Spatial heterogeneity in composite indicator: A methodological proposal

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Spatial heterogeneity in composite indicator: a methodological proposal Elisa Fusco, Francesco Vidoli, Biresh K. Sahoo PII: DOI: Reference:

S0305-0483(17)30202-5 10.1016/j.omega.2017.04.007 OME 1779

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Omega

Received date: Revised date: Accepted date:

1 March 2017 27 April 2017 29 April 2017

Please cite this article as: Elisa Fusco, Francesco Vidoli, Biresh K. Sahoo, Spatial heterogeneity in composite indicator: a methodological proposal, Omega (2017), doi: 10.1016/j.omega.2017.04.007

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Highlights • This research considers the spatial external factors into the robust BoD

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model

• Spatial conditioning to consider a multitude of territorial contextual variables

• Application on simulated data and on social supply offered by Italian Mu-

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nicipalities

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Spatial heterogeneity in composite indicator: a methodological proposal

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Elisa Fusco∗, Francesco Vidoli, Biresh K. Sahoo

Department of Statistical Science, University of Rome La Sapienza, Italy Department of Political Science, University of Roma Tre, Italy Xavier Institute of Management, Xavier University, Bhubaneswar, India

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Abstract

Most of the existing literature on the construction of composite indicator (CI) in the Benefit of the Doubt (BoD) framework fails to consider the exogenous external (tangible or intangible) factors, which might have influences on individual com-

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ponents of the CI. The present research proposes an original method to estimate the CI by considering the external factors directly into the robust BoD model.

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More specifically, a particular focus has been devoted to the “spatial conditioning” in which much of the underlying observed differences in CI scores may be attributed to a multitude of territorial conditional variables. The proposed method

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is illustrated using two data sets: one containing simulated data and the other one relative to real data covering the social supply delivered by the Italian Munici-

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palities. Results stress the importance of properly taking into account the local non-independence of the units in order to better separate the performances of in-

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dividual units by the contribution of their territories. ∗ Corresponding

author Email addresses: [email protected] (Elisa Fusco), [email protected] (Francesco Vidoli), [email protected] (Biresh K. Sahoo)

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Keywords: Data envelopment analysis, Composite indicator, Benefit of the Doubt, Spatial heterogeneity.

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1. Introduction

Economic and social complex systems often exhibit extremely rich behaviour

and strong interaction among agents. In order to better understand the "macroscopic" system behaviour and its territorial patterns, it is hence necessary to "re-

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duce the entropy", to decrease the number of parameters or - in other words to replace a "simplistic" representation with a "simpler" one on the basis of the properties of the single aspects (Foster, 2005).

Composite indicator (CI)1 (or synthetic index) may be effectively used to measure or to approximate the level of a complex phenomenon, which is not directly

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measurable, but can be obtained by using a suitable combination of simple noncommensurate indicators. However, this approximation may fail to capture the

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reality especially when the exogenous factors, which have ex-ante generated or conditioned the simple indicators, are not considered in the construction of CI

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itself2 . The main objective of this paper has been, therefore, to construct a CI taking into account not only the relationships/preferences that exist among the simple

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indicators, but also the exogenous external factors that might have affected them. 1 Without

loss of generality, in this paper composite indicators will be treated within the non-

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parametric production frontier estimation models; some terms, therefore, have to be understood as equivalent. 2 Other disturbances or distortion issues can affect the reliability of the composite indicator as the non-inclusion of the interactions among indicators (see e.g. Rogge et al., 2017); these matters

will not be explicitly taken into consideration in this paper even if the proposed framework can easily be adapted to this.

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Though greater variability among units is the desiderata final product of an index construction, the homogeneity versus heterogeneity issue among units has

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only been explicitly treated in a few research works (see, e.g. Aristei & Perugini, 2010 where heterogeneous inequality aversion parameters are estimated on the basis of country-specific tax structures; and Karagiannis & Karagiannis, 2017

where intra and inter group relationships among units are considered). The OECD handbook (Nardo et al., 2008) has also dealt with the correlation issues among the

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simple indicators, but not on the units’ heterogeneities.

In particular, the heterogeneity issue among units turns out to be important especially when some external (tangible or intangible) factors have a noticeable impact on individual components of the CI. In fact, in the production frontier framework, as suggested by Wang & Schmidt (2002) and Simar & Wilson (2007),

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the distribution of CI scores across units may be flawed when the CI estimates are influenced by conditional external factors in unknown directions. This issue,

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known as "[non]separability condition" in the two-stage nonparametric models (Daraio et al., 2015) may turn the CI estimates biased when the conditional exter-

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nal factors (that are generally used in the second stage to explain variations in the CI scores) are not factored into the computation of CI scores in the first stage. It

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is, therefore, imperative that the second-stage independent variables - in frontier terms - or the external factors - in index terms - have to be necessarily considered into the construction of CI in the first stage itself so as to better understand the

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shift direction of frontier technology. Moreover, it is to be noted that the underlying heterogeneity issue is very im-

portant especially in all the methodological frameworks, including the Data Envelopment Analysis (DEA) (or, equivalently, Benefit of the Doubt (BoD)) in which

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the CI score of each unit is calculated taking into account its relative distance from the frontier/benchmark units.

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Take into account for the heterogeneity is, therefore, a crucial task in all practical applications; ignore or neglect it, in fact, could lead to invalid, inaccurate and open to criticism CIs prone to generate wrong incentives. Consider, for example, the optimal allocation problem of the public spending among Regions (see e.g.

Vidoli, 2012 for a more extensive discussion) where all the local demand and sup-

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ply factors must be included in the analysis3 in order to overcome the principle of historical expenditures and correctly estimate the standard needs. Underestimating systematically, in fact, a single area with respect to the allocative transfer or to the public service objectives may have a strong impact on

the final objectives of public service (Rogge et al., 2017), may not allow to bal-

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ance regional differences (see paragraph 4.2 for a practical application) and finally may have a deep impact on equity and efficiency - the two public goals that are

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often difficult to achieve together.

In this paper, an original estimation framework has been proposed considering

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the external conditionings/constraints directly into the BoD framework. More specifically, a particular focus has been devoted to the “spatial conditioning” in

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which much of the underlying observed differences in CI scores may be attributed to a multitude of territorial conditional variables. Here the location of each unit is considered as the conditional Z of an underlying spatial point pattern process,

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without imposing any additional functional hypothesis on Z. The remainder of this paper proceeds as follows. Section 2 provides the

3 But

it is not always possible due to lack of reliable or sufficiently disaggregated data.

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background information underlying the CI framework. Section 3 addresses the methodological and logical issues concerning the definition and measurement of

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spatial heterogeneity underlying the CI. Subsections 4.1 and 4.2 offer, respectively, applications based on simulated data on regular grid and on Italian public social sector. Section 5 concludes with some remarks.

2. From simple to composite indicators: a literature survey

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A comprehensive measure of the overall performance of a complex phenomenon requires the aggregation of multiple non-commensurate individual simple indicators into a composite indicator (CI). While developing such CI, all the individual simple indicators might not be equally diagnostic of overall performance (Greenberg & Nunamaker, 1987, Barrow & A. 1989). In other terms, to be meaningful,

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the CI requires setting of unknown weights, depending upon the relative importance of the single chosen indicators. If weights fail to capture the relative pri-

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orities, the resulting CI might become questionable, in terms of its unintended consequence, especially when the CI is used as incentive becoming a tool in prac-

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tical evaluation settings. For example, a poorly constructed CI resulting from a wrongly weights structure, used in an assessment setting, may reward a badly

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performing decision making unit (DMU) and penalize a good performing DMU. See, e.g., studies where CIs are used as incentive in various evaluation processes:

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Public Administrations equalization (see paragraph 4.2), reward/control of citizens/firms (e.g. the taxpayers’ risk assessment indicator used in Italy by the Revenue Agency - OECD, 2016) or, more in general, the introduction of incentives in order to reduce disparities (Rogge et al., 2017). Data Envelopment Analysis (DEA) and the econometric approach may be con6

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sidered two ways of endogenously generating unknown weights (Cherchye, 2001; Cherchye et al., 2004, 2007b,a, 2008b,a; Sahoo & Acharya, 2010, 2012). Because

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of the identification of an efficient frontier, DEA seemed to have an advantage over the traditional econometric approach in generating the impartial BoD weighting

set (Melyn & Moesen, 1991). That is, if a DMU has high performance according to one indicator, then its relative weight should be correspondingly high. Since the

CI estimated from the DEA measures the maximum performance of a DMU, the

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BoD weighting reflects the underlying assumption that good performance reflects high priority.

