Ludovic Dibiaggio
email: ludovic.dibiaggio@skema.edu

Benjamin Montmartin
email: benjamin.montmartin@skema.edu

Lionel Nesta
email: lionel.nesta@gredeg.cnrs.fr

Regional Alignment and Productivity Growth

Keywords: JEL Classification: 030, 040, R11 Regional Alignment, β-convergence, productivity growth, multi-regional model

We propose the concept of regional alignment to suggest that synergistic relations among the scientific expertise, technological specialization and industry composition of regions affect regional productivity growth. In this paper, we test an extended conditional β-convergence model using data on 94 French departments (NUTS3) for the period 2001-2011. Our results indicate that a conditional β-convergence is associated with a σ-divergence process in the total factor productivity (TFP) growth of French regions.

This process is strongly affected by the level of regional alignment. Indeed, we find evidence that regional alignment both directly and indirectly influences regional productivity growth. The indirect effect of regional alignment materializes through its leverage on R&D investment, which is one of the most important drivers of productivity growth. Moreover, using a heterogeneous coefficients model, we show that the positive effect of regional alignment on TFP growth increases with the industrial and technological diversity of regions, which suggests that regional alignment increases the value of Jacobs externalities more than Marshall-Arrow-Romer (MAR) externalities.

Introduction

In recent years, there has been increasing interest in persistent regional differences in the economic performance of regions in almost all studied countries. While the dispersion of GDP per capita among countries has narrowed over the last thirty years, within-country differences have widened almost everywhere (OECD, 2009(OECD, , 2016)).

Unlike predictions popularized by Friedman's book "The World is Flat" (2005), the digital revolution and the decline in transportation and communication costs have not led the world toward a general convergence. The world has, in fact, never been so spiky [START_REF] Florida | Cities and the creative class[END_REF][START_REF] Moretti | Are cities the new growth escalator? The World Bank Policy Working Paper[END_REF], which raises serious policy issues since national economic integration is an essential part of governments' agendas.

The view described above goes against the traditional view assuming an absolute (unconditional) βconvergence across countries and regions with growth rates declining with the level of GDP per capita [START_REF] Solow | A Contribution to the Theory of Economic Growth[END_REF], Barro and Salai-i-Martin, 1991[START_REF] Barro | Economic Growth[END_REF][START_REF] Sala-I-Martin | The Classical Approach to Convergence Analysis[END_REF]. Empirical results confirm absolute βconvergence across OECD countries but do not find convergence across US states (e.g., see [START_REF] Evans | Convergence revisited[END_REF][START_REF] Sala-I-Martin | The Classical Approach to Convergence Analysis[END_REF][START_REF] Evans | How Fast Do Economies Converge?[END_REF], US counties [START_REF] Higgins | Handbook of Australian, New Zealand and Antarctic Birds[END_REF][START_REF] Young | Sigma Convergence versus Beta Convergence: Evidence from U.S. County-Level Data[END_REF] or across European NUTS 3 regions [START_REF] Paas | Regional Income Inequalities and Convergence Processes in the EU-25[END_REF][START_REF] Simeonescu | The Beta-convergence Analysis and Regional Disparities in EU-28[END_REF]. However, absolute β-convergence only tells us whether and at what pace economies converge toward a steady state in time but not whether and to what extent there are disparities among regional economies. The steady state of a region may depend on characteristics that are specific within that region and therefore may vary and even diverge over time. Regions may converge, conditional on other region-specific variables being held constant, but to different steady states.

To better account for the evolution of regional heterogeneity in countries, σ-convergence is more interesting and is generally measured as the temporal dynamic of the standard deviation of regional GDP per capita [START_REF] Quah | Empirical cross-section dynamics in economic growth[END_REF]. Persistent within-country σ-divergence is striking because analysts generally assume that institutional conditions are homogeneous and that there is higher capital, labor and knowledge mobility within countries than between countries. One convincing explanation of this phenomenon is that a positive relationship exists between agglomeration processes and growth, which are essentially spurred by spatially-mediated knowledge externalities [START_REF] Baldwin | Agglomeration and regional growth[END_REF] 1 . Knowledge spillovers induce complementarities in firms' R&D investment by facilitating access to external knowledge but often require social ties based on frequent face-to-face interactions. The literature suggests that there are two opposing views of localized knowledge spillovers [START_REF] Glaeser | Growth of cities[END_REF]. The Marshall-Arrow-Romer (MAR) framework and Porter's framework emphasize that industrial specialization in a single industry facilitates knowledge flows across firms sharing similar or related technological knowledge. In contrast, the opposite prescription promotes Jacob (1969)'s externalities resulting from the agglomeration of firms from different industries and where knowledge diversity increases the likelihood of cross-fertilization. This process may explain why industrial diversity contributes to the long-term growth of regions only when they rely on similar or related technologies [START_REF] Frenken | Related variety, unrelated variety and regional economic growth[END_REF]: Boshma, 2015).

However, divergence also arises among similar types of regions exhibiting significant growth-rate heterogeneity despite their similar initial conditions [START_REF] Garcilazo | The Contribution of Regions to Aggregate Growth in the OECD[END_REF]. This divergence suggests that regional productivity differences may be due to heterogeneous access to specific resources and infrastructures or, more importantly, how regions organize, allocate, and develop their resources [START_REF] Rosenthal | Evidence on the nature and sources of agglomeration economies[END_REF][START_REF] Aghion | Science, technology and innovation for economic growth, linking policy research and practice in STIG Systems[END_REF]. This argument echoes [START_REF] Saxenian | Regional Advantage: Culture and Competition in Silicon Valley and Route 128[END_REF]'s explanation of the different growth paths of the Silicon Valley and Boston Route 128, which is based on the ability of local industrial and innovation systems to promote interdependence and exchanges among individuals and institutions. Thus, the relations among localized knowledge spillovers, agglomeration and growth may rely on the underlying mechanisms in traditional growth models that have not yet been explored [START_REF] Breschi | Knowledge spillovers and local innovation systems, a critical survey[END_REF].

We argue that regional alignment is a critical determinant of the differences in regional growth paths.

Regional alignment reflects the level of synergies among the scientific domains, technological fields, and industrial sectors in which a region has expertise. In this article, we focus on science and technology synergies as an indicator of the effectiveness of interactions among universities, firms, and local political institutions or agencies.

Universities have been shown to have a significant role in both producing basic research stimulating technological innovation through collaborations with firms, licensees or spinoffs and generating spillovers through the creation of high skilled workers who are trained by experts. However, spillovers should be more significant and impactful in aligned regions if local firms and startups can, either directly or indirectly, exploit the knowledge and human capital produced by universities. This paper makes several contributions. First, it proposes a definition and measurement of regional alignment as an indicator of the effectiveness of synergies among complementary agents in a region, thus enabling systematic comparisons of regional characteristics. Second, we extend the conditional β-convergence model of total factor productivity (TFP) growth [START_REF] Ha | Accounting for Trends in Productivity and R&D: A Schumpeterian Critique of Semi-Endogenous Growth Theory[END_REF] to take into account the existence of both localized knowledge spillovers between regions and the theoretical influence of regional alignment on productivity growth. Third, using French firm-level data aggregated at the NUTS 3-level for the period 2001-2011, we estimate an extended conditional β-convergence model of productivity growth. We use different econometric models (Simultaneous Equations Model and Heterogenous Coefficients Model) and consistent estimators (IV) to address the complexity of the interrelations among our explanatory variables and provide a better estimation of the influence of regional alignment on the dynamics of regional productivity growth.

Our results highlight the conditional β-convergence that is associated with a σ-divergence of the regional productivity/growth processes in France. In other words, we obtain evidence that the heterogeneity of French regions in terms of TFP increased during the last decade. We also find that regional alignment matters for explaining this heterogeneity. Regardless of the estimation method used, regional alignment positively influences regional TFP growth. Furthermore, our simultaneous equation model reveals that regional alignment has a significant indirect effect on R&D investment, which is a driver of TFP growth. Our last main empirical finding highlights that the nonlinear and heterogeneous effects of regional alignment are conditional on the industrial diversity of the region. Indeed, we show that the effect of regional alignment on productivity growth is negatively related to industrial specialization, which seems to indicate that the positive effect of regional alignment plays a greater role in Jacobs' externalities than in MAR externalities.

