Assessing health efficiency across countries with a twostep and bootstrap analysis *


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1 Assessng health effcency across countres wth a twostep and bootstrap analyss * Antóno Afonso # $ and Mguel St. Aubyn # February 2007 Abstract We estmate a semparametrc model of health producton process usng a twostage approach for OECD countres. By regressng data envelopment analyss output effcency scores on nondscretonary varables, both usng Tobt analyss and a sngle and double bootstrap procedure, we show that neffcency s strongly related to GDP per head, the educaton level, and health behavour such as obesty and smokng habts. The bootstrappng procedure corrects lkely based DEA output scores takng nto account that envronmental varables are correlated to output and nput varables. JEL: C14, C61, H52, I11 Keywords: techncal effcency, health, DEA, bootstrap, semparametrc * The opnons expressed heren are those of the authors and do not necessarly reflect those of the author s employers. # UECE Research Unt on Complexty and Economcs; Department of Economcs, ISEG/TULsbon Techncal Unversty of Lsbon, R. Mguel Lup 20, Lsbon, Portugal, emals: UECE s supported by FCT (Fundação para a Cênca e a Tecnologa, Portugal), fnanced by ERDF and Portuguese funds. $ European Central Bank, Drectorate General Economcs, Kaserstraße 29, D Frankfurt am Man, Germany, emal:
2 Contents 1. INTRODUCTION MOTIVATION AND LITERATURE ANALYTICAL METHODOLOGY DEA FRAMEWORK NONDISCRETIONARY INPUTS AND THE DEA/TOBIT TWOSTEPS PROCEDURE NONDISCRETIONARY INPUTS AND BOOTSTRAP EMPIRICAL ANALYSIS DATA AND INDICATORS PRINCIPAL COMPONENT ANALYSIS DEA EFFICIENCY RESULTS EXPLAINING INEFFICIENCY THE ROLE OF NONDISCRETIONARY INPUTS CONCLUSION APPENDIX 1 SINGLE AND DOUBLE BOOTSRAP PROCEDURES APPENDIX 2 POTENTIAL YEARS OF LIFE NOT LOST REFERENCES ANNEX DATA AND SOURCES TABLES AND FIGURES
3 1. Introducton In ths paper we systematcally compare the output from the health system of a set of OECD countres wth resources employed (doctors, nurses, beds and dagnostc technology equpment). Usng data envelopment analyss (DEA), we derve a theoretcal producton fronter for health. In the most favourable case, a country s operatng on the fronter, and s consdered as effcent. However, most countres are found to perform below the fronter and an estmate of the dstance each country s from that border lne s provded the socalled effcency score. Moreover, by estmatng a semparametrc model of the health producton process usng a twostage approach, we show that neffcency n the health sector s strongly related to varables that are, at least n the short to medum run, beyond the control of governments. These are GDP per capta, the educaton level, and unhealthy lfestyles as obesty and smokng habts. In methodologcal terms, a twostage approach has become ncreasngly popular when DEA s used to assess effcency of decsonmakng unts (DMUs). The most usual twostage approach has been recently crtcsed n statstcal terms. 1 The fact that DEA output scores are lkely to be based, and that the envronmental varables are correlated to output and nput varables, recommend the use of bootstrappng technques, whch are well suted for the type of modellng we apply here. Therefore, we employ both a more usual DEA/Tobt approach and sngle and double bootstrap procedures suggested by Smar and Wlson (2007). Our paper s one of the frst applcatons of ths very recent technque. 2 Our results followng ths procedure are compared to the ones arsng from the more tradtonal one. The paper s organsed as follows. In secton two we provde motvaton and brefly revew some of the lterature and prevous results on health provson effcency. Secton three outlnes the methodologcal approach used n the paper and n secton four we present and dscuss the results of our effcency analyss. Secton fve provdes the conclusons. 1 See Smar and Wlson (2000, 2007). 2 See Afonso and St. Aubyn (2006) for an applcaton to the educaton system. 3
