Framework, data description, and sample selection

Similar to Galagedera et al. (2018), Premachandra et al. (2012), and Galagedera et al. (2016), our study explores operational efficiency and investment efficiency. However, the present study diverges slightly regarding the variables used as inputs, intermediates, outputs, and DEA methods. Although Premachandra et al. (2012) and Galagedera et al. (2016) considered net asset value (NAV) as an intermediate variable, they also included other variables such as fund size, variability in returns, and expense ratio as inputs for the portfolio (investment) process. We included transaction costs and management fees as intermediates because more fixed assets result in higher transaction costs for firms. Furthermore, we incorporated NAV into our models as outputs, such as changes in bond and equity funds. While the production process of mutual funds remains the same in any setting, the types of efficiency under consideration within the “black box” and the nature of the firms in a particular market (Taiwan Province, People’s Republic of China, in our case) denote the inputs or outputs that must be used (Nourani et al. 2021). For example, Galagedera et al. (2018) proposed a three-stage network model to evaluate mutual funds, with the three stages being operational management, resource management, and portfolio management. Given the similar environmental conditions (low-risk undertakings) in both operational and resource management processes, these two processes are combined, resulting in a three-stage network model in which the first two stages function as an allied process. Another significant difference between the two-stage production process we devised for ITCs and that of Galagedera et al. (2018) is that the latter includes total risk, downside risk, systematic risk, and NAV as inputs for the portfolio management process, with the return included as the output of the investment stage.

In comparison, for inputs, we used employees, net fixed assets, and operating expenses; for intermediates, we used transaction costs and management fees; and for outputs, we used increases in the net asset values of bond and equity funds. The proposed efficiency model represents the operational mechanism of the ITCs while remaining in line with the literature on mutual funds. Based on the rationale provided above for our choice of variables used in DEA, we outlined the production process of ITCs proposed in this study.

To assess complex production processes, an adequate number of inputs and outputs must be considered. These indicators produce a series of network structures that are linked through intermediate measures. Breaking down the production process into several divisions and subsequently opening the black box allows the respective decision-makers to act in the shareholders’ best interests. Based on the intermediation approach, this study divides the production process of ITCs into two stages: operational efficiency and investment efficiency. In the operational efficiency stage, ITCs invest in human and financial capital, specifically, employees, net fixed assets, and operating expenses, to generate the intermediates, that is, the transaction costs, management fee of the equity fund, management fee of the bond fund, and other expenses, which can be aptly used as the inputs for investment efficiency. However, it is important to note that employees, net fixed assets, and operating expenses do not produce the costs and fees for transactions and management. Instead, as ITCs grow, their inputs on employees, fixed assets, and operating expenses increase. In other words, with a higher amount of these inputs, which come from more business, ITCs are expected to also have higher amounts spent on transaction costs, the management fee of the equity fund, the management fee of the bond fund, and other expenses.

Next, in the investment efficiency stage, these intermediates are used to generate two outputs, that is, the change in equity funds and the change in bond funds. Figure 1 shows the network framework of ITCs, with their performance divided into two connected stages. In the first stage, we measured operational efficiency. We also assessed ITCs’ investment efficiency in the second stage. Table 1 presents the definitions of the input, intermediate, and output indicators and provides the relevant references used in the two-stage production process.

Fig. 1
figure 1

The two-stage production process of investment trust companies

Table 1 Variables definitions

To conduct an efficiency analysis, firm-level data from the ITCs are required. Therefore, we compiled the financial data presented in the Taiwan Economic Journal (TEJ) database to collect necessary information regarding active ITCs. In addition, to ensure the availability of data, we checked the data published by the Taiwan Province Stock Exchange Market Observation Post System.Footnote 2 Our sample included ITCs for which data were available from 2011 to 2020. Since the variables of inputs, intermediates, and outputs are intertemporal price variables, we deflated all variables according to the 2011 Consumer Price Index in Taiwan Province, PRC.Footnote 3 In addition, we ensured that all companies in the sample had the necessary data for the specified sample period. We excluded ITCs that lacked sufficient data for the sample period. The final sample comprised 34 ITCs that was divided equally between domestic and foreign operators in the industry. Table 2 presents a list of the ITCs evaluated in this study.