In the CI framework, DEA method has been considered preferable over the alternative econometric method in many respects4 . First, DEA deals with aggregating several performance dimensions as represented by their individual simple

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indicators in constructing the CI without any input. This virtue makes DEA ideal

4 Even

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when the inputs for producing the individual indicators remain largely unknown5 . there are two shortcomings underlying the nonparametric BoD approach, but which

can be overcome. The first one is that the CI scores are sensitive to extreme values, which can,

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however, be overcome by introducing more robust frontier estimation techniques. The second one is that the direction vector, when used to construct CI, are also influenced by outliers, in which case a robust version of the principal component analysis (PCA) can be used to generate such

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direction vector (Vidoli et al., 2015). 5 The general interest in the production of performance rankings in terms of aggregation of individual simple indicators without any input has been manifested in several application areas.

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For detailed references of these, please cf., among others, Greenberg & Nunamaker, 1987; Cook & Kress, 1990; Adolphson et al., 1991; Lovell, 1995; Lovell et al., 1995; Lovell & Pastor, 1997; Hashimoto, 1996, 1997; Cherchye, 2001; Cherchye et al., 2004, 2007b,a, 2008b,a; FernandezCastro & Smith, 1994; Hollingsworth & Smith, 2003; Halkos & Salamouris, 2004; Despotis, 2005; Sahoo & Acharya, 2010, 2012; Zanella et al., 2013, 2015; Fusco, 2015; Karagiannis & Lovell,

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Second, DEA is considered to be superior to econometric method in putting aside the limitations such as the choice of specific functional forms and of the stochastic

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structure in its aggregation of individual indicators. Third, DEA is now no longer considered a mere deterministic methodology. In

fact, DEA estimators of the production frontier have been found to have desirable

properties for providing a base for constructing a wide range of formal statistical tests (Banker, 1993; Banker & Natarajan, 2004). Fourth, DEA is more realistic.

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As a matter of fact, while DEA optimizes weights for each DMU involved, the econometric method uses mixtures formed from averages or other measures of central tendency. In addition, the individual DMU in the DEA framework compares its own output with the efficient performers. This aspect is in sharp contrast with the econometric approaches, which routinely assume that all DMUs perform

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efficiently, an assumption at odds with the results of a number of DEA studies under the conditions of disequilibrium (Sengupta, 2000, 2003; Sengupta & Sahoo,

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2006). Finally, economists base their studies on long-run behavior to achieve the supposed fully efficient performances whereas the data they use in their statistical

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studies is almost all for short-run behavior. In the standard CI framework based on DEA, it is usually assumed that every

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DMU has similar and consistent production technology, i.e., there is no technology heterogeneity. However, in presence of technological heterogeneity due to differences in economic development, industrial structure, resource endowment

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and/or geographical environment, CI estimates obtained from a common DEA technology (that assumes away such technological heterogeneities) might be po2016; Sahoo et al., 2017; Acharya & Sahoo, 2017.

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tentially flawed and unreliable as measures of actual performance. Given these premises, it is therefore necessary to correct the CI scores by adopting a variant

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of DEA technology structure that accounts for heterogeneity in CI scores across DMUs.

In order to make the DMUs’ CI evaluations more effective, the robust order-m

method of Cazals et al. (2002) can be applied to improve the CI estimates by con-

trolling the influences of extreme observations and measurement errors in the data.

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Furthermore, to make the comparative CI evaluation effective across the DMUs, the conditional efficiency approach by Daraio & Simar (2005, 2007b,a), De Witte

& Rogge (2010) and De Witte & Kortelainen (2013) extends the robust orderm method of Cazals et al. (2002) to model the conditional background factors (Z) representing technology heterogeneities. This conditional efficiency approach

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draws m observations with a particular probability that is obtained by estimating a nonparametric kernel around the background characteristics z of the evaluated

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DMU. Therefore, only DMUs having similar background characteristics enter into the reference group against which relative CI of the unit is estimated.

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In the recent years, heterogeneity issue in a nonparametric framework, has been widely dealt with using the conditional approaches. See, e.g., various stud-

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ies of efficiency, among others, Bonaccorsi et al. (2006); Cherchye et al. (2010); De Witte & Rogge (2011); Haelermans & Witte (2012); De Witte & Kortelainen (2013); De Witte et al. (2013) on educational sector, De Witte & Geys (2011,

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2013) on public libraries, Cazals et al. (2008) on post offices, Daraio & Simar

(2005); Daouia & Simar (2007); Souza et al. (2008); Badin et al. (2010) on banks and mutual funds, Halkos & Tzeremes (2011a); Rogge (2012); Verschelde & Rogge (2012); Cordero et al. (2015) on local services, De Witte & Marques

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(2010b,a); Vidoli (2011); Fuentes et al. (2015) on water sector, Halkos & Tzeremes (2011b, 2013) on regional welfare and environment.

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Note, however, that the aforementioned papers only control for socio-economics characteristics such as demography, income, etc., but neglect to consider the full territorial differences. Moreover, they discuss on the choice of a specific set of contextual variables that are believed to affect efficiency scores. In this frame-

work, however, an erroneous or incomplete set of determinants Z could highly risk

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influencing the efficiency scores for which no ex post modeling validation is available. Following this research stream, Vidoli & Canello (2016) recently proposed, ever in a nonparametric framework, to reconsider the conditional/heterogeneity problem as a spatial issue (Stuart & Sorenson, 2003; Baldwin & Okubo, 2006; Okubo et al., 2010) by identifying local potential competitors through an indica-

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tor of spatial dependence, thus, avoiding the necessity of choosing a multitude of determinants. They suggest an empirical procedure that aims at choosing a right

technology.

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portion of territory (i.e., spatial bandwidth) where DMUs produce with similar

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Spatial heterogeneity, in fact, is also sometimes referred to “sub-regional variation”, or to “parent contagion” or better to "uneven distribution of a trait, event,

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or relationship across a region" (Anselin, 2010). These specific kind of significances may be read as local conditioning of a unit performance. That is, spatial heterogeneity describes a patchy landscape, and spatial dependence refers to the

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local non-independence of occurrences that are near to each other. In economic terms, the neighbourhood is often not random, but describes a

common culture among enterprises or public administrations of a specific region, due to some - more or less manifest - effects, which brings the unit to

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produce/group spatially nearby. However, the question that remains to be answered is: what should be the optimum region to use as the homogeneous area

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within companies? Or more generally, which are those units that are affected by a common culture/by a conditional (manifest or not) variable? Even homogeneous environments, in fact, are likely to be heterogeneous if a different/larger scale is considered. This issue, also known as the modifiable areal unit problem (MAUP,

Wong, 2009), is, in fact, “a problem arising from the imposition of artificial units

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of spatial reporting on continuous geographical phenomena resulting in the generation of artificial spatial patterns” (Heywood et al., 1998).

The answer to this question cannot be general, but rather depends on the application at hand. For example, if membership information of a district or a production area or an institutional administrative region is available and is sufficient to

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describe a common culture, it can then be used as a conditional variable Z. Otherwise, even with some theoretical and practical limits that are still to be solved,

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spatial clustering methods (Aldstadt & Getis, 2006) or optimal radius methods (Vidoli & Canello, 2016) can be usefully used.

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3. Accounting for spatial heterogeneity in composite indicator: meaning and

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applicability

An original CI method is here suggested by introducing the possibility of con-

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sidering regional differences that are common to the neighbouring units of the DMU under evaluation. The main purpose of this proposed CI method is not to clear the differences among units, but to better understand as to whether or not these differences are due to the external factors, which affect directly or indirectly the individual simple indicators that are simultaneously considered to form the CI. 11

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As already noted, the classical BoD model is a special variant of the nonparathe boundary of the hull of technology6 Ψ: p+q

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metric DEA framework of measuring efficiency for a given unit (x, y) relative to

Ψ = {(x, y) ∈ R+ |(x, y) is feasible}

(1)

Hypothesizing convexity, free disposability and CRS postulates on Ψ, its con-

vex hull characterization, in accordance with Cherchye & Kuosmanen (2002) and

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De Witte & Rogge (2010), can be expressed as: n

b BoD = {(1, y) ∈ R1+q |y ≤ ∑ γiYi for (γ1 , ..γn ) Ψ + i=1

(2)

s.t. γi ≥ 0, i = 1, ..., n}.

The main drawback underlying this classical BoD specification is that its re-

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sulting CI estimates (as measured by the output efficiencies) are very sensitive to extreme values, outliers and measurement errors in data. In such cases, Daraio &

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Simar (2005)’s robust order-m method, which has been developed in the probabilistic framework for the estimation of production efficiency, can be considered as an appropriate CI estimation tool to overcome such aforementioned limitations.

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In robust order-m method, given a sample of m random variables with replacement Sm = {Y j }mj=1 drawn from the density of Y of n observations, the random set

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e m can be defined as: Ψ

6 In

em = Ψ

m [

1+q

{(1, y) ∈ R+ |Y j ≥ y}.

(3)

j=1

a BoD model the development of CI estimate of a unit requires maximization of the q

different individual output performance indicators without explicitly taking into account the inputs, or with a dummy input equal to unity for all the units.