4 2. Motvaton and lterature Health s one of the most mportant servces provded by governments n almost every country. Accordng to OECD (2005), OECD countres expended an average of 8.7 per cent of GDP n 2003 on health nsttutons, of whch 6.3 per cent of GDP were from publc sources. In a general sense, health provson s effcent f ts producers make the best possble use of avalable nputs, and the sole fact that health nputs weght heavly on the publc purse would call for a careful effcency analyss. A health system not beng effcent would mean ether that results (or outputs ) could be ncreased wthout spendng more, or else that expense could actually be reduced wthout affectng the outputs, provded that more effcency s assured. Research results presented here ndcate that there are cases where consderable mprovements can be made n ths respect. The fact of health spendng beng predomnantly publc s partcularly true n OECD countres. Table 1 summarses some relevant data for thrty OECD countres concernng health spendng. For nstance, publc expendture as a share of total spendng averaged 72.5 per cent n 2003, rangng from 44.4 per cent n the USA to 90.1 per cent n the Czech Republc. For the EU15, average total spendng was 8.8 per cent of GDP n 2003, whch s close to the OECD value, slghtly up from the 8.1 per cent rato observed n On the other hand, average publc expendture as a share of total expendture n health was, n 2003, lower n the EU15 than n the OECD, the correspondng ratos beng equal to 69.9 and 72.5 percent, respectvely. Furthermore, data reported n Table 1 show that total per capta health spendng s very dverse across OECD countres. Indeed, the country that spends more on health n per capta terms, the USA, expends more than two tmes the OECD average and eleven tmes more than the country that spends the least, Turkey, even though the per capta GDP rato between those two countres s roughly fve and a half. [Insert Table 1 here] 4
5 Moreover, the relevance of assessng the qualty of publc spendng and redrectng t to more growth enhancng tems s stressed, for nstance, n EC (2004) as beng an mportant goal for governments to pursue. Internatonally, there s a shft n the focus of the analyss from the amount of publc resources used by a government, to servces delvered, and also to acheved outcomes and ther qualty (see OECD, 2003). In our research, we measure and compare health output across countres usng precsely the abovementoned type of qualty measures we resort to crossnatonally comparable evdence on health varables, as reported n OECD (2005). Prevous research on the nternatonal comparatve performance of the publc sector n general and of health outcomes n partcular, ncludng Afonso, Schuknecht and Tanz (2005) for publc expendture n the OECD, and Gupta and Verhoeven (2001) for educaton and health n Afrca, has already suggested that mportant neffcences are at work. These studes use free dsposable hull analyss (FDH) wth nputs measured n monetary terms. Spnks and Hollngsworth (2005) assess health effcency for OECD countres usng DEA based Malmqust ndexes. They report a mean value of for an OECD dataset suggestng that overall, member countres have moved slghtly away from the fronter, mplyng a decrease n techncal effcency, between 1995 and Usng both FDH and DEA analyss, Afonso and St. Aubyn (2005) studed effcency n provdng health and educaton n OECD countres usng physcally measured nputs and concluded that f all countres were effcent, nput usage could be reduced by about 13 per cent wthout affectng output. Usng a more extended sample Evans et al. (2000) evaluate the effcency of health expendture n 191 countres usng a parametrc methodology. In ths paper, we estmate semparametrc models of the health producton process usng a twostage approach. In a frst stage, we determne the output effcency score for each country, usng the mathematcal programmng approach known as DEA, relatng health nputs to outputs. In a second stage, these scores are explaned usng regresson analyss. Here, we show that nondscretonary factors are ndeed hghly correlated to neffcency,.e. they are sgnfcant envronmental varables, usng 5