In Table 3, the weighted averages of inputs, intermediates, and outputs for 340 observations for the sample period are segregated into two distinct panels, which correspond to domestic and foreign samples with 170 observations each. On average, the results demonstrated that domestic ITCs hired a greater number of employees than foreign ITCs. In addition, domestic ITCs have more net fixed assets and lower operating expenses than foreign ITCs. Domestic ITCs incurred higher transaction costs and other expenses during the sample period. In contrast, domestic ITCs tended to spend less on bond funds but more on equity funds than foreign ITCs with the same indicators. This comparison emphasizes that domestic ITCs are more prudent regarding their management fee spending and tend to spend less on risky assets. Meanwhile, foreign ITCs’ higher median management fees for bond funds compared to domestic ITCs indicate that a greater number of domestic ITCs have high spending on bond funds. In addition, the output values of domestic ITCs are significantly better than those of foreign ones. The change in the equity fund is nearly 50%, while the change in the bond fund is more than double that in foreign outputs. These findings were as anticipated, with more values placed on certain inputs and intermediate quantities. Table 3 demonstrates that all values are normally distributed across the two panels.

Table 3 Descriptive statistics of inputs, intermediates, and outputs (n = 340)

Golany and Roll (1989) stated that for DEA, an isotonicity assumption is required. This assumption states that the input and output factors must have a positive correlation, showing that a proportional increase in the input indicator produces a proportional increase in the output indicator. To ensure that this assumption was met, we performed a Spearman’s rho correlation test, as shown in Table 4, which yielded satisfactory results. All values indicated positive correlations between the variables used in the DEA, except for net fixed assets (X2) and other expenses (Z4). The trivial correlation coefficient of − 0.035 obtained was negligible and unsurprising, as more fixed assets resulted in fewer additional expenses. Hence, in general, the results indicate positive correlations between the variables. Furthermore, Golany and Roll (1989) suggested that the number of DMUs should at least double the number of input and output factors. Our sample satisfied this requirement, with 34 > 2 × (3 + 4 + 2). Cooper et al. (2006) established a more restricted rule for the minimum number of DMUs; they recommended that DMUs be at least three times more than the input and output factors. This constraint was also satisfied by our sample: 34 > 3 × (3 + 4 + 2). Finally, as our ITCs operate in the same environment, our sample fulfills the homogeneity assumption. Thus, the affirmatory results for the isotonicity assumption, minimum number of DMUs, and homogeneity assumption all indicated that our model had a high level of construct validity regarding the selection of the input, intermediate, and output variables.

Table 4 Correlation matrix

Table 5 displays the results of the Kolmogorov–Smirnov test of the differences between foreign and domestic capital regarding the inputs, intermediates, and outputs used in the DEA. The use of this test is in line with prior studies such as Nataraja and Johnson (2011). The findings showed significant differences between all input and output quantities when comparing domestic and foreign ITCs. Compared with foreign ITCs, domestic ITCs had a significantly higher number of net fixed assets and employees but lower operating expenses. In addition, domestic ITCs showed a greater increase in equity funds and an increase of more than double in bond funds. Regarding intermediate factors, domestic ITCs showed significantly higher transaction costs and management fees for equity funds than for other expenses. Although domestic operators had lower management fees for bond funds than foreign operators, the statistical test of differences demonstrated that it was not significant at the 10% level. Overall, these results indicate that domestic and foreign ITCs have noticeable differences in their technologies.

