Methods for Constructing Non-compensatory Composite Indices: A Comparative Study

Transcription

1 Mehods for Consrucng Non-compensaory Compose Indces: A Comparave Sudy Maeo Mazzoa and Adrano Pareo Absrac Non-compensably and comparably of he daa over me are cenral ssues n compose ndces consrucon. The am of hs paper s o compare wo nonaddve approaches: Mazzoa-Pareo Index (MPI) and Weghed Produc (WP) mehod. The MPI s a non-lnear compose ndex whch rewards he uns wh balanced values of he ndvdual ndcaors. I normalzes he componens wh respec o he mean and sandard devaon and allows relave comparsons only. The WP mehod allows o buld, for each un, wo compose ndces closely nerrelaed: a sac ndex for space comparsons, and a dynamc ndex for me comparsons. An applcaon o ndcaors of well-beng n he Ialan regons n 2005 and 2009 s presened Key words: compose ndces, normalzaon, aggregaon 1 Inroducon Compose ndces for comparng counry performance wh respec o muldmensonal phenomena, such as developmen, povery, qualy of lfe, ec., are ncreasngly recognzed as a useful ool n polcy and publc communcaon (OECD, 2008). Consderable aenon has been devoed n recen years o he fundamenal ssue of compensably among he componens of he ndex (a defc n one dmenson can be compensaed by a surplus n anoher) and more and more ofen a non-compensaory approach has been adoped (e.g. he new Human Developmen Index calculaed by UNDP n 2010 s gven by a geomerc mean). Maeo Mazzoa, Ialan Naonal Insue of Sascs; emal: mazzo@sa. Adrano Pareo, Ialan Naonal Insue of Sascs; emal: pareo@sa.

2 2 Maeo Mazzoa and Adrano Pareo In hs work, we compare wo dfferen non-addve approaches: Mazzoa-Pareo Index (MPI) and Weghed Produc (WP) mehod. The MPI s a non-lnear compose ndex whch ransforms he ndvdual ndcaors n sandardzed varables and summarzes he daa usng an arhmec mean adjused by a penaly coeffcen relaed o he varably of each un. The am s o penalze he uns wh unbalanced values of he ndcaors n a non-compensaory perspecve. The WP mehod, also ermed as he geomerc aggregaon approach, s a classc daa aggregaon echnque n ndex number heory. An applcaon of he Jevons ndex o ndczed ndcaors s presened ha allows o buld, for each un, a sac and a dynamc ndex, for boh spaal and emporal comparsons. In Secon 2 a bref descrpon of he MPI s repored; n Secon 3 he ndces based on he WP mehod are presened; fnally, n Secon 4 an applcaon o real daa s proposed. 2 Mazzoa-Pareo Index The Mazzoa-Pareo Index s a compose ndex based on he assumpon of nonsubsuably of he ndcaors,.e., hey have all he same mporance and a compensaon among hem s no allowed (De Muro e al., 2010). The ndex s desgned n order o sasfy he followng properes: () normalzaon of he ndcaors by a specfc creron ha delees boh he un of measuremen and he varably effec; () synhess ndependen from an deal un, snce a se of opmal values s arbrary, non-unvocal and can vary wh me; () smplcy of compuaon; (v) ease of nerpreaon. Le us consder a se of ndvdual ndcaors posvely relaed wh he phenomenon o be measured. Gven he marx X={x j } wh n rows (uns) and m columns (ndcaors), we calculae a sandardzed marx Z={z j } as follow: where M x j and z j ( x = j M S x j x j ) 10 S x j are, respecvely, he mean and he sandard devaon of he j-h ndcaor. Denong wh M z and S z, respecvely, he mean and he sandard devaon of he sandardzed values of he -h un, he generalzed form of MPI s gven by: where + MPI / = M ± S z cv = S M s he coeffcen of varaon of he -h un and he sgn ± z z depends on he knd of phenomenon o be measured. If he compose ndex s ncreasng or posve,.e., ncreasng values of he ndex correspond o posve varaons of he phenomenon (e.g., he soco-economc developmen), hen MPI - s used. Vce versa, f he compose ndex s decreasng or negave,.e., ncreasng values of he ndex correspond o negave varaons of he phenomenon (e.g., he povery), hen MPI + s used (Mazzoa and Pareo, 2011). z cv

3 Mehods for Consrucng Non-compensaory Compose Indces: A Comparave Sudy 3 Ths approach s characerzed by he use of a funcon (he produc S z cv ) o penalze he uns wh unbalanced values of he ndcaors. The penaly s based on he coeffcen of varaon and s zero f all he values are equal. The purpose s o favour he uns ha, mean beng equal, have a greaer balance among he dfferen ndcaors. 3 Sac and dynamc compose ndex The weghed produc mehod s one of he major echnques n compose ndex consrucon snce represens a rade-off soluon beween addve mehods wh full compensably and non-compensaory approaches (OECD, 2008). When an unweghed geomerc mean of raos - such as he Jevons ndex - s compued, he obaned resul sasfes many desrable properes from an axomac pon of vew (Dewer, 1995). Le x j he value of he ndcaor j for he regon a me (j=1,, m; =1,, n; = 0, 1 ). A sac compose ndex may be defned as follows: SCI = m j= 1 xj xrj 100 where x rj s he reference or base value, e.g., he average. Therefore, values of SCI ha are hgher (lower) han 100 ndcae regons wh above (below) average performance. In order o compare he daa from me 0 o 1, for each un, we can consruc a dynamc compose ndex gven by: 1 m m 1 m / = xj 1 0 DCI j= 1 xj For he crculary or ransvy propery of he ndex number heory, SCI and 1 / / 0 DCI are lnked by he relaon: DCI = (SCI SCI ) DCIr. 1 4 An applcaon o he Ialan regons In order o compare he wo approaches, we consder a se of ndcaors of well-beng n he Ialan ces, a regonal level, n 2005 and The varables used are he followng: Sporng acves, Close o supermarkes, Green space, Publc ranspor, Parkng provson, Chldren s servces, Elderly home care. The MPI - s used, snce he compose ndex s posve,.e., ncreasng values of he ndex correspond o posve varaons of well-beng.