The remainder of this paper is organized as follows. In Section 2, we present the concept of regional alignment and its statistical measure. Section 3 develops an extended conditional β-convergence model of regional productivity growth. In Section 4, we describe the data and the main descriptive statistics. Section 5 introduces the first estimations of the model, which highlight the process of regional productivity growth in France and the role of regional alignment. Then, using more advanced econometric techniques, we further investigate the influence of regional alignment in Section 6 and conclude in Section 7. 2 The regional alignment concept and its empirical measure

Literature background: R&D investment and knowledge spillovers as the main drivers of productivity growth

Investing in R&D is central for economic growth [START_REF] Romer | Increasing Returns and Long-Run Growth[END_REF][START_REF] Romer | Endogenous Technological Change[END_REF][START_REF] Grossman | Innovation and Growth in the Global Economy[END_REF]Helpman, 1991, Aghion and[START_REF] Aghion | A Model of Growth Through Creative Destruction[END_REF]. The underlying assumption is that the level of R&D investment determines the likelihood of successfully exploiting technological opportunities, increasing the stock of knowledge, and generating productive innovation. As a quasi-public good, newly generated knowledge should spillover at a negligible marginal cost, reduce the overall cost of R&D, and contribute to both local and national economic growth. However, as noted by [START_REF] Lucas | On the mechanics of economic development[END_REF] and evidenced by several empirical studies (e.g., [START_REF] Hippel | Sticky information and the locus of problem solving: implications for innovation[END_REF]Maskell and Malmberg, 1999;[START_REF] Breschi | Knowledge spillovers and local innovation systems, a critical survey[END_REF][START_REF] Storper | Buzz: face-to-face contact and the urban economy[END_REF][START_REF] Laursen | The paradox of openness: appropriability, external search and collaboration[END_REF], spillovers rely on social or contractual interactions that are spatially bounded. Consequently, regional growth divergence appears when the size of regions differs due to localized knowledge spillovers; therefore, there are variations in the costs of R&D and growth paths [START_REF] Baldwin | Global Income Divergence, Trade, and Industrialization: The Geography of Growth Take-Offs[END_REF]. Recent works analyzing the link between agglomeration and spillovers confirm the economic significance of knowledge spillovers and their sensitivity to the distance among innovative firms [START_REF] Audrestch | R&D Spillovers and the Geography of Innovation and Production[END_REF][START_REF] Rosenthal | Geography, Industrial Organization, and Agglomeration[END_REF]Carlino et al. 2012, Buzard and[START_REF] Buzard | The geography of research and development activity in the U.S[END_REF]Murata et al. 2014;[START_REF] Kerr | Agglomerative forces and cluster shapes[END_REF][START_REF] Lychagin | Spillovers in Space: Does Geography Matter[END_REF][START_REF] Buzard | Localized Knowledge Spillovers: Evidence from the Spatial Clustering of R&D Labs and Patent Citations[END_REF][START_REF] Bloom | Identifying technology Spillovers and Produc Market Rivalry[END_REF][START_REF] Lucking | Have R&D Spillovers Changed?[END_REF].

The economic literature identifies the types of knowledge spillovers that vary according to the economic structure of a territory. On one hand, according to the MAR view on spillovers, by co-locating in the same area, firms in a specific industry reduce their information and transaction costs as well as their R&D effort to build absorptive capacities and benefit from local knowledge spillovers. The spatial, economic and cognitive proximities of these firms facilitate the exchange of goods and services, human capital flows, and the pooling of specific resources, thus boosting spillovers and returns on R&D (e.g., see [START_REF] Glaeser | Growth of cities[END_REF][START_REF] Cohen | Absorptive capacity, a new perspective on innovation and learning[END_REF]Levinthal,1990). MAR externalities provide a theoretical justification for the innovative benefits of the agglomeration of firms that are either in the same industry or in related industries [START_REF] Porter | The Competitive Advantage of Nations[END_REF][START_REF] Frenken | Related variety, unrelated variety and regional economic growth[END_REF][START_REF] Boschma | Related variety, trade linkages and regional growth in Italy[END_REF][START_REF] Neffke | How do regions diversify over time? Industry relatedness and the development of new growth paths in regions[END_REF]. A series of empirical works shows that regions exhibiting a high level of interdependence and relatedness among the knowledge bases of local firms tend to follow a common innovation path and therefore benefit more from knowledge externalities [START_REF] North | Institutions, Institutional Change and Economic Performance[END_REF][START_REF] Asheim | bases and regional innovation systems: Comparing Nordic clusters[END_REF][START_REF] Frenken | Related variety, unrelated variety and regional economic growth[END_REF][START_REF] Martin | Regional economic resilience, hysteresis and recessionary shocks[END_REF][START_REF] Colombelli | The emergence of new technology-based sectors in European regions: A proximity-based analysis of nanotechnology[END_REF]. However, regions with industries relying on similar or related knowledge bases may become trapped in a spiral of negative "lock-in", leading to technological obsolescence and economic decline (Marin and Sunley, 2012;[START_REF] Neffke | Agents of Structural Change, The Role of Firms and Entrepreneurs in Regional Diversification[END_REF].

On the other hand, according to the Jacobs (1969) view on spillovers, size matters because larger cities contain more variety and therefore provide more knowledge combination opportunities, thus yielding greater economic returns on local firms' innovative efforts. Indeed, cognitive diversity matters to avoid "lock-in" as firms develop new capabilities that generally result from interactions with heterogeneous and unrelated sources of knowledge (Miguelez, and Moreno, 2015;[START_REF] Neffke | Agents of Structural Change, The Role of Firms and Entrepreneurs in Regional Diversification[END_REF]. Thus, the benefits of agglomeration not only depend on scale but also rely on the composition and organization of local economic and innovative activities2 .

Agglomeration studies consistently show that knowledge externalities arise from interdependencies or synergies across complementary actors (such as suppliers, customers, universities and public institutions) that either exchange resources, human capital, products or services or learn from each other (Marshal, 1920;[START_REF] Glaeser | Growth of cities[END_REF][START_REF] Henderson | Universities as a source of commercial technology, a detailed analysis of university patenting, 1965-1988[END_REF][START_REF] Cooke | Regional Innovation Systems, Clusters, and the Knowledge Economy[END_REF][START_REF] Breschi | Knowledge spillovers and local innovation systems, a critical survey[END_REF]). An important source of knowledge variety is interaction with universities (Gibbons andJohnston, 1974, Rosenberg, 1990;[START_REF] Nelson | American universities and technical advance in industry[END_REF][START_REF] Bishop | Gaining from interactions with universities: Multiple methods for nurturing absorptive capacity[END_REF]. Although academic knowledge can be accessed through scientific publications, patents or licenses, the proximity between scientists and engineers affects the intensity and quality of collaborations (Gibbons and Johnston, 1974;[START_REF] Mansfield | The modern university: contributor to industrial innovation and recipient of industrial R&D support[END_REF][START_REF] Arundel | Proximity and the use of public science by innovative European firms[END_REF][START_REF] Bishop | Gaining from interactions with universities: Multiple methods for nurturing absorptive capacity[END_REF].

Regional alignment as a new input for productivity growth

In this paper, we argue that in addition to R&D investment and localized spillovers, regional innovative and growth performances are related to the alignment of the scientific expertise, technological capabilities and industrial specialization of regions. The concept "regional alignment" refers to the accumulation of assets and capabilities in various actors that can be mutually synergistic if effectively combined. This simple definition has three implications. First, regional alignment emphasizes the idea that in order to be productive, scientific knowledge must translate into concrete applications. This is similar to the idea of Arora and Gambardella (1994), who distinguish between general and abstract knowledge, which is found in the realm of science, and local and concrete knowledge, which belongs to the domain of technologies and industrial applications 3 . From these preliminary definitions, one can already infer that the region's endowment in both basic and applied knowledge is an essential characteristic of regional alignment. Second, a greater integration between scientific, technological and industrial expertise implies a greater accumulation of assets and their corresponding infrastructure. In other words, regional alignment emphasises the role of a minimum critical mass which translates into greater facilities, enhanced access to heavy experimental protocols, etc...For instance, universities play a significant role in creating more productive human capital and attracting talented students from other regions (Salter andMartin, 2001, Moretti, 2012). The literature has clearly established that university spillovers are geographically bounded and directly constribute to local firms' innovation [START_REF] Jaffe | Real effects of academic research[END_REF][START_REF] Acs | R&D spillovers and recipient firm size[END_REF][START_REF] Anselin | Local geographic spillovers between university research and high technology innovations[END_REF][START_REF] Laursen | Exploring the effect of geographical distance and university quality on industry-university collaboration in the UK[END_REF][START_REF] Simeonescu | The Beta-convergence Analysis and Regional Disparities in EU-28[END_REF]. Indeed, academic researchers gain from collaborations with local firms providing potential research directions and access to additional resources (e.g., see [START_REF] Lee | The sustainability of university-industry research collaboration: An empirical assessment[END_REF]D'Este and[START_REF] D'este | Why do academics engage with industry? The entrepreneurial university and individual motivations[END_REF], thus reinforcing the alignment process. Third, we believe that the concept of regional alignment may reveal potentially useful interactions among complementary agents such as public and private scientists, engineers, business communities, and policy makers, which may contribute to the innovation, the productivity and the growth of the local economy. Overall, we argue that regional alignment as defined here plays a critical role in the growth potential of regions.

The regional alignment measure

The objective of our regional aligment measure is to capture the potential synergistic relations among the scientific, technological and industrial resources in a region. The empirical estimation of regional alignment is a three-step process.

Step 1: Measuring the level of synergies among all the science domains and the technological fields at the national level

We use patent statistics and scientific publications to unravel regional alignment. We use scientific publications to unravel the scientific expertise of regions. Patent statistics span over a greater range of actors, namely public and private scientists and engineers. Therefore, we believe that patent can be used to qualify the technological and industrial expertise of the regions. We measure the level of synergies by the intensity of the combined use of technological fields and scientific domains in all French patents4 . More precisely, the level of synergy between the technological field j ∈ J and the scientific domain k ∈ K is denoted as τ jk , which results from the number of citations of scientific papers in domain k by patents associated with technological field j. We propose a parametric measure of τ jk by using a random combination of j and k that follows a hypergeometric distribution5 and define τ jk as:

τ jk = C jk -µ jk σ jk (1)
where C jk is the empirical number of co-occurrences observed between technology j and scientific domain k, µ jk is the expected (mean) value of a random technological co-occurrence and σ jk is its standard deviation.

Thus, if C jk > µ jk , then technology j and scientific domain k are highly related. Conversely, if C jk < µ jk , then j and k are poorly related. More details on the computation of τ jk are provided in Appendix B.