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This specification mitigates the impacts of outlying observations, as the unit under evaluation is not compared to peers on the entire domain, but rather to a

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subset of peers of size m. Indeed, the procedure iteratively selects B different subsets of size m with replacement from the original reference set Y of n observa-

tions. For each iteration b ∈ B, the CI score of any unit can be computed from the following linear programming problem:

(4)

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e e m }, ∀ b = 1, . . . , B. λmb (1, y) = sup{λ |(1, λ y) ∈ Ψ

It is worth noting that, in the above framework, the parameter m serves dual roles: (i) it is a trimming parameter that can be used to test the robustness of the estimated efficiencies with varying m, and (ii) it identifies the potential competitors

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for each unit and for each of these B subsets7 . Having obtained the B CI scores, the robust version of the CI estimate of any unit (e λm (1, y)) can be computed as

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the arithmetic average of the B estimates e λmb (1, y):

1 B b e λm (1, y) ≈ ∑ e λ (1, y) B b=1 m

(5)

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Note that unlike the traditional BoD method based CI estimates, the robust e λm (1, y) estimates can assume values greater than 1. This is quite possible when a

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sub-sample of m observations is drawn with replacement from the full sample Y of

n observations and the evaluated unit’s CI estimate is compared with a reference

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e m consisting of units with, on average, lower performance levels. A CI sample Ψ estimate e λm (1, y) equal to 1 implies that the evaluated unit performs on a level that is similar to the average performance level realized by the expected m peers. 7 This

is, in particular, the most interesting meaning in spatial perspective.

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An estimate of e λm (1, y) being less than 1 means that the performance exhibited by the evaluated unit’s performance is worse compared to the average performance

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by the m peers. By definition, the CI estimates for any evaluated unit based on the robust BoD (RBoD) model are less sensitive to extreme values as they are not compared to peers on the entire domain. In this framework, each unit is compared with a ref-

erence set of m peers with higher performance levels without taking into account

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any other conditional variable Z. However, ignoring the impact of contextual variables Z may be a significant limitation especially when the presence of these variables greatly influences some of the individual simple indicators along with their relationships for the whole sample and/or for specific clusters. Traditional approaches (Daraio & Simar, 2007b; Jeong et al., 2010; Badin

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et al., 2010, 2012) suggest to account for heterogeneous environmental conditions by estimating a nonparametric kernel around the conditional environmental vari-

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ables Z of the evaluated unit. Following these proposals, the conditional convex hull over which the CI estimates are computed can be defined as:

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em = Ψ z

m [

1+q

{(1, y) ∈ R+ |Y j,Z=z ≥ y}.

(6)

j=1

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where Z is the set of conditional environmental variables that is assumed to affect unit’s CI estimate. Recently, some authors (Florens et al., 2014, Mastromarco &

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Simar, 2015) have proposed an alternative flexible nonparametric two-step conditional efficiency approach that aims to not only eliminate the dependence of inputs and/or outputs on common factors, but also to avoid the use of smoothing methods. In particular, in a CI framework, Verschelde & Rogge (2012) have proposed 14

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an adjusted version of the conditional order-m model, called as the robust and environment-adjusted BoD model, which allows to consider multiple aspects of

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the operating environment into the CI estimate. They evaluated effectiveness of the local police departments by conditioning for demographic and socio-economic

variables, years and administrative regions or typology of municipalities as spatial variables. However, in our opinion and specifically from a spatially point of view, all these formulations do not properly include the tangible and intangible spatial

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conditional factors (Vidoli et al., 2016), but only consider the main global spatial trend through tangible environmental factors Z.

Given these premises, following the Vidoli & Canello (2016) approach, a Spatial Conditional BoD model is proposed in this paper by considering the spatial distribution of the points as Z and comparing each unit with its own spatial peers

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belonging to a generic portion of territory S j .8

Overall, the modified optimization problem can be formulated by introducing

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e m in equation (3) an additional constraint associated with in the random set Ψ spatial proximity as:

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em = Ψ S

m [

1+q

{(1, y) ∈ R+ |Y j, j∈S j ≥ y}.

(7)

j=1

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In this modified setting, the choice of S j - which takes the meaning of spatial bandwidth in accordance with the conditional nonparametric literature - is

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strictly linked to the applicative issues: S j can be derived from the membership of an administrative region, of a spatial homogeneous cluster or - as in both our simulation and application exercises presented in section 4 - can be linked to a 8 It

is worth noting that when the portion of territory tends to the whole one, the model turns to

the standard order-m framework.

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measure/matrix of contiguity.

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4. Application The CI estimation exercise under our proposed Spatial RBoD approach in equation (7) is carried out, respectively, on a simulated dataset (subsection 4.1), and on a real-life dataset of the social supply delivered by the Italian Municipalities (subsection 4.2). In both exercises our proposed method has been found

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robust and useful in considering the need to take into account the spatial location patterns of the units. 4.1. Simulated data

To test properties and accuracy of the Spatial RBoD model, four different spa-

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tial point patterns processes9 have been simulated10 to generate four levels of spatially auto-correlated locations (see Figure 1).

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More specifically, the first DGP (SIM1) is a homogeneous Poisson process with an average density of µ = 50 points per area (intensity) that generates no

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spatially autocorrelated random patterns at all. The second (SIM2) and the fourth (SIM4) DGPs represent the non-Poisson cluster processes - in particular, a Matérn cluster process (Matern, 1986)11 - with a constant density of µ = 50 but with

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different radius values (r = 0.08 for SIM2 and r = 0.05 for SIM4). The third spatial point process is a useful model for exhibiting a random pattern of points in a bi-

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9A

dimensional space. 10 Please see, e.g., Baddeley et al., 2015 for a detailed discussion on point patterns processes. 11 This because a Matérn cluster process is strictly comparable with any homogeneous Poisson

process as the centre point (parent) of each cluster, is supposed to fall randomly within the area.

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DGP (SIM3) is also a Matérn cluster process with a fixed r = 0.05, but with nonhomogeneous number of points for cluster, i.e. µ = f (·) (Waagepetersen, 2008). SIM3: Clustered r=0.05, mu=inhomogeneous

SIM4: Clustered r=0.05, mu=50

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SIM2: Clustered r=0.08, mu=50

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SIM1: Independent r=n, mu=50

Figure 1: Simulated data

Spatial randomness (SIM1) and spatial clusterization (SIM2, SIM3 and SIM4)

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are both confirmed by the Complete Spatial Randomness test (CSR - Diggle, 2003) (see Table 1) and the Ripley’s K-function (K(r) - Ripley, 1976, 1981) (see

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Figure 2). Table 1 reveals that the null hypothesis of homogeneous Poisson DGP is accepted for SIM1 (p-value = 0.5309) but rejected for SIM2, SIM3 and SIM4

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(p-value < 0.001).

χ2

p-value

SIM1

19.323

0.5309

SIM2

1, 018

< 0.001

SIM3

827.25

< 0.001

SIM4

1, 759

< 0.001

Simulated data

Table 1: Chi-squared test of CSR using quadrat counts (Quadrats: 5 by 5 grid of tiles)

Similarly, the plots of Figure 2 reveal that the theoretical distribution K pois (r) 17

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b and its estimated counterpart K(r) of the homogeneous Poisson DGP converge

completely for SIM1, but diverge for SIM2, SIM3 and SIM4.

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K (r )

0.05 0.00

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0.10

^ (r ) K i so K poi s (r )

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K (r )

0.15 0.05

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K (r )

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0.10 0.05 0.00 0.00

SIM4:Clustered r=0.05, mu=50

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^ (r ) K i so K poi s (r )

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^ (r ) K i so K poi s (r )

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0.15

^ (r ) K i so K poi s (r )

SIM3:Clustered r=0.05, mu=inhomogeneous

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SIM1:Independent r=n, mu=50

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Figure 2: Simulated data

Once the spatial points patterns are defined, two simple indicators (I1 and I2 )

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are constructed from the same starting value but differences are generated by adding a random term that follows a normal distribution with zero mean and a

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constant variance of 0.5 for each cluster. This will enable us to have a strong spatial autocorrelation between I1 and I2 so as to highlight mostly the impact of spatial conditioning. Figure 3 (Figure 4) shows clusters with lower values of I1

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(I2 ) (light colors), clusters with higher values of I1 (I2 ) (dark colors) or clusters

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lon

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Longitude

0.75

SIM2:Clustered r=0.08, mu=50

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0.50

0.25

0.00

SIM3:Clustered r=0.05, mu=inhomogeneous

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● ● ● ● ● ● ●●● ● ● ● ● ●● ● ● ●● ●● ●● ● ●● ● ●● ● ● ●● ●● ● ● ● ●● ●●●● ●● ● ●● ●● ● ●● ● ● ●●●● ● ●● ● ● ● ●● ● ●● ●● ● ●● ● ● ● ● ●●●● ● ●●● ● ●● ● ●●● ● ●●● ● ● ● ●● ●● ● ● ● ●● ● ●●● ● ● ●●● ● ●● ● ● ●● ● ● ●● ● ● ● ● ● ● ●● ● ● ●● ● ●● ● ● ●● ●● ● ●● ●● ●● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ●● ●● ●● ● ●● ● ● ● ● ● ●● ●