6 DEA jargon. 3 They are, however, of a fundamentally dfferent nature from nput varables, n so far as ther values cannot be changed n a meanngful spell of tme by the DMU, here a country. 3. Analytcal methodology 3.1. DEA framework DEA, whch assumes the exstence of a convex producton fronter, allows the calculaton of techncal effcency measures that can be ether nput or output orented. The purpose of an outputorented study s to evaluate by how much output quanttes can be proportonally ncreased wthout changng the nput quanttes used. Ths s the perspectve taken n ths paper. Note, however, that one could also try to assess by how much nput quanttes can be reduced wthout varyng the output. Both output and nputorented models wll dentfy the same set of effcent/neffcent producers or DMUs. 4 The descrpton of the lnear programmng problem to be solved, output orented and assumng varable returns to scale hypothess, s sketched below. Suppose there are p nputs and q outputs for n DMUs. For the th DMU, y s the column vector of the outputs and x s the column vector of the nputs. We can also defne X as the (p n) nput matrx and Y as the (q n) output matrx. The DEA model s then specfed wth the followng mathematcal programmng problem, for a gven th DMU: Max s. to δ λ, δ δ y n1' λ = 1 λ 0 Yλ x Xλ. (1) 3 Throughout the paper we use nterchangeably the terms nondscretonary, exogenous and envronmental when qualfyng varables or factors not ntally consdered n the DEA programme. 4 See Farrell (1957) semnal work, popularsed by Charnes, Cooper and Rhodes (1978). Coell, Rao, O Donnell and Battese (2005) and Thanassouls (2001) offer good ntroductons to the DEA methodology. 6
7 In problem (1), δ s a scalar satsfyngδ 1, more specfcally t s the effcency score that measures techncal effcency of the th unt as the dstance to the effcency fronter, the latter beng defned as a lnear combnaton of best practce observatons. Wthδ > 1, the decson unt s nsde the fronter (.e. t s neffcent), whle δ = 1 mples that the decson unt s on the fronter (.e. t s effcent). The vector λ s a (n 1) vector of constants that measures the weghts used to compute the locaton of an neffcent DMU f t were to become effcent Nondscretonary nputs and the DEA/Tobt twosteps procedure The standard DEA models as the one descrbed n (1) ncorporate only dscretonary nputs, those whose quanttes can be changed at the DMU wll, and do not take nto account the presence of envronmental varables or factors, also known as nondscretonary nputs. However, socoeconomc dfferences may play a relevant role n determnng heterogenety across DMUs ether schools, hosptals or countres achevements n an nternatonal comparson and nfluence outcomes. In what health s concerned, these exogenous socoeconomc factors can nclude, for nstance, household wealth, eatng habts and educaton level. As nondscretonary and dscretonary nputs jontly contrbute to each DMU outputs, there are n the lterature several proposals on how to deal wth ths ssue, mplyng usually the use of twostage and even threestage models. 5 Let z be a (1 r) vector of nondscretonary outputs. In a typcal twostage approach, the followng regresson s estmated: ˆ, (2) δ = + zβ ε where δˆ s the effcency score that resulted from stage one,.e. from solvng (1). β s a (r 1) vector of parameters to be estmated n step two assocated wth each 5 See Ruggero (2004) and Smar and Wlson (2007) for an overvew. 7
8 consdered nondscretonary nput. The fact that ˆ δ 1 has led many researchers to estmate (2) usng censored regresson technques (Tobt), although others have used OLS Nondscretonary nputs and bootstrap The twostage DEA/Tobt method s lkely to be based n small samples for two reasons. Frstly, the fact that output scores are jontly estmated by DEA mples that the error term ε n equaton (2) s serally correlated. Secondly, nondscretonary varables z are correlated to the error term ε I. Ths derves from the fact that nondscretonary nputs are correlated to the outputs, and therefore to estmated effcency scores. To surmount ths, Smar and Wlson (2007) propose two alternatves based on bootstrap methods 7. Smlarly to the DEA/Tobt procedure, the effcency score depends lnearly on the envronmental varables, but the error term s a truncated, and not a censored, normal random varable. The frst bootstrap method ( algorthm 1 ) mples the estmaton of the effcency scores usng DEA, as n the DEA/Tobt analyss. However, the nfluence of nondscretonary nputs on effcency s estmated by means of a truncated lnear regresson. Bootstrappng then assesses coeffcent sgnfcance. We have consdered 2000 bootstrap estmates for that effect. The scores derved from DEA are based towards 1 n small samples. Smar and Wlson (2007) second bootstrap procedure, algorthm 2, ncludes a parametrc bootstrap n the frst stage problem, so that bascorrected estmates for the effcency scores are produced. These corrected scores replace the DEA orgnal ones, and estmaton of envronment effects proceeds lke n algorthm 1. 