Step 2: Measuring the regional technological expertise (RTE) and regional scientific expertise

(RSE) in each technological field and scientific domain

We use the commonly-used indicators of technological and scientific specializations, i.e, the revealed technological advantage (RTA) indicator developed by Balassa (1961Balassa ( , 1969) ) and the revealed scientific advantage (RSA). The RTA of a specific technological field j is defined as the ratio of the share of regional applicants' patents associated with technological field j to the share of the country's patents associated with technological field j.

RT

A ij = P ij / j P ij i P ij / ij P ij
where P ij is the number of patents in technological field j granted in region i. We then define the RTE in technological field j for a region as a binary transformation of the RTA:

RT E ij =1 if RT A ij ≥ 1 RT E ij =0 if RT A ij < 1 (2)
The same ratio is used to define RSA in domain k and measures the share of regional patents citing scientific publications associated with scientific domain k to the share of the country's patents citing articles in journals associated with scientific domain k.

RSA ik = P ik / k P ik i P ik / ik P ik where P ik is the number of patents citing scientific domain k that were granted in region i.

RSE ik =1 if RSA ik ≥ 1 RSE ik =0 if RSA ik < 1 (3)
Step 3: Measuring regional alignment

We define the level of regional alignment (RA) as the mean of the level of regional alignment for each combination j and k. In other words, for each pair of technology field j and scientific domain k, the level of regional alignment is defined as the interactions among the RTE in j, the RTE in k and the level of synergy between j and k.

RA ijk = τ jk * RT E ij * RSE ik (4)
Where RA ijk is the regional alignment of region i in technology j and scientific domain k. Next, we compute the index of regional alignment for region i as:

RA i = jk RA ijk (J i * K i ) (5)
where J i represents the number of technological fields for which region i is active, and K i represents the number of scientific domains for which region i is active. Regional alignment increases with the effective number of related co-expertises jk (high τ jk ) but decreases with the potential number of co-expertises jk within the region (high J i and K i ).

3 A regional productivity model

The TFP dynamic equation

Our theoretical framework is based on a variety R&D-based growth models proposed by [START_REF] Grossman | Innovation and Growth in the Global Economy[END_REF]Helpman (1991, chap.3) andJones (1995). In these models, TFP is measured as a stock of knowledge, and its accumulation over time drives economic growth. The main difference in different R&D-based growth models concerns an assumption made regarding the returns of this stock of knowledge, that is, the returns to the TFP stock. If we assume constant returns to scale, then we obtain endogenous growth with an immediate adjustment to the steady state. If we assume decreasing returns to scale, then we obtain semi-endogenous growth with an adjustment path to the steady state; i.e, the short-run growth rate differs from the long-run steady state growth rate.

We assume decreasing returns to the TFP stock and thus use the approach proposed by Jones (1995).

Consequently, we can write the equation governing the dynamics of TFP for region i (noted Ṫ F P ) as:

Ṫ F P i = η i RD λ i T F P φ i (6)
where RD refers to the level of R&D investment level of the region, and η i > 0 is an exogenous productivity paremeter specific to the region. Following Jones, we assume that λ < 1, given the existence of potential duplications in R&D activities, and φ < 1 refers to the decreasing returns of the TFP stock.

Regional alignment and the exogenous productivity parameter

In the previous section, we developed the concept of regional alignment and illustrated how it could be a central explanation for the productivity dynamics of regions. We argue that regional alignment could be the theoretical source of the exogenous productivity parameter. In other words, we believe that regional alignment is an interesting candidate that could endogenously explain the value of the productivity parameter of regions.

We thus assume:

η i ≡ RA µ i ( 7
)
where RA is the level of regional alignment, and µ < 1 is the returns of regional alignment. In ( 7), we assume that regional alignment is Hicks neutral for the dynamics of productivity. Obviously, we can also imagine that regional alignment more directly influences either the returns of R&D investment (Harrod neutral) or the returns of the productivity stock (Solow neutral). In any case, this consideration is not the first importance for the purpose of this paper.

From a mono-regional to a multi-regional model of productivity

Previously, our theoretical framework considered regions such as Robinson Island. In reality, it is difficult to imagine that what happens in a particular region is totally independent of what happens in other regions, especially when they are in the same country where input mobility is strong. Indeed, due to the existence of (localized) knowledge spillovers, local trade and input mobility, it is obvious that the productivity dynamics of a particular region are driven not only by its productivity stock but also by the productivity stock of its neighboring regions. For a two-region (denoted as i and j) trade and growth model (Martin and Ottaviano, 1999), the existence of localized knowledge spillovers is modeled in the following way:

Ṫ F P i = η i RD λ i [T F P i + δT F P j ] φ (8)
where δ ∈ [0, 1[ measures the importance of interregional spillovers and thus their spatial boundary. We extend the concept proposed by Martin and Ottaviano (1999) to an N-regions model and rewrite (8) in the following way:

Ṫ F P i = η i RD λ i [W i T F P ] φ (9)
where

W i = [δ ii , δ ij , ..., δ in ]
is a 1 × n vector describing the strengh of the link between region i and the other regions. T F P is an N × 1 vector of the productivity level such that T F P = [T F P i , T F P j , ..., T F P n ]. We follow new economic geography and growth (NEGG) theory by asumming that region i can fully benefit from its productivity stock, whereas it benefits to a lesser extent from the productivity stock of its neighboring regions.

Consequently, in equation ( 8), we obtain δ ii = 1 and δ in ≤ δ ii , ∀m = i. As a short example, assume that region i has a significant link with three regions denoted respectively as j,k and l; then, the total stock of productivity that benefits region i is given by

W i T F P = T F P i + δ ij T F P j + δ ik T F P k + δ il T F P l .
Thus, in our model, the productivity growth of the region is influenced by its "usuable" productivity stock (W i T F P ), which is composed of its own productivity level (T F P i ) plus a share of other regions' productivity stock.

By inserting proposition (7) into expression (9) and dividing through by T F P i , we obtain the following equation for the productivity dynamics of a particular region i:

g i = RA µ i RD λ i T F P -1 i (W i T F P ) φ ( 10
)
where g i = Ṫ F P i /T F P i is the growth rate of the productivity of region i. Next, it is necessary to rewrite (10) as T F P i is included in W i T F P . After some manipulations, which are provided in appendix C, we are able to determine the growth rate of the productivity of a multi-region framework as:

g i = ζ i RA µ i RD λ i (W i T F P ) φ-1 ζ i ≡ [(W i -Wi )(W i W i ) -1 W i ] -1 (11)
where Wi = [0, δ ij , ..., δ in ] is a 1 × n vector describing the strength of the link between region i and the other regions. The only difference in vector W i is that in Wi , we have δ ii = 0. Consequently, (W i -Wi ) is a nonnegative (1 × n) vector of the form [1, 0, 0..., 0]. The interesting result of this simple equation is that the TFP growth rate of a region depends not only on its distance to the frontier (T F P i ) but also its distance to the frontier of its neighboring (or influencing) regions. As we assume decreasing returns of the productivity stock (φ < 1), for a region, all else being equal, the higher the neighboring levels of productivity stock are, the lower its growth rate.

From theory to empirical specification

We now assume that there is a shock ε it to the growth rate in each period. This shock is generated by a stationary process with a mean of zero. Then, following [START_REF] Ha | Accounting for Trends in Productivity and R&D: A Schumpeterian Critique of Semi-Endogenous Growth Theory[END_REF], the log-linear approximation of a discrete-time version of the generalized productivity-growth function ( 11) for region i yields:

ln

T F P it = ln ζ i + µ ln RA it + λ ln RD it + (φ -1) ln[W i T F P t ] + η t + ε it (12)
In what follows, we discuss the expected value of the main parameters from a theoretical point of view. As we assume decreasing returns of the TFP stock, i.e, φ < 1, we also assume that there is a conditional β-convergence of the regional TFP growth where β ≡ (1-φ) < 0. Based on the discussion in the previous section, we can expect a positive value for µ as a better regional alignment should increase the effects of both R&D and productivity stock. We also assume that 0 < λ < 1, implying that non-cooperative R&D investment decisions generate some duplications at the regional level.

Data and descriptive statistics 4.1 Data sources

The data collection required to evaluate the influence of regional alignment on regional productivity growth was very important and time consuming. Indeed, we needed to use various microeconomic data sources. In what follows, we describe the main databases we used to construct the empirical measures for the key variables of Model (12).

Data sources for the regional alignment measure: Patstat and the Web of Science (WOS)

As described in Section 2, the regional alignment measure aims to estimate the potential synergies between scientific domains and technological fields as well as the scientific and technological expertise of regions.

Therefore, we relied on information obtained from patents and scientific publications.

We retrieved patent data from the Patstat database compiled by the European Patent Office (EPO). We collected information from all patents granted in France for which the priority year was between 1995 and 2011 (n= 574,515). To avoid problems due to irregular patenting activity in regions, every year reports the aggregation of a backward five-year window of patenting. Using the IPC codes that each patent is associated with, we used the correspondence table provided by the EPO to allocate each patent to one of the 35 technological fields in the database (see Appendix A). To measure RTE, the usefulness of patents is measured as the number of forward citations of patents associated with each technological class.

To estimate the scientific expertise of regions, we retrieved 753,046 journal articles that were indexed by the Institute of Scientific Information's (ISI's) WOS and published between 1995 and 2011 by researchers located in France. We collected the year of publication, the name of the journal and the address of the researchers using the zip code to determine the French region they were associated with. Using the WOS's classification of journals that separates journals into scientific categories, we identified 22 scientific domains (see Appendix A).

Then, the researchers' addresses enabled us to allocate each publication to a region and measure the expertise of each region on a yearly basis.

