Evaluating the Effect of Public investment on Productivity Growth Using an Urban Economics Approach for the Spanish Provinces

Abstract

In this article, the authors examine whether variations in the level of public capital across Spain’s Provinces affect productivity growth over the 1985–2004 period. The analysis is motivated by contemporary urban economics theory, involving a production function for the competitive sector of the economy (industry), which includes the level of composite services derived from “service” firms under monopolistic competition. The authors extend the basic production function in the industry sector by explicitly introducing public capital stock and human capital in order to quantify its impact on regional economic growth. The model is also extended to include additional and necessary factors representing technological externalities and spillovers which are not represented in the basic model. The authors use spatial econometric panel data estimation techniques and when we control for temporal and regional specific effects, overall infrastructural endowment boosts regional labor productivity. The results are not significant when we disaggregate public capital stock into local and transport components, although the magnitude of the coefficient is greater in the former case.

Keywords

public capital urban economics spatial econometrics

Introduction

There has been a fair amount of debate and empirical work to date on the potential impact of public capital on private sector economic performance. Since the start of the current economic recession in August 2007, numerous renowned institutions and economists have suggested the need for an expansive fiscal policy as a means of mitigating the world’s economic recession. Macroeconomists typically emphasize three “conventional” channels through which public infrastructure may affect growth. It has a direct productivity effect on private production inputs and a complementarity effect on private investment. However in the short term, public investment could cause a crowding out effect on private spending through the financial system. An increase in the stock of public capital in infrastructure may have an adverse effect on activity, to the extent that it crowds out private investment. So, despite the direct and complementarity effects, the net effect of an increase in public infrastructure may well be to hamper, rather than raise, economic growth. The importance of each effect might depend on several factors like the initial stock of the economy, the diversity of productive structures, the degree of maturity of infrastructure systems, the quality of the capital stock in place, and on institutional and policy factors.

The empirical research conducted on the effects of public infrastructure is also inconclusive, although there is a general consensus on the need for a certain level of public infrastructural provision; the results obtained differ substantially once this level is achieved. Different perspectives and different econometric methods have been approached in the analysis. We can discern the cost function or dual approach, the estimation of autoregressive vectors (VAR) models, the frontier analysis,¹ and the production function approach. Under the dual approach, most of the analyses determine a positive impact of public investment in reducing the entrepreneur costs.² The evidence however is ambiguous under a VAR approach, where papers like Flores de Frutos, Gracia-Díez, and Pérez-Amaral (1998) and Batina (1999) find a positive effect, whereas Otto and Voss (1996) and Voss (2002) find a negative impact. The frontier analysis framework when it has been applied to the Spanish case always determined a positive effect.³ The approach with most numerous analyses is based on neoclassical production functions. The first author to emphasize the relationship between infrastructure and private productivity was Ratner (1983) for the US economy. He introduced public capital stock as an input in the aggregate production function and found that it had a small but significant effect on the level of production. Aschauer (1989a) also found a similar (albeit larger) effect, and by breaking public capital stock down into its constituent parts was able to show that the components with the greatest impact on productivity were transport infrastructure, energy, and water supply. Subsequently, Aschauer (1989b) estimated a panel data model for seven industrialized countries, obtaining similar results with a first differences specification. More recently, papers like Munell (1990, 1993), Ford and Poret (1991), Bajo-Rubio and Sosvilla-Rivero (1993), Otto and Voss (1994), and Mas et al. (1996) also determined a positive effect using this methodology. However, there are some papers that found a negative effect, such as McMillin and Smyth (1994), Otto and Voss (1996), and Voss (2002), and even papers providing no evidence of the effect of public investment on economic activity, including Tatom (1991), Batina (1999), Holz-Eakin (1994), Evans and Karras (1994), Baltagi and Pinnoi (1995), and García-Mila and McGuire (1992), among others. In recent papers, there is more consensus concerning a positive impact, but one much less significant than that estimated by Aschauer.

It has been argued in the literature that the apparently positive impact of public capital stock might be due to simultaneity problems between both variables, to the nonstationary of the series or to inadequate model specifications which could cause spurious relations or fail to appropriately control for region or country heterogeneity. Our approach has taken these considerations into account. To control for simultaneity issues, we estimate the model by instrumental variables techniques (2SLS), and as the variables in the estimated reduced form are in first differences the series are stationary. Regarding specification, we suggest that increasing returns and spillover effects are essential for a proper understanding of spatial disparities in economic development and need to be included in the specification. In order to incorporate imperfect competition, increasing returns and externalities in the form of market interdependence our model is rooted in contemporary urban economics theory.⁴ The existence of spillovers is another possible explanation for the inconclusive nature of the literature on the regional effects of public infrastructures. Most of the analysis has focused on whether public infrastructure has positive productive effects, paying relatively little attention to the existence of covariates spillovers, like the fact that productive public capital may shift economic activities from one location to another.

The article’s contribution is twofold: methodological and empirical. Its main contribution is the use of a different theoretical framework to quantify the impact of public investment on output. Our contribution to the existing literature is also methodological with regard to public capital spillovers, as they are usually incorporated into the model in an ad hoc manner, but their inclusion here is derived from the structural model itself. From an empirical perspective, as far as we are aware, this article is the first empirical application to estimate the effect of public infrastructure on output founded on the arguments of urban economics theory. Moreover, literature for the Spanish provinces (Nomenclature of Territorial Units for Statistics [NUTS] III) is very scarce and inconclusive regarding spillover effects.⁵ Although our model presents a limitation in assessing each covariate spillover, the new evidence we provide is that we control for spillovers in human capital and other sources of spatial externalities, like external shocks, ensuring the legitimacy of the covariate coefficients. Finally, our last contribution is to provide new results derived from different approach to the role of local and transport public capital investment to compare the results obtained in Moreno and López-Bazo (2007), the only paper that, for the Spanish provinces, had previously analyzed their different roles simultaneously.⁶ It is important to evaluate the effects of public investment on local and transportation infrastructures separately as they present different characteristics. Local investment is often referred to as point infrastructures, meanwhile transport infrastructures are defined as network infrastructures. The effects of each of them might be different either in terms of its own effect on productivity and in the spillovers they might produce. Positive spillovers are explained as the result of the network effects of public capital (e.g., roads, railways, etc.). However, as is stated in Puga (2002), one should not forget that roads have lanes going both ways, as transport infrastructures also makes easier for firms in richer regions to supply poorer regions at a distance, and can thus harm the industrialization prospects of less-developed areas.⁷ On the other hand, if factors of production are mobile across regions, one might observe negative interregional spillovers associated with regional infrastructure endowments. This is more likely to concern infrastructures which benefits are local, often referred to as “point infrastructures,” so this aggregate is more prone to determine negative spillovers.

Our results show that urban economics theory is a suitable theoretical approach to test Aschauer’s hypothesis. We provide new evidence of the positive effects of public investment on Spanish provincial growth, in line with Gómez-Antonio and Fingleton (2011). Changes in provincial output are positively associated with changes in public investment within the same province. When we disaggregate the effect of public capital stock among transport and local public capital stock, none of them is found to be significant although the local public capital stock coefficient is greater. In other words, with the necessary caution, the results indicate that local public stock presents a higher positive impact on productivity growth than public investment on transport. These results challenge those obtained in Delgado and Alvarez (2007) and Álvarez, Arias, and Orea (2006) for road infrastructures, but are partly consistent with Moreno and López-Bazo (2007), using a completely different approach, reinforcing the results obtained.

The article is divided into the following sections. The Model: Theoretical Background section presents the model specification for quantifying public capital stock on productivity, Data section describes the data sources, Estimation Procedure section defines the estimation procedure, Results section records and discusses the empirical results and Summary and Concluding Remarks section summarizes the main conclusions of the article.

The Model: Theoretical Background

At the core of this model is the concept of increasing returns to scale, which has become popular in recent years within both urban and geographical economics (Rivera-Batiz 1988; Abdel-Rahman and Fujita 1990; Quigley 1998; Fujita, Krugman, and Venables 1999; Fingleton 2003). All this literature enables increasing returns in the region or city while at the same time the decision problem for each actor is explicitly stated as one of profit or utility maximization. External scale economies can be derived from the increase in diversity or variety in producer inputs with increasing region size, even though firms are earning normal profits. The monopolistic competition model developed by Dixit and Stiglizt 1997 allows for an equilibrium solution in the context of competitive producers but with increasing returns to the economy as a whole.

The model in this article, following Rivera-Batiz (1988) and Abdel-Rahman and Fujita (1990), considers two sectors: industry (including manufacturing and traded services) and producer services, following the arguments of urban economic literature. The nontraded producer service sector (hereafter services) comprises local services that are not traded on national or international markets and are identified as the array of input requirements that industry demands, in terms of repair and maintenance, transportation and communication services, advertising, engineering and legal support, and so on. We will modify the production function in the industry sector by explicitly introducing public capital stock and human capital in order to quantify its impact on regional economic growth.⁸ It is assumed a monopolistic competition market structure for services, which is a direct result of the fact that the markets for services are generally highly competitive and face relatively minor entry and exit barriers, while consumers and producers have highly specialized demands, differentiating each service sector firm from others. Therefore, firms in the service sector are assumed to be typically numerous, small, independent, and heterogeneous. Industry on the other hand is assumed to have a competitive market structure and demands a wide array of different types of services performing highly specialized tasks.

We obtain a simple and empirically tractable reduced form linking growth in productivity level to output growth, human capital and public capital stock, similar to one of the extensions of the model estimated in Ciccone and Hall (1996). Due to the limitations of the basic theory, the model is extended by including additional and necessary factors representing technological externalities (e.g., knowledge spillovers) which are not represented in the basic model outlined above.

The Dixit and Stiglizt (1997) theory of monopolistic competition provides the reason why an increase in service labor determines an increase in service variety, rather than more of the same variety. There is no point in a variety’s production being split due to the existence of fixed costs, firms prefer to concentrate on producing a single variety and achieve internal economies of scale. As each firm produces its own differentiated services, the ensuing monopoly power allows prices to be a mark up on marginal cost. The number of firms supplying services is an endogenous variable in the model instead of being an ad hoc restriction. There is an equilibrium level of output and therefore equilibrium labor requirement per service firm that is a constant; these equilibrium values depend on exogenous parameters.

We follow Fingleton (2004) in order to get to an empirical tractable reduced form, and we develop his model by introducing public capital stock and human capital as additional inputs in the production function of industry sector. We derive the reduced form linking output (Q) to the intensity of activity in a unit area given by the total labor force (N), the amount of public capital stock ( $K_{P}$ ), the level of human capital (H) and land (L).

By substituting the level of composite services I in the Cobb-Douglas production function, we obtain industry production technology, where industry output (Q) is a function of the input of industry labor M, I, $K_{P}$ , H, and L. Note that industry is competitive, with constant returns to scale.

Q = (M^{β} I^{1 - β})^{α} K_{P}^{υ} H^{ψ} L^{1 - α - υ - ψ} .

Following Rivera-Batiz (1988), ample evidence tends to show that, broadly speaking, services are relatively labor-intensive. We thus assume, for simplicity, that each firm producing composite service uses only labor as an input,⁹ the requirements of which are given by

{\tilde{L}}_{i} = a i (d) + s

Where s represents the fixed labor input requirement and a the marginal input requirement. Following the Chamberlinian framework, the technology used by all firms is considered identical, implying that a and s are the same for all composite service sector firms.

Following the same steps as in Fingleton (2004), utilizing equilibrium values we get:

\begin{aligned} Q = {[{(β N)}^{β} {[{(\frac{(1 - β) N}{a i (d) + s})}^{μ} i (d)]}^{1 - β}]}^{α} K_{P}^{υ} H^{ψ} \\ Q = N^{β + μ α (1 - β)} β^{β α} i (d)^{α (1 - β)} (a i (d) + s)^{- μ α (1 - β)} K_{P}^{υ} H^{ψ} = N^{γ} Φ K_{P}^{υ} H^{ψ}, \end{aligned}

Where:

\begin{aligned} γ = β α + μ α (1 - β) = α [1 + (1 - β) (μ - 1)] \\ Φ = β^{β α} i (d)^{α (1 - β)} {(a i (d) + s)}^{- μ α (1 - β)} {(1 - β)}^{μ α (1 - β)} . \end{aligned}

In which i(d) is the “typical” output of a service variety, and there are D varieties. The level of monopoly power in the service sector is given by the exogenous parameter µ.¹⁰

Increasing returns to scale are implied by $γ > 1$ , from equation (4) this will occur if $β$ < 1 and µ > 1, provided α is not too small. With a high enough μ and a sufficient low β, the production function could have increasing returns, where the favorable effect of density prevails over the congestion effects. This means that assuming that the loss of production resulting from restricting land is not too large, we also need intermediate goods to be relevant for the output of the competitive sector (β < 1) and them to be imperfect substitutes of each other with a degree of monopoly power (μ > 1).

In order to move closer to a convenient reduced form, we log-linearize equation (3) by taking natural logarithms, hence

L o g \frac{Q}{N} = \frac{L o g Φ}{γ} + \frac{γ - 1}{γ} L o g Q + \frac{υ}{γ} L o g K p + \frac{ψ}{γ} L o g H .

Assuming that the economy is at a competitive equilibrium and workers are paid the value of their marginal product, M/N = β

L o g \frac{Q}{M} = \frac{L o g Φ}{γ} + \frac{γ - 1}{γ} L o g Q + \frac{υ}{γ} L o g K p + \frac{ψ}{γ} L o g H - L o g (β) .

In order to model the technical progress rate, we consider the labor units to be efficiency labor units, assuming that the number of labor efficiency units (M) is equal to total employment (E) times the level of efficiency (A):

M_{t} = E_{t} A_{t} = E_{t} A_{0} e^{λ t},

Where E_t is the level of total employment and A_t is the efficiency level at time t that is determined by the initial level A ₀ and the rate of technical progress λ. It is possible to express the model in terms of the level of manufacturing productivity per unit area:

Log \frac{Q}{E_{t}} = \frac{Log Φ}{γ} + \frac{γ - 1}{γ} Log {(Q)}_{t} + \frac{υ}{γ} Log {(K p)}_{t} + \frac{ψ}{γ} Log {(H)}_{t} - Log (β) + Log (A_{0}) + λ t .

Taking first difference with respect to time gives the exponential growth rates, hence:

\frac{\dot{Q}}{E} = \frac{γ - 1}{γ} \dot{Q} + \frac{υ}{γ} \dot{K p} + \frac{ψ}{γ} \dot{H} + λ .

The Rate of Technical Progress

To model the rate of technical progress, it is assumed that it depends on within-region effects, captured by a constant $b_{0}$ , and effects that spill over from “neighboring” regions. Spillovers are captured by a weights matrix, W, which is a square n × n matrix for n regions with cell values denoting the strength of interregional interaction.¹¹

Combining the outlined factors produces the following specification:

λ_{i} = b_{0} + ρ \sum_{r = 1}^{R} W_{i γ} λ_{γ} + ϵ .

Or in matrix terms, and rearranging

λ = (I - ρ W)^{- 1} (b + ϵ) .

In which

ϵ

is an independent and identically distributed disturbance term representing measurement error and exogenous shocks to the level of efficiency, hence

ϵ ~ N (0, Ω^{2})

The contribution to area i’s efficiency level is given by row i of vector W λ, which contains the sum of the weighted efficiency levels of all other provinces. Note that by making λ depend on W λ and not simply the constant, we capture all the effects influencing the efficiency level, including those represented by random shocks. This specification creates an endogenous spatial lag and therefore requires spatial econometrics methodology.

Combining equations (9) and (11) produces the following specification:

\frac{\dot{Q}}{E} = \frac{γ - 1}{γ} \dot{Q} + \frac{υ}{γ} \dot{K p} + \frac{ψ}{γ} \dot{H} + (I - ρ W)^{- 1} (b + ϵ) .

And substituting and arranging terms, we arrive at the model reduced form to be estimated:

\frac{\dot{Q}}{E} = b_{0} + ρ W \frac{\dot{Q}}{E} + \frac{γ - 1}{γ} [\dot{Q} - ρ W \dot{Q}] + \frac{υ}{γ} [\dot{K p} - ρ W \dot{K p}] + \frac{ψ}{γ} [\dot{H} - ρ W \dot{H}] + ϵ,

which can also be expressed as a model with spatial dependence in the error term, a Spatial Error model (SEM):

\begin{matrix} \frac{\dot{Q}}{E} = b_{0}^{^{'}} + \frac{γ - 1}{γ} \dot{Q} + \frac{υ}{γ} \dot{K p} + \frac{ψ}{γ} \dot{H} + u \\ u = ρ W u + ϵ \end{matrix}\},

with

b_{0}^{^{'}} = {[I - ρ W]}^{- 1} b_{0}

Data

The empirical model was fitted to the 1985–2004 period for Spanish provinces and all the variables are measured in yearly rates of growth.¹² The period analyzed is particularly relevant, as the Spanish economy has undergone a sustained period of growth over the last forty years, with a large increase in public investment figures. At the beginning of the period, the level of government capital endowment and economic activity in the Spanish regions was far below those of other European economies. However, after Spain joined the European Union, there was a very intensive period of capital investment by the Spanish government with no perceptible effect due to economic cycles. Spain benefited from growing funds assigned to finance infrastructure projects in less-developed regions in order to promote growth and cohesion within the countries of the European Union.

Province r productivity level is created as Gross Value Added, in industry sectors (including building and energy activities¹³), divided by r’s industrial employment. The data were provided until 1997 by Fundación Banco Bilbao Vizcaya Argentaria¹⁴ (FBBVA) and thereafter by Fundación de las Cajas de Ahorro Confederadas (FUNCAS) 2005 Growth of production (Q) is constructed using Gross Value Added in industry sectors.

Human capital variable (H) is the proportion of people in each province with higher education, according to information published in Human Capital in Spain and its Distribution by Provinces (1964-2004) by Instituto Valenciano de Investigaciones Económicas (IVIE).

Finally, productive public capital stock (Kpb) was taken from the publication “Capital Stock in Spain and its distribution by territories (1964–2003)” ¹⁶ which detailed work done by FBBVA in collaboration with IVIE. This variable is essentially composed of two elements, namely transport infrastructure and local public capital stock, as they are assumed to be the productive part of overall public capital stock. Transport includes airports, ports, road, and railway infrastructures, and local public capital stock comprises local government infrastructures of various kinds, infrastructure relating to water supply and management, plus other residual investments.

Figure 1 shows the evolution of average province productivity for 1985–2004. It suggests the existence of some structural breaks, at least around 1993 and/or 2000. For a proper selection of the breaks, we make use of the SupF test suggested by Andrews (1993) for a structural break test with an unknown break point. If the existence of a structural break is identified, the estimation of the break points is pursued using the work of Bai and Perron (1998, 2003). Results are shown in Figure 2.

Figure 1

Evolution of average province productivity (thousands of constant [2000] euros per worker; T = 20).

Figure 2

Detection of structural break points. A. F-chow tests for all potential change points. B. Determination of the optimal number of break points. C. Selected break points: 1987, 1990, 1993, and 2001.

First, Figure 2A shows the set of F-Chow statistics calculated for all potential change points between 1987 and 2001, together with the boundaries corresponding to a SupF test at the 5 percent significance level. As the F statistics are over their boundary, there is evidence of some structural changes, at the 5 percent significance level. Next, in order to select the number of breakpoints, Figure 2B shows the Bayesian Information Criterion (BIC) and the Residual Sum of Squares criteria of partitions corresponding with the different number of possible breakpoints. For both criteria, the minimum is reached when using four breakpoints which, as shown in Figure 2C, correspond to 1987, 1990, 1993, and 2001.

Table 1 shows the annual average of productivity growth rates and the covariates involved in our models. Results refer both to the period (1985–2004) and to the different subperiods derived when we consider the possible structural breaks in 1987, 1990, 1993, and 2001.

Table 1.

Annual Average Growth Rates for All the Variables of the Models (Percent)

(n = 47)		$- 5 p t \frac{\dot{Q}}{E}$	$\dot{Q}$	Total $\dot{K p}$	Local $\dot{K p}$	Transport $\dot{K p}$	$\dot{H}$
1985–2004	M	1.95	3.25	4.39	4.29	5.19	5.39
1985–2004	SD	0.49	0.89	0.89	0.96	1.16	1.69
1985–1987	M	3.76	4.97	3.99	4.59	3.37	4.70
1985–1987	SD	2.82	3.24	1.54	1.20	1.43	8.48
1987–1990	M	3.72	5.98	5.39	5.40	6.61	5.22
1987–1990	SD	2.49	2.23	1.38	1.28	3.22	7.44
1990–1993	M	0.92	−1.19	4.70	5.17	5.95	5.07
1990–1993	SD	1.60	2.00	1.09	0.88	2.65	6.45
1993–2001	M	1.02	3.47	4.11	3.81	4.92	6.09
1993–2001	SD	0.89	1.07	0.88	0.83	1.39	3.13
2001–2004	M	2.46	3.25	4.09	3.36	4.92	4.51
2001–2004	SD	1.23	1.35	1.49	2.17	2.34	4.17

The highest productivity growth rate corresponds to the first two subperiods, 1985–87 and 1987–90. However, it then decreases until 2001. Finally, the productivity growth rate almost returns to its previous magnitudes for the last subperiod. As Table 1 shows, public capital stock shows the highest growth rates in the second subperiod, 1987–90, while the highest human capital variable growth rate takes place between 1993 and 2001.

Figure 3 depicts the spatial distribution of annual productivity growth rates for the entire period (1985–2004) and for the different subperiods. The growth rate pattern also differs considerably among the different subperiods. Until 1990, the highest growth rates correspond to the provinces located in center of Spain. Afterward, between 1990 and 1993, on one hand, and especially between 2001 and 2004, on the other, productivity growth was higher for the southern provinces. Finally, from 1993 to 2001, the highest growth rate corresponds to the eastern provinces.

Figure 3

Spatial distribution of annual productivity growth rates between several time periods (n = 47).

In order to capture spillover and estimate the models, we must specify one or several weight matrices (W matrix) reflecting the network of cross-sectional relationships in the system of provinces. We develop a mixed neighborhood criterion based on the distance between the geometric centers of the different provinces, and qualified it by incorporating the r nearest neighbors to avoid situations of excessive imbalance. Thus, binary matrix W ₁ ^b is defined as:

w_{i j}^{b} (k, r) = \{\begin{matrix} i f min_{i \neq s} \{d_{i s}\} > k \Rightarrow \{\begin{matrix} w_{i j}^{b} (k, r) = 1 i f j \in N_{r} (i) \\ w_{i j}^{b} (k, r) = 0 i f j \notin N_{r} (i) \end{matrix} \\ i f min_{i \neq s} \{d_{i s}\} \leq k \Rightarrow \{\begin{matrix} w_{i j}^{b} (k, r) = 1 i f d_{i j} \leq k \\ w_{i j}^{b} (k, r) = 0 i f d_{i j} > k \end{matrix} \end{matrix},

where d_ij is the distance in kilometers between the centroids of provinces i and j, k the interaction radii, and N_r(i) the set of the r regions closest to region i. As usual, w_ii = 0 for all i. The resulting binary matrix W^b has been row-standardized, obtaining the weight matrix denoted as W ₁. We present the results obtained for k = 250 and r = 1, but we have confirmed that the results are consistent with other choices. Furthermore, in order to check the robustness of our results to the selected W ₁ matrix, we will also use two alternative weight matrices frequently used in spatial econometrics literature. A second weight matrix, W ₂, is obtained after row-standardizing the matrix whose weights are the inverse of the square distance between any pair of provinces, reflecting the fact that distance between provinces negatively affect interactions:

{w_{2}}_{i j}^{} = \frac{1}{d_{i j}^{2}} .

We constructed following Fingleton (2001) a slightly different weight matrix, W ₃, which takes into account not only distance between provinces but also the economic size of them. Economic size (gross domestic product [GDP]) is considered relevant because of the extensive trade and labor market that a large and diverse local economy naturally generates. With this matrix, we take into account that economic similarities between provinces can increase interactions.

{w_{3}}_{i j}^{} = \frac{Q_{j}}{d_{i j}^{2}} .

To check for spatial dependence in the variables, we calculate for the different specification of the weight matrix, the set of standardized Moran I statistics. Results are shown in Table 2, and in all cases, the null hypothesis of no spatial dependence is clearly rejected.

Table 2.

Moran I test for Testing the Null Hypothesis of No Spatial Autocorrelationa

(T = 20, n = 47)	W₁	W₂	W₃
$\frac{\dot{Q}}{E}$	19.69*	21.29*	17.89*
$\dot{Q}$	30.75*	33.69*	28.28*
Total $\dot{K p}$	43.89*	53.91*	46.60*
Local $\dot{K p}$	42.25*	51.56*	45.18*
Transport $\dot{K p}$	38.53*	47.44*	41.18*
$\dot{H}$	7.19*	8.67*	8.03*

Finally, the correlation matrix among the variables is displayed in Table 3. Productivity growth is mainly correlated with GDP growth rate. Correlations among the different public capital stock variables are reflecting the higher share of local public capital stock on the total one. Human capital rate of growth does not present a high correlation with any of the other variables.

Table 3.

