector Development,Infrastructure Economics and Finance,Macroeconomics and Economic Growth,Finance and Financial Sector Development
abstracts: cdata!: This paper offers an empirical
            evaluation of the output contribution of infrastructure.
            Drawing from a large data set on infrastructure stocks
            covering 88 countries and spanning the years 1960-2000, and
            using a panel time-series approach, the paper estimates a
            long-run aggregate production function relating GDP to human
            capital, physical capital, and a synthetic measure of
            infrastructure given by the first principal component of
            infrastructure endowments in transport, power, and
            telecommunications. Tests of the cointegration rank allowing
            it to vary across countries reveal a common rank with a
            single cointegrating vector, which is taken to represent the
            long-run production function. Estimation of its parameters
            is performed using the pooled mean group estimator, which
            allows for unrestricted short-run parameter heterogeneity
            across countries while imposing the (testable) restriction
            of long-run parameter homogeneity. The long-run elasticity
            of output with respect to the synthetic infrastructure index
            ranges between 0.07 and 0.10. The estimates are highly
            significant, both statistically and economically, and robust
            to alternative dynamic specifications and infrastructure
            measures. There is little evidence of long-run parameter
            heterogeneity across countries, whether heterogeneity is
            unconditional, or conditional on their level of development,
            population size, or infrastructure endowments.
repnme: repnme: Is infrastructure capital productive ? a
            dynamic heterogeneous approach
display_title: Is infrastructure capital productive
            a dynamic heterogeneous approach
listing_relative_url: /research/2011/06/14373600/infrastructure-capital-productive-dynamic-heterogeneous-approach
topicv3: Infrastructure and Growth m1326244 1083,Energy and Extractives m1326595 507,Transport m1326661 135,Power Systems m1327093 508,Telecommunications and Broadband Access
            m1327340 286
publishtoextweb_dt: 2010-07-01T00:00:00Z
historic_topic: Transport,Public Sector Development,Infrastructure Economics and Finance,Macroeconomics and Economic Growth,Finance and Financial Sector Development
no_of_pages: 34
model_extractions: WPS5682 Policy Research Working Paper 5682 Is Infrastructure Capital Productive? A Dynamic Heterogeneous Approach _César Calderón Enrique Moral-Benito Luis Servén_ The World Bank Development Research Group Macroeconomics and Growth Team June 2011
Policy Research Working Paper 5682

# **Abstract**

This paper offers an empirical evaluation of the output contribution of infrastructure. Drawing from a large data set on infrastructure stocks covering 88 countries and spanning the years 1960–2000, and using a panel time-series approach, the paper estimates a long-run aggregate production function relating GDP to human capital, physical capital, and a synthetic measure of infrastructure given by the first principal component of infrastructure endowments in transport, power, and telecommunications. Tests of the cointegration rank allowing it to vary across countries reveal a common rank with a single cointegrating vector, which is taken to represent the long-run production function. Estimation of its parameters is performed using the pooled mean group estimator, which allows for unrestricted short-run parameter heterogeneity across countries while imposing the (testable) restriction of long-run parameter homogeneity. The long-run elasticity of output with respect to the synthetic infrastructure index ranges between 0.07 and 0.10. The estimates are highly significant, both statistically and economically, and robust to alternative dynamic specifications and infrastructure measures. There is little evidence of longrun parameter heterogeneity across countries, whether heterogeneity is unconditional, or conditional on their level of development, population size, or infrastructure endowments.

This paper is a product of the Macroeconomics and Growth Team, Development Research Group. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://econ.worldbank.org. The author may be contacted at ccaldero@worldbank.org.

_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 views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent._ Produced by the Research Support Team
# **Is infrastructure capital productive?**

# * **A dynamic heterogeneous approach**

**César Calderón**<sup>a</sup> , **Enrique Moral-Benito**<sup>b</sup> , **Luis Servén**<sup>a</sup> a _The World Bank, 1818 H Street NW, Washington DC 20433, USA_ b _Bank of Spain, Alcalá 48, 28014, Madrid, Spain_

## **JEL Classification:** H54, E23, O40

**Keywords:** Infrastructure, panel cointegration, parameter heterogeneity.

***** We are grateful to Nicola Limodio and Stephane Straub for useful comments and suggestions. We also thank participants at the Toulouse conference on  ̳Infrastructure Economics and Development‘, the Australian National University conference on  ̳The Economics of Infrastructure‘, the IMF conference on  ̳Investment Scaling-up in Low-Income Countries‘, the 2010 Symposium of Economic Analysis, and seminars at Brookings and the Bank of Spain. However, any remaining errors are ours only. Rei Odawara and Junko Sekine provided excellent research assistance. This work was partly supported by the Knowledge for Change Program as well as the Government of Japan through the Consultancy Trust Fund Program (CTF) of the World Bank. The views expressed in this paper are only ours and do not necessarily reflect those of the World Bank, its Executive Directors, or the countries they represent.
# **1. Introduction**

The macroeconomic literature has long been interested in the contribution of infrastructure capital to aggregate productivity and output. Numerous theoretical papers have approached it using an aggregate production function including public capital as an additional input, first in the context of Ramsey-type exogenous growth models (e.g., Arrow and Kurz 1970) and later in endogenous growth models (Barro 1990, Futagami, Morita and Shibata 1993). This analytical literature has grown enormously in the last fifteen years, exploring a multitude of variants of the basic models, such as alternative financing schemes, simultaneous consideration of public capital and productive current spending flows, utility-yielding public capital, or public infrastructure congestion.<sup>1</sup> Quantitative assessments of the contribution of infrastructure are critical for many policy questions – such as the output effects of fiscal policy shocks instrumented through public investment changes (e.g., Leeper, Walker and Yang 2010; Ilzetzki, Mendoza and Végh 2010), or the extent to which public infrastructure investment can be self-financing (Perotti 2004). The empirical literature offering such quantitative assessments took off with the seminal work of Aschauer (1989) on the effects of public infrastructure capital on U.S. total factor productivity. The literature has boomed over the last two decades, with dozens of papers using a large variety of data and empirical methodologies, and with widely contrasting empirical results.<sup>2</sup> For example, Bom and Ligthart (2008) report that in a large set of empirical studies using industrialcountry data in a production function setting, estimates of the output elasticity of public capital range from -0.175 to +0.917.

