1. Introduction

The construction of a regional I-O matrix can be actually constructed using regional primary data (survey method), although it is costly and time consuming and therefore, rarely used at the regional level. More frequently, non-survey methods, which rely on various techniques that adjust a national I-O matrix, are used. According to Richardson (1972), the derivation of these methods is based on various secondary sources of statistical data applied to the national model. Several methods have already been proposed and are nowadays applied for the construction of regional matrices. However, the use of hybrid methods is stressed since they combine both survey and non-survey approaches through the use of specific data and information from small scale surveys and general statistical data.

This last approach is currently considered the most feasible to derive regional I-O matrices (Lahr, 2001), thus, numerous versions of hybrid regionalization methods have been developed (Lahr, 1993). Furthermore, the hybrid approach is classified into two categories, depending on the origin of the information: 1. The top-down approach if the construction of the regional matrix is based on a national I-O matrix and 2. The bottom-up approach, when the construction of the regional matrix uses information coming from the respective region. (West, 1990).

As already mentioned, the construction of regional I-O matrices is usually modeled after national matrices, and they are considered as an analytical tool for the formulation of growth and regional development policies. However, their results are highly aggregated, leaving aside the spatial interdependencies that characterize the real behavior of a regional economy. Furthermore, top-down results do not capture the spatial heterogeneity of the intra-regional structure and its fragmented and partial economic interactions, normally taking place in different locations within a region.

The international literature on regional matrices construction, despite showing an interest for regional matters and their efforts to apply it to the regional field, lacks a spatialized regional approach, since their first applications. The first studies were made by Isard (1951) and Leontief (1955). Then came the studies of Leontief and Strout (1963), Morrison (1974), Morrison and Smith (1974), Round (1983) and Richardson (1985) Miller and Blair (1985), Hewings and Jansen (1986).³

The main focus of analysis of these models is the differentiation of the number of regions: An isolated region, two regions and multiple regions, and then focusing in the challenge of creating multiple regional models. They assume that the spatial disaggregation of a region and the use of political spatial units, as representative of economic spatial units are not central problems for the construction of regional I-O matrices.

Nowadays, the problem remains very similar. Given the restricted available regional information⁴, during the last 20 years, the debate in the literature has been centered on techniques and alternative methodologies for the application of the top-down approach for the construction of regional I-O matrices.

The debate is then focused only in how a regional I-O matrix should be constructed with hybrid methods from a top-down perspective, focusing on the one hand, on the estimation of regional purchases and sales, relying mainly on the 1995 and 1997 Flegg`s location quotient. On the other hand, the need to improve the accuracy of Stone´s 1969 RAS algorithm and the traditional location quotients is stressed. As Michael Lahr (1993), states, hybrid model construction should pursue a most accurate non-survey model of a region as possible -using adequate regional purchase coefficients and minimizing data aggregation.

In a similar fashion, the studies of this topic in México, have been oriented to the application of the top-down approach mainly by using Flegg and RAS methods for the construction of regional I-O matrices, basically to most of Mexican states, and other regions and municipalities. Among the most important researches, we have the following: (Dávila, 2002), (Fuentes and Brugués, 2001); (Calicó et al, 2003), (Fuentes, 2003 y 2005), (Armenta, 2007), (Cruz, 2008), (Chapa, 2009), (Rosales, 2010), (Aroche Fidel, 2013) and ( Dávila, 2015).

Without any doubt, these works, just like those at the international level are very important applied researches that try to improve the knowledge related to the construction of regional I-O matrices in Mexico, through a top-down approach. However, it is evident for an academic and policy-making standpoint, that there is a need, both nationally and internationally, for methodologies that could integrate spatial elements of the economic behavior of regions in the construction of I-O matrices, in order to have a more precise analysis and a closer view of the real performance of a region, as well as to promote better policy-making decisions in terms of regional growth and development

Consequently, to address the challenge of the construction of a spatialized regional input-output matrix we propose a spatial, theoretical and methodological approach from a bottom-up perspective, first, with the use of spatial economic units, as the foundations of the construction of regional matrixes, instead of administrative political entities, such as states, municipalities, provinces or counties.