Correlation Matrix of the Variables of the Model

(T = 20, n = 47)	$- 5 p t \frac{\dot{Q}}{E}$	$\dot{Q}$	Total $\dot{K p}$	Local $\dot{K p}$	Transport $\dot{K p}$	$\dot{H}$
$\frac{\dot{Q}}{E}$	1.000	0.597	0.143	0.159	0.017	0.024
$\dot{Q}$	0.597	1.000	0.216	0.092	0.077	0.013
Total $\dot{K p}$	0.143	0.216	1.000	0.826	0.507	−0.018
Local $\dot{K p}$	0.159	0.092	0.826	1.000	0.152	−0.012
Transport $\dot{K p}$	0.017	0.077	0.507	0.152	1.000	0.002
$\dot{H}$	0.024	0.013	−0.018	−0.012	0.002	1.000

Estimation Procedure

In this section, we focus on defining the estimation procedure for the SEM specified in equation (14), considering that we are working with a panel data set. From this perspective, we can refer to the most ample panel data model, called the two-way model, which, in matrix notation, has the following expression:

\begin{aligned} \begin{matrix} y_{t} = x_{t} β + δ + {θ_{t} + u}_{t} \\ u_{t} = ρ W u_{t} + ϵ_{t} \\ ϵ_{t} ~ N [0, σ_{}^{2} I_{R}] \end{matrix}\} \\ w i t h y_{t} = [\begin{matrix} y_{1 t} \\ y_{2 t} \\ y_{3 t} \\ ⋮ \\ y_{R t} \end{matrix}]; x_{t} = [\begin{matrix} 1 & x_{21 t} & \dots & x_{k 1 t} \\ 1 & x_{22 t} & \dots & x_{k 2 t} \\ 1 & x_{23 t} & \dots & x_{k 3 t} \\ ⋮ & ⋮ & \dots & ⋮ \\ 1 & x_{2 R t} & \dots & x_{k R t} \end{matrix}]; β = [\begin{matrix} β_{1} \\ β_{2} \\ β_{3} \\ ⋮ \\ β_{k} \end{matrix}]; δ = [\begin{matrix} δ_{1} \\ δ_{2} \\ δ_{3} \\ ⋮ \\ δ_{R} \end{matrix}], \end{aligned}

Where

y = \frac{\dot{Q}}{E}

x = [\dot{Q}, \dot{K p,} \dot{H}]

, i is an index for the spatial unit (a province in our case), with i = 1,…, R, and t is an index for the time dimension, with t = 1,…, T. In relation to panel data, model (18) includes the terms δ_i, θ_t which denote, respectively, the unobservable spatial and time specific effect. In fact, δ_i, θ_t represent the effect of omitted variables peculiar to each spatial unit and time period, respectively. The spatial dependence among spatial units is considered in model (18) through the error term, by including the spatial autocorrelation coefficient, ρ. This specification implies that the value of the residuals in one province depends on the value of the residuals in the neighboring provinces, and this process is seen as nuisance spatial dependence.

Other restricted versions of the complete model (18) are one-way models consisting of considering only the unobservable spatial unit, δ_i, or only the unobservable time specific effect, θ_t, as follows:

\begin{aligned} \underline{S p a t i a l F E} \underline{T i m e F E} \\ \begin{matrix} y_{t} = x_{t} β + δ + u_{t} \\ u_{t} = ρ W u_{t} + ϵ_{t} \\ ϵ_{t} ~ N [0, σ_{}^{2} I_{R}] \end{matrix}\} \begin{matrix} y_{t} = x_{t} β + {θ_{t} + u}_{t} \\ u_{t} = ρ W u_{t} + ϵ_{t} \\ ϵ_{t} ~ N [0, σ_{}^{2} I_{R}] \end{matrix}\} \\ (19) (20) \end{aligned}

One important issue in modeling panel data sets refers to the nature of those effects omitted from the specification (Mundlak 1978). Discussion about random or fixed effects appears routinely in all panel estimations but it is really important if we are using spatial data. The situation may be summarized according to two different positions.

On one hand, models including a spatial structure need a very large sample (a large R, number of provinces, in our case), because the convergence results are obtained with R tending to infinite. But, on the other hand, if the omitted effects are nonrandom, a problem of incidental parameters appears (the number of parameters grows at the same rate as the number of observations); in that case, a large T and small R are preferable. The last observation leads Anselin, Le Gallo, and Jayet (2008) to discard the use of fixed effects in mechanisms of spatial dependence: “Since spatial models rely on asymptotics in the cross-sectional dimension (…), this would preclude the fixed effects model from being extended with a spatial lag or spatial error term.” These authors prefer the random effect framework; where the inference is conditional and we only need a very large R (the improvements with T are of minor importance).

Elhorst (2003) does not share that view when he states that “The spatial units of observation should be representative of a larger population, and the number of units should potentially be able to go to infinity in a regular fashion. Moreover, the assumption of zero correlation between μ_r and the explanatory variables is particularly restrictive. Hence, the fixed effects model is compelling, even when R is large and T is small.”

Both approaches are contradictory and reflect strong methodological positions with evident implications. In our article, we support Elhorst’s (2008) point of view, as we believe that the unobservable effects (or the omitted variables they are representing) are probably correlated with our explanatory variables. Hence, we will directly specify and estimate the Fixed Effect (FE) panel data models with spatial dependence in the error term (FE-SEM model).

Two main approaches have been suggested in the literature to estimate FE-SEM models. One is based on the maximum likelihood (ML) principle and the other on instrumental variables or generalized method of moments (IV/GMM) techniques. We briefly describe the main advantages and disadvantages of both techniques.

ML estimation makes two main important assumptions regarding the normality of the errors, on one hand, and the exogeneity of the explanatory variables of the model, on the other. However, it assures that the spatial autocorrelation parameter, ρ, is restricted to its parameter space by the Jacobian term in the log likelihood function. ML estimation is based on the maximization of the following log likelihood function:

\begin{aligned} - \frac{R T}{2} log ({2 π σ}_{}^{2}) + T log |I_{R} - ρ W| - \frac{1}{2 σ_{}^{2}} \sum_{t = 1}^{T} {e_{t}}^{'} e_{t}^{} \\ w i t h e_{t}^{} = (I_{R} - ρ W) [y_{t}^{*} - x_{t}^{*} β], \end{aligned}

Where

y_{t}^{*}

and

x_{t}^{*}

refer to the demeaned variables, which are calculated differently depending on which FE panel data model we are estimating (models 16, 19 or 20). Following, we refer to the procedure for obtaining the demeaned variable

y_{t}^{*}

in each of the three FE panel data models, and the technique can be extended to derive the demeaned explicative variables comprising the

x_{t}^{*}

matrix.

Elements for the demeaned variables $y_{t}^{*}$ in model (16) are calculated as follows:

y_{i t}^{*} = y_{i t} - \frac{1}{T} \sum_{t = 1}^{T} y_{i t} - \frac{1}{R} \sum_{i = 1}^{R} y_{i t} + \frac{1}{R T} \sum_{i = 1}^{R} \sum_{t = 1}^{T} y_{i t} .

While the elements of

y_{t}^{*}

and

x_{t}^{*}

when estimating spatial fixed effect in equation (19) read as:

y_{i t}^{*} = y_{i t} - \frac{1}{T} \sum_{t = 1}^{T} y_{i t} .

And finally, demeaned variables when considering the time specific fixed effect panel data model in equation (20) are obtained through the following transformation:

y_{i t}^{*} = y_{i t} - \frac{1}{R} \sum_{i = 1}^{R} y_{i t} .

The IV/GMM technique presents the disadvantage of possibly ending up with a coefficient estimate for ρ outside its parameter space. The main advantage of this technique comes when the estimated model includes any endogenous variables. In this case, the three-stage procedure proposed by Kelejian and Prucha (1998) can be applied to estimate the SEM (without FE). In the first stage, the model is estimated by 2SLS taking care of the endogenous variables of the model. The second stage uses the resulting 2SLS residual to estimate ρ and σ² using a GM procedure. In the third stage, the estimated ρ is used to perform a Cochrane-Orcutt transformation to account for the spatial dependence in the residuals. Finally, as regards the estimation procedure for the FE-SEM, a similar three-stage procedure can be carried out. However, in this case, it is necessary to make use of the 2SLS procedure for FE panel data set described, for instance, in Baltagi (2005), which also makes use of demeaning transformation as described above.

Results

In The Model: Theoretical Background section, we theoretically derived a model for explaining industrial productivity growth ( $\frac{\dot{Q}}{E}$ ) through the growth rates of industrial Gross Value Added ( $\dot{Q}$ ), public capital stock ( $\dot{K p}$ ), and human capital ( $\dot{H}$ ). Furthermore, public capital stock has been broken down into two main components, local and transportation. We estimated one model with each of these variables and another one with both of them included.