However, much of the empirical literature on the contribution of infrastructure to aggregate output is subject to major caveats. Studies based on time-series have often ignored the non-stationarity of aggregate output and infrastructure capital, which typically display stochastic trends. This has sometimes led to implausibly high estimates of the productivity of infrastructure, owing to spurious correlation between both variables (Gramlich 1994).<sup>3</sup> In addition, empirical

> 1 See for example Turnovsky (1997), Glomm and Ravikumar (1997), Baier and Glomm (2001), and Ghosh and Roy (2004).

> 2 See for example Sánchez-Robles (1998); Canning (1999); Demetriades and Mamuneas (2000); Röller and Waverman (2001); Esfahani and Ramirez (2003); Calderón and Servén (2004). A recent overview of relevant empirical literature is provided by Romp and de Haan (2007).

> 3 One example is Aschauer‘s original estimate of the output elasticity of public capital, which was so high that the implied marginal product of infrastructure capital was close to 100% per year.

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studies also have to deal with potential simultaneity between infrastructure and income levels. For example, richer or faster-growing countries are likely to devote increased resources to infrastructure development. Failing to control for these and similar forms of reverse causality implies that estimates of the output elasticity of infrastructure may be confounded with the income elasticity of the demand for infrastructure services, and hence may suffer from upward biases.<sup>4</sup> Finally, studies using cross-section or panel macroeconomic data typically fail to account for the potential heterogeneity in the output elasticity of infrastructure across countries or states, which could arise from technological features such as network effects, scale economies and other factors that may affect the output elasticity of public capital.<sup>5</sup> This paper estimates the contribution of infrastructure to aggregate output using a production function framework including as inputs infrastructure assets, human capital, and noninfrastructure physical capital. We use a large cross-panel dataset comprising 88 countries and over 3,500 country-year observations, drawn from countries with very different levels of income and infrastructure endowments.

One distinguishing feature of our approach is that, in contrast with much of the earlier literature, we use physical measures of infrastructure rather than monetary ones – such as a public investment flow or its accumulation into a public capital stock. We do this for two reasons. First, as an abundant literature has argued, public expenditure can offer a very misleading proxy for the trends in the public capital stock, as the link between spending and capital is mediated by the extent of inefficiency and corruption surrounding project selection and government procurement practices, which can vary greatly across countries and over time (e.g., Pritchett 2000; Keefer and Knack 2007). Second, our interest here is infrastructure capital, rather than broader public capital, and in many countries the two can be very different owing to the involvement of the public sector in non-infrastructure industrial and commercial activities (a common occurrence in virtually all countries over a good part of our sample period), and due

> 4 A way out of this problem is to use a full structural model in the empirical estimation. In this vein, some empirical studies have used stripped-down versions of Barro‘s (1990) framework (e.g., Canning and Pedroni 2008). An alternative is to use some kind of instrumental variable approach, ideally featuring outside instruments for infrastructure. For example, Calderón and Servén (2004) employ demographic variables as instruments -- alone or in combination with internal instruments -- in a GMM panel framework. Roller and Waverman (2001) follow a similar approach.

> 5 In this vein, Gregoriou and Ghosh (2009) estimate the growth effects of public expenditure in a panel setting, and find that they exhibit considerable heterogeneity across countries.

3
also to the increasing participation of the private sector in infrastructure industries worldwide, especially since the 1990s.

The paper‘s approach allows us to tackle some of the main methodological problems of earlier literature and extend it in several dimensions. First, we take account of the multidimensionality of infrastructure<sup>6</sup> and, in contrast with the abundant literature that measures infrastructure in terms of an investment flow or its cumulative stock, or in terms of a single physical asset (such as telephone density), we consider three different types of core infrastructure assets – in power, transport and telecommunications – summarized into a synthetic infrastructure index, constructed through a principal-component procedure. Second, we use a panel cointegration approach to deal with the non-stationarity of the variables of interest and avoid the ̳spurious regression‘ problem of much of the earlier time-series literature. Third, to address concerns with identification and reverse causality, we establish that only one cointegrating relation exists among the variables, and that this applies to all the countries in the panel. We interpret this relationship as the aggregate production function, and verify that our infrastructure index and the other productive inputs are exogenous with respect to its parameters – the parameters of interest in our context, in the terminology of Engle, Hendry, and Richard (1983). We estimate these parameters using the Pooled Mean Group estimator of Pesaran, Shin and Smith (1999), which allows for unrestricted cross-sectional heterogeneity of the short-run dynamics while imposing homogeneity of the long-run parameters. Fourth, we deal explicitly with potential cross-country heterogeneity of the (long-run) parameters of the production function through individual and joint Hausman tests of parameter homogeneity, as well as through additional experiments that let the output elasticity of infrastructure vary with selected country characteristics.

Our estimates of the output elasticity of infrastructure lie in the range of 0.07 to 0.10.<sup>7</sup> Moreover, our estimates are very precise, and robust to the use of alternative econometric specifications and alternative synthetic measures of infrastructure. Likewise, the estimated elasticities of the other inputs – human and non-infrastructure physical capital – are in line with

> 6 Canning (1999) also considers the multidimensionality of infrastructure using three different physical measures; however, reverse causality issues are not addressed and the single cointegration rank hypothesis is imposed and not tested.

> 7 This is very close to the value that emerges from the meta-study by Bom and Ligthart (2008) of the output elasticity of public capital. After adjusting for publication bias, they place the output elasticity of public capital at 0.086.

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those found in the empirical macroeconomic literature (Bernanke and Gurkaynak, 2001; Gollin, 2002). They are also highly significant and robust to the various experiments we perform.

We also find little evidence of heterogeneity across countries in the output elasticities of the inputs of the aggregate production function. Specifically, the output elasticity of infrastructure does not seem to vary with countries‘ level of per capita income, their infrastructure endowment, or the size of their population.

The rest of the paper is organized as follows: Section 2 describes the dataset. Section 3 lays out the methodological approach. Section 4 describes the empirical results. Finally, section 5 concludes.