0.00

●

0.25

1.00

●●●● ● ●

0.50

0.75

0.75

lon

SIM1: Independent r=n, mu=50

1.00

0.50

0.25

0.00

1.00

Latitude

0.00

0.25

0.50

Latitude

Figure 3: Simulated data: I1

18

SIM4:Clustered r=0.05, mu=50

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1.00

●

0.75

1.00

0.50

0.25

●● ●●●● ● ● ● ●● ●● ● ● ● ● ●● ● ●●● ●●●●●● ● ● ● ● ●● ●● ● ● ● ●●●●● ●●● ●● ● ● ●● ●● ● ● ●● ●● ● ● ● ●● ● ●● ●● ● ● ●●● ● ● ●● ● ● ●● ● ● ●● ●● ●● ● ●●● ● ● ● ●●● ●● ●●● ●● ● ● ●●

●● ● ●●● ●● ● ● ●● ● ● ● ● ●● ●● ●●● ● ● ● ●● ●● ●●● ● ●● ● ●●● ● ●●● ●● ● ●●●● ● ●● ●● ●● ● ● ●●● ● ● ● ●● ●● ● ● ● ●● ●●● ●

0.75

lon

CE

with neither high nor low values of I1 (I2 ).

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●

●● ●●● ● ● ●● ●● ●●●

0.00 0.00

0.25

●● ●● ● ● ● ● ● ● ●●● ●●● ●● ●● ● ● ● ●

● ●● ● ●● ● ● ●● ●●

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● ● ● ● ● ●● ●● ● ●●● ●● ● ●● ● ●● ● ● ● ● ● ● ●● ● ● ● ●●

0.50

Latitude

0.75

1.00

ACCEPTED MANUSCRIPT

0.25

0.00

0.00

0.25

0.50

0.75

1.00

0.50

0.25

0.00

0.00

●

1.00

●●●● ● ●

0.25

0.50

Latitude

0.75

0.75

0.50

0.25

0.00

1.00

Latitude

SIM4:Clustered r=0.05, mu=50

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1.00

●

0.00

0.25

0.50

Latitude

Figure 4: Simulated data: I2

0.75

0.50

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●● ● ●●● ●● ● ● ●● ● ● ● ● ●● ●● ●●● ● ● ● ●● ●● ●●● ● ●● ● ●●● ● ●●● ●● ● ●●●● ● ●● ●● ●● ● ● ●●● ● ● ● ●● ●● ● ● ● ●● ●●● ●

0.75

lon

0.75

SIM3:Clustered r=0.05, mu=inhomogeneous

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CR IP T

0.50

1.00

lon

Longitude

0.75

SIM2:Clustered r=0.08, mu=50

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lon

SIM1:Independent r=n, mu=50 1.00

0.25

●

●● ●●● ● ● ●● ●● ●●●

0.00

1.00

0.00

0.25

0.50

0.75

1.00

Latitude

AN US

After having generated the simulated data, the CI scores are computed using

both RBoD and Spatial RBoD methods12 . In order to test whether the Spatial RBoD model is able to identify the previously constructed clusters, k = 50 neighbours are considered in the construction of the K-nearest neighbours’ spatial-

M

weight matrix (see Figure 5). SIM2:Clustered r=0.08, mu=50

SIM3:Clustered r=0.05, mu=inhomogeneous

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CE

PT

ED

SIM1:Independent r=n, mu=50

SIM4:Clustered r=0.05, mu=50 ● ●● ● ● ●● ● ●●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ●●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ●●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

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Figure 5: K-nearest neighbours links (k = 50)

AC

The results are found highly satisfactory. As expected, in the case of SIM1,

12 In

order to test the robustness of our proposed method, a comparison with the classical BoD

has been done. The results are not reported here due to space constraint, but are available upon request from the authors.

19

ACCEPTED MANUSCRIPT

both RBoD and Spatial RBoD methods provide the same results (see Figure 6) given that spatial regularities cannot be found and so all units are compared against

CR IP T

the global frontier. In this way, units with lower I1 and I2 values obtain lower CI scores (red points), while units with higher I1 and I2 values obtain higher CI scores (dark green points). Independent r=n, mu=50

(0.126,0.252]

●

(0.252,0.377]

●

(0.377,0.503]

●

(0.503,0.629]

●

(0.629,0.755]

●

(0.755,0.881]

●

(0.881,1.01]

0.75

0.50

0.25

0.00

0.00

0.25

0.50

0.75

Latitude

●

(8.11e−05,0.126]

●

(0.126,0.252]

●

(0.252,0.377]

0.75

●

(0.377,0.503]

●

(0.503,0.629]

●

(0.629,0.755]

●

(0.755,0.881]

●

(0.881,1.01]

Longitude

(8.11e−05,0.126]

●

Longitude

●

1.00

Spatial RBoD

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AN US

1.00

RBoD

Independent r=n, mu=50

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0.50

0.25

0.00

1.00

0.00

0.25

0.50

0.75

1.00

Latitude

M

Figure 6: RBoD and Spatial RBoD estimates (SIM1)

ED

In the case of SIM2 and SIM3 (Figures 7 and 8), the RBoD and the Spatial RBoD-based CI estimates are though still highly correlate but some differences begin to emerge highlighting that in the proposed method units are compared with

PT

different local benchmarks.

Clustered r=0.08, mu=50 1.00

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CE RBoD

(0.138,0.265]

●

(0.265,0.391]

●

(0.391,0.518]

●

(0.518,0.644]

●

(0.644,0.771]

●

(0.771,0.897]

●

(0.897,1.02]

0.75

Longitude

(0.0116,0.138]

●

AC

●

0.50

0.25

0.00

0.00

0.25

0.50

0.75

Spatial RBoD ●

(0.0116,0.138]

●

(0.138,0.265]

●

(0.265,0.391]

●

(0.391,0.518]

●

(0.518,0.644]

●

(0.644,0.771]

●

(0.771,0.897]

●

(0.897,1.02]

0.75

Longitude

1.00

Clustered r=0.08, mu=50

0.50

0.25

0.00

1.00

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0.00

0.25

Latitude

0.50

Latitude

Figure 7: RBoD and Spatial RBoD estimates (SIM2)

20

0.75

1.00

ACCEPTED MANUSCRIPT

Clustered r=0.05, mu=inhomogeneous

●

(0.0237,0.147]

0.75

(0.147,0.271]

●

(0.271,0.395]

●

(0.395,0.519]

●

(0.519,0.643]

●

(0.643,0.766]

●

(0.766,0.89]

●

(0.89,1.01]

Longitude

●

0.50

0.25

0.00

0.00

0.25

0.50

0.75

Spatial RBoD ●

(0.0237,0.147]

●

(0.147,0.271]

●

(0.271,0.395]

●

(0.395,0.519]

●

(0.519,0.643]

●

(0.643,0.766]

●

(0.766,0.89]

●

(0.89,1.01]

0.75

0.50

0.25

0.00

1.00

Latitude

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0.00

CR IP T

RBoD

Clustered r=0.05, mu=inhomogeneous 1.00

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Longitude

1.00

0.25

0.50

0.75

1.00

Latitude

AN US

Figure 8: RBoD and Spatial RBoD estimates (SIM3)

These differences are, however, more pronounced in the case of SIM4 wherein most units, by construction, are well clustered (see Figure 9). In this case two distinct situations, zoomed in both (a) and (b) of Figure 9, emerge. In the first magnification of both figures (a) and (b), wherein a well-defined spatial cluster

M

is considered, units receive lower CI scores when evaluated against the global frontier (RBoD), but altogether different levels of CI scores if evaluated against the

ED

local benchmarks (Spatial RBoD). So Spatial RBoD allows to bring out inherent local differences by identifying spatial clusters.

PT

In the second magnification of both figures (a) and (b), however, there is an overlapped set of clusters wherein both RBoD and Spatial RBoD methods yield

AC

CE

seemingly similar CI estimates.