Table 5 Kolmogorov–Smirnov test of differences between foreign and domestic capitals

The DDF-based metafrontier and group frontiers

Following ODonnell et al. (2008) to measure the impact of technological heterogeneity, we grouped N ITCs into two groups (Gg, g = 1, 2). The sample of the Gg group is Ng, where N1 + N2 = N. These ITCs used (m) inputs to generate (d) intermediates ( first stage), which are ultimately transformed into (s) outputs ( second stage). In our study, in the first stage, we input ({mathbf{x}} in R_{ + }^{m}), intermediate outputs (the first stage), input (the second stage) ({mathbf{z}} in R_{ + }^{d}), and final outputs ({mathbf{y}} in R_{ + }^{s}). We assumed a convexFootnote 4 production possibility set and defined the DDF-based two-stage network framework, both metafrontier (M) and group-specific (G), using the following equations:

$$begin{aligned} {overset{lower0.5emhbox{$smash{scriptscriptstylerightharpoonup}$}}{D}}^{M} left( {{mathbf{x,z,y}};{mathbf{g}}_{{mathbf{x}}} ,{mathbf{g}}_{{mathbf{y}}} } right) & = Maxleft{ {alpha + beta :left( {{mathbf{x}} – alpha {mathbf{g}}_{{mathbf{x}}} ,{mathbf{z,y}} + beta {mathbf{g}}_{{mathbf{y}}} } right) in T^{M} left( {{mathbf{x,z,y}}} right)} right}. \ {overset{lower0.5emhbox{$smash{scriptscriptstylerightharpoonup}$}}{D}}^{G} left( {{mathbf{x,z,y}};{mathbf{g}}_{{mathbf{x}}} ,{mathbf{g}}_{{mathbf{y}}} } right) & = Maxleft{ {gamma + tau :left( {{mathbf{x}} – gamma {mathbf{g}}_{{mathbf{x}}} ,{mathbf{z,y}} + tau {mathbf{g}}_{{mathbf{y}}} } right) in T^{G} left( {{mathbf{x,z,y}}} right)} right},G = G_{1} ,G_{2} . \ end{aligned}$$

The respective technology sets are thus detailed as follows:

The ITCs used (T^{M} left( {{mathbf{x,y,z}}} right)):({mathbf{x}} in R_{ + }^{m}) to generate the intermediate outputs ({mathbf{z}} in R_{ + }^{d}) in the first stage. Meanwhile, they used ({mathbf{z}} in R_{ + }^{d}) to make the final outputs ({mathbf{y}} in R_{ + }^{s}) in the second stage.

ITCs used (T^{G} left( {{mathbf{x,,y,z}}} right)):({mathbf{x}} in R_{ + }^{m}) in Group Gg in the first stage to yield ({mathbf{z}} in R_{ + }^{d}). They used ({mathbf{z}} in R_{ + }^{d}) to produce ({mathbf{y}} in R_{ + }^{s}) in the second stage.

Furthermore, the meta-technology set consists of the G-group-specific technology set (T^{M} left( {{mathbf{x,y,z}}} right) = left{ {T^{G1} left( {{mathbf{x,y,z}}} right) cup T^{G2} left( {{mathbf{x,y,z}}} right)} right}). Fried et al. (2008) claimed that the direction vector ({mathbf{g}} = left( {{mathbf{g}}_{{mathbf{x}}} ,{mathbf{g}}_{{mathbf{y}}} } right)) should be selected before the DDF can be evaluated. In the present study, we considered the direction to be ({mathbf{g}} = left( {{mathbf{g}}_{{mathbf{x}}} {mathbf{ = x,g}}_{{mathbf{y}}} {mathbf{ = y}}} right)) (Chiu et al. 2012). In this case, the inefficiency measure of the (ITC_{o}) of meta-technology and group-specific technology sets under convex constraints can be represented by the following two linear programs:

$$begin{array}{*{20}l} {overset{lower0.5emhbox{$smash{scriptscriptstylerightharpoonup}$}} {D}^{M} = Max,alpha _{o}^{M} + beta _{o}^{M} } hfill & {} hfill \ {sumnolimits_{{g = 1}}^{{G_{g} }} {sumnolimits_{{j = 1}}^{{N_{g} }} {lambda _{j}^{g} x_{{ij}}^{g} le x_{{io}}^{g} – alpha _{o}^{M} g_{{iox}} } } ,} hfill & {i = 1, ldots ,m,} hfill \ {sumnolimits_{{g = 1}}^{{G_{g} }} {sumnolimits_{{j = 1}}^{{N_{g} }} {lambda _{j}^{g} z_{{hj}}^{g} ge z_{{ho}}^{g} } } ,{text{ }}} hfill & {h = 1, ldots ,d,} hfill \ {sumnolimits_{{g = 1}}^{{G_{g} }} {sumnolimits_{{j = 1}}^{{N_{g} }} {eta _{j}^{g} z_{{hj}}^{g} le z_{{ho}}^{g} } } ,{text{ }}} hfill & {h = 1, ldots ,d,} hfill \ {sumnolimits_{{g = 1}}^{{G_{g} }} {sumnolimits_{{j = 1}}^{{N_{g} }} {eta _{j}^{g} y_{{rj}}^{g} ge y_{{ro}}^{g} + beta _{o}^{M} g_{{roy}} } } ,} hfill & {r = 1, ldots ,s,} hfill \ {sumnolimits_{{g = 1}}^{{G_{g} }} {sumnolimits_{{j = 1}}^{{N_{g} }} {lambda _{j}^{g} } } = 1,} hfill & {} hfill \ {sumnolimits_{{g = 1}}^{{G_{g} }} {sumnolimits_{{j = 1}}^{{N_{g} }} {eta _{j}^{g} = 1,} } } hfill & {} hfill \ {lambda _{j}^{g} ,eta _{j}^{g} ge 0,g = 1, ldots ,G_{g} .} hfill & {} hfill \ end{array}$$

(1)

$$begin{array}{*{20}l} {overset{lower0.5emhbox{$smash{scriptscriptstylerightharpoonup}$}} {D}^{g} = Max{text{ }}gamma _{o}^{g} + tau _{o}^{g} } hfill & {} hfill \ {sumnolimits_{{j = 1}}^{{N_{g} }} {lambda _{j}^{g} x_{{ij}}^{g} le x_{{io}}^{g} – gamma _{o}^{g} g_{{iox}} } ,} hfill & {i = 1, ldots ,m{text{,}}} hfill \ {sumnolimits_{{j = 1}}^{{N_{g} }} {lambda _{j}^{g} z_{{hj}}^{g} ge z_{{ho}}^{g} } ,{text{ }}} hfill & {h = 1, ldots ,d,} hfill \ {sumnolimits_{{j = 1}}^{{N_{g} }} {eta _{j}^{g} z_{{hj}}^{g} le } z_{{ho}}^{g} ,{text{ }}} hfill & {h = 1, ldots ,d,} hfill \ {sumnolimits_{{j = 1}}^{{N_{g} }} {eta _{j}^{g} y_{{rj}}^{g} ge y_{{ro}}^{g} + tau _{o}^{g} g_{{roy}} } ,} hfill & {r = 1, ldots ,s,} hfill \ {sumnolimits_{{j = 1}}^{{N_{g} }} {lambda _{j}^{g} } = 1,} hfill & {} hfill \ {sumnolimits_{{j = 1}}^{{N_{g} }} {eta _{j}^{g} = 1,} } hfill & {} hfill \ {lambda _{j}^{g} ,eta _{j}^{g} ge 0,} hfill & {} hfill \ end{array}$$

(2)

where (lambda_{j}^{g}) and (eta_{j}^{g}) represent the intensity variables corresponding to the first and second processes, respectively, and N1 + N2 = N.

