*2.1. Linkage Analysis and the Hypothetical Extraction Method*

In the context of multiple industries interacting within the economic system, the relevance of production sectors, especially local ones and their contributions to economic growth, requires a measure that delineates both forward and backward linkages in order to empirically analyze the complete role of an industry and to evaluate the technological connections between economic sectors (Kay et al. 2007). Within this framework, intersectoral linkages are regarded as techno-economic connections between industries that are embodied in the exchange of tangibles and intangibles (Hauknes and Knell 2009). Intersectoral linkages, therefore, estimate interdependencies between sectors, which affect the paths of vertical specialization (Reis and Rua 2009). In the literature on regional economics, different indicators have been suggested and widely adopted for the identification of key sectors. These sectors, characterized by relevant intermediate purchases (backward linkages) and sales (forward linkages), are more likely than the other sectors to propagate growth impulses all over the economy (OECD 2021). Golan et al. (1994) integrated this definition to include three further conditions to appropriately define a key cluster, i.e., (i) a well-developed domestic market, (ii) a competitive local business climate, and (iii) efficient production factors.

This broader definition takes the size of the linkages as an approximation for the potential benefits of stimulating the sector. This should imply that the taxpayer cost of acquiring these benefits would be the same among sectors and among forward and backward linkages. In reality, stimulating wider sectors is more expensive than smaller ones; hence, key sector measures have to be corrected in consideration of their dimensions to more accurately achieve this objective (Temurshoev and Oosterhaven 2014). In addition, industries of the same size do not necessary require similar policy measures. This fact has been considered in the net backward linkages, which correct the standard (gross) backward linkages for the size of final demand, assuming that a relatively wide final demand is more easily stimulated than a small-sized one (Oosterhaven 2004, 2007). Lastly, the creation of benefits of sizeable backward linkages requires demand-stimulating measures, whereas the generation of benefits of wide forward linkages requires a further improvement in productivity, i.e., price reducing policies to strengthen output growth. Obviously, the cost of these quite different policy measures per unit of potential benefit, i.e., per linkage measures, will not be the same. Hence, selecting key sectors requires much more analysis than just establishing which sectors have the largest forward and backward linkages. To address this issue, many key sectors' measures have been proposed in the literature (Golan et al. 1994).

On the one side, the various measures result from methodological enhancements, such as the substitution of the direct backward linkages (Chenery and Watanabe 1958) with the total backward linkages originated by the column sums of the Leontief-inverse (Rasmussen 1956), or the substitution of the row sums of the Leontief-inverse (Rasmussen 1956) with the row sums of the Ghosh-inverse (Beyers 1976), in the case of total forward linkages (Jones 1976). On the other side, these measures originate from different labelling of the same measure in different works. Among these measures, it is worth mentioning the output-tooutput multiplier (Miller and Blair 2009), analogous to the total flow multiplier (Szyrmer 1992), is comparable to the earlier Hypothetical Extraction Method (HEM) of whole sectors (Strassert 1968; Schultz 1977; Temurshoev 2010), which was later reformulated by Meller and Marfán (1981), Cella (1984), and Clements (1990).

In the present work, we adopted the latter, straightforward and flexible version of the HEM, since it allows for the extraction of any subset of transactions, instead of a mere removal of full rows and columns, (Miller and Lahr 2001; Gallego and Lenzen 2005). The HEM identifies the 'keyness' of a sector through the hypothetical output loss in the economic system due to the abrupt stop of the related activity, i.e., assuming all sales to (and purchases from) the other sectors are set to zero. The Hypothetical Extraction Method (HEM) was adopted in this study to assess the position occupied by various economic sectors within a given (country or regional) economy. This method is regarded as an improvement of the Classical Multiplier Method (Rasmussen 1956), which measures the 'keyness' of a sector only in terms of simple averages of direct and indirect technical coefficients. HEM, indeed, weights the 'keyness' of a sector by assuming its external linkages, i.e., sales and purchases from all other sectors, as null (Guerra and Sancho 2010). The output loss deriving from this extreme condition quantifies the underlying network of economic linkages (Miller and Lahr 2001) and provides a measure of 'keyness' (Miller and Blair 2009). Therefore, the HEM evaluates the extent at which the total output of the economy would change (e.g., decrease) if a j-th sector is removed from the economic system.