21

ACCEPTED MANUSCRIPT

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● ● ● ●●●● ● ●● ● ● ●● ● ● ● ●● ● ●●●● ● ●● ●● ●● ●● ● ● ●●● ● ● ●● ● ●● ● ●● ● ● ●●● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ●● ●●● ● ● ● ●● ● ●● ● ● ● ●● ● ●● ● ●● ● ● ● ●● ●● ● ● ●●● ● ● ● ●●●● ●● ● ● ●●● ● ● ● ● ● ● ● ● ●● ●● ● ● ●● ● ● ● ●● ● ● ● ●● ● ●● ● ● ● ● ●●●●● ● ●● ● ● ●● ● ●●●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ●●● ●● ● ●● ● ● ● ● ● ● ● ●● ● ●●● ● ●●● ● ●● ● ● ●● ●● ●● ● ● ● ●● ● ●● ● ● ● ● ● ●●● ●● ● ● ● ● ●● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ●●●● ●● ● ●● ● ● ● ●● ● ● ●● ● ●● ● ● ●● ● ●● ●● ●●●●● ● ●● ●●● ● ●● ● ● ●●● ●● ●● ● ● ●●● ●● ● ● ●● ● ● ● ●● ● ● ●●●● ● ● ●● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ●● ● ● ● ●● ● ● ● ●● ●

●

●

●● ●● ● ● ● ●● ●● ●● ●

0.75

0.50

0.25

●● ● ● ●●● ● ●● ●● ●● ● ● ●●● ● ● ●● ●● ●● ● ●●● ●●● ●

●

●

CE

●

Longitude

●

●

0.25

0.25

● ● ●● ● ●● ●●● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ●● ● ● ● ●● ● ●● ●● ●● ● ●● ●● ● ●● ●● ●● ●● ● ●● ● ● ● ●● ● ● ● ● ●● ● ● ●● ●● ● ●●● ●● ●● ● ● ●●● ● ● ● ●● ● ● ●● ● ●

●

●● ●●●●●●● ● ●● ● ● ●● ●● ● ● ● ● ● ●● ●● ●● ●● ● ●● ● ● ● ●●● ●●●● ● ● ●●

● ● ● ●●●● ● ●● ● ● ●● ● ● ● ●● ● ●●●● ● ●● ●● ●● ●● ● ● ●●● ● ● ●● ● ●● ● ●● ● ● ●●● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ●● ●●● ● ● ● ●● ● ●● ● ● ● ●● ● ●● ● ●● ● ● ● ●● ●● ● ● ●●● ● ● ● ●●●● ●● ● ● ●●● ● ● ● ● ● ● ● ● ●● ●● ● ● ●● ● ● ● ●● ● ● ●● ● ● ●● ● ● ● ● ●●●●● ● ●● ● ● ●● ● ●●●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ●●● ●● ● ●● ● ● ● ● ● ● ● ●● ●●● ● ● ●●● ● ●● ● ● ●● ●● ●● ● ● ● ●● ● ●● ● ● ● ● ● ●●● ●● ● ● ● ●● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ●● ●●●● ●● ● ●● ● ● ●● ● ● ● ● ●● ● ●● ● ● ●● ● ●● ●● ●●●●● ● ●● ●●● ● ●● ● ● ●●● ●● ●● ● ● ●● ●● ● ● ●● ● ●● ● ● ● ● ● ●●●● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ●●●● ●● ● ● ● ● ● ● ●● ● ●● ● ●● ●

●

●● ●● ● ● ● ●● ●● ●● ●

0.00

0.50

Latitude

PT

●

●

0.00

● ● ●● ●●● ●● ● ● ● ● ● ●●●●● ●●● ● ● ●● ●● ●●● ●●● ●● ● ●●● ● ●●●

ED

0.00

M

0.00

0.00

● ●● ● ●●● ● ●● ● ● ● ● ● ● ● ●●● ● ● ●● ●●● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ●● ●●● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ●● ● ● ● ● ●

AN US

0.50

● ●● ● ●● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ●● ● ●● ●● ●● ● ● ●●● ● ●● ●● ● ●● ●● ●●● ● ● ● ● ● ●● ● ● ● ●●● ● ● ● ●● ● ● ●● ●● ● ●● ●● ● ● ● ● ● ● ●● ● ●●●● ● ● ●●● ● ● ● ● ●● ●● ● ● ●● ● ● ●● ● ●● ● ●● ● ●●●● ● ● ● ●● ●●● ●● ● ●● ● ● ●● ●● ●● ● ●●● ● ●● ● ● ● ● ●● ●● ● ● ●●●●●● ●● ●● ● ● ●● ● ● ● ● ● ●●● ●● ● ● ● ●● ●●●● ●● ● ● ●

●● ● ● ● ● ●●●● ● ●● ● ● ● ● ●●● ● ● ● ● ●●● ●● ●● ●● ● ●●● ●● ● ● ●● ●● ●● ● ● ● ●● ● ● ● ●●●● ● ● ● ● ●● ● ● ● ● ●● ●●● ● ● ●● ●●●●●●● ● ●● ●● ● ●● ●●● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ●● ●●● ● ● ● ● ●● ●● ● ●● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ●●●●●●● ●● ● ● ● ● ●● ● ● ●● ● ●

● ●●●● ●● ● ● ●●● ●● ● ● ● ● ●●●● ● ● ●● ●● ●● ●● ●● ●● ●● ● ●● ● ●● ●● ●● ●● ● ● ●● ● ●● ●● ● ● ●● ●● ● ● ●● ● ●● ●● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ●● ● ●●● ● ●●● ●●● ●● ● ●● ● ● ●●

● ● ●● ●● ●● ● ● ●● ●● ● ● ● ●● ● ● ● ● ● ●● ●● ●●● ● ●●●●● ●● ● ●● ● ● ●● ● ●●● ●● ●●●● ● ● ●● ●● ●● ●● ● ● ●● ●●●● ● ●● ● ● ● ●● ● ●● ●●●●● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ●●● ● ● ●● ● ● ● ● ● ●●●● ● ● ● ●●●● ● ● ● ●● ● ●● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ●●●● ● ● ● ●●● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ●● ●● ● ●● ● ●● ● ●● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●●● ●●●● ● ● ● ●●● ● ● ● ● ●● ● ●● ● ● ● ● ●● ● ● ● ●

● ● ● ● ● ● ● ●● ● ● ● ●●● ● ●● ●● ● ● ● ●

CR IP T

● ● ●● ●●● ●● ● ● ● ● ● ●●●●● ●●● ● ● ●● ●● ●●● ●●● ●● ● ●●● ● ●●●

Longitude

Longitude

0.75

AC

]

●●●●● ●●● ● ● ●● ●● ●●● ●●● ●● ● ●●● ● ● ● ●

Longitude

]