We measured the level of interdependence between each pair of scientific domain-technological field by using the so-called "non-patent literature references" (NPLR) and selected references to scientific journal literature (Narin et al. 1997, Perko andNarin, 1997;[START_REF] Cockburn | Absorptive capacity, coauthoring behavior and the organization of research in drug discovery[END_REF][START_REF] Fleming | Science as a map in technological search[END_REF]. In other words, we retrieved all scientific publications that have been used in French patents. Scientific publications can be identified in NPLR by extracting titles between quotation marks in patents. However, titles are often incomplete, preventing us to find the right reference systematically. We used Google Scholars to find the full title of the article, as well as the name of the journal publishing the article, which allowed us to allocate each article to a WOS scientific category. We obtained 13,838 articles published between 2000 and 2011.

Data sources for the TFP measure: FICUS and FARE

The FICUS and FARE databases contain the financial statements of all enterprises (with the exception of microenterprises and agricultural holdings) with turnover that exceeded 75000 euros and that were active between 1997 and 2011. All nominal variables are deflated using various deflators made available online by INSEE, the National statistical office in France, including deflators for production, value added, intermediate consumption, investment, and hours worked. It is from these deflated data, and therefore by volume, that the levels of TFP are calculated. Although they contribute to GDP, companies with no employees are excluded from the analysis because it is not possible to compute their productivity index. Of the 32 million observations for the study period, the database includes approximately 16 million observations after the exclusions. This significant reduction in the number of observations is equivalent to excluding a mass of companies representing 7% of the total value added.

One important potential bias that occurs when using firm-level data is related to location. Indeed, the location of firms is not necessarily equivalent to the location of production activities; the latter pertains to the establishments themselves. Although the vast majority of companies have only one establishment (93.5% of the companies in our sample), in our base, multi-establishment firms represent 53% of total value added and 56% of total employment. Hence, these multi-establishment companies represent a sizeable bias toward heavily agglomerated territories. Larger companies tend to settle their headquarters close to major administrative, political and economic centers. By way of consequence, we would tend to overestimate the economic activities of agglomerated areas and underestimate the economic activities of more rural areas. To correct for this geographical bias, we use establishment-level data (the annual Declarations of Social Data, i.e., DADS data).

Such data make it possible to know, for each company, the location of manpower by establishment. Since these establishments are geographically identified by municipality and assuming that there is a proportional relationship between the proportion of staff per establishment and all other production variables (turnover, value added, investment, capital stock, and intermediate consumption), it is possible to correct the aggregation bias mentioned above.

Overall, the regional TFP measure is based on data gathered on more than 3.5 million establishments from 1997 to 2011 and includes more than 18 million observations. Finally, the establishments are aggregated at the departmental level. The methodology used in this paper to calculate the TFP at the firm level is described in appendix D.

Data sources for the R&D measure: the R&D survey

Growth theory explains TFP growth by means of R&D investments. Although the translation of R&D investments into observed TFP growth may be diffused over time, it is necessary to account for the intangible investments that must eventually translate into product or process innovation, that is, into TFP growth.

Therefore, we use the French R&D survey that is collected each year by the French Ministry of Higher Education, Research and Innovation. This database provides firm-level information on R&D activities and, particularly, on domestic R&D expenditures (DERD) and the sources of R&D financing. This survey database is organized into three files: the first provides firm-level information allowing us to characterize the firms, the second provides R&D sector-level information on the financial sources used by the firms to develop their R&D activities and the third provides NUTS 3-level information on the R&D executed in each department in each firm (expenditure and staff). The sample used to calculate aggregate R&D expenditures at the regional level includes 11,000 firms.

Descriptive statistics and a spatial analysis

In this section, we conduct a descriptive analysis of the variables we use to estimate our extended conditional β-convergence Model ( 14). To produce our "usuable" productivity stock variable (W TFP), we construct a spatial matrix describing the links among the French regions. This spatial matrix uses two criteria to weight the link between the two regions. First, we generate a matrix of economic similarity between regions. The economic similarity between region i and region j is measured by the inverse of the euclidean distance of their share of valued added in agriculture, industry and services. Second, we generate a contiguity matrix, and we multiply the two matrices.

Consequently, in our final spatial matrix, two regions are linked if they are geographically contiguous, and the weight of this link depends on the economic similarity between these two regions. Consequently, if a region has three contiguous regions, then we add the weighted average productivity stock of these three regions to the region's own productivity stock. This calculation explains why the descriptive statistics for the "usuable" stock of productivity (W TFP) are higher than those for the stock of productivity (TFP). The main characteristics of the distribution of "usuable" productivity stock (W TFP) are similar to those of productivity stock. Nevertheless we note that by taking into account the spatial dependence between the regions and their capacity to benefit from external knowledge, we slightly reduce the heterogeneity of the distribution.

Concerning R&D investment, the distribution is also highly heterogeneous, with a minimum of 0 for the Lozere region in 2005 and 2008 and a maximum of 3.74 billion euros for the Hauts-de-Seine region in 2011.

We can see considerable heterogeneity in R&D investment as the standard deviation is more than two times higher than the mean. Finally, and most importantly, we focus on our measure of regional alignment. Again, our data on the French regions indicate there is considerable heterogeneity as the minimum value is -0.049 for the Creuse region in 2001, and the maximum value is 152.655 for the Lot region in 2004. The distribution of regional alignment is highly skewed to the right with considerable excess Kurtosis, implying that most levels of regional alignment are above the mean of 9.173. Due to the small size of certain French regions, some did not report any patents and thus have a value of 0 for regional alignment. Among our 1,034 observations, 10%

(103 observations) have a zero value. These 103 observations are split among 37 regions, but 40% of those with zero values concern only 7 French regions: Cantal, Aude, Lozere, Creuse, Meuse, Alpes-de-Haute-Provence and Haute-Alpes. Nevertheless, no region has a zero value over the entire study period. Maps (c) and (d) represent the geographical disribution of R&D investment and regional alignment, respectively. Although the interpretation of such maps is a matter of taste, we do observe that the map of regional alignment (Figure 2(b)) differs from the three other maps, which do exhibit some degree of overlap.

5 The link between productivity growth and regional alignment

The conditional β-convergence of TFP growth and regional Alignment

To estimate a conditional β-convergence model, estimators that can address endogeneity must be used as the productivity stock is in the right-hand side of equation ( 12). In addition, the causal relationships existing among productivity levels, R&D investment and regional alignment could be sources of additional endogeneity.

Indeed, as Myrdal (1957) notes, the dynamics of regions, and especially their inequalities, are driven by circular cumulative causation between the variables. Consequently, the ordinary least squares (OLS) estimator is inconsistent and inefficient for estimating a conditional β-convergence model.

To address these endogeneity problems, our econometric strategy includes two processes: 1) directly estimating equation ( 12) using consistent IV estimators and 2) estimating equation ( 12) using a simultaneous equation model (SEM) with a three-stage least squares (3SLS) estimator. Both methods are able to address endogeneity problems related to temporal dependence and causality, but the advantage of the SEM approach is that it provides results on the complex interrelationships existing among productivity stock, R&D investment and regional alignment.

We start with the first part of our econometric strategy by presenting the estimation of equation ( 12) with three different estimators: least squares dummy variables (LSDV) (OLS on panel), IV-2SLS and IV-GMM. The LSDV estimation is inconsistent but provides a benchmark compared with the two IV consistent estimators. The results presented in Table 2 provide evidence of a β-convergence process in the French regions' productivity growth. More precisely, the efficient estimators (IV-2SLS and IV-GMM) estimate the speed of convergence to be between 14.5% and 17.4%. Table 2bis in Appendix E provides the estimation of the same equation using the productivity stock of the region (T F P ) instead of its usable productivity stock (W i T F P ). For this estimation, the speed of convergence is estimated to be between 10.3% and 12.1%, which is significantly lower than the results presented in Table 2. Thus, the results imply that the β-convergence process is influenced by spatial dependence among regions. Consequently, geography matters for explaining local productivity dynamics and implies that the empirical estimation using an a-spatial framework, i.e, using the productivity stock of a region instead of its "usuable" stock, is likely to understimate the true speed of convergence. The same results occur if we do not control for endogeneity as the LSDV estimator clearly underestimates the speed of convergence (less than 9%; see Table 2).

Table 2 also confirms that the β-convergence process is conditional on the level of R&D investment and regional alignment. Indeed, the consistent estimators presented in Table 2 show that both R&D investment and regional alignment positively and significantly influence the productivity growth of a region. Consequently, French regions naturally converge toward different steady states according to their behavior in terms of R&D investment and regional alignment. The heterogeneity in French regions' R&D profiles (see map (a) in Figure 2) can thus explain why regions with the highest productivity experienced a high productivity growth, and viceversa. Indeed, as R&D investment is highly concentrated in a few regions that are also the leading productivity regions, naturally, these regions have high productivity growth potential. Nevertheless, the differences in R&D investment cannot explain why some regions with relatively high productivity stock and low R&D profiles experienced high productivity growth, such as the Aude, Lot-et-Garonne and Pyrénées-Orientales regions. It seems that regional aligment is able to provide one explanation for the dynamics of these regions, and this is a very important implication. Indeed, if the level of R&D investment is strongly correlated with industries (and thus cannot be strongly influenced by political strategies), then the level of regional alignment can be more easily influenced by public authorities especially because government and local authorities strongly support scientific activities. Consequently, regional alignement could be very important in the political strategy of local authorities (especially for low-intensive R&D regions) to boost TFP growth in both the short and long run.