In this approach we consider that the traditional assumptions in the construction of regional I-O matrices imply that the spatial dimension of a regional economy is not determinant for the construction of I-O matrixes and their models. Consequently the regional economy is economically interpreted without space, merely as a sectorial aggregate in a point of a space, which means either that the link between spatial economic location between production and consumption areas in a region does not affect the regional economic behavior or that production and consumption are located in the same place, which is completely inadequate and unreal. Furthermore, it is assumed that the economic areas are bounded to the political spatial units. Consequently, the construction of regional I-O matrixes is based on the assumptions of sectorial aggregation and its location on a spatial unit, without considering their economic spaces.

Therefore, we promote the development of a new research agenda in the construction of regional I-O matrices based on the spatialization approach that we mentioned above. However, it is important to mention that it is still an exploratory research carried out by the application of different methodologies based on bottom-up approaches, as an attempt to build a spatial economic methodology for the construction of regional I-O matrices based on the approach of the spatial dimension of the economy.

This essay is part of that research agenda, which in this case, is based on the application of the Flegg´s Webber and Elliot bottom-up methodology (1995 and 1996) ⁵, with a complementary spatial and statistical assessment of their advantages and a comparison with the top-down approach⁶.

Hence, our main concern in this article, is to develop and implement a methodology for the construction of a spatialized regional I-O matrix, in order to give some empirical evidence of the spatial differences and their effects on the economic linkages of a region when using both approaches. As a case study we used the state of Sonora, where we compare both matrices.

The spatial disaggregation has the region as its basis, through the identification and delineation of ten spatial economic functional units (SEFUs), which we define as economic sub-regions. This analysis required the construction of ten sub-regional input-output matrices, one for each sub region, with a size of 20 x 20 economic sectors. This lead to the creation of 100 matrices: 10 sub-regional and 90 inter sub-regional matrices, while the construction of the top-down regional input - output matrix (constructed by the Government of Sonora), is done in an aggregate way, using Flegg`s methodology⁷, without considering the intra - regional structure of the state.

It is important to mention that, despite the fact that the analysis is done using the state of Sonora, a political spatial unit, the construction of the regional I-O matrix under bottom-up approach was created through the functional integration of the 10 economic sub-regions and their economic behavior, that were identified in the area. As already mentioned, we constructed 10 sub-regional I-O matrices, integrated by their economic interdependencies, therefore the Sonora bottom-up regional matrix is an outcome of the aggregation of its economic sub regions, whereas, the regional top-down matrix was created based on political area of Sonora.

For the comparison of both matrices, we used two comparative analyses. The first one is spatial, oriented to the comparison between the economic linkages identified in both matrices with those that exist at the sub-regional level. For this purpose, we use the traditional approach of Chenery and Watanabe⁸ (1958), and we compare their classification on both matrices, with the economic specialization and diversification to economic specialization and diversification of the economic sub regions, so as to find a contrast between the economic linkages arising from both approaches and the existence of spatial economic linkages within the economic sub-regions with the expectation to have a preliminary evidence of the no coincidence between the classified regional aggregated economic linkages coming from the top-down approach and those identified at spatial level.

The second comparative analysis is sectorial, and complements the first one, given that it attempts to find statistical evidence to support our hypothesis concerning the inefficacy of the top-down approach to capture observed economic linkages in a region. We do this, first, by applying the methodology of Feser and Bergman (2009) that identifies clusters and the economic branches that integrate them, as well as a statistical measurement of their economic linkages. The first part of the analysis is concerned with the presence and economic link of the identified clusters and the second one, is related to the multiplier effects of the type of economic linkages that arise from their economic performance, using multipliers indexes of output and demand. (Miller and Blair, 2009) and (K. Burgos, 2007). Both analyses are based on the principal components analysis (PCA) whose main features are presented in attachment 4.