Attention has been paid to the causality relationships between the variables involved in the model. To this respect, we calculate the statistic proposed by Hurlin (2007) for testing the null of Granger noncausality between pair of variables. Results are gathered in Table 4.

Table 4.

Testing the Null of Granger Noncausality for Panel Data^a

(T = 20, n = 47)	Included lags = 1	Included lags = 2	Included lags = 3
H₀: $\frac{\dot{Q}}{E}$ not cause $\dot{Q}$	4.82*	7.41*	6.80*
H₀: $\dot{Q}$ not cause $\frac{\dot{Q}}{E}$	8.30*	9.58*	13.96*
H₀: $\frac{\dot{Q}}{E}$ not cause total $\dot{K p}$	2.96*	0.51	0.61
H₀: Total $\dot{K p}$ not cause $\frac{\dot{Q}}{E}$	10.20*	6.58*	4.84*
H₀: $\frac{\dot{Q}}{E}$ not cause local $\dot{K p}$	1.85	−0.11	0.77
H₀: Local $\dot{K p}$ not cause $\frac{\dot{Q}}{E}$	6.39*	5.33*	2.82*
H₀: $\frac{\dot{Q}}{E}$ not cause Transport $\dot{K p}$	3.60*	2.78*	12.32*
H₀: Transport $\dot{K p}$ not cause $\frac{\dot{Q}}{E}$	7.37*	5.31*	8.95*
H₀: $\frac{\dot{Q}}{E}$ not cause $\dot{H}$	2.69*	2.11*	0.63
H₀: $\dot{H}$ not cause $\frac{\dot{Q}}{E}$	6.45*	3.84*	3.87*

^aAn asterisk means that the null hypothesis is rejected at the 5 percent level of significance

Figures on Table 4 refer to the statistics denoted by Hurlin (2007) as ${\tilde{Z}}_{N, T}^{H n c}$ , which follows the standard normal distribution N(0,1), for a large R sample and under the homogenous noncausality null hypothesis. Such statistic is built using the standard Wald statistics for testing the Granger noncausality relationship for each of the R regions. As a consequence, if the absolute value of ${\tilde{Z}}_{N, T}^{H n c}$ is superior to the normal corresponding critical value for a given level of risk, the homogenous noncausality hypothesis is rejected. From results displayed in Table 4, past values of total and local public capital growth rates are useful when one comes to forecast productivity growth in at least one region of the panel, but not the opposite. The same result is obtained for the human capital variable when using three lags in the calculation of the statistic. As a consequence, it is expected that total and local public capital together with human capital variables affect/cause the level of productivity growth. On the contrary, there seems to be a bidirectional causality relationship between productivity and gross value added, and between productivity and transportation public capital. These results imply that standard estimation techniques would offer biased estimated parameters. Consequently, we will estimate the models by Instrumental Variables (IV) methods, and, more precisely, by Two Stages Least Square estimation method (2SLS). Both endogenous variables are instrumented using one temporal lag of the variable as well as, the contemporaneous and one-lagged spatial lag of the variable.

We first estimate model (14) for the cross-sectional data referred to the growth rates of the variables for the whole period 1985–2004, as well as for the different subperiods. Results are gathered in Table 5. The first block of the table shows results when applying Ordinary Least Square Method (OLS) and the IV method, 2SLS. In this case, we instrumented GDP by its contemporaneous spatial lag variable. The second and third blocks of Table 5 show several diagnostics to check the validity of the instruments and the spatial dependence in the residual of the 2SLS estimation. From results displayed in the table, we can see that, in general, instruments available for cross-sectional data in this context are weak as can be inferred from the F statistic that tests the null of nonsignificance of instrument parameter at the first stage of the estimation process. Consequently, the Hausman test does not support the 2SLS estimation in most of the cases.

Table 5.

Estimated Models for Explaining Industrial Productivity Growth (Q.E) between the Different Time Periods^a,b

	1985–2004		1985–1987		1987–1990		1990–1993		1993–2001		2001–2004
	OLS	2SLS	OLS	2SLS	OLS	2SLS	OLS	2SLS	OLS	2SLS	OLS	2SLS
Constant	0.012*	0.016*	−0.005	−0.001	−0.013	−0.046	−0.004	0.006	0.003	0.002	0.001	−0.001
Constant	(3.574)	(2.807)	(−0.77)	(−0.16)	(−1.47)	(−1.13)	(−0.42)	(0.42)	(0.47)	(0.19)	(0.20)	(−0.33)
$\dot{Q}$	0.396*	0.017	0.711*	0.628*	0.979*	2.458	0.392*	0.931	0.392*	0.487	0.845*	0.921*
$\dot{Q}$	(5.777)	(0.048)	(10.38)	(4.35)	(9.13)	(1.54)	(4.01)	(1.93)	(3.15)	(1.04)	(13.58)	(10.25)
Total $\dot{K p}$	−0.056	0.112	0.231	0.244	−0.197	−1.267	0.302	0.218	−0.168	−0.212	−0.090	−0.110
Total $\dot{K p}$	(−0.811)	(0.655)	(1.64)	(1.69)	(−1.14)	(−1.05)	(1.68)	(0.89)	(−1.11)	(−0.82)	(−1.58)	(−1.83)
$\dot{H}$	−0.053	−0.043	−0.033	−0.040	0.052	0.089	0.068*	0.080*	0.007	0.004	0.004	0.005
$\dot{H}$	(−1.640)	(−0.983)	(−1.26)	(−1.39)	(1.79)	(1.14)	(2.25)	(1.96)	(0.18)	(0.10)	(0.20)	(0.26)
Diagnostic IV: Instrumented variable: $\dot{Q}$ ; Instruments: $W \underset{t}{Q}$
F statistic (First stage)		3.16		13.05*		1.08		3.25		3.31		42.30*
Hausman		1.28		0.43		0.86		1.30		0.04		1.37
Critical value Hausman		9.49		9.49		9.49		9.49		9.49		9.49
Spatial diagnostics: Testing the null of no spatial dependence on the residuals
LM test no spatial lag		1.89		1.48		0.00		1.38		9.90*		4.48*
robust LM test no spatial lag		3.51		1.13		1.80		0.59		13.26*		1.57
LM test no spatial error		0.09		3.06		1.88		2.75		7.80*		5.92*
robust LM test no spatial error		1.72		2.71		3.68		1.97		11.17*		2.01

Note: ^aIn parentheses are the t ratios.

^bAn asterisk means that the null hypothesis is rejected at the 5 percent level of significance.

As explained before, a spatial error specification version of the model can be derived when considering that the level of efficiency depends on within-region effects and other effects that spill over from “neighboring” regions. The same procedure will be repeated using different alternatives for the public capital stock variable: total public capital, local public capital, transport public capital and, simultaneously, local and transport public capital. We use a bottom-up approach, starting with the simple OLS estimation (pool models) which ignores both the nature of the panel data set and the possible spatial dependence. Estimation results for the different specifications in relation to public capital stock are shown in Table 6.

Table 6.

Estimated Models for Explaining Industrial Productivity Growth $(\frac{\dot{Q}}{E})$ using the Pool Models and the Different Alternatives in Relation to Public Capital^a,b

	Total public capital		Local public capital		Transport public capital		Both type of public capital
	OLS	2SLS	OLS	2SLS	OLS	2SLS	OLS	2SLS
Constant	0.002	0.004	−0.004	0.002	0.005*	0.011*	−0.002	0.003
Constant	(0.831)	(1.428)	(−1.524)	(0.755)	(2.735)	(5.144)	(−0.675)	(1.107)
$\dot{Q}$	0.441*	0.200*	0.437*	0.216*	0.447*	0.218*	0.440*	0.216*
$\dot{Q}$	(18.097)	(5.503)	(18.554)	(6.279)	(18.797)	(6.249)	(18.610)	(8.674)
Total $\dot{K p}$	0.051	0.183*
Total $\dot{K p}$	(0.896)	(2.942)
Local $\dot{K p}$			0.188*	0.227*			0.198)*	0.269*
Local $\dot{K p}$			(3.997)	(4.587)			(4.159)	(5.383)
Transport $\dot{K p}$					−0.022	0.007	−0.042	0.056
Transport $\dot{K p}$					(−0.698)	(0.201)	(−1.340)	(1.722)
$\dot{H}$	0.007	0.008	0.007	0.008	0.006	0.008	0.007	0.008
$\dot{H}$	(0.939)	(1.110)	(0.996)	(1.133)	(0.925)	(1.047)	(1.011)	(1.161)
Diagnostic IV: Instrumented variable: $\dot{Q}$ ; Instruments: $\underset{t - 1}{Q}, W \underset{t}{Q}, W \underset{t - 1}{Q}$
F statistic (First stage $\dot{Q}$ )		280.94*		304.18*		303.70*		299.02*
F statistic (First stage $\dot{K p}$ )						366.02*		350.60*
Hausman		79.52*		78.84*		80.56*		841.00*
Critical value Hausman		9.49		9.49		9.49		11.07

Spatial diagnostics: Testing the null of no spatial dependence on the residuals^c
LM test no spatial lag		192.33*		180.69*		193.59*		182.41*
Robust LM test no spatial lag		57.90*		41.96*		57.50*		40.22*
LM test no spatial error		423.96*		386.49*		424.77*		384.66*
Robust LM test no spatial error		289.53*		247.75*		288.68*		242.47*

Note: ^aIn parentheses are the t ratios.