# **2. Data**

Our goal is to estimate the contribution of infrastructure capital to output in a large panel data set using an infrastructure-augmented aggregate production function framework, in which aggregate output is produced using non-infrastructure physical capital, human capital, and infrastructure. The data set is a balanced panel comprising annual information on output, physical capital, human capital, and infrastructure capital for 88 industrial and developing countries over the period 1960-2000, thus, totaling 3,520 observations. The Appendix lists the sample countries used in the analysis.<sup>8</sup> Real output is measured by real GDP in 2000 PPP US dollars from the Penn World Tables 6.2 (Heston, Summers, and Aten, 2006). The data on physical capital was constructed using the perpetual inventory method. To implement it, the initial level of the capital stock was estimated using data on the capital stock and real output from PWT 5.6 for those countries for which such data is available. We extrapolate the data for countries without capital stock information in PWT 5.6 by running a cross-sectional regression of the initial capital-output ratio on (log) real GDP per worker.<sup>9</sup> As a robustness check, we also constructed an alternative ̳back-cast‘ projection of capital per worker. Specifically, to construct an initial capital stock for the year 1960, we assume a zero capital stock in the distant past. Using the average in-sample growth rate of real investment

> 8 Country coverage is dictated by the availability of information. In particular, time coverage is limited by the human capital indicator, which is not available after 2000.

> 9 The regression used for extrapolation is: K/Y = -1.1257+0.2727*ln(Y/L), where K is the capital stock, Y is real GDP, and L is the labor force. The depreciation rate employed in the perpetual inventory calculations is 6%.

5
(1960-2000), we project real investment back to 1930. Next, ignoring the capital stock that may have existed in that year, we accumulate the projected real investment forward into a capital stock series. The resulting level of the latter in 1960 is then taken as the initial capital stock, and the in-sample capital stock series is constructed accumulating observed investment.<sup>10</sup> Our preferred measure is the capital stock series obtained from the PWT data. Assuming, as we do for the back-casting, that the pre-sample growth rate of real investment was equal to the average of the 1960-2000 period could be misleading, in view of the severe global shocks of the 1930s and 1940s (e.g. the Great Depression and World War II), which likely had a nonnegligible reflection on the rates of growth capital stocks around the world. Nevertheless, we also report estimates using the capital stock series constructed through back-casting, In turn, the stock of human capital is proxied by the average years of secondary schooling of the population, taken from Barro and Lee (2001). Finally, the labor input is proxied by the total labor force as reported by the World Bank‘s World Development Indicators.

Measuring physical infrastructure poses a challenge. Typically, the empirical literature on the output effects of infrastructure has focused on a single infrastructure sector. Some papers do this by design,<sup>11</sup> while others take a broad view of infrastructure but still employ for their empirical analysis an indicator from a single infrastructure sector.<sup>12</sup> In reality, ̳physical infrastructure‘ is a multi-dimensional concept that refers to the combined availability of several individual ingredients – e.g., telecommunications, transport and energy. In general, none of these individual ingredients is likely to provide by itself an adequate measure of the overall availability of infrastructure. For instance, a country may have a very good telecommunications network and a very poor road system or a highly unreliable power supply. In such situation, the availability of telecommunications services alone would provide a misleading indicator of the status of overall physical infrastructure.

However, attempting to capture the multi-dimensionality of infrastructure by introducing a variety of infrastructure indicators as inputs in the production function also poses empirical

> 10 The rationale behind this calculation is that the assumed level of the capital stock in 1930 has only a very minor effect on the capital stock that results for 1960, and hence it is immaterial whether we set the level of capital stock at zero or some other arbitrary level in 1930. 11 For example, Röller and Waverman (2001) evaluate the growth impact of telecommunications infrastructure, and Fernald (1999) analyzes the productivity effects of changes in road infrastructure.

> 12 In the empirical growth literature, for example, the number of telephone lines per capita is usually taken as the preferred indicator of overall infrastructure availability; see for example Easterly (2001) and Loayza, Fajnzylber and Calderón (2005).

6
difficulties. It could lead to an over-parameterized specification, and hence to imprecise and unreliable estimates of the contribution of the individual infrastructure indicators. In our framework this is a concern not only for the usual reasons of multicollinearity – indeed, several of the infrastructure indicators we shall use are fairly highly correlated --<sup>13</sup> but also because, as described below, we shall use a nonlinear procedure to estimate the parameters of the production function. In these conditions, a parsimonious specification with relatively few regressors is much more likely to result in stable estimates robust to alternative choices of initial values.

For these reasons, we follow a different strategy. We use a principal component procedure to build a synthetic index summarizing different dimensions of infrastructure.<sup>14</sup> We focus on three key infrastructure sectors: telecommunications, power and road transport. This choice is consistent with previous literature on the output impact of infrastructure, which has typically focused on one of these individual sectors, most often telecommunications. The synthetic infrastructure index is the first principal component of three variables measuring the availability of infrastructure services in these three sectors. Specifically, the variables underlying the index are:

- (a) _Telecommunications_ : Number of main telephone lines, taken from the International Telecommunications Union‘s World Telecommunications Development Report CDROM. As a robustness check, we also experiment with an alternative measure , namely the total number of lines (main lines and mobile phones), from the same source.

- (b) _Electric Power_ : Power generation capacity (in Megawatts), collected from the United Nations‘ _Energy Statistics_ , the United Nations‘ _Statistical Yearbook_ , and the U.S. Energy Information Agency‘s _International Energy Annual_ .<sup>15</sup>

- (c) _Roads_ : Total length of the road network (in kilometers), obtained from the International Road Federation‘s World Road Statistics, and complemented with information from

> 13 For instance, in our panel data set the full-sample correlation between the total number of phone lines (main and mobile) and overall power generation capacity is 0.92, while the correlation between total road length and overall power generation capacity is 0.65, and that between road length and main telephone lines is 0.61. In turn, the correlation between paved (as opposed to total) road length and power generation capacity is 0.83, while that between paved road length and main telephone lines is 0.84.

> 14 A similar approach is employed by Alesina and Perotti (1996) in their analysis of investment determinants, and by Sánchez-Robles (1998) to assess the growth effects of infrastructure.