Consequently, the operational efficiency of the first stage in the meta-technology and group-specific technology sets is defined as (OE_{o}^{M} = 1 – alpha_{o}^{M}) and (OE_{o}^{g} = 1 – gamma_{o}^{g}), respectively, which is the operational efficiency with values between 0 and 1. The efficiency of the second stage in these sets is defined as (IE_{o}^{M} = {1 mathord{left/ {vphantom {1 {left( {1 + beta_{o}^{M} } right)}}} right. kern-nulldelimiterspace} {left( {1 + beta_{o}^{M} } right)}}) and (IE_{o}^{g} = {1 mathord{left/ {vphantom {1 {left( {1 + tau_{o}^{g} } right)}}} right. kern-nulldelimiterspace} {left( {1 + tau_{o}^{g} } right)}}), or investment efficiency. To make the efficiency measure consistent, investment efficiency takes a derivative between 0 and 1. The target (ITC_{o}) is regarded as efficient in both stages if (OE_{o}^{M}),(OE_{o}^{g}),(IE_{o}^{M}), and (IE_{o}^{g}) have a value of 1. (TE_{o}^{M} = {{OE_{o}^{M} } mathord{left/ {vphantom {{OE_{o}^{M} } {IE_{o}^{M} }}} right. kern-nulldelimiterspace} {IE_{o}^{M} }}) denotes the technical efficiency of the overall stage in a metafrontier setting.

Decompositions of metafrontier inefficiency

The frontier of metafrontier operational efficiency ((MOE_{o})) is smaller than that of group-specific operational efficiency ((GOE_{o})). However, the frontier of meta-frontier investment efficiency ((MIE_{o})) is larger than that of group-specific investment efficiency ((GIE_{o})). Mathematically, we have

$$begin{aligned} & MOE_{o} left( {{mathbf{x,z,y}}} right) le GOE_{o} left( {{mathbf{x,z,y}}} right) \ & MIE_{o} left( {{mathbf{x,z,y}}} right) le GIE_{o} left( {{mathbf{x,z,y}}} right). \ end{aligned}$$

(3)

The ratio between (MOE_{o}) and (GOE_{o}) is known as the operational technology gap ratio (OTGR), whereas that between (MIE_{o}) and (GIE_{o}) is known as the investment technology gap ratio (ITGR). In equations, we have:

$$begin{aligned} & OTGR({mathbf{x,z,y}}) = {{MOE({mathbf{x,z,y}})} mathord{left/ {vphantom {{MOE({mathbf{x,z,y}})} {GOE({mathbf{x,z,y}})}}} right. kern-nulldelimiterspace} {GOE({mathbf{x,z,y}})}} le 1 \ & ITGR({mathbf{x,z,y}}) = {{MIE({mathbf{x,z,y}})} mathord{left/ {vphantom {{MIE({mathbf{x,z,y}})} {GIE({mathbf{x,z,y}})}}} right. kern-nulldelimiterspace} {GIE({mathbf{x,z,y}})}} le 1. \ end{aligned}$$

(4)

(GOE_{o}) ((GIE_{o})) is closer to (MOE_{o}) ((MIE_{o})) if the values of OTGR and ITGR are closer to one.

Given that the ratios of the two frontiers are unable to explore the source of meta-frontier inefficiency (Chiu et al. 2012, 2013), we obtained the operational technology gap inefficiency (OTGI) and operational technical inefficiency (OTI) of the operational stage, and the investment technology gap inefficiency (ITGI) and investment technical inefficiency (ITI) of the investment stage as follows:

$$begin{aligned} & OTGIleft( {{mathbf{x,z,y}}} right) = GOEleft( {{mathbf{x,z,y}}} right) times left( {1 – OTGR({mathbf{x,z,y}})} right) \ & ITGIleft( {{mathbf{x,z,y}}} right) = GIEleft( {{mathbf{x,z,y}}} right) times left( {1 – ITGR({mathbf{x,z,y}})} right). \ end{aligned}$$