^bAn asterisk means that the null hypothesis is rejected at the 5 percent level of significance.

^cThe critical value for testing the null of no spatial effect is $χ_{0.05}^{2} (1) = 3.84$ .

The model selection procedure begins with the analysis of results and some diagnostics referred to those models. Instruments are relevant and the rejection of the null hypothesis of the Hausman test indicates that the 2SLS method is better than OLS. However, the null of no spatial autocorrelation is rejected in all the cases and, according to the magnitudes of the respective statistics, spatial autocorrelation should be modeled by means of the SEM specification. Therefore, so far, we can conclude on the convenience of the SEM specifications, which must be estimated by 2SLS. Results for different SEM models and for all alternative specifications in relation to public capital are shown in Tables 7 –10.

Table 7.

Results for the 2SLS Estimation of Several SEM Models for Explaining Industrial Productivity Growth $(\frac{\dot{Q}}{E})$ with Total Public Capital Variable^a,b

(T = 20, n = 47)	SEM	SEM + 17 time dummies^c (θ₈₈ … θ₀₄)	SEM + 8 time dummies^c	Province FE + SEM+ 8 time dummies^c
Constant	0.003	0.021*	0.016*	0.003
Constant	(1.80)	(3.35)	(3.51)	(0.46)
$\dot{Q}$	0.024	0.229	0.313*	0.467*
$\dot{Q}$	(0.25)	(1.61)	(3.18)	(4.38)
Total $\dot{K p}$	0.250*	0.156*	0.122*	0.132
Total $\dot{K p}$	(3.54)	(2.20)	(1.99)	(1.90)
$\dot{H}$	0.012	0.011	0.010	0.009
$\dot{H}$	(1.62)	(1.80)	(1.64)	(1.67)
Dum88_91	—	—	−0.011*	−0.010*
Dum88_91	—	—	(−3.03)	(−2.93)
Dum9293	—	—	−0.007	0.002
Dum9293	—	—	(−1.01)	(0.32)
Dum94	—	—	0.003	0.008
Dum94	—	—	(0.65)	(1.55)
Dum95	—	—	−0.009	−0.006
Dum95	—	—	(−1.92)	(−1.34)
Dum96_00	—	—	−0.016*	−0.017*
Dum96_00	—	—	(−5.87)	(−6.62)
Dum01	—	—	−0.020*	−0.017*
Dum01	—	—	(−4.27)	(−3.70)
Dum02	—	—	−0.004	−0.002
Dum02	—	—	(−0.85)	(−0.52)
Dum0304	—	—	−0.013*	−0.009*
Dum0304	—	—	(−2.93)	(−2.15)
ρ	0.516	0.184	0.266	0.271
F statistics to test
H₀: θ₈₈ = … = θ₀₄ = 0		14.58*
H₀: θ₈₈ = θ₈₉ = θ₉₀ = θ₉₁		2.51
H₀: θ₉₂ = θ₉₃		0.03
H₀: θ₉₆ = θ₉₇ = θ₉₈ = θ₉₉ = θ₀₀		1.61
H₀: θ₀₃ = θ₀₄		1.07
H₀: δ = … = δ₄₇ = 0				0.76

Note: ^aIn parentheses are the t ratios.

^bAn asterisk means that the null hypothesis is rejected at the 5 percent level of significance.

^cThe reference category is the year 1987, the first data of our estimation sample taking into account that we are analyzing growth rates and introducing one temporal lag of some growth rate variables as instruments.

Table 8.

Results for the 2SLS Estimation of Several SEM models for Explaining Industrial Productivity Growth $(\frac{\dot{Q}}{E})$ with Local Public Capital Variable^a,b

(T = 20, n = 47)	SEM	SEM + 17 time dummies^c (θ₈₈…θ₀₄)	SEM + 8 time dummies^c	Province FE + SEM+ 8 time dummies^c
Constant	0.005*	0.0225*	0.017*	0.005
Constant	(2.72)	(3.50)	(3.66)	(0.71)
$\dot{Q}$	0.065	0.243	0.336*	0.478*
$\dot{Q}$	(0.76)	(1.77)	(3.66)	(4.71)
Local $\dot{K p}$	0.158*	0.087	0.057	0.070
Local $\dot{K p}$	(2.93)	(1.58)	(1.19)	(1.32)
$\dot{H}$	0.011	0.011	0.010	0.009
$\dot{H}$	(1.62)	(1.79)	(1.63)	(1.65)
Dum88_91	—	—	−0.010*	−0.009*
Dum88_91	—	—	(−2.90)	(−2.80)
Dum9293	—	—	−0.006	0.003
Dum9293	—	—	(−0.85)	(0.42)
Dum94	—	—	0.004	0.008
Dum94	—	—	(0.72)	(1.54)
Dum95	—	—	−0.009	−0.006
Dum95	—	—	(−1.90)	(−1.35)
Dum96_00	—	—	−0.016*	−0.017*
Dum96_00	—	—	(−6.06)	(−6.73)
Dum01	—	—	−0.019*	−0.016*
Dum01	—	—	(−4.18)	(−3.56)
Dum02	—	—	−0.003	−0.001
Dum02	—	—	(−0.70)	(−0.35)
Dum0304	—	—	−0.012*	−0.008*
Dum0304	—	—	(−2.83)	(−2.03)
ρ	0.501	0.178	0.262	0.267
F statistics to test
H₀: θ₈₈ = … = θ₀₄ = 0		14.18*
H₀: θ₈₈ = θ₈₉ = θ₉₀ = θ₉₁		2.50
H₀: θ₉₂ = θ₉₃		0.01
H₀: θ₉₆ = θ₉₇ = θ₉₈ = θ₉₉ = θ₀₀		1.78
H₀: θ₀₃ = θ₀₄		1.15
H₀: δ₂ = … = δ₄₇ = 0				0.74

Note: ^aIn parentheses are the t ratios.

^bAn asterisk means that the null hypothesis is rejected at the 5 percent level of significance.

Table 9.

Results for the 2SLS Estimation of Several SEM models for Explaining Industrial Productivity Growth $(\frac{\dot{Q}}{E})$ with Transport Public Capital Variable^a,b

(T = 20, n = 47)	SEM	SEM + 17 time dummies^c (θ₈₈ … θ₀₄)	SEM + 8 time dummies^c	Province FE + SEM+ 8 time dummies^c
Constant	0.008*	0.025*	0.020*	0.009
Constant	(4.54)	(3.81)	(4.11)	(1.24)
$\dot{Q}$	0.072	0.262*	0.351*	0.480*
$\dot{Q}$	(0.81)	(2.02)	(3.96)	(4.76)
Transport $\dot{K p}$	0.005	0.014	0.027	0.028
Transport $\dot{K p}$	(0.15)	(0.46)	(0.96)	(0.93)
$\dot{H}$	0.011	0.011	0.009	0.009
$\dot{H}$	(1.52)	(1.73)	(1.61)	(1.62)
Dum88_91	—	—	−0.009*	−0.008*
Dum88_91	—	—	(−2.64)	(−2.49)
Dum9293	—	—	−0.005	0.004
Dum9293	—	—	(−0.69)	(0.49)
Dum94	—	—	0.004	0.008
Dum94	—	—	(0.81)	(1.54)
Dum95	—	—	−0.008	−0.006
Dum95	—	—	(−1.86)	(−1.33)
Dum96_00	—	—	−0.016*	−0.017*
Dum96_00	—	—	(−6.09)	(−6.63)
Dum01	—	—	−0.019*	−0.017*
Dum01	—	—	(−4.20)	(−3.64)
Dum02	—	—	−0.003	−0.002
Dum02	—	—	(−0.73)	(−0.43)
Dum0304	—	—	−0.012*	−0.009
Dum0304	—	—	(−2.87)	(−2.15)
ρ	0.516	0.180	0.257	0.266
F statistics to test
H₀: θ₈₈ = … = θ₀₄ = 0		15.29*
H₀: θ₈₈ = θ₈₉ = θ₉₀ = θ₉₁		2.40
H₀: θ₉₂ = θ₉₃		0.01
H₀: θ₉₆ = θ₉₇ = θ₉₈ = θ₉₉= θ₀₀		1.84
H₀: θ₀₃ = θ₀₄		1.13
H₀: δ₂ = … = δ₄₇ = 0				0.70

Note: ^aIn parentheses are the t ratios.