6 See Smar and Wlson (2007) for an extensve lst of publshed examples of the two step approach. 7 See Appendx 1, where the method s exposed n more detal. We mplemented these algorthms n Matlab. Programmes and functons are avalable on request. 8
9 4. Emprcal analyss 4.1. Data and ndcators OECD (2005) s our chosen health database for OECD countres. 8 Typcal nput varables nclude medcal technology ndcators and health employment. Output s to be measured by ndcators such as lfe expectancy and nfant mortalty, n order to assess potental years of added lfe. It s of course dffcult to measure somethng as complex as the health status of a populaton. We have not nnovated here, and took two usual measures of health attanment, nfant mortalty and lfe expectancy. 9 Effcency measurement technques used n ths paper mply that outputs are measured n such a way that more s better. Ths s clearly not the case wth nfant mortalty. Recall that the Infant Mortalty Rate (IMR) s equal to: (Number of chldren who ded before 12 months)/(number of born chldren) We have calculated an Infant Survval Rate, ISR, IMR ISR = 1000, (3) IMR whch has two nce propertes: t s drectly nterpretable as the rato of chldren that survved the frst year to the number of chldren that ded; and, of course, t ncreases wth a better health status. We have consdered a thrd output measure, whch we call Potental Years of Lfe Not Lost, PYLNL. Ths varable was computed on the bass of the ndcator Potental 8 The data and the sources used n the paper are presented n the Annex. 9 These health measures, or smlar ones, have been used n other studes on health and publc expendture effcency see Afonso, Schuknecht and Tanz (2004), and Gupta and Verhoeven (2001). 9
10 Years of Lfe Lost, PYLL, reported by OECD (2005). Ths last varable, PYLL, equals the number of lfe years lost due to all causes before the age of 70 and that could be, a pror, prevented. Therefore, and for our subsequent DEA analyss, and smlarly to the Infant Mortalty Rate, a transformaton had to be done, n order to provde an ncreasng monotonc relaton between the varable, number of years not lost, and health status. Our transformed varable s: PYNLL = λpyll, (4) where λ= s an estmate of the number of potental years of lfe for a populaton under 70 years. 10 Therefore, our fronter model for health s based upon three output varables:  the nfant survval rate,  lfe expectancy,  and potental years of lfe not lost. We compare physcally measured nputs to outcomes. Quanttatve nputs are the number of practsng physcans, practsng nurses, acute care beds per thousand habtants and hghtech dagnostc medcal equpment, specfcally magnetc resonance magers (MRI). 11 Table 2 reports the relevant statstcs for the set of OECD countres. [Insert Table 2 here] From Table 2 one notces that practsng nurses per one thousand persons, n the perod , ranged from 1.6 n Korea to 14.7 n Ireland. For the same perod there was also a hgh range of practsng physcans per one thousand persons, from n Turkey and n Korea to n Italy and n Greece. Addtonally, the 10 See detals n Appendx A commonly used ndcator of medcal technology; see, for nstance, RetzlaffRoberts et al. (2004). 10