Therefore, this paper consists of five parts: (1) Interpretation and methodology; (2) the description of the main natural features of Sonora (3) the comparative analysis of both regional matrices, taking into account their economic linkages and pointing out their differences and associations with the spatial location of the economic activities within the region; (4) The spatial and statistical measurement of the economic linkages and their multiplying effects of the identified clusters according to both approaches; and (5) Conclusions

2. Interpretation and methodology

According to the theoretical and methodological approach of the economic concentration of the spatial dimension of the economy, a perspective that we have been developing, it is assumed that economic concentration in space causes the formation of economic spatial units that determine and characterize the spatial structure and the behavior of the economy in a geographic space. In a generic way, these spatial units are defined as functional spatial economic units⁹, which are the result of the economic growth and development in space (Asuad, 2001, 2007 and 2014).

We assume that economic growth in the national and international space is not homogeneous or politically bounded to states nor municipalities; on the contrary, the spatial distribution of economic activity has created its own economic spatial units, economic regions and territories, which basically conform a set of market areas that theoretically are integrated by an economic center and its periphery. The economic center or node is highly concentrated with the main economic activities and most of the regional population lives there. Therefore, it is relevant since most of economic interactions are carried out there, as well as production, exchange and consumption of a region. Thus, a node or hub is defined as a site or place, whose economy is characterized by its economic dominance and connection with a set of minor economic sites that interact and compete with each other. An economic site is defined as a place on the economic space, where economic activities are highly concentrated and from which a set of economic impulses are exerted through economic exchanges; this guides the spatial economic behavior as a whole.

The economic nodes are spatial economic subunits distributed in a given geographic or political space, and are characterized by their extremely dense economic activity and their population concentration. Indeed, they behave as the centers of a given market area where most of the spatial concentration of production and consumption is located. Furthermore, they are related through the economic activities carried out with their areas of influence, represented by the economic flows established among them. The economic importance of nodes depends on their economic interaction, which is an outcome of the type of connection and market relationships they established. These can be characterized as economic complementarities or competition among themselves, or just a mixture of both economic interactions. So, if these interactions were relevant, they would lead to structure sub economic spaces, which as a whole integrate the economic space, or in other words a set of economic sites and their economic interactions in a given geographic space.

Economic space, in order to exist, requires at least the existence of a pair of economic sites interacting with each other. Of course, this does not coincide with any geographic or political space, despite their influence on the economic decision-making process. Therefore, economies based on market behavior and territorial development defines how the economy as a whole is structured in space. The role of the state as a normative institution that regulates and guides market economic behavior, depends on its political and economic actions as well as its orientation. Thus, the behavior of the market economy, territorial development and eventually, the political economy of the state, are the main factors that define how the economy in space is structured.

This interpretation is related to the approach of the spatial dimension of the economy where the concept of economic space and its derivative categories: economic territory and economic region, which are the essential ingredients of our theoretical perspective.

The economic territory is defined as the product of the origin and development of the economic activity on space and it is expressed through the economic uses of land, which in an aggregated way can be synthetized in the creation of cities and their development. On the other hand, economic regions are related to the spatial functional economic behavior based on the center-periphery model. This is the result of the creation of market areas, and it is characterized by the spatial economic distribution on space, based on the creation of an economic center, where a great part of the production and consumption takes place and interacts with its area of influence, conformed by a set of economic sites. It is evident then, that, in territorial terms, the economic center is characterized by the existence of the main city in the region, while the territorial attributes of its periphery are the minor cities or localities that are economically connected through the network of transportation and communication systems. Finally, an economic region is an economic space integrated by the main city and the system of localities that interact with each other. Therefore, we assume that the economic interactions on space are the result of market transactions, characterized by sectorial economic behavior and their synergy with the natural and territorial space in a given area. This leads to the formation of the economic space and in turn develops a system of cities and localities that are economically related through the network of transportation that links them.

According to the above interpretation the methodology for the construction of a regional input-output matrix from bottom-up, is presented in diagram 1.

3. The identification of the spatial economic functional units (SEFU) and of the economic sub-regions

The identification and delimitation of the spatial economic functional units are based on the identification of the economic and population nodes (already done in the first step). Then, it was necessary to identify and select the node’s area of influence. This can be done relying on the assumption that an economic node is also the economic center of a market area, whose periphery or area of influence depends on physical proximity or distance, and on the economic attraction that the node exerts over it.

It is important to recall that these economic spatial units lead us to identify the economic space, in which the economic activity takes place. Of course, there is no coincidence with political or geographical units. According to our interpretation, their delineation is absolutely essential, in order to understand the economic behavior in space. We delimitate these units using the Reilly Index ¹⁵ , which essentially, measures the economic distance between a pair of nodes and their possible area of influence, based on the assumption that a pair of nodes that are spatially located near each other may be in competition for the attraction of the nearby areas of influence, then it is draw their borders in order to establish who is attracting those areas.

According to the results of the index, nine spatial economic functional units were identified. However, in order to integrate all economic areas that form the region, another SEFU, characterized by the extreme dispersion of their localities was also identified. Consequently, the economic space of Sonora is structured by ten SEFU’s, Therefore, the most important economic characteristics of the economic sub-regions of Sonora reflect a great heterogeneity, given that only four of its sub-regions account for almost the total production of the region. The economic sub-regions Hermosillo, Agua Prieta, Obregon and Nogales account for 83% of the regional production, 80% of the total value added and 72% of total regional employment. At the same time, as a whole they account for 65% of total population of the region. The other six economic sub regions share 17% of the regional production, 20% of the total value added and 28% of the employment, with only 35% of the total population of the region. (See Map 3 and Table 3 below.)

4.The construction of the regional bottom-up input-output matrix of Sonora

The methodology for the construction of the regional matrix of Sonora from the bottom up approach, comprises the following steps:

1. The identification of the dominant economic sectors by economic sub region, applying the Pareto distribution criteria; 2. The estimation of the economic specialization index by economic sub region ; 3. The built of the sub regional input-output tables, which is done firstly through an adjustment to the Flegg´s methodology, firstly by taking into account the economic specialization coefficient of the sub regions, secondly by estimating lambda λ, as a definition of the economic size of the sub region compare to regional and national production by applying the following index:

Where:

Y_sr =TGP Sub-regional

Y_{r =} TGP Regional

Y_n= TGP National

Then, the input-output table by economic sub region is calculated, by multiplying the economic specialization coefficient by economic sub region, using the technical coefficient of state production and the gross production of the economic sub region, which is specified as:

Where:

<mml:msub> <mml:mrow> <mml:mi>F</mml:mi> <mml:mi>L</mml:mi> <mml:mi>Q</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>i</mml:mi> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub>

is the modified Flegg’s coefficient;

<mml:msub> <mml:mrow> <mml:mi>C</mml:mi> <mml:mi>I</mml:mi> <mml:mi>L</mml:mi> <mml:mi>Q</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>i</mml:mi> <mml:mi>j</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> </mml:msub>

is the crossed-holding location coefficient;

λ, is an algorithm that takes into account the economic size of the sub-region;

<mml:msub> <mml:mrow> <mml:mi>a</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>i</mml:mi> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub>

represents the national technical coefficients.

According to Davila 2002, the value of λ^δ increases as the size of the region those it. Then, the greater the value of lambda, the lower the value of the regional adjustment of imports and consequently, the higher the level of regional self-sufficiency. Flegg and Webber have found, through various studies in England, that a value of δ=0.3 can minimize the differences between the multipliers obtained by location coefficients and those calculated by direct observation.

Therefore, following Flegg´s proposition, it is estimated the percentage of the technical coefficients of production supplied within the region, based on the interpretation of the modified location coefficient, as follows:

𝐹𝐿𝑄_𝑖𝑗≥1∴𝑡_𝑖𝑗≡1

𝐹𝐿𝑄_𝑖𝑗≤1∴𝑡𝑖𝑗=𝐹𝐿𝑄_𝑖𝑗

Nevertheless, given that Flegg´s proposition does not have any proposal for the inter regional trade coefficients analysis, we first identified the sub region´s commercial function as buyer or seller, based on those performances at sectorial level in the nation, through the analysis of the national input-output tables. Thus, we identified the sectors of the sub regions, which belong to horizontal or vertical vectors in the regional matrix, taking as a reference the national commercial function that the sector plays as buyer or seller. Consequently, these sectors were located in the sub regions through census data, allowing their classification of the sub regions as sellers and buyers.

Given the lack of inter-sub regional trade information and to the great amount of technics for their estimation (Miller and Blair, 2009, Chapter 8, we made a preliminary estimation of sales and purchases between sub regions, taking into account their differences in economic specialization and the value of economic interaction among them, that we estimated through the application of an economic interaction index. (Attachment III). Finally, we estimated the multiple sub regional matrix of Sonora, created by the distribution of the gross production value, that include the trade coefficients of the sub regions by the adjustment to Flegg´s methodology and also the ones that come about from the application of the RAS method to the values of the economic interactions index among sub regions. ((Attachment IV).

5. The comparative analysis of the regional economic linkages and their multiplier effects between the top-down and bottom-up approaches.

As we mentioned before, in this step we applied both a spatial and a sectorial analyses of the economic linkages and the multiplier effects, derived from the regional matrices constructed by both approaches.

6. Conclusions

The results have shown that, at both spatial and sectoral levels, contrary to the top-down regional matrix the sectors and economic structure in the bottom-up regional matrix are theoretically and statistically coherent. This allows us to conclude the importance of considering the economic space as a fundamental element in the construction of regional input-output matrices.

A key element in this methodology is the type of regionalization that divides the economic sub regions and explains how they are integrated. The spatial economic functional units (SEFU, which do not consider political or natural spaces, as representations of the economic behavior on space, are essential. A naïve view of economic space is that in which the economic behavior is bounded to a political, natural or administrative space, leading us to misleading view of the real economic behavior on regions and cities. This view also alters the effectiveness and efficiency of the decision-making process and attempts to impulse regional economic growth and social development.

We consider this research has demonstrated the importance of taking into account the spatial distribution of the economic sectors within the region, through sub regions for the construction of a regional input-output matrix using a bottom-up approach, even with the lack of regional data, in order to create a closer view of the real regional economic performance and development

This approach, compared to the top-down, offers richer and more realistic information of the behavior of the analyzed sectors within sub-regions and cities, not only through the number of observed productive chains in our analysis, but also through its meaningful industrial and services activities, classifying their economic linkages, in accordance to the existing economic specialization and diversification of the economic sub-regions of Sonora. Furthermore, this analysis is also consistent with the sectorial economic structure and its linkages and multiplier effects.

It is obvious the need to identify the location, of production and consumption of the economic activities taking place in order to define the economic linkages of the regional structure and at a sub-regional level. Without their inclusion, the analysis produces an aggregated region without space under the assumption that it behaves similar to the nation.

Even though this is an exploratory essay that tries to find alternative methodologies to the traditional top-down approach by using bottom-up perspectives, it is still under development. We believe we have presented solid elements to support our view and the advantages for continuing our research and for the achievement of a methodological proposal from a regional and territorial development perspective.

This methodology needs to be improved, especially through the elaboration of regional economic accounts so as to make them coherent with the national accounts of Mexico. It is also evident the need to construct a regional data base at a national level, to broaden the availability of information and thus expand the available tools of regional analysis. This should be done as the literature recommends, especially when it comes to the construction of regional make and use tables.

Besides, we also need a deeper analysis and estimation of the economic interactions, or, in other deepen into the estimation of inter-regional trade coefficients to get a sound estimate of sales and purchases between sub regions and regions, given the limitation of information and specific data for this type of analysis. Thus, the economic interaction index according to the economic space interpretation has to be reviewed. The existence of methodologies and techniques and also a new regional model have to be developed through a spatial perspective, (a multiregional mode) which implies taking into account the intra-regional and inter regional economic interdependencies.. It is likely that the incorporation of a principal components analysis and of spatial econometrics might improve it.

Furthermore, it is also needed to have a more robust statistical analysis to prove the advantages of our approach compared to the traditional one. We already have important and favorable advances concerning the improvement of this methodology, in terms of the mentioned needs. We expect them to be published soon.

Finally, it is important to stress out that this work is part of a line of research developed by the CEDRUS organization over a long period of time, and it has been applied to the construction and analysis of regional input-output matrices. This has already opened new research proposals for spatial and economic applied studies in different areas in economics.