● ●● ●● ● ●● ● ●● ●●● ● ● ● ●● ● ● ● ●● ●● ● ●●● ● ●● ●● ● ● ● ●● ● ●● ● ● ●● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ●● ● ●● ● ● ● ●● ● ●● ●●●● ● ●● ● ●● ● ● ●● ● ● ● ● ● ●●●●● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ●●●● ●● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●●●●● ● ● ● ● ● ●●● ● ● ● ● ● ●● ● ● ● ● ●● ●● ● ●● ● ● ●●● ● ●●● ●● ●● ● ● ●●● ●●● ●● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ●●● ● ● ● ●●● ● ● ● ●● ●● ● ●● ●● ●●● ●●● ●● ● ● ● ● ●● ● ● ●● ●● ● ● ●●● ● ● ● ●● ●● ● ● ● ● ● ● ●●● ● ●●● ●● ● ● ●● ● ● ● ●● ● ● ●● ●● ● ● ● ●● ● ●● ● ●●●●● ●● ● ●● ● ● ● ● ●● ●●● ●● ●● ● ● ● ●● ● ● ● ● ● ● ●●●●● ●●● ●● ● ● ● ● ●●● ● ●● ● ●●● ● ● ●● ●●● ● ●● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ●● ● ● ●● ●● ● ● 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●●● ● ● ●●● ● ●●● ● ● ● ●● ● ● ●● ●● ● ● ●● ● ●● ● ● ●● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● Clustered ●●● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● r=0.05, mu=50 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 1.00 ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●●● ● ●● ●● ●● ● ●● ● ● ●●● ● ● ● ●●● ● ● ● ● ●● ● ● ●● ●●● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ●●● ● ● ● ●● ● ● ● ●● ● ● ●●● ● ● ● ● ● ● ●●●●●● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ●●● ●●● ● ● ● ●●●● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ●●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ●●● ● ●●● ●●●● ● ● ● ● ●● ● ●● ● ● ● ●● ● ●● RBoD ● ● ●●●● ●● ● ●● ● ●● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● (0.00433,0.131] ● ● ● ●● ●● ●● ● ●●● ● ● ●● ● ● ●● ●● ● ● ● ●●● ● ●●● ●● ●● ● ● ● ●● ● ● ●● 0.75 ●● ● ●● ●●● ●● ● ●● ● ● ●●●●● ●●● ●●● ● ●●● ● ● ●● ●●● ● ●●● ●● ● ● ●● ● ● ●● ● ● ● ●● ●●● ●●● ● ●●● ●●●● ●● ● ●● ● ● ● ●● ●● ● ● ●●● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● (0.131,0.257] ●●● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ●● ● ● ● ● ● ●●● ● ●● ● ● ● ● ● ●● ● ●● ●● ●● ●●● ●●● ● ●● ● ● ● ● ● ● ● ● ● ●● ●● ● ●●● ●● ● ● ●● ●●● ● ● ● ● ●●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● (0.257,0.384] ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●●●● ● ● ●● ●● ●●● ●●●●●●● ● ● ● ●● ● ● ● ●● ●● ● ● ●● ● ● ● ● ●● ● ● ● ●● ● ● ● ●● ● ● ●● ●● ●●● ● ● ● ● ●● ● ●● ● ● ● ● ●● ● ● ● ● ● ●● ●●●● ● ● ●●●● ● ● ●● ● ●● ● ●● ●● ● ● ● ● ●●● ●● ●● ●● ● ● ●● ● ●●● ● ●● ● ● ● (0.384,0.51] ●●● ● ●● ●● ● ● ●●● ● ● ● ●●●●●● ● ● ● ● ●●●● ● ● ● ● ●● ●● ● ● ●● ● ●● ● ●●● ● ●● ● ●● ● 0.50 ● ● ● ●● ●● ● ● ●●● ● ●● ●●● ● ● ●● ●● ●● ● ● ●●● ● ●● ●● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ●● ● ● ●● ●●●● ●● ● ● ●●●● ● ● ● ●● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ●●●●● ●● ● ● ● ●● ●●● ● ● ● ●●●● ● ●● ● ● ●● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ●● (0.51,0.637] ●● ● ●● ●● ●● ● ● ● ● ● ● ● ● ●●● ●● ● ● ●● ●● ● ●● ● ● ● ●● ● ● ● ●● ● ● ● ●● ● ● ● ●● ● ● ● ● ●● ●● ●● ●● ● ● ●● ● ●● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ●● ●● ● ●● ● ● ● ● ● ● ● ●●● ● ●● ● ● ● ● ●● ●● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ●● ●●● ●● ● ● ●● ● ● ● ● ● ● ● ●● ● ●● ● ●● ●● ● ● ●● ● ● ● ● ● ●● ● ● ●● ● ● ●● ●● ● ● ● ● ● ●● ●● ● ● ● ●● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ●● ●● ● ● ●● ● ● ● ●● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ●● (0.637,0.763] ●● ● ●● ●● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ● ● ●● ● ● ●●●● ● ●● ●● ●●● ● ●● ●●●● ● ●● ●● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ●● ●● ● ● ● ● ● ● ● ● ●● ●● ●● ● ● ● ● ●● ●● ● ● ● ● ●● ● ●● ● ● ●● ●●● ● ● ● ● ● ● ●●● ● ●●●● ● ●● (0.763,0.89] ● ●● ● ●● ● ●● ● ● ● ● ● ● ● ●● ● 0.25 ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ●● ● ●● ●● ● ● ● ● ● ● ●●●● ● ●●● ●● ● ●● ● ● ● ● ●● ● ●● ● ●● ● (0.89,1.02] ● ●● ● ● ● ● ●● ● ● ● ●● ● ●● ●● ●●● ●● ● ● ●● ● ●● ● ● ● ●● ●● ● ● ● ●● ● ● ● ● ● ● ●●● ● ● ●● ● ● ●●● ● ● ●●● ● ●● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ●●● ●● ● ●● ●● ● ● ● ● ●● ● ● ● ● ●● ● ●●● ● ●● ● ●● ●● ●● ● ● ● ●● ●● ● ●● ●● ● ● ● ●●●●● ●● ●● ● ●● ●● ● ● ● ● ●● ●●● ●● ●●● ●● ●● ●● ● ● ●● ●● ●● ● ● ● ● ● ● ● ● ● ●●●●● ● ●● ● ●●● ● ● ● ● ●● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● 0.00 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ●●●● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ●●●● ● ●● ● ● ●● ●● ● ●● ● ● ● ● ●● ● ● ● ● ●● ●● ● ● ● ● ● ●● ●● ● ● ● ●●● ● ● ● ● ● ●●0.50 ● ● ●●●● ● ●● ● ● ● ● ●● ● ● ● ●●●● ●●●● ● ● ● ● ● ● ● ●● ● ● ●●● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ●● ● ●●●●● 0.00 0.25● ●● 0.75 1.00 ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ●● ●●● ● ● ● ●● ● ●● ● ● ●● ● ●●●● ●● ●● ●● ● ● ● ●● ● ● ● ● ●● ● ● ●●● ● ●● ●●● ●●● ● ●●●●●● ●● ●● ● ●● ● ● ●● ● ● ● ● ● ● ●●●● ● ●Latitude ● ● ● ●● ●● ●● ● ● ● ● ●● ●●● ● ●● ● ● ● ●● ● ●●● ● ● ●●● ● ● ● ●●● ● ● ●● ● ● ●● ● ● ● ●●●● ● ●● ●● ●● ● ● ● ●● ● ● ● ● ●●●●● ● ● ● ●● ●●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ●● ● ● ● ● ● ●● ●● ●● ● ●● ● ● ●●● ●● ● ● ●● ● ● ● ●● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ●● ● ●● ● ● ● ● ● ●● ● ●● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● (a) ● ●● ●●●● ● ● RBoD ● ●●● ● ● ● ● ● ●● ●●●●●●● ●● ●●●● ●● ● ●● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ●● ●● ● ● ●● ● ● ●● ●●● ● ● ● ●● ●● ● ●●● ● ●●● ● ●● ● ●● ● ● ● ● ● ●● ● ●●● ● ● ● ● ● ● ● ●● ● ●● ●● ● ●● ● ● ● ●● ●● ● ● ● ●● ● ● ●● ●● ● ●● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ●● ● ●●● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●●●● ●● Clustered ● ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ●● ● ●● ● ● ● ●● ● r=0.05, mu=50 ● ●● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ●● ●●● ●● ● ● ●● ● ● ●● ● ● ●● ● ● ● ●●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 1.00 ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ●●● ● ●● ● ●● ● ●● ● ● ●●●●●●●● ● ● ●●● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●●● ● ● ● ● ●● ● ● ●●● ● ● ● ● ● ● ●●● ● ● ●● ● ● ● ● ● ●●● ● ● ● ● ●● ● ●●● ●●●●●●● ● ● ● ● ●●●● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● Spatial RBoD ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ●● ● ● ● ●●● ●●● ● ● ● ● ● ● ●● ● ●● ● ● ● ●● ●● ● ● ●● ● ● ● ●●● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●● ●●● (0.00433,0.131] ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●●● ●● ●● ●● ● ●●● ● ●● ●● ●● ● ● ● ●●● ● ● ● ●● ●● ● ● ●●● ● ● ●● 0.75 ● ●● ● ●● ●●● ●● ● ●● ● ●●● ●● ● ●● ● ● ●●●●● ● ●●● ● ● ●● ●●● ● ●●● ●● ● ● ●● ● ● ●● ● (0.131,0.257] ●● ●●● ●●● ● ● ●●● ●●● ●●●● ●● ● ●● ● ● ● ●● ● ● ● ●●● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ●● ● ● ●● ● ●● ● ● ● ● ● ●● ● (0.257,0.384] ●● ●● ●● ●●● ●●● ● ● ●● ● ● ● ● ● ● ● ● ●●●● ●● ● ●● ●● ● ● ●● ●●● ● ● ● ● ● ●●● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● (0.384,0.51] ● ● ● ●● ● ●●●● ● ●●● ●● ●●● ● ●●●●●●● ● ●● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ●●● ● ● ●● ● ● ●● ● ●●● ●● ● ● ● ●● ● ● ● ● ● ●●● ● ● ● ● ●●●● ●● ● ● ● ●● ● ●● ● ●● ●● ● ● ● ●● ● ●●● ●● ● ● ●● ●● ● ● ● ●●● ● ●●● ● ● ● ● ● ● ● ● ●●●● ●●●● ●● ●● ● ● ● ● ●● ●● ● 0.50 ● ● ● ● ●● ●● ● ●●● ● ●●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● (0.51,0.637] ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ●● ● ● ●● ● ●● ●● ● ●● ● ● ●● ●● ● ● ● ● ●●●●● ● ●● ●● ● ●● ● ●● ●● ● ● ● ● ●● ● ● ● ●●● ● ● ● ● ● ●● ● ●● ● ● ● ● ●● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ●●● ●● ● ● ● ●● ●● ● ●● ●● ● ● ● ●● ● ● ● ● ●● ●● ●● ● ● (0.637,0.763] ●● ● ● ●●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ●● ●● ● ● ● ● ●● ● ●● ● ●● ● ● ● ●● ● ● ● ●●●● ●● ●● ● ●● ●● ● ● ● ● ● ● ● ● ● ●●● ● ●● ● ● ● ●● ● ●● ●● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ●● ●● ● ●●●● ●●● ● ● ● ● (0.763,0.89] ● ● ●● ● ● ● ●● ●● ● ● ● ●● ● ● ● ●● ● ● ● ● ●● ● ● ●● ●●● ●● ●● ● ● ●●● ● ● ● ●● ● ● ● ● ● ●● ● ● ●● ● ● ● ●● ● ● ●● ●● ●●● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.25 ●● ● ● ●●● ●● ● ● ● ●●● ● ● ●● ● ● ● ●● ●● ●●● ●● ●● ● ●● ●● ●● ●● ●● ●● ●● ● ● ● ●● ● ●● ●● ●●● ●● ● ● ● ●● ● ● ● ●● ● ● ● ● ●● ● ●● ● ● ●● ● ● ● ● ●● ● ● (0.89,1.02] ● ● ●●● ●●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ●●● ● ●● ●● ● ● ●●●● ● ● ● ● ●●●●● ●●● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●●●●● ● ● ●●● ● ●● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ●● ●●● ●● ●● ●● ● ●● ● ● ●●●● ●●● ● ●● ● 0.00 ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ●● ●● ●●● ● ● ●● ● ●●●●● ●● ● ● ● ●●● ●● ● ● ● ●● 0.00 0.25 0.50 0.75 1.00 ● ● ●● ● ● ● ●● ●● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Latitude ● ●● ●● ● ●● ● ● ● ●● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●●● ● ●● ● ● ● ● ● ● ●●● ●● ●●●● ● ●● ● ● ● ●● ● ● ●●● ● ● ● ● ●● ● ●● ●●● ●● ● ●● ● ● ●● ● ● ● ●● ● ● ● ●● ●● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● (b) Spatial RBoD ●●●●● ● ● ● ●● ● ●●●●●● ● ●● ● ●●● ●●● ● ● ●● ● ● ●● ● ●● ●● ●● ● ●● ● ● ● ●●● ● ● ● ●● ● ● ● ●● ●● ●● ● ●● ● ● ● ● ● ● ● ●● ●● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●●● ● ● ● ● ● ●●●● ● ●● ● ●● ● 1.00 ● ●● ●●● ●● ● ● ● ● ●