> 15 The International Energy Annual (IEA) is the Energy Information Administration‘s main report of international energy statistics, with annual information on petroleum, natural gas, coal and electricity beginning in the year 1980. See webpage: http://www.eia.doe.gov/iea/

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national statistical agencies and corresponding national ministries.<sup>16</sup> To conduct robustness checks, we use two alternative measures of transport infrastructure: the length of the paved road network, collected from the same sources, and the combined total length of the road and railway network. The railway information is obtained from the World Bank‘s Railways Database and complemented with data from national sources.<sup>17</sup> As we shall impose constant returns to scale in the estimations (see below), the three variables underlying the index (phone lines, power generation capacity and the length of the road network) are measured in per-worker terms, and expressed in logs.<sup>18</sup> Their first principal component accounts for 82 percent of their overall variance and, as expected, it is highly correlated with each of the three individual variables.<sup>19</sup> More specifically, the correlation between the first principal component and main telephone lines per worker is 0.96, its correlation with power generation capacity is 0.97, and its correlation with the total length of the road network is 0.74. In addition, all three (log-standardized) variables enter the first principal component with approximately similar weights: where _z_ is the synthetic infrastructure index, _(Z1/L)_ is the number of main telephone lines (per 1,000 workers), _(Z2/L)_ is the power generation capacity (in GW per 1,000 workers), and _(Z3/L)_ represents the total length of the road network (in km per 1,000 workers).

Table 1 shows descriptive statistics for output, physical and human capital, and the various infrastructure indicators, for the cross-section corresponding to the year 2000. Output and the capital stock are expressed in PPP US dollars at international 2000 prices, while infrastructure variables are expressed in per worker terms.

> 16 One caveat regarding these data, as noted by Canning (1999), is that they may exhibit significant variations in quality. In particular, they do not reflect the width of the roads nor their condition. 17 The railways database can be found at http://go.worldbank.org/13EP3YJVV0

> 18 Before applying principal component analysis, the underlying variables are standardized in order to abstract from units of measurement.

> 19 As shown below, the individual infrastructure measures display stochastic trends, and hence we obtain the first principal component by computing the weights from the (stationary) first-differenced series.

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# **3. Econometric Methodology**

The core of our empirical analysis consists in estimating the following production function: where _y_ denotes real output, _k_ and _h_ represent physical capital and human capital, respectively, and _z_ denotes the infrastructure capital. All variables (except human capital) are expressed in log per worker terms (e.g. _kit_ = ln( _Kit_ / _Lit_ ) where _Lit_ represents the workforce) and, in keeping with the majority of earlier literature, constant returns to scale have been imposed. The subscripts _i_ and _t_ index countries and years, respectively;  i and  t capture country-specific and time-specific productivity factors, and  it is a random disturbance that will be assumed uncorrelated across countries and over time.<sup>20</sup>

## **3.1 Panel unit root testing**

Empirical assessments of the output contribution of infrastructure using time-series data have often failed to deal adequately with the non-stationarity of the variables. Here we address the issue using panel unit root and cointegration tests developed in the recent panel time-series literature. Unlike the traditional panel literature, which deals with samples in which the crosssectional dimension _N_ is large but the time dimension _T_ is small, the panel time-series literature is concerned with situations in which both _N_ and _T_ are of the same order of magnitude; see Breitung and Pesaran (2008) for an overview. As a considerable literature has shown, the panel time-series (or multivariate) approach to integration and cointegration testing yields higher test power than separate, conventional tests for each unit in the panel; see, for example, Levin, Lin and Chu (2002).

The first step is to test for the stationarity of the variables under consideration, namely, output, the stocks of physical and human capital, and the composite index of infrastructure capital, all (except for human capital) measured in logs per worker. As a preliminary step, we remove the cross-sectional means from the data, to render the disturbances cross-sectionally

> 20 As noted by Canning (1999), infrastructure appears twice in (1): first as _z_ , and then as part of overall physical capital _k_ . Hence the total elasticity of output with respect to infrastructure capital can be approximated as     , where  is the share of infrastructure in the overall physical capital stock. Evaluation of  requires data on the price of infrastructure, which are not widely available. Nevertheless, calculations based on the data of Canning and Bennathan (2000) suggest that it is a small number. For the countries in our sample with available data, its median is 0.08, and its standard deviation is 0.05.

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independent. To test for the presence of a unit root in each panel series, we employ the unit root test of Im, Pesaran and Shin (2003) ( _IPS_ ), which allows for heterogeneous short-run dynamics for different cross-sectional units. Specifically, the testing procedure averages the individual unit root test statistics.<sup>21</sup> The basic regression framework is the following: with the null hypothesis of non-stationarity H0:  i=1, for all _i_ , and the alternative H1:  i<1, for some _i_ . The test is based on a _t_ -statistic, defined as the average of the individual _ADF-t_ statistics, (3). The critical values are tabulated by Im, Pesaran and Shin (2003).<sup>22</sup>

# **3.2 Panel cointegration testing**

If the null of a unit root fails to be rejected, we next proceed to test for cointegration among the variables of interest. Several tests have been proposed in the literature for this purpose —e.g. McCoskey and Kao (1998), Kao (1999), and Pedroni (2004). However, all these tests simply evaluate the presence of cointegration and do not account for the potential existence of more than one cointegrating relationship. To assess the cointegration rank, we follow the approach of Larsson and Lyhagen (2000). Assume that the _p_ -dimensional vector _Yit_  ( _yit_ , _kit_ , _hit_ , _zit_ )' for country _i_ (where _p=_ 4 in our case and _i = 1,...,N)_ has an error correction model (ECM) representation (if the Granger representation theorem holds). We first test the hypothesis that each of the _N_ countries in the panel has at most _r_ cointegrating relationships among the _p_ variables. In other words, we test the null _H_ 0<sup>:</sup>  _i ri_  _r_ for all _i=1,...,N_ , against the alternative _Ha_ :  _i p_<sup>for all</sup><sup>_i=1,...,N_,where</sup>  _i_ is the number of cointegrating

> 21If the data are statistically independent across countries, under the null we can regard the average t-value as the average of independent random draws from a distribution with known expected value and variance (that is, those for a non-stationary series). This provides a much more powerful test of the unit root hypothesis than the usual single time series test. In particular, this panel unit root test can have high power even when a small fraction of the individual series is stationary. In this context, Karlsson and Lothgren (2000) find that the power of the IPS test increases monotonically with: (i) the number _N_ of cross-sectional units in the panel; (ii) the time dimension _T_ of each individual cross-sectional unit, and (iii) the proportion of stationary series in the panel.