The sigma divergence of productivity growth and regional alignment

Since the previous subsection highlights the existence of a conditional convergence process in the TFP growth of French regions, a natural question is whether the β-convergence is associated with a σ-convergence, i.e, if heterogeneity in TFP among the French regions has increased over time. To test the presence of the σconvergence, we use two different indicators measuring the cross variations in TFP stocks: the cross standard deviation of TFP stock and the cross coefficient of variation (which corresponds to the standard deviation divided by the mean).

These two graphs clearly highlight that the conditional β-convergence is associated with a strong σ-divergence of productivity among the French regions. Indeed, the cross standard deviation increased by 5% over the study period (basis year: 2001), and the coefficient of variation increased from 98% in 2001 to 111% in 2011. It is also important to note that the 2008 financial crisis seems to have increased the σ-divergence process. These results suggest that the variables conditioning the β-convergence process, i.e., R&D investment and regional alignment, are at the heart of this process of increasing heterogeneity among the French regions. We thus decide to study the time evolution of the cross standard deviation of R&D investment and regional alignment (Figure increased by 2% between these two dates. Thus, if heterogeneity in R&D investment and in regional alignment both affect an increase in the heterogeneity of TFP heterogeneity across French regions, it suggests that the heterogeneous dynamics of regional alignment play a predominant role in the observed σ-divergence process.

6 A deeper discussion of the influence of regional alignment

The indirect effect of regional alignment on productivity growth

As explained in the previous section, the estimation of our conditional β-convergence Model ( 12) with complex interrelationships among the explanatory variables suggests that it is necessary to develop a simultaneous equation model. This model will allow us to better understand the direct and indirect effects of regional alignment on TFP growth by taking into account potential reverse causality and the cumulative causation mechanisms of the variables.

As we do not want to impose any restrictions regarding the causality existing among productivity stock, R&D investment and regional alignment, we start by defining a general simultaneous equation system for each explanatory variable of the core equation ( 12). In what follows, F refers to a linear function, and all variables are expressed in logarithm. A detail explanation of the variables used in the SEM approach is provided in Appendix F.

                 T F P it = F (RA it , RD it , W i T F P t , u it ) W i T F P t = F (W i T F P t-1 , RD it , RA it , HI it , HT it , HS it , EN it , EX it , u it ) RD it = F (RD it-1 , RA it , W i T F P t , SU B it , HI it , HS it , HT it , u it ) RA i = F (RA it-1 , RD it , W i T F P t , SU B it , HI it , HS it , HT it , u it )
where u it = α i +η t +ε it includes both the idiosyncratic error term and the individual and time fixed effects. The first equation of the system is the TFP growth equation we defined in ( 12). The second equation corresponds to the usuable TFP stock for the region that we explain by a set of variables. The first (W i T F P t-1 ) is a temporal lag that takes into account the strong time dependency of TFP stock. The second (RD) and third (RA) are the R&D investment and regional alignment of the region, respectively, which are the main drivers of conditional TFP growth convergence. We also include three different Herfindahl indices that measure the specialization of the region in terms of industries (HI), technologies (HT) and sciences (HS). Finally, in this second equation, we include the dynamics of the entry (EN) and exit (EX) of firms in the region. The third equation of our system explains the level of R&D investment in regions. As R&D investment is strongly time dependent, we include the temporal lag of R&D investment as the first explanatory variable (RD it-1 ). We add the level of regional alignment (RA), usable productivity stock (W T F P ), the amount of R&D subsidies received (SUB) and our three Herfindhal specialization indices. Finally, the last equation of the system explains RA. As explanatory variables, we include the temporal lag of regional alignment, R&D investment and usuable productivity stock (W T F P ). We also take into account R&D subsidies (SUB) to see if public support for R&D drives regional alignment. Finally, our three Herfindhal specialization indices are used as controls.

In what follows, we present the 3SLS results obtained from the SEM that was previously developed. To check for robustness, we also present, for each equation, the results we would obtain if nonsignificant causality among productivity stock, R&D investment and regional alignment were not considered. In the following tables, the estimations of the entire SEM is called the "full system", whereas the estimations of the SEM without significant causality is called the "restricted system".

The results for the core equation (Table 3) are consistent with our previous findings using IV-2SLS and IV-GMM estimators and are presented in Table 2. Indeed, we find evidence of a β-convergence process of TFP growth that is conditional on a positive effect of both regional alignment and R&D investment. The speed of Table 5: R&D investment model with the SEM approach Our model for R&D investment fits well with our data with an R 2 of nearly 80%. Most of the results obtained are comparable to those in the literature. Indeed, we find evidence that R&D investment is serially correlated in time with a coefficient related to the lagged value of approximately 0.34. We also find that R&D subsidies have a positive influence. A more important result concerns the influence of the usable productivity stock and regional alignment. First, we do not find a significant relationship between the productivity stock and the level of R&D investment, which implies that the causal relationship between R&D investment and productivity stock is clear: investment in R&D drives productivity, but the inverse does not hold. Second, we find that RA has a strong positive effect on the level of R&D investment. This result reveals that regional alignment has a more complex influence on productivity growth. Indeed, all the previous estimates show that regional alignment can be seen as a regional input that directly drives productivity growth. This last equation highlights that regional aligment has an indirect effect on productivity growth through its leverage effect on R&D investment (which is the other input of productivity growth). Consequently, this last result implies that regional alignment is at the heart of the σ-divergence process of productivity that we observe among the French regions. Concerning the controls, our results show that the specialization of regions in terms of industries and technologies limits R&D investment, whereas specialization in science has a positive effect.

Finally, another reason for using the SEM approach is that it allows us to better understand what creates regional alignment. We obtain the following results from the regional alignment equation ( Our model for regional alignment, which takes into account both time and regional fixed effects, is clearly not satisfactory because it has an R 2 of approximately 37%. Indeed, only the industrial and technological specialization of regions seem to positively drive the regional alignment, but the significance remains low (pvalue > 5%). This result clearly highlights the complexity of explaining the (res)sources that create the regional alignment. Unfortunately, our dataset does not allow us to test a richer model, and it is clear that a better understanding of the elements at the source of regional alignment is needed.

The heterogenous effect of regional alignment

In the previous subsection, we highlighted that regional alignment has an indirect effect (which is cumulative with its direct effect) on regional productivity growth. Another important question that arises is whether the impact of regional alignment is homogenous or heterogeneous among French regions. To investigate this question, we need to estimate equation ( 12) by including a heterogenous coefficient; i.e, the impact of the explanatory variables are region specific. To retain the logic of the conditional β-convergence model, we still assume that there is a common β parameter for the usable productivity stock, but we allow for the heterogeneous impacts of both regional alignment and R&D investment. Thus, we estimate the following model

ln T F P it = α i + µ i ln RA it + λ i ln RD it + (φ -1) ln W i T F P t + η t + ε it (13)
Therefore, we use the common correlated effects estimator (CCE) proposed by Peasaran ( 2006) that takes into account the unobserved common factors of the regions, i.e, a cross-section correlation. Due to the small temporal dimension of our data, we apply two corrections for the mean group estimates: jacknife and recursive. As we can see, the estimated pooled and mean group values are similar to those obtained with the IV and SEM methods when we apply the jacknife correction, whereas they are very different when we apply a recursive correction. We thus decide to focus our analysis on (1) the results provided by the jacknife correction and (2)

the heterogenous effect of regional alignment.

In the previous section, we highlight that regional alignment has a positive indirect effect on R&D investment.

As discussed in Section 2, localized knowledge spillovers induced by R&D activities are important drivers of productivity growth. Thus, a natural question arises as to whether the positive (direct and indirect) effect of regional alignment on productivity growth plays a role in Jacob or MAR externalities. To provide some answers to this question, we analyze whether the reaction of regions with respect to regional alignment (µ i ) is influenced by their level of specialization at three different levels: industry, technology and science. This level of specialization is computed using the Herfindahl index. If Marshallian externalities dominate, concentration of either scientific (HS), technological (HT ) or industrial (HI) activities would increase the effectiveness of regional alignment (µ i ) on productivity growth. If instead Jacobian externalities dominate, such concentration measures (HS, HT , HI) would have a negative effect on µ i .

To start this analysis, we compute and test the Pearson's correlation coefficient between the heterogeneous effects of regional alignment and our three measures of specialization. We find a systematic negative correlation coefficient between the effect of regional alignment and the the value of the Herfindhal indices. Nevertheless, the negative correlation is only (strongly) significant for the Herfindhal index related to technology (ρ = -0.2128 with a p-value of 0.039), whereas it is less significant for the Herfindhal indices related to industry (ρ = -0.0997 with a p-value of 0.339) and science (ρ = -0.121 with a p-value of 0.246). To delve deeper into the analysis and because one limitation of the Pearson's correlation coefficient is its linearity, we decide to run a simple OLS regression. The Table 8 shows our main findings.

Dependent Variable: µ i (see equation ( 13)) When we test for the linear effect of our Herfindhal measures (Model 1), we find that technology specialization has a significant negative effect on the effect of regional alignment on productivity growth but only at the 10% level of significance. Specialization in terms of industry and science do not seem to have a significant influence.

Nevertheless, we can easily imagine that a non-linear relation exists. Model 2 introduces non-linear effects, and this change strongly affects our findings. Indeed, Model 2 indicates that there is a negative convex relation between industrial specialization and the effect of regional alignment on productivity growth. The two others measures of specialization (technology and science) have no significant impact. Model 3 confirms the results of Model 2. Using results from Model 3, we compute ∂µ i /∂HI = 0 in order to identify the value HI * which provides us with the threshold below (above) which Jacobs (MAR) externalities dominate. We find a value HI * = 0.058, and empirically observe that for 91% of regions, the Jacobs externality dominates (diversity enhances the effect of regional alignment) whereas in the remaining 9% of regions, the MAR externalities dominates (specialization enhances the effect of regional alignment). Consequently, we find evidence that in most cases, industrial specialization reduces the positive effect of regional alignment on productivity growth. This result suggests that regional alignment plays more of a role in Jacob's spillovers rather than MAR externalities.