11 number of MRI per mllon persons ranged from 0.2 n Mexco to 32.2 n Japan, and the hosptal acute care beds per one thousand persons ranged from 1.0 n Mexco to 9.1 n Japan. Table 2 also shows that for the perod lfe expectancy at brth ranged form 68.4 years n Turkey to 81.5 n Japan, and nfant mortalty ranged form 2.4 n Iceland to 36.3 n Turkey. In addton, the potental years of lfe not lost per populaton was 73 per cent above the average n Hungary and 29 per cent below average n Japan Prncpal component analyss In order to go around the eventual dffcultes posed to the DEA approach when there are a sgnfcant number of nputs and/or outputs, we used prncpal component analyss (PCA) to aggregate some of the ndcators. The use of PCA reduces the dmensonalty of multvarate data, whch s what we have regardng health status, and the health care resources used. The dea of PCA s to descrbe the varaton of a multvarate data set through lnear combnatons of the orgnal varables (see, for nstance, Evertt and Dunn, 2001). Generally, we are nterested n seeng f the frst few components portray most of the varaton of the orgnal data set, for nstance, 80 per cent or 90 per cent, wthout much loss of nformaton. In a nutshell, the prncpal components are uncorrelated lnear combnatons of the orgnal varables, whch are then ranked by ther varances n descendng order. Ths provdes a more parsmonous representaton of the data set and avods that n the DEA computatons too many DMUs are labelled effcent by default. Usually one apples PCA by mposng that the orgnal varables are normalzed to have zero mean. Ths means that the computed prncpal components scores also have zero mean, and therefore some of the results from PCA are negatve. Snce DEA nputs and outputs need to be strctly postve, PCA results wll be ncreased by the most negatve value n absolute value plus one, n order to ensure strctly postve data (see, for nstance, Adler and Golany, 2001). 11
12 We appled PCA to the four nput varables, doctors, nurses, beds and MRI unts. The results of such analyss (see Table 3) led us to use the frst three prncpal components as the three nput measures, whch explan around 88 per cent of the varaton of the four varables. Ths also mples that we only take nto account the components whose assocated egenvalues are above 0.7, a rule suggested by Jollfe (1972). Applyng PCA also to the set of our selected output varables, lfe expectancy, nfant survval rate and potental number of years of lfe not lost, we selected the frst prncpal component as the output measure snce t accounts for around 84 per cent of the varaton of the three varables (see Table 3). [Insert Table 3 here] We report n Table 4 the abovementoned prncpal components, to be used n the subsequent secton n DEA computatons. [Insert Table 4 here] 4.3. DEA effcency results In Table 5 we report results for the standard DEA varablereturnstoscale techncal effcency output scores and peers of each of the consdered countres. The specfcaton used ncludes as nputs the frst three components of the PCA performed to the base varables doctors, nurses, beds and MRI unts. As output we use the frst component of the PCA appled to the base varables nfant survval rate, lfe expectancy, and potental years of lfe not lost, as explaned n the prevous secton. [Insert Table 5 here] It s possble to observe n Table 5 that seven countres would be located on the theoretcal producton possblty fronter wth the standard DEA approach: Canada, 12
13 Fnland, Japan, Korea, Span, Sweden and the USA 12. Canada, Fnland, Japan, Span and Sweden are located n the effcent fronter because they perform qute well n the output ndcator, gettng above average results. On the other hand, Korea and the USA are generally below average regardng the use of resources n all the frst three components selected. Another set of three countres s located on the opposte end Hungary, the Slovak Republc and Poland. DEA analyss ndcates that ther output could be substantally ncreased f they were to become located on the effcency fronter. On average and as a conservatve estmate, countres could have ncreased ther results by 40 per cent usng the same resources Explanng neffcency the role of nondscretonary nputs Usng the DEA effcency scores computed n the prevous subsecton, we now evaluate the mportance of nondscretonary nputs. We present results both from Tobt regressons and bootstrap algorthms. Even f Tobt results are possbly based, t s not clear that bootstrap estmates are necessarly more relable. In fact, the latter are based on a set of assumptons concernng the data generaton process and the perturbaton term dstrbuton that may be dsputed. Takng the pros and cons of both methods nto account, t seems sensble to apply both of them. If outcomes are comparable, ths adds robustness and