> 22 It has been shown that the empirical size of the IPS test is fairly close to its nominal size when _N_ is small, and that is has the most stable size among the various panel unit root tests available (Choi, 2001). However, when linear time trends are included in the model, the power of the test declines considerably (Breitung, 2000; Choi, 2001).

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relationships present in the data for country _i_ . To conduct this test, Larsson and Lyhagen (2000) define the _LR-bar_ statistic as the average of the _N_ individual trace statistics of Johansen (1995),

> where _LRNT_ ( _H_ ( _r_ )| _H_ ( _p_ )) is the average of the individual trace statistics and _E_ ( _W_ ) and _Var_ ( _W_ ) are the mean and the variance of the variable _W_ , whose asymptotic distribution is the same as that of the individual trace statistic. The cointegrating rank suggested by the testing procedure based on the standardized _LR_ - _bar_ statistic equals the maximum of the N individual ranks. Further, under fairly general conditions, Larsson et al. (2001) show that the standardized LR-bar statistic for the panel cointegration rank test is asymptotically distributed as a standard normal.

Since the time dimension of our sample is too short for the asymptotic properties of the individual trace statistics to hold, in our empirical application we use Reimers‘ (1992) smallsample correction — _i.e._ we multiply the individual trace statistics by  _T_  _Lp_  / _T_ , where _L_ is the lag length used to construct the underlying VAR, and _p_ is the total number of variables.<sup>23</sup> Next, we follow Larsson and Lyhagen (2000) and test for the smallest cointegration rank in the panel using the panel version of the principal component test developed by Harris (1997). Specifically, we test the null hypothesis _H_ 0<sup>:</sup>  _i ri_  _r_ against the alternative _Ha_ :  _i ri_  _r_ **.** Thus, this hypothesis is the opposite of that used for the _LR-bar_ test in the sense that the alternative is that there are more than _r_ cointegrating vectors. For this purpose, Larsson and Lyhagen (2000) developed the standardized _PC-bar_ statistic: ˆ  2 _T_ ˆ ' ˆ *  1 ˆ where<sup>_~~c~~_</sup> _r_<sup>is the mean of the individual test statistics</sup> _cr_  _T_  _St_  _nn St_ , _E(W)_ and _Var(W_ ) are _t_  1 the mean and the variance respectively of the variable _W,_ whose asymptotic distribution is the

> 23 As done for the panel unit root tests, we remove the cross-sectional means from the data prior to implementing the panel cointegration tests.

11
ˆ ˆ _t_ ˆ ˆ * _T_  1  _<u>j</u>_  ˆ same as that of _cr_ , and we have defined _St_   _j_  1 _n j_ and  _nn_   _j_  _T_ 1 _k_   _ab_ ( _j_ ) , where _a_  _m_  and _b_ are any time series, _k(.)_ is a lag window, _m_ is the bandwidth parameter and ˆ  1 _T_  _ab_ ( _j_ )  _T_  _t_  1 _at_  _jbt_ . In large samples, the PC-bar test follows a standard normal distribution. If at least one of the individual ranks is less than the hypothesized value, the test asymptotically rejects the null. Hence, the _PC-bar_ statistic gives the minimum cointegration rank amongst all the cross-sectional units.

In short, we first use the _LR-bar_ test to estimate the maximum number of cointegration relations, and then we use the _PC-bar_ test to assess if for any country the number of cointegrating relations is less than the maximum given by the LR-bar test. If in the second step the null hypothesis cannot be rejected, the conclusion is that the number of cointegrating relations is the same for all cross-sectional units.

# **3.3 Heterogeneous panel data techniques**

As we report below, the panel cointegration tests indicate a common unit cointegration rank among GDP, physical capital, human capital and the composite index of infrastructure. We interpret the single cointegration vector (whose parameters may vary across countries) as a longrun production function. To estimate its coefficients, we adopt a single-equation approach. If there were more than one cointegrating relation, single-equation estimation would only determine a suitable combination of the various cointegrating relations. However, in the presence of a single cointegrating vector, Johansen (1992) shows that if the equations of the marginal model have no cointegration, the single-equation estimator is equivalent to the estimator resulting from system estimation of all the equations.<sup>24</sup> To estimate the coefficients in equation (1), we use the pooled mean group ( _PMG_ ) estimator developed by Pesaran, Smith, and Shin (1999).<sup>25</sup> In practical terms, we embed the production function equation (1) into an ARDL( _p,q_ ) model:

> 24 The single-equation analysis could be inefficient under certain circumstances (see Johansen 1992).

> 25 This estimator has been previously implemented in different contexts, for example Cameron and Muellbauer (2001) analyze the relationship between earnings, unemployment and housing in a panel of UK regions, while Égert et al. (2006) consider exchange rates, productivity and net foreign assets in a panel of countries.

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> where _i_  1,..., _N_ denotes the cross section units, and we impose homogeneity of the long-run ' coefficients  _i_    _i_ . Here _yi_   _yi_ 1<sup>,...,</sup> _yiT_  is the _T_ x _1_ vector that contains the _T_ observations of GDP for unit _i_ in the panel, _Fi_ = ( _ki_ , _hi_ , _zi_ ) is the _T_ x _3_ matrix of inputs (physical capital, human capital and infrastructure), and<sup></sup> _i_<sup>are the coefficients that measure the speed of adjustment</sup> towards the long-run equilibrium. Also,  is a _T_ x _1_ vector of ones and<sup></sup> _i_<sup>represents a country-</sup> ' specific fixed effect. The disturbances  _i_   _i_ 1<sup>,...,</sup>  _iT_  are assumed to be independently distributed across _i_ and _t_ , with zero means and country-specific variances  _i_ 2 0 . As before, all the variables are cross-sectionally de-meaned prior to estimation in order to remove common factors, as required by the assumption of cross-sectional independence.