Thus, the diversified regions could strongly increase their potential growth by increasing their regional alignment between science and technology. In the figure below, we represent the convex relationship between industrial specialization and the effect of regional alignment. Figure 5 (except for the three most specialized regions)

clearly indicates that regional alignment has a greater impact on the productivity growth of regions that are more industrially diversified. In a sense, this is not a surprising result because increased alignment between the scientific and technological capabilities in diversified regions will have a more positive effect positive effect on productivity growth due to Jacobs spillovers. In less diversified regions, potential Jacobian spillovers are lower, implying that regional alignment has less of an effect on productivity growth, all else being equal.

-

Conclusion

Since the pioneering works of [START_REF] Barro | Economic Growth[END_REF], the economic literature has widely discussed the concept of regional convergence. A large consensus on conditional convergence suggests that there are persistent or increasing regional differences in terms of economic performance. Although the literature can clearly explain why "weak" regions become weaker and strong regions become stronger, economists remain unable to explain why regions with similar conditional factors (such as the level of R&D investment and human capital) experience different growth paths. The aim of this paper is to provide a theoretical and empirical explanation. This paper makes several contributions. First, we propose the concept of regional alignment as a measure of the level of synergies among the scientific fields, technological domains, and industrial sectors in which a region has expertise. We argue that regional alignment can theoretically reflect regions' specific characteristics and contributes to the development of a productivity parameter in growth theory. Therefore, regional alignment is a critical determinant of differences in regional growth paths. Second, we develop a spatial extension of the traditional convergence model of productivity growth [START_REF] Ha | Accounting for Trends in Productivity and R&D: A Schumpeterian Critique of Semi-Endogenous Growth Theory[END_REF] by including both localized knowledge spillovers and regional alignment as an input of productivity growth. Third, using French firmlevel data aggregated at the NUTS 3 level over the period 2001-2011, we estimate our extended conditional convergence model and test our hypothesis about the influence of regional alignment. Using consistent IV estimators, we develop a simultaneous equations model and a heterogeneous coefficient model to better understand the influence and causality of regional alignment in the dynamics of regional productivity growth.

Our results confirm most of our hypotheses about the importance of regional alignment as an essential driver of the productivity growth of a region. First, we obtain evidence that regional productivity growth in France follows a conditional convergence process (regions tend to converge toward different steady states), which is associated with increasing heterogeneity across regions. All the results indicate that regional alignment matters for explaining this productivity growth process. Indeed, regional alignment has a direct effect on productivity growth and can be seen as a regional input. Moreover, our simultaneous equations model shows that regional alignment has an indirect effect by leveraging the role of private R&D investment in local productivity growth.

The last empirical finding emphasizes the heterogeneous effect of regional alignment across the French regions.

We find evidence that a negative relationship exists between the effect of regional alignment on productivity growth and the level of industrial specialization of regions. In other words, regional alignment matters more for the productivity growth in diversified regions than that in specialized regions. This result suggests that regional alignment tends to increase the value of Jacob's externalities and thus materializes if the regional industrial structure is sufficiently diversified.

In addition to the simple synergies between science and technology, we argue that regional alignment contributes to productivity growth because it actually reveals the efficiency of coordination mechanisms, which [START_REF] Aghion | Science, technology and innovation for economic growth, linking policy research and practice in STIG Systems[END_REF] 2) call "the intimate and multiple connections of technological change and innovation with advances in science, on the one hand, and the set of socio-economic institutions operating in a given context, on the other." (Aghion et al. 2009, p. 2). Synergistic relations between scientists and engineers can only be effective if there are spaces for dialogue contributing to a shared understanding among different communities, which allow knowledge to be shared despite high cognitive distance [START_REF] Sabel | Work and Politics: The Division of Labour in Industry[END_REF][START_REF] Lester | Innovation, The Missing Dimension[END_REF][START_REF] Lowe | Constructing entrepreneurial advantage: consensus building, technological uncertainty and emerging industries[END_REF][START_REF] Cohendet | Epistemic communities, localization and the dynamics of knowledge creation[END_REF]. This study may explain why, everything else being equal, despite a low level of R&D investment, aligned regions can change their steady state growth path better than other regions. Further works should analyze the micro-processes that lead to effective regional alignment.

It follows that regional alignment could be an important guide for policy makers (especially for low-intensive R&D regions) who are willing to boost TFP growth in both the short and long run. The question, then, is to discuss possibilities to generate and reinforce alignment. In this respect, regional alignment is not orthogonal to smart specialization policy design [START_REF] Foray | Smart specialization. The Concept[END_REF]2011;McCann and Ortega-Argilés, 2015). They both recognize the importance of taking into account the heterogeneity of regional trajectories based on industrial, institutional, cultural and historical specificities. However, unlike the related variety interpretation of smart specialization [START_REF] Balland | Smart specialization policy in the European Union: relatedness, knowledge complexity and regional diversification[END_REF], regional alignment does not focus on a technology driven policy. Rather, synergies between technological fields and scientific domains only reveal systemic dynamics, the effectiveness of decisions made by heterogeneous actors and the level of efficiency in resource allocation processes. It would be interesting to understand better the mechanisms underlying the emergence of those synergies, in particular to design policies dedicated to laggard regions. This number of joint occurrences is used to construct our measure of synergy by relating it to some measure of its expected frequency Ĉjk under the hypothesis of random joint occurrence. There is no authoritative measure of Ĉjk , but we propose a parametric-based measure in this paper. More precisely, we assumre that the number C jk of patents assigned to j and k is a hypergeometric random variable. Thus, the probability of drawing C patents with both technology j and scientific domain k follows the hypergeometric density function.

P (X jk = x) =    C j x       N -C j C k -x       N C k   
where is X jk is the number of patents assigned to both technology j and scientific domain k, x is the hypergeometric random variable and N is the total number of patents. The mean value (expected frequency)

and variance of random co-occurrence are:

Ĉjk = µ jk = E(X jk = x) = C j C k N σ2 jk = µ jk N -C j N N -C k N -1
If the actual number C jk of co-occurrences observed between j and k greatly exceeds the expected value µ jk of random co-occurrences, then j and k are synergistic; that is, they are more productive than a random association.

Conversely, when C jk < µ jk , then technologies j and k are poorly synergistic (their combined use produces fewer patents than expected when drawing a random combination). Hence, the level of synergy between j and k is defined as:

τ jk = C jk -µ jk σ jk
Equation ( 4) has two interesting features. First, τ jk is a real number that can be either positive or negative, with no lower or upper bounds: τ jk ∈ R :] -∞; +∞[ the sign is a straightforward and intuitive interpretation.

Second, τ jk is similar to a t-student, so that if τ jk ∈] -1.96; +1.96[, then the null hypothesis H0 (no synergistic relations exist between technology j and k) can be safely accepted. Third, τ jk is a matrix of science technological relations that can be seen as an approximation of scientific and technological knowledge represented as a hierarchical tree (Popper, 1972).

Appendix C: The productivity growth rate in a multi-regional model

We start with equation ( 10)

g i = RA µ i RD λ i (T F P i ) -1 (W i T F P ) φ
Rewriting (W i T F P ) = T F P i + Wi T F P , we thus have

T F P i = [W i -Wi ]T F P
Next, we rewrite the level of productivity in region i as:

T F P i = [W i -Wi ][(W i W i ) -1 W i W i ]T F P = [W i -Wi ][(W i W i ) -1 W i ][W T F P ]
Inserting this last expression of T F P i into equation (10) leads to:

g i = RA µ i RD λ i [W i -Wi ][(W i W i ) -1 W i ][W i T F P ] -1 (W i T F P ) φ = RA µ i RD λ i [W i -Wi ][(W i W i ) -1 W i ] -1 (W i T F P ) -1 (W i T F P ) φ = RA µ i RD λ i [W i -Wi ][(W i W i ) -1 W i ] -1 (W i T F P ) φ-1
where Y zt denotes the real gross output produced by firm z at time t using the set of n inputs X nzt , where input X is alternatively capital stocks (K) and labor, in terms of hours worked (L) and intermediate inputs (M ). S nzt is the cost share of input X nzt in the total cost. Subscripts τ and n are indices for time and inputs, respectively. Symbols with an upper bar correspond to the measures for the reference point (the hypothetical firm), which are computed as the means of the corresponding firm level variables, over all firms in year t. Note that ( 14) implies that the reference points ln Y and ln X are the geometric means of the firm's quantities and input quantities respectively, whereas the cost shares of inputs of the representative firms S is computed as the arithmetic mean of the cost share of all firms in the dataset.

Equation ( 14) allows us to estimate the productivity stock of each firm in our sample. To produce a regional measure of productivity, we calculate the average level of productivity for firms located in region i:

T F P it = z=N z=1 T F P zt N ( 15
)
and then apply the following formula:

T F P it = T F P it × L it L t ( 16
)
where L it is the employment level in region i at time t, and L t is the employment in France at time t.

All nominal output and inputs variables are available at the firm level. Industr-level data are used for price indexes, hours worked and depreciation rates.

Output

Gross output deflated using sectoral price indexes published by INSEE (French system of national accounts).