^bAn asterisk means that the null hypothesis is rejected at the 5 percent level of significance.

Table 10.

Results for the 2SLS Estimation of Several SEM Models for Explaining Industrial Productivity Growth $(\frac{\dot{Q}}{E})$ with Local and Transport Public Capital Variable^a,b

(T = 20, n = 47)	SEM	SEM + 17 time dummies^c (θ₈₈ … θ₀₄)	SEM + 8 time dummies^c	Province FE + SEM+ 8 time dummies^c
Constant	0.005*	0.023*	0.018*	0.006
Constant	(2.65)	(3.61)	(3.81)	(0.88)
$\dot{Q}$	0.066	0.245	0.342*	0.479*
$\dot{Q}$	(0.77)	(1.76)	(3.71)	(4.72)
Local $\dot{K p}$	0.159*	0.088	0.058	0.072
Local $\dot{K p}$	(2.95)	(1.60)	(1.23)	(1.35)
Transport $\dot{K p}$	0.011	0.016	0.028	0.029
Transport $\dot{K p}$	(0.32)	(0.52)	(1.00)	(0.97)
$\dot{H}$	0.011	0.011	0.010	0.009
$\dot{H}$	(1.62)	(1.79)	(1.64)	(1.66)
Dum88_91	—	—	−0.009*	−0.009*
Dum88_91	—	—	(−2.72)	(−2.60)
Dum9293	—	—	−0.005	0.003
Dum9293	—	—	(−0.76)	(0.46)
Dum94	—	—	0.004	0.008
Dum94	—	—	(0.77)	(1.57)
Dum95	—	—	−0.008	−0.006
Dum95	—	—	(−1.86)	(−1.32)
Dum96_00	—	—	−0.016*	−0.017*
Dum96_00	—	—	(−6.10)	(−6.73)
Dum01	—	—	−0.019*	−0.016*
Dum01	—	—	(−4.10)	(−3.49)
Dum02	—	—	−0.003	−0.001
Dum02	—	—	(−0.63)	(−0.29)
Dum0304	—	—	−0.012*	−0.008*
Dum0304	—	—	(−2.78)	(−1.99)
ρ	0.501	0.178	0.259	0.266
F statistics to test
H₀: θ₈₈ = … = θ₀₄ = 0		14.17*
H₀: θ₈₈ = θ₈₉ = θ₉₀ = θ₉₁		2.41
H₀: θ₉₂ = θ?₉₃		0.01
H₀: θ₉₆ = θ₉₇ = θ₉₈ = θ₉₉ = θ₀₀		1.78
H₀: θ₀₃ = θ₀₄		1.18
H₀: δ₂ = … = δ₄₇ = 0				0.72

Note: ^aIn parentheses are the t ratios.

^bAn asterisk means that the null hypothesis is rejected at the 5 percent level of significance.

The first column in each table shows the results for the SEM specification applied to the pool data set. We then begin to consider the panel nature of the data through estimation of the time FE-SEM specification specified in equation (20). The results are shown in the second column of each table. The time FE-SEM models include a set of seventeen time dummy variables, with 1987 as the reference.¹⁷ At the bottom of column 2, we offer results for some F statistics. The first set of results indicates that time FE-SEM models outperform previous models since such temporal effect are statistically significant at the 5 percent level. The results for the remaining F statistics show evidence in favor of some simplifications of the previous model, as specific effects for several years are statistically equal.¹⁸ Therefore, the next step consists of estimating the restricted models which only include eight time dummy variables. The results are shown in the third column of the tables (SEM + 8 time dummies). Finally, we complete these models with the incorporation of fixed province effects (Province FE + SEM + 8 Time dummies). As the models obtained are nested in the previous models, the selection strategy will be based on calculation of the respective F statistic for testing the nonsignificance of the fixed effects included in the last models. All the results point in the same direction. The inclusion of fixed province effects does not improve the previous models, due to the surprising insignificance of the province dummy variables. These models do not statistically differ from the previous ones and, consequently, the SEM specification with eight time dummies is the best specified model for the four analyzed cases associated to the definition of public capital stock.

Important conclusions can be drawn from our selected models:

The results indicate the existence of increasing returns to scale in the manufacturing sector for the Spanish provinces. The estimated model is similar to an expansion of the dynamic Verdoon’s law. The manufacturing sector has great potential for the dynamic use of scale economies, a substantial proportion of productivity growth is determined by output growth; differences in regional productivity are caused by disparities in output growth. A faster growth of output in a region will lead to an increase in productivity growth, which in a subsequent phase, leads to an increase in this region competitiveness, causing a faster growth of the region’s exports and increasing the overall rate of growth. This mechanism reinforces the disequilibrium in regional economic growth. Nevertheless, unless some assumption is made about the evolution of total capital stock, the degree of the returns to scale cannot be directly obtained from the estimations.

Public infrastructure investment has a significant and positive effect on productivity as a whole; we show it to be an important variable, with a 1 percent increase inducing a 0.12 percent increase in productivity. This is particularly significant, as it shows that, in spite of the increase in capital stock in the Spanish economy in the last twenty years, there is still room for public investment to continue to improve mean labor productivity. For the Spanish economy, the direct effect of public investment on productivity and its complementarity with private investment seems to be greater than the crowding-out effect on private investment. This means that provincial public capital stock has not yet reached its optimal level. This result is in line with previous findings in the literature, for instance, Munell (1992), reviewing several papers, reports an average elasticity of output to public capital of about 0.15. The magnitude of the elasticity obtained for public infrastructure is also within the range of variation of the elasticities obtained in other papers that find a positive effect referred to the Spanish regions. In Mas et al. (1994, 1996), the elasticity associated with productive public infrastructures is 0.23 and 0.08, respectively. Goerlich and Mas (2001) reported an elasticity of 0.02 and Boscá, Cutanda, and Escribá (2006) obtained output elasticities for public infrastructures of 0.026 (0.035 in the long run). It is interesting to note that the very few papers that have analyzed Spanish provincial infrastructure effects, such as Delgado and Álvarez (2007), Álvarez, Arias, and Orea (2006), Moreno and Lopez-Bazo (2007) and Gómez-Antonio and Fingleton (2011), all found a positive provincial effect. However, Delgado and Álvarez (2007) conclude that the existence of negative spillovers for the industrial sector counterbalances the positive effect of the infrastructure coefficient. The elasticity we obtained is similar to that found in Gómez-Antonio and Fingleton (2011), which is particularly relevant as it constitutes a close theoretical approach based on New Economic Geography (NEG), which shares similar postulates to urban economics theory. Still, as has been stated in previous sections, Gómez-Antonio and Fingleton (2011) incorporate public capital as a factor shaping technical progress, with results pointing in the same direction, so we can conjecture that the way public capital is introduced into the model does not affect the main results obtained.

There are reasons to believe that public investment in neighboring provinces might have a positive impact in own-region productivity, resulting from the connecting feature of network infrastructures, such as roads, railways, or airports; but we might also have reasons to consider the existence of negative spillovers which would arise from the migration of factors to locations with better infrastructure stocks. Network infrastructures may alter location decisions of firms, increasing investments and outputs in some provinces while causing disinvestments and possible job losses for others. Public investment in one region could have a negative effect on other regions that are its closest competitors for labor and mobile capital. The sign of spatial spillovers in infrastructures is also inconclusive in the empirical literature. Holtz-Eakin and Schwartz (1995) for the United States, provide no evidence of infrastructure spatial spillovers. By contrast, Pereira and Roca-Sagalés (2003), Cohen and Morrison (2004), and Bronzini and Piselli (2009) find significant positive spatial spillovers for Spain, the United States, and Italy, respectively. Interregional spatial spillovers could also have a negative sign. Boarnet (1998) using data for California’s counties found that the output of counties is negatively affected by neighboring counties' infrastructure. Sloboda and Yao (2008) for the United States, and Pereira and Andraz (2006) for Portuguese regions, determined that public capital provided in a particular region raises the comparative advantage of that region over the others, and could therefore attract production factors from other locations where output or productivity might decrease. For the Spanish provinces, the literature incorporating spillover effects is very scarce and inconclusive. Delgado and Álvarez (2007) analyze the effects of high-capacity roads on different economic sectors and conclude that, for the industrial sector, the existence of negative spillovers counterbalances the positive effect of the infrastructure coefficient. Álvarez, Arias, and Orea (2006) consider the effects of roads infrastructure on output without finding any significant coefficient for spillovers. On the other hand, Moreno and Lopez-Bazo (2007) and Gómez-Antonio and Fingleton (2011) determined a negative spillover for transport and total infrastructure, respectively. Our results challenge those obtained in Delgado and Alvarez (2007) and Álvarez, Arias, and Orea (2006), as the existence of spillovers does not determine a negative global effect for public infrastructure and we do find spillovers to be significant. In our model, the existence of spillovers are associated to technical progress, so public and human capital are not supposed to affect changes in technology, but are included as additional inputs in the production function, alas under our specification we cannot assess the effects of public infrastructures spillovers. Our model allows for the estimation of spillover effects in the residual term when we estimate an SEM. This imposes a constraint that involves the rest of covariates spillovers embedded in the estimated reduced form. We recognize spillovers importance in our analysis and control for their existence but unfortunately we cannot untangle which part of them is due to each covariate. Essentially, the advantage of this approach is that the possible existence of human capital spillovers and spatial externalities from other sources, such as external shocks, are captured in our specification, guaranteeing the robustness of covariate coefficients.