As equation (4) makes explicit, the _PMG_ estimator restricts the long-run coefficients to be equal over the cross-section, but allows for the short-run coefficients, speed of adjustment and error variances to differ across cross-sectional units.<sup>26</sup> We therefore obtain pooled long-run coefficients and heterogeneous short-run dynamics. Thus, the _PMG_ estimator provides an intermediate case between full parameter homogeneity, as imposed by the dynamic fixed effects estimator, and unrestricted heterogeneity, as allowed by the mean group ( _MG_ ) estimator of Pesaran and Smith (1995), based on separate time-series estimation for each cross-sectional unit.

Estimation of the long-run coefficients in (4) is based on the concentrated log-likelihood function under normality. The pooled maximum-likelihood estimator of the long-run parameters is computed using an iterative non-linear procedure. Once the long run parameters have been computed, both the short-run and the error-correction coefficients can be consistently estimated running individual OLS regressions of <sup>_y_</sup> _i_ on _yi_ ,  1  _Fi_ ,  1  .

To test the validity of the long-run parameter homogeneity restrictions, we use Hausman tests of the difference between MG and PMG estimates of the long-run coefficients. These are preferable to likelihood ratio tests owing to the ̳large N‘ setting, which would cause the number of parameter restrictions to be tested by the likelihood ratio test to rise with sample size.

As described, the empirical strategy adopted in the paper is based on the single-equation estimation of the only cointegrating vector present in the data, which we interpreted as the

> 26 In the context of country-level production functions, it seems reasonable to allow for heterogeneity of the shortrun-dynamics due to, for instance, differences in adjustment costs across countries.

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aggregate production function. Our estimates of the output elasticity of physical capital, human capital and infrastructure, and the associated inference, are obtained from an equation describing the time path of GDP per worker, with the time path of the three inputs determined by some unspecified marginal model. For this approach to be valid, the inputs have to be weakly exogenous -- in the sense of Engle, Hendry, and Richard (1983) -- for the parameters of the cointegrating relation. In this context, weak exogeneity means that changes in the inputs (e.g. infrastructure) do not react to deviations from the long-run equilibrium, although each input may still react to lagged changes of both GDP per worker and the other inputs of the production function. If changes in the inputs did react to deviations from the estimated long-run equation, the implication is that the single equation used in the analysis could be capturing the demand for physical capital, human capital or infrastructure rather than the production function – or a combination of both.

The requirement that the inputs be weakly exogenous can be verified through a standard variable-addition test. Specifically, as shown by Johansen (1992) and Boswijk (1995), weak exogeneity for the long-run parameters can be checked by testing the significance of the cointegrating vector in a reduced-form regression of each input on its own past and those of output and the other inputs of the production function.

Formally, weak exogeneity of the inputs amounts to the requirement that the  coefficients not be significantly different from zero in the following system of equations (Johansen 1992): ˆ where  _ít_ ( )  _yi_ , _t_  1  _fi_ , _t_  1  is the estimated long-run equilibrium error term from equation (4) above, and  _fit_  1  ( _kit_  1,  _hit_  1,  _zit_  1) . We estimate the system of equations country by country using the SURE estimator proposed by Zellner (1962). Once we have all the country-specific SURE estimates we compute the Mean Group (MG) estimator (Pesaran and Smith 1995) and we carry out a Wald test under the null that the three coefficients on the added error terms are jointly zero. If soothe null is not rejected, we can conclude that the three inputs are weakly exogenous with respect to the parameters of the cointegrating relation.

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# **4. Empirical Results**

Our empirical implementation starts by checking the order of integration of the different variables and testing for the existence of cointegration among them. Then we turn to the estimation of the parameters of the cointegrating relation(s).

## **4.1 Integration and cointegration**

Table 2 reports the panel integration and cointegration tests. Panel A in the table shows the results of applying the panel unit root test of Im, Pesaran and Shin (2003) to each of the model‘s variables. In every case, the test statistic lies well below the 5% critical level, thus failing to reject the null of a unit root. Individual tests for each country (not shown) yield a similar verdict – they fail to reject the null in the overwhelming majority of cases. After taking first differences, however, the panel test (not reported) rejects the null of nostationarity for each of the variables. From this we conclude that all the variables are I(1).

We next test for cointegration. A battery of residual-based panel tests (not reported to save space), whose alternative hypotheses variously include homogeneous and heterogeneous cointegration (Kao, 1999; Pedroni, 1995, 1999), strongly support the view that the model‘s variables are cointegrated. However, as already noted, these tests are uninformative about the number of cointegrating relations, and with more than two I(1) variables under consideration, the possibility of multiple cointegration vectors cannot be ruled out. Further, in a panel context the possibility that different cross-sectional units may have different orders of cointegration cannot be dismissed either.

To assess the cointegration rank, we turn to the LR-bar test of Larsson, Lyhagen and Lothgren (2001) and the panel version of the PC-bar test of Harris (1997) proposed by Larsson and Lyhagen (2000). As already mentioned, we proceed in two stages. We first use the LR-bar test to establish the maximum cointegrating rank – i.e., the maximum number of cointegrating relations present in any of the panel‘s cross-sectional units (countries). We then use the panel version of the PC-bar test to establish the minimum cointegrating rank.

As Panel B of Table 2 reports, the LR-bar test overwhelmingly rejects the null that the maximum rank is zero (the test statistic of 9.03 is far above the 5% critical value of 1.96), but cannot reject a maximum rank of one. In turn, panel C shows that the PC-bar test cannot reject a 15
minimum cointegrating rank of one – the computed test statistic of 1.21 is well below the critical 5% value of 1.96.

Since the maximum cointegration rank from the LR-bar test and the minimum cointegration rank from the PC-bar test coincide, the null hypothesis of a common cointegrating rank for all countries in the panel cannot be rejected. Hence, taken together the test results imply that, for each of the sample countries, there exists one single cointegrating vector among the four variables in our study, which we shall interpret as the infrastructure-augmented production function.

# **4.2 Estimation results**

The next step is to estimate the parameters of the single cointegrating vector, which in principle might differ across countries. As discussed, we opt for the PMG estimator of Pesaran, Shin and Smith (1999), which estimates the coefficients of the long-run relation along with those characterizing the short-term dynamics. We use Hausman specification tests to assess the validity of the homogeneity restrictions imposed by PMG on the long-run parameters.