Labor

Labor input is obtained by multiplying the number of effective workers (i.e., the number of employees plus the number of outsourced workers minus workers taken from other firms) by average number of hours worked.

The annual series for hours worked are available at the 2-digit industry level and provided by the Groningen Growth Development Center (GGDC)). This choice was made because there are no data on hours worked in the EAE survey.

Capital input

Capital stocks are computed from the investment and book value of tangible assets following the traditional perpetual inventory method (PIM):

K t = (1 -δ t-1 ) K t-1 + I t (17)
where δ t is the depreciation rate, and I t is real investment (deflated nominal investment). Both the investment price indexes and depreciation rates are available at the 2-digit industrial classification level from the INSEE data series.

where s is firm z's market share in the NUTS 3 region i, including all sectors. A value close to unity indicates market concentration, the polar case being unity, where all the market is supplied by a unique monopolistic company. Hence, the HHI is usually used to account for the degree of competition in an industry, most often within a country or geographic unit. In our case, however, we aim to determine the weight of large firms within a region across all productive activities. Hence, this measure of competition considers the local industrial structure rather than the degree of competition that most often spans over single NUTS 3 regions.

To compute the HHI index for technologies (denoted as HT), we use the PATSTAT database and calculate the following statistics:

HT i = j s 2 ij
where s is technology field j's share in the NUTS 3 region i, including all the patents that have been granted.

A value close to unity indicates strong technology concentration, the polar case being unity, where all the patents granted in a region belong to one technological field. Hence, the HHI for technology indicates the local technological structure of NUTS 3 regions.

To compute the HHI index for sciences (denoted HS), we use the WOS database and calculate the following statistics:

HS i = k s 2 ik
where s is scientific domain k's share in the NUTS 3 region i, including all publications. A value close to unity indicates a strong scientific concentration, the polar case being unity, where all the publications produced in a region belong to one scientific domain. Hence, the HHI for science indicates the local scientific structure of NUTS 3 regions.

R&D subsidies (SUB)

We use the second and third files of the R&D survey to calculate the level of public R&D subsides received by firms at the NUTS 3 level.The second file of the R&D survey describes the financial sources used by firms to develop their R&D activities. From this file, we can calculate the share of R&D expenditures financed by public funds by firm and sector. However, this file only includes 11,000 R&D firms. To correctly geo-localize the R&D subsidies in each NUTS 3 region, we had to match the information calculated from the second file of the R&D survey with the information contained in the third file, i.e, the R&D executed by firms at the NUTS 3 level. Thus, we are able to redistribute the subsidies proportionally to the R&D executed in each NUTS 3 region (taking into account the size and sectorial composition of the firms in the region) .

Figure 1 Figure 1 :

 11 Figure 1 below provides two maps representing the means of TFP growth and TFP stock for the period 2001-2011. These two maps highlight a strong geographical concentration of both TFP growth and TFP stock

Figure 2

 2 Figure2provides two maps representing the geographical disribution of average R&D investment and regional alignment for the period 2001-2011.

Figure 2 :

 2 Figure 2: TFP, R&D and RA in French regions

Figure 3 :

 3 Figure 3: Evolution of the TFP heterogeneity of the French regions

Figure 4 :

 4 Figure 4: Cross standard deviation of regional alignment and R&D investment

Figure 5 :

 5 Figure5: Link between the effect of regional alignment and industrial specialization

Table 1 :

 1 productivity growth is skewed to the left with excess kurtosis, implying a high proportion of negative productivity growth within the French regions. We also see important heterogeneity in the productivity stock (TFP) of the French regions. The lowest productivity stock value is obtained for the Lozere region in 2009 (0.349), and the highest is obtained for the Paris region in2007 (20.117). The distribution of productivity stock is strongly skewed to the right with a high level of excess kurtosis, which means that most of the productivity stock observed is well over the average productivity stock (2.752). Descriptive Statistics

	Variable	Obs.	Mean	S.D	Min	Max	Skewness Kurtosis
	∆ ln T F P 1034	-0.004	0.036	-0.158	0.094	-1.235	5.656
	T F P	1034	2.752	2.475	0.349	20.117	3.371	20.469
	W T F P	1034	4.939	3.216	1.401	25.908	2.757	15.355
	RD	1034 251 381 527 109	0	3 738 882	4.181	22.220
	RA	1034	9.173	10.346 -0.049 152.6547	4.525	46.761

Table 1 below shows that, on average, productivity growth in the French regions during the last decade has been negative (-0, 4% per annum) with strong heterogeneity as the minimum value is -15.8% (Savoie region in 2008) and the maximum value is 9.4% (Paris region in 2007). Moreover, the empirical distribution of

Table 2 :

 2 Conditional β-convergence model

			LSDV			IV-2SLS			IV-GMM
	Variable	Coeff.	s.e.	P-Value	Coeff.	s.e.	P-Value	Coeff.	s.e.	P-Value
	ln[W i T F P ] -0.089 0.027	0.001	-0.145 0.043	0.001	-0.174 0.042	0.000
	ln[RD]	0.002 0.002	0.389	0.014 0.006	0.017	0.015 0.005	0.003
	ln[RA]	0.002 0.001	0.001	0.014 0.004	0.000	0.017 0.003	0.000

Table 6

 6

	):

Table 6 :

 6 Regional alignment model with the SEM approach

Table 7 :

 7 The results of the estimations are provided in the table 7. Conditional β-convergence with the CCE approach

		Jacknife Correction		Recursive Correction
	Variable Coeff. S.E. t-Stat P-Value Coeff. S.E. t-Stat P-Value
			Pooled
	W T F P	-0.168 0.986 -0.17	0.865	-0.416 0.149 -2.78	0.005
			Mean Group Estimates
	RD	0.013 0.004	3.35	0.001	0.010 0.004	2.49	0.013
	RA	0.013 0.003	4.79	0.000	0.007 0.002	3.88	0.000

Table 8 :

 8 Effect of regional alignment and the specialization of regions

			Model 1			Model 2			Model 3
	Variable Obs Coeff. S.E. P-Value	Coeff.	S.E.	P-Value	Coeff.	S.E.	P-Value
	H	94	-0.112 0.316	0.723	-3.326	1.466	0.026	-3.388	1.445	0.021
	HT	94	-0.048 0.028	0.092	-0.114	0.078	0.150	-0.031	0.027	0.261
	HS	94	-0.012 0.019	0.528	-0.021	0.089	0.818
	H 2	94			28.363 12.391	0.025	29.053 12.305	0.020
	HT 2	94			0.188	0.121	0.124
	HS 2	94			0.020	0.131	0.876
	cons	94	0.027 0.013	0.036	0.114	0.037	0.003	0.105	0.036	0.004
			R 2 : 0.049			R 2 : 0.1319		R 2 : 0.1080

  assigned to the same patent document, then C j ∩ C k = and thus the number C jk of the observed co-occurences of j and k is C t jk = n P t nj P t nk . Applying the latter to all possible pairs of technologies, we obtain a matrix Ω (J * K) with J = 35 technological fields and K = 22 scientific domains.

	Let P t nj = 1 if patent n is assigned to technology j and 0, otherwise. The total number of patents assigned
	to technology j in t is thus C t j = n P t nj . Let P t nk = 1 if patent n mentions an article published in a journal
	associated with the scientific domain k in the non-patent citations section and 0, otherwise. The total number
	Mathematics & Statistics nk . Since technology j and scientific domain k = n P t of patents assigned to scientific domain k in t is thus C t
	6 k may be	Clinical Medicine	17	Philosophy & Theology
	7	Communication & Textual Studies	18	Physics & Astronomy
	8	Earth & Environmental Sciences	19	Psychology & Cognitive Sciences
	9	Economics & Business	20	Public Health & Health Services
	10	Enabling Strategic Technologies	21	Social Sciences
	11	Engineering	22	Visual & Performing Arts
		List of Technological field
	Number	Name	Number	Name
	1	Electrical machinery, apparatus, energy	19	Basic materials chemistry
	2	Audio-visual technology	20	Materials, metallurgy
	3	Telecommunications	21	Surface technology, coating
	4	Digital communication	22	Micro-structural and nano-technology
	5	Basic communication processes	23	Chemical engineering
	6	Computer technology	24	Environmental technology
	7	IT methods for management	25	Handling
	8	Semiconductors	26	Machine tools
	9	Optics	27	Engines, pumps, turbines
	10	Measurement	28	Textile and paper machines
	11	Analysis of biological materials	29	Other special machines
	12	Control	30	Thermal processes and apparatus
	13	Medical technology	31	Mechanical elements
	14	Organic fine chemistry	32	Transport
	15	Biotechnology	33	Furniture, games
	16	Pharmaceuticals	34	Other consumer goods
	17	Macromolecular chemistry, polymers	35	Civil engineering
	18	Food chemistry

see Ciccone,

for empirical evidence of the link between the job density and growth of European NUTS 3 regions. A series of studies have tested the sensitivity of knowledge spillovers to distance to determine whether they can explain agglomeration (e.g., see[START_REF] Jaffe | Geographic Localization of Knowledge Spillovers as Evidenced by Patent Citations[END_REF][START_REF] Ciccone | Productivity and the Density of Economic Activity[END_REF] Audretsch and Feldman, 1996;[START_REF] Combes | Economic Structure and Local Growth, 1984-1993[END_REF][START_REF] Rosenthal | Evidence on the nature and sources of agglomeration economies[END_REF]

[START_REF] Porter | The Economic Performance of Regions[END_REF][START_REF] Carlino | The agglomeration of R&D labs[END_REF][START_REF] Buzard | The geography of research and development activity in the U.S[END_REF][START_REF] Bloom | Identifying technology Spillovers and Produc Market Rivalry[END_REF][START_REF] Lychagin | Spillovers in Space: Does Geography Matter[END_REF][START_REF] Buzard | Localized Knowledge Spillovers: Evidence from the Spatial Clustering of R&D Labs and Patent Citations[END_REF]).