The human capital variable presents the expected sign but is slightly less significant than the others variables in the model. This can be explained by the temporal lag that this variable needs to affect productivity growth rate.

It is also of interest to break down public capital stock into different categories in order to investigate the contribution of different types of infrastructures to productivity growth. Different kinds of infrastructures might have different effects on productivity growth. For instance, the role of transport and communication infrastructures is primarily to integrate a region with the rest of the network, as well as develop links with rest of the economy. This is in contrast to the role of local infrastructures, which might help develop comparative regional advantages. The results obtained when analyzing public capital stock in local and/or transport infrastructures are shown in Tables 8 –10. All the estimated coefficients are positive, but in the case of local public capital stock the coefficient presents a higher magnitude. These results challenge previous contributions that analyzed transport infrastructures for the Spanish provinces, such as Álvarez, Arias, and Orea (2006) and Delgado and Alvarez (2007), who found a positive and significant effect for road infrastructures and high-capacity roads, respectively. However, they are partially consistent with Moreno and López-Bazo (2007). They find a positive effect for public investment for both types of infrastructures, but the effect of the spillover in transport investments is negative and they do not find spillovers for local infrastructures. This could explain why we do not find the public transportation investment coefficient significantly different from zero in our specification and the reason why it is smaller than that of local infrastructure. Although we cannot assure that the two coefficients are significantly different from zero at the 5 percent level, the sign and values of the estimated coefficients of the other explanatory variables remain similar, supporting the robustness of the results. With the necessary caution, our results suggest, consistent with Moreno and López-Bazo (2007), that more resources should be used to increase provinces' local infrastructure endowments as this component has a greater impact on productivity. Once the spillovers are taken into account, policy implications are interesting. Most of the relevant literature has been aiming at promoting economic growth through the recommendation of policies that finance transportation infrastructures when the returns of increasing investment in local infrastructures might be more effective to increase productivity in lagged provinces.

According to Moreno and López-Bazo (2007), the lower nonsignificance of disaggregate public capital stock may also be due to a high degree of complementarity between the two types of infrastructure; when considered individually, their effect on productivity growth is not suitably reflected. Another possible explanation could be the existence of decreasing returns on public capital stock; the returns on local and transport public capital stock for Spanish provinces are decreasing and present very low values since 1985.

Finally, robustness of the obtained results from the selected weights matrix is analyzed through results reported in Table 11. The obtained results are very similar to the ones we would have obtained using the alternative weights matrices denoted as W ₂ (based on the inverse of the square of distances) or the W ₃ matrix defined by Fingleton (2001).

Table 11.

Robustness of Results of the Selection of Alternative Weight Matrices^a,b

(T = 20, n = 47)	SEM + 8 time dummies^c
(T = 20, n = 47)	W ₁	W ₂	W ₃
Constant	0.016*	0.017*	0.016*
Constant	(3.51)	(3.71)	(3.52)
$\dot{Q}$	0.313*	0.305*	0.316*
$\dot{Q}$	(3.18)	(3.30)	(3.44)
Total $\dot{K p}$	0.122*	0.116*	0.111*
Total $\dot{K p}$	(1.99)	(1.98)	(1.97)
$\dot{H}$	0.010	0.010	0.009
$\dot{H}$	(1.64)	(1.64)	(1.58)
Dum88_91	−0.011*	−0.011*	−0.009*
Dum88_91	(−3.03)	(−3.08)	(−2.72)
Dum9293	−0.007	−0.007	−0.007
Dum9293	(−1.01)	(−1.07)	(−1.01)
Dum94	0.003	0.003	0.003
Dum94	(0.65)	(0.62)	(0.64)
Dum95	−0.009	−0.010*	−0.012*
Dum95	(−1.92)	(−2.07)	(−2.58)
Dum96_00	−0.016*	−0.016*	−0.016*
Dum96_00	(−5.87)	(−5.92)	(−5.84)
Dum01	−0.020*	−0.020*	−0.019*
Dum01	(−4.27)	(−4.31)	(−4.11)
Dum02	−0.004	−0.004	−0.002
Dum02	(−0.85)	(−0.81)	(−0.51)
Dum0304	−0.013*	−0.012*	−0.012*
	(−2.93)	(−2.94)	(−2.78)
ρ	0.266	0.266	0.266

Note: ^aIn parentheses are the t ratios.

^bAn asterisk means that the null hypothesis is rejected at the 5 percent level of significance.

^cThe reference category is year 1987, the first data of the sample taking into account that we are analyzing growth rates and introducing one temporal lags of variables as instruments.

Summary and Concluding Remarks

The relationship between public infrastructures and productivity has been analyzed using a model based on the theoretical arguments of urban economics theory. As far as we know, we are the first to estimate the effect of public capital on productivity under this theoretical approach. We extended the basic production function in the industry sector by explicitly introducing public capital stock and human capital in order to quantify its impact on regional economic growth. The model is also extended to include additional and necessary factors representing technological externalities and spillovers which are not represented in the basic model. The model has been estimated using spatial panel fixed time and province effects to control for the existence of public investment spillovers, so that unmodeled heterogeneity is not responsible for the effects reported.

The substantive conclusion we reach as a result of this analysis is that there is evidence supporting a positive relationship between productivity growth, human capital, and public capital, with significant positive effects on productivity.

The results indicate the existence of increasing returns to scale in the manufacturing sector for the Spanish province. The manufacturing sector is a leading sector and it has great potential for the dynamic use of scale economies. Differences in regional productivity are caused by disparities in output growth. A faster growth of output in a region will lead to an increase in productivity growth, which in a subsequent phase, leads to an increase in this region competitiveness, what generates a faster growth of the region’s exports and increase the overall rate of growth. This mechanism reinforces the disequilibrium in regional economic growth. However, unless some assumption is made about the evolution of total capital stock, the degree of the returns to scale cannot be directly obtained.

Public capital stock is shown to be an important variable, with a 1 percent increase inducing around a 0.12 percent increase in productivity. This result remains unchanged with different model specifications. In spite of the public sector’s investments in Spain over the last twenty years, public investment has a positive effect on economic growth.

When we disaggregate capital stock into local and transport components, the results are not significant, although the sign of the estimated coefficients are positive, and in the case of local public capital stock the coefficient presents a higher magnitude. This can be explained as a result of a combination of two factors. First, the high degree of complementarity between the two is lost when the variables are considered individually and, second, there are decreasing returns on public capital stock in Spanish provinces. There seems to be a threshold in the public/private capital stock ratio, above which returns decrease sharply. This justifies further research on the spatial effects of different types of infrastructure investment.

In terms of policy involvement on a national scale, in the absence of more detailed information on how the spillover effects of each type of public investment will balance out, we still have to be prudent about the global effect on economic activity. In the presence of negative spillovers, provinces might be competing with each other, attempting to obtain more infrastructures than would otherwise be provided. By altering investments in infrastructure relative to those of the neighboring regions, each region has the ability to modify the size of its infrastructure stock at the expense of its neighbor. If some regions follow a “beggar-thy-neighbor” policy, all regions will be dragged into fiscal competition. If local and national governments ignore the negative spillovers and overestimate the positive effects of public policies, this can lead to inefficiency.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors would like to express their thanks to the Ministerio de Ciencia e Innovación del Reino de España for its funding through the project ECO2009-10534/ECON.

Notes

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