Table 3 reports a variety of PMG estimates of the long-run parameters, using alternative dynamic specifications – _i.e._ , different orders of the ARDL formulation of the equation of interest– and including time dummies to account for common factors (columns 1-5) or excluding them (column 6). The first thing to note is that, with the exception of the last column in the table, the parameter estimates in the different columns are very similar to each other. They are also very precisely estimated, which is unsurprising given the large number of observations (over 3,500) and the relatively parsimonious model employed.

In the first column, the order of the ARDL specification is determined (separately for each country) using the Schwarz criterion, subject to a maximum of two lags for both the dependent and independent variables. The estimated coefficient of the capital stock is 0.34, very close to the values commonly encountered in the empirical macroeconomic literature. The coefficient of the human capital variable is 0.10, likewise in the range of previous estimates in the literature **,** while that of the synthetic infrastructure index equals 0.08.<sup>27</sup> All three estimates

> 27 The estimated coefficient of human capital ranges from 0.08-0.085 in Bloom et al. (2004), to 0.11-0.13 in Temple (1998), and 0.06-0.11 in Miller et al. (2002). On the other hand, our estimated coefficient of infrastructure capital is similar to those reported by Le and Suruga 2005 (0.076 ), Eisner 1991 (0.077), Duffy-Deno et al. 1991 (0.081), and Mas et al. 1996 (0.086).

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are significantly different from zero at the 1 percent level. Further, the Hausman tests of parameter homogeneity, reported also in the table, show little evidence of cross-country heterogeneity of any of the individual parameters (all the p-values exceed 0.20). The same applies to the Hausman test of the joint null of homogeneity of all parameters, reported at the bottom of the table, whose p-value equals 0.44. The second column of Table 3 uses the Akaike information criterion rather than the Schwarz criterion to determine lag length, still subject to a 2-lag maximum. As is customary, this choice leads to somewhat more generous lag specifications, with a majority of countries selecting longer lag lengths than under the Schwarz criterion. However, it causes little change in the size or significance of the parameter estimates in the table, and it does not affect the Hausman tests of parameter homogeneity.

In turn, column 3 of the table imposes equal lag length (two lags) for all variables and countries, instead of allowing it to be determined by information criteria. Relative to columns 1 and 2, this results in a further loss of degrees of freedom, to an extent that depends on the country-specific number of lags that were being selected by the information criteria, and a slight deterioration of the precision of the estimates. However, it is of little consequence for the values of the estimates, their overall significance, or the verdict of the Hausman tests.

We next assess the effect of alternative choices of maximum lag length, using again the Schwarz criterion. Column 4 restricts the maximum lag length to 1. Relative to column 1, this adds 88 observations to the estimation sample, but is otherwise of little consequence for the coefficient estimates, their precision, and the Hausman tests. Column 5 summarizes the opposite exercise, raising maximum lag length to 4. This leads to the loss of 176 observations relative to column 1, but again there is no material change in any of the results.

Lastly, column 6 in Table 3 examines the role of common factors by re-estimating the specification in the first column omitting the time dummies. This does cause major changes in the parameter estimates: the coefficient of the capital stock rises above 0.40, and that of the infrastructure synthetic index becomes negative and insignificant. This confirms the importance of taking into account common factors (i.e., GDP and productivity shocks correlated across countries) in the estimation.

In Table 4 we explore the robustness of the results to the use of alternative measures of infrastructure and the capital stock. In all cases we employ the Schwarz criterion with a maximum lag length of 2 to select the dynamic specification. For ease of comparison, column 1 17
just reproduces the results from the first column of Table 3. In column 2, we replace the indicator of telephone density underlying the synthetic infrastructure index, using total phone lines (fixed plus mobile) instead of main lines, which is the variable conventionally employed in the growth literature. We recalculate the synthetic index as the first principal component of total phone lines, roads, and power generation capacity, all expressed in log per worker terms. This causes fairly modest changes in the estimates: the human capital parameter falls from 0.10 to 0.07, and overall precision declines somewhat, but there is little change in the infrastructure coefficient estimate and the results of the homogeneity tests.

Column 3 replaces road density with the density of land transport lines, including both roads and railways. As before, this leads to a new synthetic infrastructure index, but the estimation results obtained with it are virtually identical to those in the first column. In column 4, we use a narrower measure of roads, namely paved roads. The only noticeable change concerns the estimated coefficient of the human capital variable, which declines by half, while its standard error doubles. However, there is virtually no change in the other estimates. In turn, the Hausman tests now show some borderline evidence against the cross-country homogeneity of the coefficient of the human capital stock. Column 5 presents the results obtained replacing the principal-component index with an average index of infrastructure in which all the three infrastructure variables (roads, phone lines and electricity generating capacity) receive the same weight. Compared with our baseline specification in Column 1, the use of the average index causes practically no changes in the estimates.

Lastly, column 6 assesses the robustness of the results to the use of an alternative capital stock series, constructed through the back-casting method described earlier. Once again, the estimation results – including remarkably the coefficient on the capital stock itself -- are virtually indistinguishable from those in the first column of the table, although now there is some indication of cross-country heterogeneity of the capital stock coefficient.

Overall, these experiments suggest that the parameter estimates of the infrastructureaugmented production function are fairly robust to alternative specifications concerning the short-run dynamics as well as the precise choice of explanatory variables. Moreover, the experiments also reveal little evidence of cross-country heterogeneity in the output elasticity of infrastructure.

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However, the tests reported so far are concerned with unconditional heterogeneity, and it might be possible to gain power by testing for more specific forms of parameter heterogeneity. For example, it could be argued that, owing to network effects, the elasticity of output with respect to infrastructure should be higher in countries with larger infrastructure endowments than the rest.<sup>28</sup> Alternatively, the elasticity could vary with the level of development – as captured for example by GDP per worker – reflecting the fact that poorer countries are less able to use infrastructure effectively. As another hypothesis, the output elasticity of infrastructure could depend negatively on the size of the overall population, owing to congestion effects.