confdence to the results we are nterested n. In order to explan the effcency scores, we regress them on GDP per capta, Y, educatonal level, E, obesty, O, and tobacco consumpon, T, as follows 13 δ = β + β Y + β E + β O + β T + ε. (5) ˆ One can brefly compare our results wth the ones reported by Afonso and St. Aubyn (2005) that addressed health effcency for 2000 usng a smlar set of nformaton but wthout prncpal component analyss. Interestngly, they reported that countres labelled as effcent were: Canada, Denmark, France, Japan, Korea, Norway, Portugal, Span, Sweden, the Unted Kngdom and the Unted States, rather along the lnes of our results. 13 Educatonal level s gven by the percentage of populaton that acheved tertary educaton n , GDP per capta refers to PPP USD n 2003, obesty refers to the percentage of obese populaton n 2002, and smokng refers to the percentage of populaton that consumed tobacco n 2003 (see the Annex for detals). 13
14 We frst report n Table 6 results from the censored normal Tobt regressons for several alternatve specfcatons of equaton (5). [Insert Table 6 here] Ineffcency n the health sector s strongly related to the four varables that are, at least n the short to medum run, beyond the control of governments: the economc background, proxed here by the country GDP per capta, the level of educaton, smokng habts, and obesty. The estmated coeffcents of the frst two nondscretonary nputs are statstcally sgnfcant and negatvely related to the effcency measure. For nstance, an ncrease n educaton achevement reduces neffcency, mplyng that the relevant DMU moves closer to the theoretcal producton possblty fronter. Therefore, the better the level of educaton, the hgher the effcency of health provson n a gven country. The same reasonng apples to GDP, wth hgher GDP per capta resultng n more effcency. On the other hand, effcency s lower the stronger smokng habts are and the hgher the percentage of obese populaton s. We also consdered other varables as nondscretonary nputs: ncome nequalty va the Gn coeffcent, the rato of publctototal expendture n health, spendng on pharmaceutcals as a percentage of health expendture, percentage of populaton over 65 years, per capta alcohol and sugar consumpton, and total calores ntake. However, none of these varables prove to be statstcally sgnfcant and the estmaton results are not reported for the sake of space. Table 7 reports the estmaton results from the bootstrap procedures employng algorthms 1 and 2, as descrbed n subsecton 3.3. Estmated coeffcents are essentally smlar rrespectve of the algorthm used to estmate them. Moreover, they are also close to the estmates derved from the more usual Tobt procedure, and, very mportantly, they are hghly sgnfcant. [Insert Table 7 here] Sgnfcance across dfferent model formulatons and estmaton methods s mportant and consttutes robust emprcal evdence that effcency n health depends drectly on 14
15 a country s wealth and on educaton levels, and nversely on tobacco consumpton and obesty. In a nutshell, populaton of poorer countres where educaton levels are low tend to under perform, so that results are further away from the effcency fronter. The same reasonng apples to the other two envronmental factors, wth hgher smokng habts and obesty levels drawng countres away from health related effcent performance. Equaton (5) can be regarded as a decomposton of the output effcency score nto two dstnct parts: the one that s the result of a country s envronment, and gven by β + β Y + β E + β O + β T ; the one that ncludes all other factors havng an nfluence on effcency, ncludng therefore neffcences assocated wth the health system tself, and gven byε t. We choose models 2 and 4 from Table 7 for our exercse of correctng for envronmental varables n order to use versons wth and wthout educaton as an exogenous factor 14. The frst column n Table 8 ncludes the bas corrected scores for Model 2, the one wth the best ft usng bootstrap algorthms (as can be seen by the lower estmated standard devaton of ε). Algorthm 2 mples a bas correcton after estmatng output effcency scores, takng nto account the correlaton between these scores and the envronmental varables. We also present score correctons for the three envronmental varables. GDP, obesty, and tobacco consumpton correctons were computed as the changes n scores by artfcally consderng that Y, O, and T vared to the sample average n each country. Fully corrected scores, presented n column fve, are estmates of output scores purged from envronmental effects and result from the summaton of the prevous four columns, truncated to one when necessary. [Insert Table 8 here] 14 Models 2 and 4 dffer from models 1 and 3 because ncome s ntroduced n logs. Ths formulaton seems to provde a better ft, as checked by comparatve values of σˆ. ε 15