0.50

0.75

● ● ● ● ● ● ● ●● ● ● ● ●●● ● ●● ●● ● ● ● ●

0.75

1.00

Latitude

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Figure 9: RBoD and Spatial RBoD estimates (SIM4)

0.00

0.25

0.25

0.50 22

Latitude

0.50

0.75

Latitude

0.75

1.00

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Figure 10 shows a comparative picture on the distributions of CI estimates by RBoD and Spatial RBoD methods for each of the four simulated data sets.

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As expected, in the case of SIM1, both the methods generate almost the same distribution of CI estimates as the ranges of both boxplots are very similar. In the other three cases (SIM2, SIM3 and SIM4), however, while RBoD model presents

almost the same distribution of CI scores due to similarly positioned boxplots, the Spatial RBoD model shows CI estimates with increasing median due to differently

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positioned boxplots.

This finding is due to the fact that the Spatial RBoD method enables units to find their own local benchmarks in the neighborhood, and hence, yield on average higher CI estimates. However, there are other possible scenarios: (i) the CI estimates obtained from the Spatial RBoD method are lower than those from the

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RBoD method when a local benchmark with very high values in both simple indicators exists in the chosen neighborhood; (ii) the CI estimates by the Spatial RBoD

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method are very similar to those by the RBoD method when a local benchmark does not exist in the chosen neighborhood; and (iii) the CI estimates by the Spatial

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RBoD method are higher than those by the RBoD method for those units who all exhibit higher values in both simple indicators in the chosen neighborhood. Clustered r=0.08, mu=50

Clustered r=0.05, mu=inhomogeneous

Clustered r=0.05, mu=50

0.75

0.75

0.75

0.75

0.50

0.50

value

1.00

value

1.00

value

1.00

0.50

0.50

0.25

0.25

0.25

0.25

0.00

0.00

0.00

0.00

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Value

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Independent r=n, mu=50 1.00

RBoD

SpRBoD

RBoD

SpRBoD

RBoD

SpRBoD

Figure 10: RBoD and Spatial RBoD boxplot

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RBoD

SpRBoD

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Finally, the global Geary’s C test13 is applied to statistically verify whether or not our proposed method is capable of removing spatial trends in the distribution

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of the CI scores. Results are reported in Table 2. In the case of SIM1 Geary’s C value is near to 1 in both methods, implying the desired absence of spatial dependence among units. In the other three cases, however, while the RBoD-based CI estimates suffer strongly from the presence of positive spatial autocorrelation (p < 0.001), the Spatial RBoD-based estimates are

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not more spatially autocorrelated (as the Geary’s C is very close to 1 and it is not statistically significant, as expected).

Simulated data

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SIM2 SIM3

CE

PT

SIM4

13 The

RBoD

Spatial RBoD

0.994

1.028

(0.108)

(0.642)

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SIM1

Geary’s C

0.562

1.202

(< 0.001)

(0.917)

0.534

0.946

(< 0.001)

(0.351)

0.315

1.105

(< 0.001)

(0.767)

Table 2: RBoD and Spatial RBoD global Geary’s C test

value of Geary C lies between 0 and 2, i.e., 0 ≤ C ≤ 2. If C < 1, it means increasing

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positive spatial autocorrelation. If C > 1, it indicates increasing negative spatial autocorrelation.

Finally, if C = 1, it is then consistent with no spatial autocorrelation.

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4.2. Estimating spatial conditioned BoD composite indicator of the social services in Italy

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The Italian Law No. 42/09 on fiscal federalism, and the subsequent implementation provisions issued by the Legislative Decree No. 216/10 regarding the

evaluation of standard expenditure needs of the Local Authorities, marked the beginning of a radical reform policy of the inter-governmental public relationship in

Italy. The evaluation of standard expenditure needs is, in fact, a first and neces-

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sary step toward building a new mechanism for efficient distribution of the grants regarding funding of essential functions for the Municipalities and Provinces. In order to overcome the problems associated with the computation of standard needs using only administrative information, from 2010 onwards, a multitude

of information regarding input, output and prices have been collected by the Ital-

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ian Ministry of Economy and Finance through specific questionnaires for each municipality. These data are now available at http://www.opencivitas.it/,

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distinctly by function, for about 6,700 municipalities. In this application, a general social service CI delivered by the municipalities

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is computed in order to evaluate the effective service performed on a territory. The service is here linked not only with the monetary resources, but also with the reg-

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ulatory constraints imposed by the higher administrative level (i.e., the Regions), which can independently allocate more or less resources to a specific function. Moreover, municipalities may suffer from the institutional constraints, which may

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exogenously affect the level of service offered within the same jurisdiction. From an application point of view, however, a CI that fails to take these constraints into consideration assumes homogeneous cluster of municipalities among different areas of Regional Local Authorities, and hence, masks the real differences among 25

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municipalities within the same area. Given these premises, three simple indicators are selected for the social ser-

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vice level for the year 2013 involving children, youth, seniors, families, disabled, people addicted to alcohol, drugs or with mental health problems, immigrants and nomads, and socio-economically disadvantaged adults. They are: (i) services re-

garding the social emergency and home care (including front and back office load factors), (ii) day care centers (e.g. recreational social and cultural activities) and

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early childhood (e.g. other educational and recreational services for early childhood), and (iii) activities related to residential care, income support and social

inclusion. All the dimensions are measured in terms of number14 of measures or actions per inhabitant that is put in place to address a specific task, and are then  normalized using the min-max criteria.

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Figure 11 plots the RBoD scores of 3,591 municipalities offering social services. Two situations may be worth highlighting here. While in the first magnifica-

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tion (upper one), relative to a northern region, municipalities exhibit a non-regular spatial pattern where the level of service seems not to have a homogeneous distri-

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bution, in the lower one, relative to Campania, municipalities with similar levels of service are close to each other.

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14 The qualitative differences among different services offered are,

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for the obvious reason, i.e., due to lack of data.

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however, not considered here

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●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

● ●

●

● ● ●

● ●

●

●

● ●

●

●

● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ●● ● ● ● ●

●

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●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

● ● ●●

●

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●

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●

●

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● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ●

(0.667,1]

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●

●

●

●

●

●

●●

● ● ●● ●● ● ● ● ●● ● ● ● ● ●● ● ●●● ● ● ●● ●● ● ● ●●

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●

● ●● ● ● ●●● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

● ● ● ● ● ● ●

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0

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● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●

M

4400000

ED

4000000

500000

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●●

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750000

1000000

1250000

Latitude

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●

PT

Figure 11: RBoD scores of social services in Italy

CE

The qualitative inferences drawn from Figure 11 are further statistically con4000000

firmed by the Geary’s C statistic. C statistic score equal to 0.65 obtained on the RBoD-based CI estimates has been found significant (p < 0.001), confirming the

AC

0

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●

●

AN US

●

CR IP T

●

Longitude

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Longitude

0

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● ● ● ● ● ● ● ● ●●● ●● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ●● ●● ● ● ●●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ●● ● ● ● ●● ● ●●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●●● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ●●● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●●● ● ● ● ● ● ●● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●● ●● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ●● ● ● RBoD ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●●●●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ●●●●●● ● ● ●● ●● ●●●●●●●● ●●● ● ● ● ● ● ● ● ● ● (0,0.333] ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ● ● ●● ● ● ● ●● ●● ● ● ● ● ●●● ● ● ● ●●● ● ● ● ● ● ●●●● ● ● ● ● ●● ● ●●●● ● ● ●● ● ● ● ● ● ●●● ● ● ●● ● ● ● ●● ● ● ●●● ●● (0.333,0.667] ● ●● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ●● ● ● ● ● ●● ●●● ●●● ● ● ● ● ●● ● ● ● ● ● ●●● ●● ● ●●● ●●● ● ● ●● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ●●●● ● ● ●●● ● ●● ● ● ● ●● (0.667,1] ● ● ● ● ● ● ● ●● ● ●● ● ● ● ●● ●●●● ● ● ●● ● ●●● ● ● ● ●●●● ● ●● ● ●●●● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ●●● ● ●● ● ●● ● ● RBoD ●● ● ● ● ● ● ● ●●● ● ● ● ● ● ●● ● ● ● ● ● ●●● ●● ● ●● ● ● ● ● ● ●● ●● ● ●●●●● ● ●●● ● ●● ● ●● ●● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● (0,0.333] ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ●● ●●●● ● ● ● ● ● ● ● ●● ● ● ● (0.333,0.667] ●●● ● ● ● ● ● ● ● ● ● ●●● ● ●●● ●● 4400000 ●● ● ● ● ● ●● ●● ●● ●● ●●● ●● ● ● ● ● ●●● ● ●● ● ● ● ● ● ● ● ●

500000

750000

existence of global spatial dependence. Moreover, a counter-proof of a higher level of local spatial dependence is also shown in Figure 12.