(5)

Furthermore, we have ITI and OTI as the managerial inefficiency of (ITC_{o}) in group-specific best practices based on the input excess in the operational stage or output shortfall in the investment stage as follows:

$$begin{aligned} & OTIleft( {{mathbf{x,z,y}}} right) = 1 – GOE({mathbf{x,z,y}}) \ & ITIleft( {{mathbf{x,z,y}}} right) = 1 – GIE({mathbf{x,z,y}}). \ end{aligned}$$

(6)

Therefore, we have the metafrontier operational and investment inefficiencies as follows:

$$begin{aligned} & MOIleft( {{mathbf{x,z,y}}} right) = OTGI^{g} ({mathbf{x,z,y}}) + OTI^{g} ({mathbf{x,z,y}}) \ & MIIleft( {{mathbf{x,z,y}}} right) = ITGI^{g} ({mathbf{x,z,y}}) + ITI^{g} ({mathbf{x,z,y}}). \ end{aligned}$$

(7)

Network-based ranking approach

Following Liu and Lu (2010), the two-stage network DDF and radial network-based approaches were combined to rank the sample ITCs and the most important inputs or outputs for benchmarking purposes. The related steps are discussed in the following paragraphs.

Step 1. Under the assumption of variable returns to scale, the efficiency scores of all ITCs are estimated, considering all possible sets of inputs, intermediates, and outputs. Each specification (k) represents one possible set. The linear programming for each two-stage network DDF specification (k) is as follows:

(8)

The solution to (lambda_{j}^{g,k^{*}}) indicates whether ITC (j) is a benchmark for the observed ITC in the first stage for each DEA specification (k). The solution of (eta_{j}^{g,k^{*}}) has the same definition as that in the second stage.

Step 2. The two-stage network DDF results are transformed into a network structure using the solution of (lambda_{j}^{g,k^{*}}) in the first stage. Each ITC was regarded as a network node. The corresponding (lambda_{j}^{g,k^{*}}) is taken as the weight of the endorsement; if ITC (j) is a paragon of the observed ITC (o) and has the corresponding (lambda_{j}^{g,k^{*}}), then a directed link of weight (lambda_{j}^{g,k^{*}}) pointing from node (o) to node (j) can be created. (eta_{j}^{g,k^{*}}) uses the same definition as in the second stage.

Step 3. (lambda_{j}^{g,k^{*}}) is normalized to solve scale effects. (E_{k}) denotes the index set for the reference set of the observed ITC. The contribution of the (i{text{ – th}}) input of the (o{text{ – th}}) ITC to the (j{text{ – th}}) ITC in the reference set under DEA specification (k) is defined as:

$$Ix_{ij}^{g,k} = {{lambda_{j}^{g,k^{*}} x_{ij}^{g,k} } mathord{left/ {vphantom {{lambda_{j}^{g,k^{*}} x_{ij}^{g,k} } {sumlimits_{{j in E_{k} }} {lambda_{j}^{g,k^{*}} x_{ij}^{g,k} } }}} right. kern-nulldelimiterspace} {sumlimits_{{j in E_{k} }} {lambda_{j}^{g,k^{*}} x_{ij}^{g,k} } }},quad {0} < Ix_{ij}^{g,k} le 1,quad i = 1, ldots ,m.$$

(9)

Similarly, the contribution of the (h – th) intermediate of the observed ITC to the (j{text{ – th}}).