16 Comparng the ranks n the last column of Table 8, resultng from correctons for both bas and envronmental varables, wth the prevously presented rankng from the standard DEA analyss (see Table 5 above), t s apparent that sgnfcant changes occurred. For the purpose of such comparson one should notce that the number of countres consdered dropped from twentyone n the DEA calculatons to nneteen n the twostep analyss, snce tobacco consumpton data was not avalable for Austra and Portugal. Some countres poorly ranked prevously are now closer to the producton possblty fronter ths s the case of Denmark, the Czech Republc, Hungary, Luxembourg, Poland, the Slovak Republc, and the UK. On the other hand, other countres see a worsenng n ther relatve poston after takng nto account envronmental varables, namely Canada, Sweden, and the US, and to a less extent, Japan. At last, countres lke Korea and Span keep ther good postonng. Addtonally, by lookng at GDP, obesty and tobacco consumpton correctons n Table 8, t s apparent that n some countres, envronmental harshness essentally results from low GDP per head, as n the Czech Republc, Korea, Poland and Span. For nstance, for the US, lower than average tobacco consumpton s offset by above average obesty, whle for Japan, Korea, Luxembourg, and Swtzerland we see an opposte pattern. Fnally, note that n countres lke Germany and Italy, all three envronmental varables push down performance, whle an nverse result can be observed for Hungary. Alternatvely, a smlar analyss can be conducted for Model 4, where we now have four envronmental varables: GDP, educaton, obesty, and tobacco consumpton (see Table 9). [Insert Table 9 here] From the results n Table 9 t s possble to conclude that educaton correcton s not benefcal for countres such as Canada, the US, Japan or Korea. Indeed, and as results from both Tobt and bootstrap analyss ndcate, the percentage of populaton wth 16
17 tertary educaton s a relevant exogenous varable n explanng health effcency scores. On the other hand, the below average results n ths varable for several other countres, such as the Czech Republc, Italy and Luxembourg, allow for an mprovement n ther effcency rankngs after makng the correctons related to all four nondscretonary factors used n Model Concluson In ths paper, we have evaluated effcency n health servces across countres by assessng outputs (lfe expectancy, nfant survval rate, potental years of lfe not lost) aganst nputs drectly used n the heath system (doctors, nurses, beds, MRI unts) and envronment varables (wealth and country educaton level, smokng habts and obesty). In methodologcal terms, we have employed a twostage semparametrc procedure. Frstly, output effcency scores were estmated by solvng a standard DEA problem wth countres as DMUs. Secondly, these scores were explaned n a regresson wth the envronmental varables as ndependent varables. Results from the frststage mply that neffcences may be qute hgh. On average and as a conservatve estmate, countres could have ncreased ther results by 40 per cent usng the same resources. Countres lke Hungary, the Slovak Republc and Poland dsplay sgnfcant room for mprovement. The fact that a country s seen as far away from the effcency fronter s not necessarly a result of neffcences engendered wthn the health system. Our second stage procedures shows that GDP per head, educatonal attanment, tobacco consumpton, and obesty are hghly and sgnfcantly correlated to output scores a wealther and more cultvated envronment are mportant condtons for a better health performance, whle a more obese populaton and prevalence of smokng habts worsen health performance. Moreover, t becomes possble to correct output scores by consderng the harshness of the envronment where the health system operates. Country rankngs and output scores derved from ths correcton can be substantally dfferent from standard DEA results. 17