For example, the innovative and economic performances of regions vary with the diversity of industry composition(Vernon, 1960; Jacob; 1969;[START_REF] Porter | The Competitive Advantage of Nations[END_REF] Duranton and Puga, 2001, Helsley and[START_REF] Helsley | Coagglomeration, clusters, and the scale and composition of cities[END_REF], the intensity of competition among local firms[START_REF] Porter | The Competitive Advantage of Nations[END_REF][START_REF] Bloom | Identifying technology Spillovers and Produc Market Rivalry[END_REF][START_REF] Grebel | Technological spillovers, product market rivalry and R&D investment[END_REF], the level of the maturity of industries[START_REF] Neffke | How do regions diversify over time? Industry relatedness and the development of new growth paths in regions[END_REF], the average or diversity of firm size[START_REF] Rosenthal | Geography, Industrial Organization, and Agglomeration[END_REF][START_REF] Delgado | Clusters and entrepreneurship[END_REF] Agarwal et al., 2014), the intensity of entrepreneurship[START_REF] Glaeser | Entrepreneurship and urban growth: an empirical assessment with historical mines[END_REF], and the overlap of resources across industries[START_REF] Porter | The Economic Performance of Regions[END_REF][START_REF] Neffke | The impact of aging and technological relatedness on agglomeration externalities, a survival analysis[END_REF][START_REF] Delgado | Clusters and entrepreneurship[END_REF][START_REF] Delgado | Clusters, convergence, and economic performance[END_REF].

Abstract knowledge means the ability to represent a range of phenomena by means of a limited number of variables. General knowledge is the ability to relate distant elements of knowledge. Conversely, local and concrete knowledge is applied to concrete experiments, a process which relies primarily upon tacit abilities and trial-and-error.

The scientific domains and technological fields are presented in Appendix A

The hypergeometric distribution, which stems from a binomial distribution, describes the probability of successfully drawing x out of N draws without replacement (while the binomial distribution assumes replacement).

convergence is estimated at 12.6%, which is slightly lower than the results obtained with other IV estimators.

The quality of our model is relatively good with an R 2 of 0.67. As the SEM approach provides similar results for our core equation of productivity growth, we focus our analysis on the three other equations of the system. We start with the usuable TFP stock equation (W T F P ) for which we obtain the following results ( Our model for the usable TFP stock equation is very good with an R 2 of 99.98%. Most of the results obtained are comparable to those in the literature. Indeed, the usuable TFP stock is serially correlated in time with a coefficient related to its lag value of 0.803. We find evidence that the level of TFP stock of a region is positively related with its level of R&D investment. In contrast, the level of regional alignment does not seem to significantly influence the usable TFP stock. This last result is due to the fact that the usuable stock of productivity also takes into account the productivity stock of neighboring regions. Indeed, if we replicate the SEM using the productivity stock instead of the "usable" stock, then we find that regional alignment has a positive and significant effect on the productivity stock. Nevertheless, dropping this nonsignificant relation between regional alignment and usable productivity stock does not change the other results (see the results for the restricted system). Concerning the control variables, the dynamics of the entry and exit of firms matter and suggest that regional barriers to firms' mobility hinders the development of regional productivity stock.

Concerning our Hefindahl indices, the usable TFP stock seems to be influenced by only the regional specialization in science. Indeed, specialization in terms of technologies and industries is not satistically significant.

We now discuss the third equation of our system related to the R&D investment level of regions. We obtain the following results for the R&D equation ( Appendix B: The parametric measure of synergies λ jk

Number of patents with technology j

To smooth the importance of patents associated with a particular technology j (or a scientific domain k), all patents are counted using a five year moving sum.

Appendix D. The TFP measure

The productivity measure

We compute total factor productivity (TFP) by using the so-called multilateral productivity index, which was first introduced by [START_REF] Caves | The economic theory of index numbers and the measurement of input, output, and productivity[END_REF] and extended by [START_REF] Good | Index number and factor demand approaches to the estimation of productivity[END_REF]. Contrary to Olley-Pakes (1996) measure, this is a non-parametric measure of TFP that does not impose a functional form for the production function. This methodology consists of computing the TFP index for firm z at time t as follows:

ln

Intermediate inputs

Intermediate inputs include the purchase of materials, merchandise, transport, traveling, and miscellaneous expenses. Intermediate inputs are deflated using sectoral price indexes for intermediate inputs published by INSEE.

Appendix E: Estimated β-convergence in a mono-regional model

Appendix F. Details on the variables used

In this appendix, we provide details on the variables used in the simultaneous equation model (SEM).

Entry (EN) and exit (EX) rates of establishments at the NUTS 3 level

We compute these measures using the FICAS and FARE databases. The measures of industry churning account for the capacity of a region to rejuvenate, that is, to devote private resources to new entrepreneurial projects. We do this by measuring firm entry into and exit from the local area. The entry rate is simply the ratio of the number of establishments new to a NUTS 3 region over the overall number of establishments in that region. For each year, we count the number of new establishments in a given NUTS 3 region using the SIRET administrative number. Entry in the database may not accurately trace entry in the region, due to the fact that inclusion in the FICUS and FARE databases implies that company revenues exceed the minimum threshold of 75,000 euros. Likewise, the exit rate is the ratio of the number of establishments exiting a NUTS 3 region over the overall number of establishments in that region. Exit from the database may not accurately trace liquidation or relocation, due to the threshhold of 75,000 euros.

Concentration indices for industry, technology and science (H, HT and HS): the computation of HHI

In this paper, we also account for the industry, technological and scientific structure of region i by computing the Hirschman-Herfindahl Index (HHI).

To compute the HHI index for indusry (denoted H), we use the FICUS and FARE databases and calculate the following statistics:


Martin Pruvost
email: martin.pruvost@ensam.eu

Pierre Mialocq
email: pierre.mialocq@saf-rangroup.com

Fakhreddine Ababsa
email: fakhreddine.ababsa@en-sam.eu

Augmented Reality assistance for R&D assembly in Aeronautics

Keywords: aeronautics, augmented reality, factory of the future, assembly

Introduction

Augmented reality (AR) allows integration of digital information in the real world through the use a device. As a brick of the Factory of the Future, it begins its deployment throughout the industry and especially in aeronautics industry where several implementations are being tested. This paper presents an AR system architecture for assisting complex assembly work by adding visual information superimposed on the physical assembly parts. The traditional approach for complex assembly assistance is to use assembly drawings and instruction manuals to describe the content of each work task. This approach is not adapted to the complexity and the small series of R&D assembly. The main objectives in this paper are to expose the benefits of AR for complex assembly and to offer an application for the instrumentation of aeronautics parts. The organization of this paper is as follows. First we present a brief overview of related work on AR assistance for assembly operations. Our use case is explained next. Then we will expose our demonstrator, explaining both its hardware and software, and then the conclusions of the work so far.

Related Work

AR can be used in several ways to assist complex assembly. [Gonzalez-Franco & al.] showed that an assembly training using an immersive Augmented Reality setup with see-through headset can give the same level of knowledge than a conventional face-to-face training. [Suarez-Warden & al.] lead an experiment using a monitor-based system for the assembly of an aircraft structure. They found the mean time for AR assisted assembly to be shorter than the time using traditional instructions. It has to be noted that none of the subjects had any prior knowledge about assembly. French start-up Diota provides an AR software and hardware solution for the manufacturing industry. A system based on projection was deployed to help the assembly of large fuselage parts, and it reduced the error rates by 90% as well as the cycle time by 35%.

Use case

Safran Helicopter Engines is a French manufacturer of gas turbine turboshaft engines for helicopters. When developing a new engine or an evolution of an existing one, they have to perform several kind of engine tests on test benches. In order to determine the performances of the engine, a large number of sensors (strain gauges, thermocouples, pressure strobes,…) are disposed on critical parts. This instrumentation is made by specialized workers who read blueprints, defer the position asked by the R&D development and then set the sensors to the requested position. When instrumenting complex-shaped parts such as centrifugal impeller, it becomes difficult to defer the dimensions.

A centrifugal impeller Thermocouples and their wires and guards

This project's aim is to use an Augmented Reality system displaying in real time a sight standing for the requested position while positioning a sensor.

The device shall use a remote monitor and a mobile camera in order to be hands-free. The instrumentist will align the sensor on the part while looking at the screen instead of his hands. The target accuracy is 1/10 mm and the parts shall not be altered by any means. In a second time, it could be useful to add the pattern of the wire's pathway and to include a measurement function that measure the offset and create position reports after the positioning.

Demonstrator

The demonstrator has been made using a 720p webcam, Unity, Vuforia and its Model Target feature. It focuses on the laying of strain gauges on a centrifugal impeller. After creating the Model Target file, the digital mock-up of the part is superimposed on the real part and a sight represents the requested position of the gauge.

The work station of the demonstrator Sights that are used to position the gauges

Despite the relatively good tracking especially in motion, the precision is inadequate. The overlay's mismatch can reach up to 4mm, as it can be seen on the edge in the right figure above. This mismatch is too important to carry a real experiment and gather significant data to conclude. However, it seems that Diota could reach the target precision of 1/10 mm in a similar project, using its software and a higher resolution camera.