4 4 Maeo Mazzoa and Adrano Pareo Daa marx s repored n Table 1 and resuls are presened n Table 2. Table 1: Indvdual ndcaors of well-beng n he Ialan regons (Years 2005, 2009) Regons Sporng acves Close o supermarkes Green space Publc ranspor Parkng provson Chldren's servces Elderly home care Sporng acves Close o supermarkes Green space Publc ranspor Parkng provson Chldren's servces Elderly home care Pemone Valle d'aosa Lombarda Trenno-Alo Adge Veneo Frul-Veneza Gula Lgura Emla-Romagna Toscana Umbra Marche Lazo Abruzzo Molse Campana Pugla Baslcaa Calabra Scla Sardegna Iala Source: hp://www3.sa./ambene/coneso/nfoerr/ass/assev.xls Table 2: MPI, SCI and DCI of well-beng (Years 2005, 2009) Mazzoa-Pareo Index Weghed Produc mehod Regon MPI05 MPI09 MPI09- MPI05 SCI05 SCI09 DCI09/05 Pemone Valle d'aosa Lombarda Trenno-Alo Adge Veneo Frul-Veneza Gula Lgura Emla-Romagna Toscana Umbra Marche Lazo Abruzzo Molse Campana Pugla Baslcaa Calabra Scla Sardegna Noe ha he base value of he sac ndces (SCI 05 and SCI 09 ), for each regon, s he naonal average (Ialy), whle he base of he dynamc ndex (DCI 09/05 ) s he value for he year 2005.

5 Mehods for Consrucng Non-compensaory Compose Indces: A Comparave Sudy 5 As we can see from Table 2, no necessarly each relave ncrease corresponds o an absolue one and vce versa. For example, from 2005 o 2009, Toscana shows a reducon of he level of well-beng compared o he average (MPI 09 -MPI 05 =-1.8; SCI 05 =113.4 vs. SCI 09 =107.3), hough he values of he ndvdual ndcaors, on he whole, are ncreased (DCI 09/05 =102.6). Ths s due o a greaer rse of he performances of he oher regons whch has produced a large ncrease of he naonal average n Overall, he regon n whch s possble o record he hghes ncrease of he wellbeng ndcaors, over he fve years, s Abruzzo (MPI 09 -MPI 05 =+6.1; DCI 09/05 =159.3). From he pon of vew of he decrease, nsead, he resuls are conflcng: he greaes relave decrease, n fac, s for Molse (MPI 09 -MPI 05 =-2.8), alhough he values of he ndvdual ndcaors are, on average, slghly ncreased (DCI 09/05 =100.5); whle he larges absolue decrease s observed n he Lgura Regon (DCI 09/05 =99.6). In Fgure 1, he comparson beween he scores obaned by MPI and SCI s presened, for he year 2005 (ρ=0.84) and he year 2009 (ρ=0.88); n general he resuls are concordan and he man dfferences are due o he dfferen mehod of normalzaon of he wo mehodologes. The MPI, n fac, assgns he same wegh o all he componens, whle he SCI assgns dfferen weghs dependng on he varably. Fgure 1: Comparng MPI and SCI (Years 2005, 2009) 160 SCI Calabra Campana Campana Scla Calabra Scla Abruzzo Baslcaa Pugla Pugla Baslcaa Pemone Sardegna Sardegna Marche Lgura Toscana Frul-V.G. Toscana Marche Trenno-A.A. Lombarda Trenno-A.A. Lgura Frul-V.G. Lombarda Lazo Pemone Lazo Valle d'aosa Valle d'aosa Umbra Umbra Emla-Romagna Abruzzo Emla-Romagna Veneo Veneo 40 Molse Molse MPI

6 6 Maeo Mazzoa and Adrano Pareo 5 Conclusons Non-compensably and comparably of he daa over me are cenral ssues n compose ndces consrucon. Non-compensaory compose ndces may be obaned by non-addve approaches; whle he queson of comparably manly depends on he normalzaon mehod. In hs paper, a comparson beween wo dfferen approaches s proposed. The MPI s based on a sandardzaon wh respec o he mean and sandard devaon ha makes he ndcaors ndependen of he varably. Therefore, all he varables are assgned equal weghs, and only relave me comparsons are allowed. The wo ndces based on he WP mehod mplcly gve more wegh o he componens ha exhbs he larges varably, and he DCI allows absolue me comparsons oo. References 1. De Muro, P., Mazzoa, M., Pareo, A.: Compose Indces of Developmen and Povery: An Applcaon o MDGs. Soc. Indc. Res., 104: 1-18 (2011) 2. Dewer, W.E.: Axomac and Economc Approaches o Elemenary Prce Indexes. NBER Workng Papers 5104, Naonal Bureau of Economc Research, Inc. (1995) 3. Mazzoa, M., Pareo, A.: Un ndce sneco non compensavo per la msura della doazone nfrasruurale: un applcazone n ambo sanaro. Rvsa d Sasca Uffcale, 1 (2011) 4. OECD: Handbook on Consrucng Compose Indcaors. Mehodology and user gude. OECD Publcaons, Pars (2008)