18 Nondscretonary outputs consdered here cannot be changed n the short run. For example, educatonal attanment s essentally gven n the comng year. However, contemporaneous educatonal and socal polcy wll have an mpact on future educatonal attanment. A smlar reasonng apples to smokng habts, whch are dffcult to change, but where, for nstance, tax measures are usually consdered and mplemented by the governments. Obesty problems also mpnge negatvely on the performance of the health system, and may be related to cultural tradtons. Fnally, note that we have appled both the usual DEA/Tobt procedure and two very recently proposed bootstrap algorthms. Results were strkngly smlar wth these three dfferent estmaton processes, whch brng ncreased confdence to obtaned conclusons. 18
19 Appendx 1 Sngle and Double Bootsrap Procedures Ths appendx brefly descrbes the sngle and double procedure proposed by Smar and Wlson (2007) and appled n ths paper. By assumpton, the true effcency score depends on the envronmental varables z, so that δ = zβ + ε 1, (A1.1) where β s a vector of parameters. ε s a truncated normal random varable, 2 dstrbuted N (0, ) wth lefttruncaton at 1 ψ ( z, β ) σ ε 15. The effcency score that solves problem (1) n the man text (the DEA problem), δˆ, s then consdered as an estmate for δ, and ths s the frst stage n the procedure. The second stage s desgned to assess the nfluence of nondscretonary nputs on effcency. The frst algorthm nvolves the followng steps: [1] The computaton of δˆ for all n decson unts by solvng (1). [2] The estmaton of equaton (A1.1) by maxmum lkelhood, consderng t s a truncated regresson (and not a censored or Tobt regresson). Denote by βˆ and the maxmum lkelhood estmates of β and σ ε. [3] The computaton of L bootstrap estmates for β and σ ε, n the followng way: For = 1,..., n draw ε from a normal dstrbuton wth varance σˆ ε 2 ˆ σ ε and left truncaton at * 1 zβˆ and compute δ = z ˆ β + ε. Then estmate the truncated 15 In a truncated normal dstrbuton, ε s not observed when t would fall below 1 β z. In a censored model (the Tobt model), ε s always observed, even f there s some nformaton loss (t s exactly equal to 1 β z when t would fall below ths value). 19
20 * regresson of δ on z by maxmum lkelhood, yeldng a bootstrap estmate * * ( ˆ β, ˆ ). σ ε Wth a large number of bootstrap estmates (e.g. L=2000), t becomes possble to test hypotheses and to construct confdence ntervals for β and σ ε. For example, suppose that we want to determne the pvalue for a gven estmate ˆ1 β < 0. Ths wll be gven by the relatve frequency of nonnegatve * 1 ˆβ bootstrap estmates. It can be shown that the estmate δˆ s based towards 1 n small samples. Smar and Wlson (2007) second bootstrap procedure, algorthm 2, ncludes a parametrc bootstrap n the frst stage problem, so that bascorrected estmates for the effcency scores are produced. The producton of these bascorrected scores s done as follows: [1] Compute δˆ for all n decson unts by solvng problem (1); [2] Estmate equaton (A1) by maxmum lkelhood, consderng t s a truncated regresson. Let βˆ and σˆ ε be the maxmum lkelhood estmates of β and σ ε. [3] Obtan L 1 bootstrap estmates for each δ, the followng way: For = 1,..., n draw ε from a normal dstrbuton wth varance 2 ˆ σ ε and left truncaton at ˆ * 1 zβˆ and compute δ = z ˆ * δ β + ε. Let y = y *, be a δ modfed output measure. Compute * * * man text, where Y s replaced by [ y y ] ˆ* δ by solvng the DEA problem (1) n the Y = 1... n. (But note that y s not replaced by y * n the lefthand sde of the frst restrcton of the problem.) [4] Compute the bascorrected output neffcency estmator as ˆ ˆ ˆ * δ = 2. δ δ, where ˆ* δ s the bootstrap average of ˆ* δ. Once these frst stage bascorrected measures are produced, algorthm 2 contnues by replacng δˆ wth Wlson (2007), we set L 1 =100. δˆ n algorthm 1, from step 2 onwards. Followng Smar and 20
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