27 500000

750000

1000000

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Figure 12: Moran plot - RBoD scores of social services in Italy

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proposed model turns to the standard order-m framework. Figure 13 shows that, in terms of spatial dependence, as has been confirmed by both Moran and Geary global tests, the two models tend to yield almost similar CI estimates with the increase in the number of neighbours considered, viz., the

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Figure 13: Spatial global measure by number of neighbours

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QZ value by Daraio & Simar, 2007a, a measure can be derived from the comparison between the conditional model and the non-conditional one by increasing the

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number of neighbours (see Figure 14). Figure 14 shows that a maximum distance between conditional and non-conditional

model has been found in correspondence with n equal to 35. This cut-off may be seen as the result of the trade-off between a greater neighbourhood size (in order to not consider too small local correlation - lower n) and a greater distance 29

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between conditional and non-conditional estimates (higher n). Corresponding to this value15 , therefore, the advantage of using a method that

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Figure 14: Difference between BoD and Spatial BoD in terms of global spatial test by number of

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already noted in the literature, Moran and Geary test provide similar measures, but not

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ber of neighbours has been fixed equal to 35 in order to better take into account the contextual neighbourhood effects. Figure 15, in fact, confirms that the local

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interactions succeeded in conditioning the single municipality score by the neighbourhood effects have been successfully detected, allowing to obtain the most accurate local comparisons.

Furthermore, local conditioning allows to correctly estimate the pure effect of

single indicators on a smaller scale. In other terms, the spatial conditioning affects

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●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

● ●

●

● ● ●

● ●

●

●

● ●

●

●

● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ●● ● ● ● ●

●

● ●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

● ● ●●

●

● ●

● ● ● ● ● ●

●

●● ● ●● ● ●●

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● ● ● ● ● ● ●●●● ●●●● ● ● ●

●

●

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●● ●

● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ●

(0.667,1]

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●

●

●

●

●

●

●

●●

● ● ●● ●● ● ● ● ●● ● ● ● ● ●● ● ●●● ● ● ●● ●● ● ● ●●

● ● ●

●

● ●● ● ● ●●● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

● ● ● ● ● ● ●

●● ●

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0

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● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●

M

4400000

ED

4000000

500000

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●●

● ● ●● ●● ● ● ● ● ● ● ●● ● ●●● ● ● ●● ● ●●

750000

1000000

1250000

Latitude

● ● ● ●● ●

●

PT

Figure 15: Spatial RBoD scores of social services in Italy

CE

In this setting, Geary’s C statistic test (equal to 0.815, p-value< 0.001) is 4000000

surely better compared to the RBoD one (0.742 and, as previously shown, it can be considered optimal), but it is still significantly different from 1 showing yet

AC

0

●

●

●

AN US

●

CR IP T

●

Longitude

●

Longitude

0

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● ● ● ● ● ● ● ● ●●● ●● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ●● ●● ● ● ●●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ●● ● ● ● ●● ● ●●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●●● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ●●● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●●● ● ● ● ● ● ●● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●● ●● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ●● ● ● Spatial RBoD ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●●●●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ●●●●●● ● ● ●● ●● ●●●●●●●● ●●● ● ● ● ● ● ● ● ● ● (0,0.333] ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ● ● ●● ● ● ● ●● ●● ● ● ● ● ●●● ● ● ● ●●● ● ● ● ● ● ●●●● ● ● ● ● ●● ● ●●●● ● ● ●● ● ● ● ● ● ●●● ● ● ●● ● ● ● ●● ● ● ●●● ●● (0.333,0.667] ● ●● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ●● ● ● ● ● ●● ●●● ●●● ● ● ● ● ●● ● ● ● ● ● ●●● ●● ● ●●● ●●● ● ● ●● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ●●●● ● ● ●●● ● ●● ● ● ● ●● (0.667,1] ● ● ● ● ● ● ● ●● ● ●● ● ● ● ●● ●●●● ● ● ●● ● ●●● ● ● ● ●●●● ● ●● ● ●●●● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ●●● ● ●● ● ●● ● ● Spatial RBoD ●● ● ● ● ● ● ● ●●● ● ● ● ● ● ●● ● ● ● ● ● ●●● ●● ● ●● ● ● ● ● ● ●● ●● ● ●●●●● ● ●●● ● ●● ● ●● ●● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● (0,0.333] ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ●● ●●●● ● ● ● ● ● ● ● ●● ● ● ● (0.333,0.667] ●●● ● ● ● ● ● ● ● ● ● ●●● ● ●●● ●● 4400000 ●● ● ● ● ● ●● ●● ●● ●● ●●● ●● ● ● ● ● ●●● ● ●● ● ● ● ● ● ● ● ●

500000

750000

slight spatial dependence among CI scores (see also Figure 16). This result may probably ascribed to the different spatial point density among the Municipality clusters, especially between the Northern and the Southern regions.

32 500000

750000

1000000

Latitude

10

Latitude

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Figure 16: Moran plot - Spatial RBoD scores of social services in Italy

The fixed rule (k = 35) imposed to identify homogeneous portions of territory

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(S j in eqn. 7), may, in fact, in some application contexts, be a suboptimal choice. In this case a variable neighbourhood policy will surely be a better alternative.

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The investigation of a variable neighbourhood policy is beyond the scope of our

CE

current paper but it can be considered for potential future research subject. 5. Final remarks

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Composite indicators have been increasingly used in recent years to describe

complex systems by assessing the performance of individual units across space and time. The present paper stresses the need to properly consider the spatial heterogeneity of the units in the computation of the relative CI. Otherwise, the CI estimates not based on spatial conditioning may often yield erroneous inferences. 33

ACCEPTED MANUSCRIPT

In this regard, an original optimization problem has been proposed (see eqn. (7)) with an aim to introduce - in the robust BoD framework - an additional constraint

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associated with spatial proximity. Two empirical illustrations have been proposed. The first one is based on

a simulated data set (see subsection 4.1) wherein our proposed method is illustrated to highlight the heterogeneous contributions of external factors in explain-

ing performance differential of a unit among different points in space. The second

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one, based on the social supply real data delivered by the Italian Municipalities (see subsection 4.2), highlights the effect of regional administrative constraints that may restrict the level of social and health supply of each municipality. In both illustrations, "spatial point patterns" are intended as non-random locations of production units, stressing thus the need to not neglect the effect of the terri-

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torial tangible and intangible factors (Vidoli & Canello, 2016) on the aggregate measure of CIs.

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Finally, the detection of spatial homogeneous optimal cluster of points that is linked to either a measure/matrix of contiguity or territorial membership, may be

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considered one of the most important stream of future research subjects in order to enhance methods in the robust BoD framework.

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References

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References

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Springer. Chap. Dynamic Macroeconomic Performance of Indian States: Some Post Reform Evidence. 34

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Adolphson, D. L., Cornia, G. C., & Walters, L. C. 1991. Operational research ’90. Oxford: Pergamon Press. Chap. A unified framework for classifying DEA

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models, pages 647–657. Aldstadt, J., & Getis, A. 2006. Using amoeba to create a spatial weights matrix and identify spatial clusters. Geographical analysis, 38, 327–343.

Anselin, L. 2010. Thirty years of spatial econometrics. Papers in regional science,

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Methodology and applications with R. London: Chapman and Hall/CRC Press. Badin, L., Daraio, C., & L., Simar. 2010. Optimal bandwidth selection for con-

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Banker, R. D. 1993. Maximum likelihood, consistency and data envelopment analysis: A statistical foundation. Management science, 39, 1265–1273. 35

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Banker, R. D., & Natarajan, R. 2004. Handbook on data envelopment analysis. Boston: Springer. Chap. Statistical tests based on DEA efficiency scores.

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