To verify this, we re-estimate the model in column 1 of Table 3 without imposing homogeneity across countries of the long-run parameter of the infrastructure synthetic index, and then look for patterns of heterogeneity in the individual-country estimates of that parameter along the three dimensions just mentioned – GDP, infrastructure endowment (both in per worker terms), and population size .

Figure 1 plots the resulting country-specific estimates of the infrastructure long-run coefficient against each of the three variables just mentioned. While there are some obvious outliers, the conclusion from all three graphs is clear: there is no relationship between the country-specific coefficient estimates and the three variables considered. This points to the absence of cross-country heterogeneity of the output elasticity of infrastructure along any of these dimensions.

Table 5 presents the results of more formal tests of parameter heterogeneity along these dimensions, using the country-specific estimates of the output elasticity of infrastructure obtained above. The first two columns of the table test if the output contribution of infrastructure varies across countries with their respective level of income per worker. We divide the countryspecific estimates into two groups, one consisting of countries with high income and the other of countries with low income. In column 1, the groups are drawn using the World Bank‘s list of ̳high income‘ countries; the low-income group is made up by all other countries in the sample. In column 2, the grouping is based instead on the sample median income per worker in the year 2000. In each case, the table reports the simple average of the parameter estimates of each of the two groups, along with the _p_ -value of the test of difference in group means. In both columns, the

> 28 This is similar in spirit to Gregoriou and Ghosh (2009), who let the growth contribution of public expenditure vary with countries‘ average level of expenditure.

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mean estimate is slightly larger in the low-income group, but the difference is small and statistically insignificant.

Column 3 of Table 5 reports a similar experiment distinguishing between countries with high and low infrastructure endowments, again defined by the sample median. The mean estimates of the infrastructure elasticity in the two groups are numerically and statistically very similar. Thus, there is little evidence that the output elasticity of infrastructure varies systematically with the degree of infrastructure development. Lastly, column 4 defines the two country groups according to country size, as given by population, with the sample median as the relevant dividing line. The output contribution of infrastructure might be expected to be larger in countries with smaller population, owing to congestion effects. The pattern of the mean estimates of the two groups seems to accord with this view: the mean estimate is much higher for small countries than for large ones (where it is actually negative and close to zero). However, the difference between the two falls well short of statistical significance.

In summary, our results indicate that the elasticity of GDP per worker with respect to the synthetic infrastructure index is around 0.08. This finding is robust to alternative econometric specifications and alternative definitions of the synthetic infrastructure index, as well as alternative measures of the capital stock. In addition, we find little evidence of heterogeneity of such elasticity across countries. Further experiments also suggest that the output contribution of infrastructure does not vary with countries‘ population, their level of income, or their infrastructure endowment. In light of our empirical specification, this suggests that cross-country variation in the marginal productivity of infrastructure is solely driven by variation in the ratio of infrastructure to output. In other words, the marginal product of infrastructure is higher wherever the (relative) infrastructure stock is lower.

Finally, we turn to the test of weak exogeneity of the production inputs described in section 3. The Wald test statistic computed from estimation of the system of equations (5) equals 5.97. Under the null of weak exogeneity of the three inputs, it follows a chi-square distribution with 3 degrees of freedom, and hence the test yields a p-value of 0.12, failing to reject the null<sup>29</sup> . Therefore we conclude that physical capital, human capital and infrastructure are weakly exogenous for the parameters of the cointegrating vector. This supports our

> 29 This corresponds to a baseline specification including two lags of all the variables. However, similar results were obtained with different lag specifications.

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interpretation that we are in fact estimating the production function instead of, for instance, an infrastructure demand equation, or a combination of both relations.

Our estimates of the output contribution of infrastructure are significant not only statistically, but also economically. To illustrate this, consider an increase in the level of infrastructure provision from the cross-country median in the year 2000 (an index of -4.65, roughly similar to the value observed in Tunisia in 2000) to the 75<sup>th</sup> sample percentile in the same year (an index of -3.69). This would translate in a 7.7 percent (=0.08*(-3.69+4.65)) increase in output per worker.<sup>30</sup> Similar calculations show that: (a) an increase in infrastructure provision from the median level observed among lower-middle income countries (-4.67, roughly equivalent to Bolivia in the year 2000) to that of the median upper-middle income country (- 4.02, Uruguay) would yield an increase in output per worker of 5.2 percent, and (b) raising the level of infrastructure provision from the value observed in the median upper-middle income country to that of the median high-income country (-2.93, corresponding to Ireland) would raise output per worker by 8.7 percent.

# **5. Conclusions**

This paper adds to the empirical literature on the contribution of infrastructure to aggregate output. Using an infrastructure-augmented production function approach, the paper estimates the output elasticity of infrastructure on a large cross-country panel dataset comprising over 3,500 annual observations. The paper addresses several limitations of earlier literature. It uses a multi-dimensional concept of infrastructure, combining power, transport and telecommunications infrastructure into a synthetic index constructed through a principal component procedure. The econometric approach deals explicitly with the non-stationarity of infrastructure and other productive inputs, reverse causality from output to infrastructure, and potential cross-country heterogeneity in the contribution of infrastructure (or any other input) to aggregate output.

The empirical strategy involves the estimation of a production function relating output per worker to non-infrastructure physical capital, human capital, and infrastructure inputs. Our estimates, based on heterogeneous panel time-series techniques, place the output elasticity of infrastructure in a range between 0.07 and 0.10, depending on the precise specification

> 30 Note that 0.08 is the estimated coefficient of the infrastructure index in regression [1] of Table 3.

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employed. The estimates are highly significant and robust to a variety of experiments involving alternative econometric specifications and different synthetic measures of infrastructure. Some illustrative calculations show that the output contribution of infrastructure implied by these results is also economically significant. Moreover, our estimates of the output contribution of human capital and non-infrastructure physical capital are likewise significant and broadly in line with those reported by earlier literature.

Lastly, tests of parameter homogeneity reveal little evidence that the output elasticity of infrastructure varies across countries. This is so regardless of whether heterogeneity is unconditional, or conditional on the level of development, the level of infrastructure endowments, or the size of the overall population. The implication is that, across countries, observed differences in the ratio of aggregate infrastructure to output offer a useful guide to the differences in the marginal productivity of infrastructure.

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