ITC in the reference set under DEA specification (k) is defined as:

$$MIz_{hj}^{g,k} = {{lambda_{j}^{g,k^{*}} z_{hj}^{g,k} } mathord{left/ {vphantom {{lambda_{j}^{g,k^{*}} z_{hj}^{g,k} } {sumlimits_{{j in E_{k} }} {lambda_{j}^{g,k^{*}} z_{hj}^{g,k} } }}} right. kern-nulldelimiterspace} {sumlimits_{{j in E_{k} }} {lambda_{j}^{g,k^{*}} z_{hj}^{g,k} } }},quad {0} < MIz_{hj}^{g,k} le 1,quad h = 1, ldots ,d.$$

(10)

The contribution of the (h – th) inputs of the second stage of the observed ITC to the (j{text{ – th}}) ITC in the reference set under DEA specification (k) is defined as:

$$MOz_{hj}^{g,k} = {{eta_{j}^{g,k^{*}} z_{hj}^{g,k} } mathord{left/ {vphantom {{eta_{j}^{g,k^{*}} z_{hj}^{g,k} } {sumlimits_{{j in E_{k} }} {eta_{j}^{g,k^{*}} z_{hj}^{g,k} } }}} right. kern-nulldelimiterspace} {sumlimits_{{j in E_{k} }} {eta_{j}^{g,k^{*}} z_{hj}^{g,k} } }},quad {0} < MOz_{hj}^{g,k} le 1,quad h = 1, ldots ,d.$$

(11)

Similarly, the contribution of the (r{text{ – th}}) output of the observed ITC to the (j{text{ – th}}) ITC in the reference set under the DEA specification (k) is defined as

$$Oy_{rj}^{g,k} = {{eta_{j}^{g,k^{*}} y_{rj}^{g,k} } mathord{left/ {vphantom {{eta_{j}^{g,k^{*}} y_{rj}^{g,k} } {sumlimits_{{j in E_{k} }} {eta_{j}^{g,k^{*}} y_{rj}^{g,k} } }}} right. kern-nulldelimiterspace} {sumlimits_{{j in E_{k} }} {eta_{j}^{g,k^{*}} y_{rj}^{g,k} } }},quad {0} < Oy_{rj}^{g,k} le 1,quad r = 1, ldots ,s.$$

(12)

Step 4. A relationship network was established. Efficiency analysis was run numerous times to enrich the network in terms of different specifications. The results from all DEA specifications are aggregated into one network for all ITCs. The adjacency matrix of the network is developed as follows:

$${mathbf{R}} = left[ {sumlimits_{k = 1}^{K} {left( {sumnolimits_{i = 1}^{m} {Ix_{ij}^{g,k} } + sumnolimits_{h = 1}^{d} {MIz_{hj}^{g,k} } + sumnolimits_{h = 1}^{d} {MOz_{hj}^{g,k} } + sumnolimits_{r = 1}^{s} {Oy_{rj}^{k} } } right)} } right]_{n times n}$$

(13)

where ({mathbf{R}}) is a squared matrix of order (n) and (K) is the total number of DEA specifications. All possible input, intermediate, and output combinations were (K = left( {2^{m} – 1} right)left( {2^{d} – 1} right)left( {2^{s} – 1} right)). Each element (R_{oj}) is the aggregated weight for the link directed from the observed node (o) to node (j). The principal diagonal elements of the matrix ({mathbf{R}}) were all 0.

Step 5. Eigenvector centrality (Liu et al. 2015) was computed to rank each ITC. The scores to measure the importance of each ITC, represented as a column vector ({mathbf{I}}), can help understand how the eigenvector centrality value can be used to rank the importance of a network ITC. The rank score for each ITC should be proportional to the importance of all nodes referencing it but weighted by the link weights:

$$c cdot {mathbf{I}} = {mathbf{R}} cdot {mathbf{I}},$$

(14)

where (c) is the proportionality constant. In its matrix notation, Eq. (14) is an eigenvector system with (n) solutions. The largest eigenvalue and its corresponding eigenvector provide the most meaningful outcomes. Each element in this eigenvector is a measure of the importance of the corresponding node. In the research area of social networks, this method, which is usually called Bonacich centrality, offers two additional conditions: the endorsing weight and the importance of the endorsing peer. Readers can refer to “Appendix” for the abbreviations of the DEA-related terms and their expositions in this study.

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