The bulk of this approach lies in the inverse Leontief matrix, i.e., L = (I − An) <sup>−</sup><sup>1</sup> in the first case and L = (I <sup>−</sup> R)−<sup>1</sup> in the second case, where An and R, are the matrices of national and regional direct input coefficients, respectively. The generic Lij entry of the L matrix measures the total requirement (multiplier), both direct and indirect, of goods and services produced by the i-th industry needed to satisfy one unit of final use of the j-th sector. Consequently, the j-th column-sum (Lj) of L measures the total requirements of the j-th sector to produce one (final) production unit. In other words, it is the extent to which a unitary increase in the final demand of the j-th sector causes a production increase in all sectors. On the contrary, the row-sum of the L matrix (Li.) measures the total production requirements of the i-th sector needed to off-set a unitary increase in the final uses of each product. In other words, output magnitude increases in the i-th sector if the final demand of all sectors increases by one unit. Initially, this was modelled in an input–output context by deleting row and column j from the A matrix of the technical coefficients (Ali et al. 2019).

To this regard, let A(j) be the (k − 1) × (k − 1) matrix without the sectors j and f(j) in the correspondingly reduced final demand vector (see Miller and Blair 2009 for details), then, the total output in the 'reduced' economy reads as x(j) = I − A(j) −1 f(j). On the contrary, in the full k-sector model, the total output is x =[I − A] −1 f. Consequently, *i* x−*i* x(j) (where *i* is a column vector of ones) is an aggregate measure of the economy loss (reflecting a decrease in gross output value) if sector j disappears, and is in turn an indirect, overall estimate of the multi-dimensional linkages of the j-th sector.

Normalization by total gross output (*i* x) and multiplication by 100 provides an estimate of the percent loss in total economic activity. The hypothetical extraction approach can also be used to measure backward and forward linkage components separately. We assume that the j-th sector buys no intermediate inputs from any production sector by replacing the j-th column in A with zeroes. Then, the following index is a candidate measure of (aggregate) backward linkage for the j-th sector:

$$\overline{\rm BL}\_{\text{j}} = \frac{\mathbf{i}' \mathbf{x} - \mathbf{i}' \overline{\mathbf{x}}\_{\text{(j)}}}{\mathbf{i}' \mathbf{x}} 100 \tag{1}$$

Similarly, replacing the i-th row of the output coefficients matrix (B = xij xi ) with zeroes and denoting this matrix as B(I) makes x (i)= v I − B(i) −1 the row vector whose entries are the sectoral production when the i-th sector is removed from the economy, i.e., the total production of all other sectors if the i-th sector sells no intermediate input. An aggregate measure of a given sector (i) forward linkage is:

$$\overline{\rm FL}\_{\rm i} = \frac{\mathbf{i}^{\prime}\mathbf{x} - \overline{\mathbf{x}}\_{\rm (i)}^{\prime}\mathbf{i}}{\mathbf{i}^{\prime}\mathbf{x}} 100 \tag{2}$$

For the sake of comparison, the previous indices were normalized as follows:

$$\overline{\text{BL}\_{\text{j}}} = \frac{\overline{\text{BL}\_{\text{j}}}}{\frac{1}{\text{k}} \sum\_{j=1}^{k} \overline{\text{BL}\_{\text{j}}}} \tag{3}$$

$$\overline{\overline{\rm FL}}\_{\text{i}} = \frac{\overline{\rm FL}\_{\text{i}}}{\frac{1}{\underline{k}} \sum\_{i=1}^{k} \overline{\rm FL}\_{\text{i}}} \tag{4}$$

The index reported in Equation (1), known as the 'backward linkage' (or 'dispersion power'), measures the activation degree of a given economic sector. Values greater than 1 indicate the importance of a given sector in the regional economy, because it requires a production level from the other sectors above the average. By contrast, the more the index falls below 1, the less important the sector considered is.

The index reported in Equation (2), known as the 'forward linkage' (or 'dispersion sensitivity'), measures the level at which the output of one sector is used as input for the remaining production sectors, and thus measures the degree of reaction characteristic of a given economic sector. As in the previous case, the greater the index is than 1, the more important the corresponding sector is, because it supplies its production to the other sectors at a level which exceeds the general average. By contrast, the more the index falls below 1, the less important the sector considered is. The joint analysis of these two indices makes it possible to determine how an individual sector is woven into the economic structure of a region, as well as its relative importance. Based on these premises, we define:

