Distinguishing between heterogeneity and inefficiency: stochastic frontier analysis of the World...

HEALTH ECONOMICS

Health Econ. 13: 959–980 (2004)

Published online 20 September 2004 in Wiley InterScience (www.interscience.wiley.com). DOI:10.1002/hec.938

ECONOMETRICS AND HEALTH ECONOMICS

Distinguishing between heterogeneity and ine⁄ciency:stochastic frontier analysis of theWorld Health Organization’spanel data on national health care systemsWilliam Greene*Department of Economics, Stern School of Business, New York University, New York, USA

Summary

The most commonly used approaches to parametric (stochastic frontier) analysis of efficiency in panel data, notablythe fixed and random effects models, fail to distinguish between cross individual heterogeneity and inefficiency. Thisblending of effects is particularly problematic in the World Health Organization’s (WHO) panel data set on healthcare delivery, which is a 191 country, 5-year panel. The wide variation in cultural and economic characteristics of theworldwide sample produces a large amount of unmeasured heterogeneity in the data. This study examines severalalternative approaches to stochastic frontier analysis with panel data, and applies some of them to the WHO data. Amore general, flexible model and several measured indicators of cross country heterogeneity are added to the analysisdone by previous researchers. Results suggest that there is considerable heterogeneity that has masqueraded asinefficiency in other studies using the same data. Copyright # 2004 John Wiley & Sons, Ltd.

JEL classification: C1; C4

Keywords panel data; fixed effects; random effects; random parameters; technical efficiency; stochastic frontier;heterogeneity; health care

Introduction

The World Health Report 2000 (WHR) [1] is aworldwide assessment of the effectiveness of healthcare delivery. The study presents a rankings basedcomparison of the productive efficiency of thehealth care systems of 191 countries. Predictably,the attention focused on these rankings has beenconsiderably out of proportion to the space thissection occupies in the larger report.

The rankings were produced using a form of the‘fixed effects’, stochastic frontier methodologyproposed by Schmidt and Sickles [2] and Cornwell

et al. [3] (see [4,5]) (ETML). The data analyzed inthis econometric study were a 5-year (1993–1997)panel. This section of the WHR has been heavilycriticized for numerous reasons related to theoverall objectives, the quality and validity of theeffectiveness measures, the input data used, andthe appropriateness of the methodology (see, e.g.Gravelle et al. [6,7] (GJJS), Williams [8] andHollingsworth and Wildman [9] (HW)). OnJanuary 8, 2001, the authors of the WHO studyconvened a panel of researchers specifically todiscuss the econometric methodology.a The focusof the meeting was the use of panel data, such asthose in the WHR study, for measurement of

Copyright # 2004 John Wiley & Sons, Ltd.Received 30 September 2003

Accepted 1 June 2004

*Correspondence to: Department of Economics, Stern School of Business, New York University, 44 West 4th St., New York,NY 10012, USA. E-mail: [email protected]. URL: www.stern.nyu.edu/�wgreene

efficiency in health care delivery. This paperreports subsequent research undertaken by theauthor to study some of the issues raised at thatmeeting.

One criticism of the fixed effects methodologyused is that the model fails to distinguish betweencross country heterogeneity unrelated to ineffi-ciency and the inefficiency itself. This ambiguity islikely to be especially problematic in these data, asthey are based on 191 sometimes vastly differentcountries; France, England and Australia appearin the sample on equal terms with Oman, SriLanka, Zimbabwe, the Seychelles, Colombia andBangladesh. We undertook to examine this issue,to reanalyze the WHO data and the methods usedin the study, and to propose alternative stochasticfrontier-based methods with greater flexibility thatwill allow the analyst to segregate individual,unmeasured heterogeneity and technical or costinefficiency.

The paper is organized as follows: The followingsection reviews the WHO methodology and takesa cursory look at their results. Next, the stochasticfrontier model and strategies for modeling paneldata are reviewed.b The WHO data that were usedin this study as well are then described. The severalstudies of the WHO data [4,5], HW [9], GJJS [6,7]that we examined were based on two outputmeasures, a composite measure of health caredelivery (COMP) and disability adjusted lifeexpectancy (DALE) and two inputs, health careexpenditure and education levels. In this study, weconsider how to use additional country specificcovariates including per capita income, incomedistribution, government effectiveness, and theallocation of health care expenditure between thepublic and private sectors to account for some ofthe heterogeneity noted earlier. The empiricalresults are then presented. We begin with anexamination of the production function used, andpropose some results that are in broad agreementwith others already in the literature regarding theimpact of income and the distribution of incomeon health care outcomes. The second set of resultswill compare fixed and random effects estimates oftechnical inefficiency. We find that concerns ofETML [4,5] notwithstanding, for these data, aform of the random effects model appears to be auseful framework for analyzing the WHO data.We then incorporate the country specific hetero-geneity in the estimated distribution of technicalinefficiency and in the production function itself.Lastly, some conclusions are drawn.

We note at the outset, this paper is intended todocument some methodological developments inthe estimation of technical inefficiency and topresent some specific empirical results to becompared to the WHO study. It is not obviousor certain that this production function approachis the best way to assess the effectiveness of thedelivery of health care (see [8] for an example).Moreover, even if so, it is unclear that the actualoutcomes measured do indeed result from a‘production function’ process as described instandard microeconomics, in which health care isthe output, spending and education are the inputs,and a process of optimization underlies theobserved data. Our main purpose in this study isindicated in the title. The WHO study provides aninteresting and well-positioned application inwhich to examine the econometric issues wepresent. Although we will be using the ETMLmodel throughout, we take no position on whetherthis is the optimal model for this purpose. To alarge extent, the model used is for illustrativepurposes. Future research should provide moreevidence on how best to approach the WHO’sobjective of assessing the performance of nationalhealth systems.

TheWHO studies of health careattainment

Evaluation of the effectiveness of health carepolicies and reforms faces two large obstacles,quantifying goals and objectives so that outcomescan be measured and enumerating inputs in a waythat resources can be directed toward them so as toachieve those objectives. The effectiveness study inthe WHR is an attempt to measure health careeffectiveness in a production function framework.

ETML’s [4,5] preferred methodology was a‘panel data’, production function estimator basedon the framework proposed by Schmidt andSickles [2] and Cornwell et al. [3]. The centralfeature of the estimator is a fixed effects linearregression model. The production function isdenoted as

yit ¼ aþ x0

itbþ vit � ui ð1Þ

where yit is the (log of the) output of the system, xitis (logs of) the set of inputs, vit is the randomcomponent representing stochastic elements as

Copyright # 2004 John Wiley & Sons, Ltd. Health Econ. 13: 959–980 (2004)

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well as any country and time specific heterogene-ity, ui is the inefficiency in the system, and i and tdenote country and year, respectively. It isassumed that ui>0. The equation is rewritten as

yit ¼ ða� uiÞ þ x0

itbþ vit

¼ ai þ x0

itbþ vit ð2Þ

Assuming that vit has the familiar stochasticproperties of a regression disturbance and isuncorrelated with other components of the model,the parameters can be estimated by least squares,using the ‘within’, or dummy variable estimator.The country specific constants embody the techni-cal inefficiency. The inefficiencies are estimated inturn by shifting the function upward so that eachconstant term is measured as a deviation from thebenchmark level

#uui ¼ maxið#aaiÞ � #aai � 0 ð3Þ

(Note that by this construction, at least onecountry is measured as 100% efficient.) Technicalefficiency is now measured by

TEit ¼E½yit j xit; ui�

E½yit j xit; ui ¼ 0�ð4Þ

Overall efficiency is constructed in ETML [4,5] bynormalizing this measure to a constructed mini-

mum level of output that would roughly corre-spond to a system with zero inputs.

Empirical analysis in the two studies used asinputs per capita public and private health careexpenditure (HEXP) and average years of educa-tion of the population (EDUC). Two measures ofhealth care attainment were analyzed, disabilityadjusted life expectancy (DALE) and a compositemeasure of health care delivery (COMP). Theproduction function employed in both cases was

yit ¼ ai þ b1 log EXPit þ b2 log EDUCit

þ b3 log2 EDUCit þ vit ð5Þ

where yit is the log of DALE in ETML [4] and thelog of COMP in [5]. Table 1 shows some of theestimated results from the two studies. The specificnumeric values for DALE, in years, have a readyinterpretation, but those for COMP have no clearnumeraire. In principle, each TEi gives thepercentage of maximal output above the minimumthat is attained in the country. Based on thenumeric values of the attainment measures, thesewould give the proportional potential for improve-ment.

These data have been reanalyzed by severalauthors. A variety of strident criticisms were raisedin William [8], who questioned the methodologyand objectives of the entire study as well as the

Table 1. Selected WHO estimates of overall efficiency

DALEa COMPb

Rank Country TEc Country TEc

1 Oman 0.992 France 0.9942 Malta 0.989 Italy 0.9913 Italy 0.976 San Marino 0.9884 France 0.974 Andorra 0.9825 San Marino 0.971 Malta 0.97825 Costa Rica 0.882 Germany 0.90250 Uruguay 0.819 Poland 0.793100 Jordan 0.711 St. Kitts and Nevis 0.643150 Afghanistan 0.517 Nepal 0.457187 Malawi 0.196 Nigeria 0.176188 Botswana 0.183 Dem. Rep. of Congo 0.171189 Namibia 0.183 Central African Rep. 0.156190 Zambia 0.112 Myanmar 0.138191 Zimbabwe 0.080 Sierra Leone 0.000

United Kingdom (24) 0.883 United Kingdom (18) 0.925United States (72) 0.774 United States (37) 0.838

aFrom Evans et al. [4, Appendix].bFrom Evans et al. [5, Annex, Table 1].cNormalized to a minimum value at which health ‘inputs’ are approximately zero.

Stochastic Frontier Analysis of WHO Panel Data 961


quality of the data set and the appropriateness ofthe outcome measures. Several econometric stu-dies have placed the first (DALE) study undernarrower scrutiny. The second (COMP) has, untilnow, not been similarly examined.

Gravelle et al. (GJJS) [6,7] observed that in thesample of 191 countries, 99.8% of the variation inthe log of DALE is between, rather than within thegroups (countries). The counterparts for the logs ofhealth expenditure (HEXP) and education (EDUC)are 98.9 and 99.8%, respectively. Thus, there isvery little actual ‘panel data’ variation in these data– it differs only slightly from a cross section. GJJSproceeded to fit several models based on the‘between’ estimators (country means) and com-puted alternative adjusted measures of efficiency.They found varying degrees of correlation betweentheir rankings and those in ETML [4,5], rangingfrom 0.97 down to about 0.39 (see their Table 2).This suggests, as we find below, that the specifica-tion can make a considerable amount of differencein the results (GJJS also argued for inclusion oftime effects and other expenditure (income minushealth expense) in the model. We will return to themodel specification in the section on technicalefficiency estimates from the WHO data).

The stochastic frontier model

The authors of the WHO analyses questioned thedistributional assumptions in the stochastic fron-

tier formulation. However, both GJJS and HWfound considerable similarity in the results across anumber of different formulations that they speci-fied, though theirs sometimes differed markedlyfrom ETML’s results. This suggests that specificassumptions about the distribution of efficienciesmay be less restrictive than these views suggest. Inaddition, we submit that the formulations exam-ined previously were narrower than they couldhave been, and the stochastic frontier modelallows the incorporation of cross country hetero-geneity in several ways that are likely to bring largebenefits in analyzing data as disparate as these.

Pooled data or cross section variants

The essential form of the stochastic productionfrontier model (see [10,11]) is

yi ¼ aþ x0

ibþ vi � ui ð6aÞ

vi � N½0; s2v � ð6bÞ

ui ¼ jUij ð6cÞ

Ui � N½0;s2u� ð6dÞ

A central parameter in the model is the asymmetryparameter, l ¼ su=sv; the larger is l, the greater isthe inefficiency component in the model. Para-meters are estimated by maximum likelihood,rather than least squares. Estimation of ui is thecentral focus of the analysis. With the model

Table 2. Descriptive statistics for variables, 1997 observations

Non-OECD OECD All

Mean Std dev. Mean Std dev. Mean Std dev.

DALE 54.32 11.73 70.27 3.01 56.83 12.29COMP 70.30 10.96 89.42 3.97 73.30 12.34HEXP 249.17 315.11 1498.27 762.01 445.37 616.36EDUC 5.44 2.38 9.04 1.53 6.00 2.62GINI 0.399 0.0777 0.299 0.0636 0.383 0.0836VOICE �0.195 0.794 1.259 0.534 0.0331 0.926GEFF �0.312 0.643 1.166 0.625 �0.0799 0.835TROPICS 0.596 0.492 0.0333 0.183 0.508 0.501POPDEN 757.9 2816.3 454.56 1006.7 710.2 2616.5PUBFIN 56.89 21.14 72.89 14.10 59.40 20.99GDPC 4449.8 4717.7 18199.07 6978.0 6609.4 7614.8

Sample 161 30 191

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estimated in logarithms, ui would correspond (to asmall degree of approximation) to 1�TEi givenearlier. Individual specific efficiency is typicallyestimated with exp(� #uui). Alternatively, #uui, itself,provides an estimate of proportional inefficiency.With parameter estimates in hand, one can onlyobtain a direct estimate of ei ¼ yi � a� x0ib ¼vi � ui. This is translated into an estimate of uiusing Jondrow et al.’s (JLMS) [12] formula,

E½ui j ei� ¼sl

1þ l2zi þ

fðziÞFðziÞ

� �; zi ¼ �eil=s ð7Þ

where s ¼ ðs2u þ s2vÞ1=2 and fðzÞ and FðzÞ are the

density and CDF of the standard normal distribu-tion, respectively.

The narrow assumption of half normality is aviewed as a significant drawback in this model.c

HW and others (see [13]) have extended it to atruncated normal model by allowing the mean ofUi in (6d) to be nonzero. A major shortcoming ofthe truncated normal model is that the specifica-tion of the distribution of ui still suppressesindividual heterogeneity in inefficiency that isallowed, at least in principle, by the fixed effectsformulation. We have several observed indicatorsof this heterogeneity, such as income distribution,per capita GDP, OECD membership, the publicshare of health care expenditures, etc., and thesecan be incorporated into the distribution of ui inways that the other methods already discussedcannot accommodate. Letting hi denote a set ofvariables that measure the group heterogeneity, wewrite

E½Ui� ¼ mi ¼ d0 þ h0

id ð8Þ

The Jondrow et al.’s result in (7) is now changedby replacing zi with z�i ¼ zi � mi=ðslÞ.

Panel data formulations

The fixed effects model. The Schmidt and Sickles[2] formulation

yit ¼ ða� uiÞ þ x0

itbþ vit

¼ ai þ x0

itbþ vit ð9aÞ

ui ¼ maxiðaiÞ � ai � 0 ð9bÞ

has been used in a number of applications (see[14]). There are two important assumptions built

into this model. First, any time invariant hetero-geneity will be pushed into ai and ultimately intothe estimate, #uui. The WHO data span a tremen-dous range of cultures, economies, and policysettings. This is likely to be a particularlyinfluential aspect of the model for these data.Second, the model assumes that inefficiency is timeinvariant. For short time intervals, this may be areasonable assumption. But, five years may belong enough for this to be questionable. HW didfind evidence to suggest that this assumption maybe inconsistent with the data.

Both of these restrictions can be relaxed byplacing country specific constant terms in thestochastic frontier model – we call this a ‘true’fixed effects model,

yit ¼ ai þ x0

itbþ vit � uit ð10Þ

where uit has the specifications in (6), (8) for thestochastic frontier model. Superficially, thisamounts simply to adding a full set of countrydummy variables to the stochastic frontier model.The model is still fit by maximum likelihood, notleast squares. This approach is generally avoided,both for the practical difficulty of computing thelarge number of parameters (the dummy variablescannot be conditioned out of the model) andbecause of a presumption that the incidentalparameters problem would appear – a persistentbias in estimates of the main parameters in ‘true’fixed effects models such as this. Surprisingly, thisapproach has hardly been used previouslyd in spiteof the fact that most of the received panel dataapplications involved fairly small panels, and thereexists almost no actual evidence on the incidentalparameters problem in stochastic frontier estima-tion.e

This true fixed effects model places the unmea-sured heterogeneity in the production function;with a loglinear model, it produces a neutral shiftof the function, specific to each country. Onemight, instead, have the heterogeneity reside in theinefficiency distribution. This could be accom-plished with the modification of (6d) and (10),

mi ¼ d0i þ h0

id ð11Þ

that is, by placing the country specific dummyvariables in the mean of the truncated normaldistribution, rather than in the production func-tion. Once again, in a moderate sized sample, thisis a minor reformulation of the familiar model.The problems noted in the next paragraph will



appear, but how serious these are is an empiricalissue, not a foregone conclusion.

The true fixed effects model has the virtuesmentioned above. Weighing against it are, first,the incidental parameters problem and second, thepossibility that the model is now overspecified. Theincidental parameters problem [20] is a persistentbias that arises in nonlinear fixed effects modelswhen the number of periods is small (five is small).It has been widely documented for binary choicemodels (see, e.g., [21,22]) but not systematicallyexamined in stochastic frontier models. In Greene[Reconsidering heterogeneity in panel data esti-mators of the stochastic frontier model. J Econ2004, forthcoming], we found that the biases incoefficient estimates were surprisingly small anddid not appear in the patterns predicted byreceived results for other models, and, moreover,that there appeared to be only minor biasestransmitted to the estimates of technical ineffi-ciency. The second problem now is that the modelmay be overspecified. If there is persistent ineffi-ciency, it is now completely absorbed in thecountry specific constant term which is alsocapturing any time invariant heterogeneity.Whereas the earlier fixed effects form would tendto overestimate the inefficiency component, it ispossible that this form will underestimate it.Unfortunately, this blending of the two effects isinherent in the modeling problem, and there is nosimple solution that will be entirely satisfactory.Ultimately, ai þ vit � uit contains both countryspecific heterogeneity and inefficiency, and bothmay have invariant and time varying elements.The intent of this paper is to suggest thataccommodating the possibility of time invariantheterogeneity, to the extent possible, is preferableto ignoring it altogether, as in model (9).

The random effects model. The random effectsmodel is obtained by assuming that ui is timeinvariant and uncorrelated with the includedvariables in the model

yit ¼ aþ x0

itbþ vit � ui ð12Þ

In the linear regression case analyzed by ETML,the parameters are estimated by two step general-ized least squares (see [23, Chapter 11]). On thebasis of a Hausman specification test, ETML [4,5]concluded that the random effects model wouldnot be appropriate for their data. Their test wasconducted in a model that did not include any of

the observed country specific effects, so the resultmay have been more convincing than appropriate.A significant problem of interpretation wouldemerge in that test, as the real interest is inwhether the inefficiency term could be treated as arandom or fixed effect, but the Hausman test couldreject the former because of the presence ofheterogeneity correlated with the regressors, butnot necessarily related to inefficiency in the modelas such. This complication would be a specialfeature of the stochastic frontier model. However,even without this difficulty, the linear regressionbased random effects model would have a sig-nificant drawback for present purposes even if itwere not rejected; there is no implied estimator ofinefficiency in this model, that is, no estimator ofTEi as in the fixed effects case.f

Pitt and Lee [24] showed how the time invariantcomposed error model could be extended to apanel data version of the stochastic frontier model.The direct extension would be of limited usefulnesshere, first because of the assumption of uncorre-latedness of ui and xit and, second, because of theassumption of time invariance of the inefficiency.The first of these can be remedied in the samefashion as suggested earlier. Estimation of therandom effects model with heterogeneity in E[Ui],see (8), is straightforward (see [25]). With thisextension, the JLMS estimator becomes

E½ui j e1; e2; . . . ; eT ; hi� ¼ Zi þ cfðZi=cÞFðZi=cÞ

� �ð13Þ

where Zi ¼ gmi � ð1� gÞ%eei, g ¼ 1=ð1þ Tl2Þ,c2 ¼ gs2u, and %eei ¼ ð1=TÞSieit.

Battese and Coelli (BC) [26,27] have proposed anow widely used modification of the truncationmodel that can also accommodate some systematicvariation in the inefficiency. In their formulation,

uit ¼ ZtjUi j ð14Þ

where Zt ¼ 1þ Z1ðt� TÞ þ Z2ðt� TÞ2 andUi � N½m;s2u�. Various other forms of the functionZt have been proposed, such as Zt ¼ exp½�Zðt�TÞ� (see BC [26,27] for discussion). Kumbhakarand Orea [28] suggest a more general form, Zit ¼expðg0itpÞ where git is a set of observable covariatesand p is another parameter vector to be estima-ted.g This extension subsumes BC’s formulation aswell as many others. Greene [29] added theheterogeneous truncation form in (11) to this aswell. Let gi ¼ ðZi1; Zi2; . . . ; ZiT Þ and ei ¼ ðei1; ei2; . . . ;eiT Þ. Estimates of technical inefficiency based onthis model follow the same form as those in the

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Pitt and Lee model in (13), where Zi is now

Z�i ¼

s2vmi þ s2ug0

ieis2v þ s2ug

0

igið15Þ

and c becomes

c�i ¼

s2us2v

s2v þ s2ug0

igið16Þ

The initial assumptions of homogeneity and timeinvariance are obviated in this model.

A random effects counterpart to the true fixedeffects model in (10) would be a ‘true randomeffects’ stochastic frontier model,

yit ¼ ðaþ wiÞ þ x0

itbþ vit � uit ð17Þ

The time invariant, random constant term embo-dies the cross country heterogeneity in theproduction function (not the mean of inefficiency).The one sided inefficiency component now variesfreely across time and country. This form of themodel overcomes both of the drawbacks notedearlier. Estimation of this model by simulatedmaximum likelihood methods is discussed inGreene [19,30]. Measured heterogeneity (incomedistribution, public contribution to health carefinancing, etc.) could enter this model through twoavenues. Simple cross country heterogeneity mayaffect the location of the frontier, which would bemodeled in the form

wi ¼ z0

ihþ oi ð18Þ

This produces a ‘hierarchical’ or ‘multilevel’model. The heterogeneity may also enter thedistribution of uit which can, as before, have meanmi or, in principle, even mit with time variation inthe covariates. Country specific estimates ofinefficiency are computed using the JLMS [12]formulation. We note that this extension of themodel, in which the heterogeneity enters both theproduction relationship and the mean of theinefficiency, can also be accommodated in boththe Pitt and Lee [24] model and the Battese andCoelli [26,27] model, in both cases, as specifiedearlier.

TheWorld Health OrganizationWHRdata set

The data set used in this analysis were used inETML [4,5].h The full data set is a panel of data

observed for 191 member countries of the WHO.The panel data are observed for 5 years, 1993–1997, though 51 of the 191 countries are observedin only one year.i The data are more fullydescribed in the WHR and in numerous publica-tions that can be obtained from the WHO website.

Two outcome variables were observed,

DALE disability adjusted life expectancyCOMP composite measure of success in 5

health goals, overall health, healthdistribution, responsiveness, responsive-ness in distribution, fairness in finan-cing. This variable was gathered bysurvey. See [1] for details and discussion

Natural logs of both outcome variables were usedin the analysis to follow (all references to these inregression results are based on logs). GJJS [6,7]expressed some skepticism about using logs for theDALE variable. In the interest of comparability,we have maintained the forms used by theresearchers at WHO in this study. The first ofthe outcome variables is the familiar outputmeasure that was analyzed by HW, GJJS andWilliams in addition to ETML [4,5]. The secondvariable was analyzed in [5], but was not analyzedby the other authors mentioned.

Two variables are modeled as the inputsvariables to the production process of health careattainment:

HEXP health expenditure per capita in 1997ppp$

EDUC average years of schooling

Both input variables entered the productionfunction in log form. The translog model, withsquares and the cross product was also considered,but not used by ETML. A restricted form of thismodel that has appeared in the earlier studies isdiscussed below. In order to maintain compar-ability with their study, we have adopted theirsimpler functional form. Several variables thatwere observed only for 1997 provide indicators ofcross country heterogeneity were:

GINI Gini coefficient, income inequal-ity. This measure ranges fromzero (perfect equality) to one(complete inequality). Detailsmay be found at the World Bank



data website on income andgrowth, www.worldbank.org/re-search/growth/dddeisqu.htm.The ‘Deininger and Squires DataSet’ from which it is derived isalso described there

VOICE World Bank measure of demo-cratization and freedom of poli-tical unit. This variable indicatesaspects of the political process,civil liberties, political rights andthe extent to which citizens parti-cipate in the selection of thegovernment. It is measured bysurveys

GEFF measure of government effective-ness, World Bank measure. Thisrefers to the quality of publicservice provision, bureaucracy,competence of civil servants, in-dependence of civil servants frompolitical pressure and the cred-ibility of government commit-ment to service policies

TROPICS dummy variable for tropicallocation

POPDEN population density, peopleper km2

PUBFIN percentage of health care paid bythe government

GDPC per capita GDP in 1997 ppp$OECD dummy variable for OECD mem-

bershipj

The population density and per capita GDPvariables appear in logs in all model results tofollow. Further details on the data constructionare provided in Appendix A in [31]. Descriptivestatistics for the 1997 values of the data used hereare given in Table 2.

The original data set contained missing valuesfor approximately 40 countries for the VOICE,GEFF and POPDEN variables. Recent updates ofthe World Bank website enabled us to completedata for VOICE and GEFF for all but sevencountries for VOICE (Cook Islands, San Marino,Niue, Monaco, Nauru, Palau, Tuvalu) and fourfor GEFF (Cook Islands, Niue, Palau, Laos.)Proxies for these remaining values were obtainedby computing fitted values from a regression of therespective values of these variables on all otherobserved data for the country using the completedata. This represents only 7 and 4 missing values

out of 191, so the effect should be minimal in thecomputed results. Data on population densitywere missing for 18 countries. The website http://www.geography.about.com/library/cia/blcindex.htmcontains year 2000 data on land area, populationand many other variables for all of the countriesfor which our data were missing. The populationdensity was calculated by drawing the year 2000estimated population back to 1997 using theestimated population growth rate for 3 years, thencomputing the density in persons per squarekilometer. The analysis below is thus based ondata for all 191 countries.k

Technical ine⁄ciency estimates fromtheWHO data

The production function

Health expenditure is the most visible input to thehealth care process, and public health expenditureis a major component of health care policy. Thereis striking variation in the public share of financingof health expenditure across countries and, asnoted by Berger and Messer [BM, 32], across timeas well. As can be seen in Table 2, the mean andstandard deviation of 59.4 and 21.0 for PUBFIN,respectively, suggest a range of variation of at least20–100%. Some specific values for the largereconomies include 72% for Canada, 25% forChina, 92% for the Czech Republic, 77% forFrance, 78% for Germany, 13% for India, 57%for Italy, 82% for Norway, 78% for Sweden, 97%for the United Kingdom and 44% for the UnitedStates. It is unclear, however, how variation in thepublic financing of health care will translate tovariation in health outcomes. Berger and Messer[32] suggest, for example, that the degree of publicfinancing could lead to either improvement orworsening of the efficiency of health care delivery.As they note, this aspect has not received muchattention in the empirical literature. The data andmodels framed for this study will allow us toexamine this issue.

A number of researchers have examinedthe responsiveness of health expenditure toincreases in income. Newhouse [33] reportedestimates of the income elasticity of healthexpenditures in the range of 1.15–1.31. Subsequentresearchers have examined cross section and

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pooled time series-cross section data sets withsimilar results (see [32] for a survey). Our ownresults based on the WHO data for 1997, shown inTable 3, are consistent, with elasticity estimates of1.09 for the full sample and 1.23 for the OECDcountries. The values in excess of 1.0 suggest thatpopulations value health care as a normal good.

BM recount a series of cross section studies thathave found small and insignificant relationshipsbetween income levels and health outcomes.Subsequent analyses of income distribution as analternative explanation have likewise concluded

that the distribution of income adds little expla-natory power. Our results of linear least squaresregressions of our two health outcomes, DALEand COMP (in logs) on health care expenditure,education, the Gini measure of income inequality,and the log of per capita GDP and its square,shown in Table 4, do not agree with these findings.The results suggest that for the poorer, non-OECDcountries, there are significant relationships in theexpected directions both for per capita GDP andfor the income distribution. The results alsosuggest that the relationships are stronger for

Table 3. Income responsiveness of health care expenditure

Constant Income Education R2

Non-OECD �3.67 (0.251)n 1.02 (0.037)n 0.275 (0.616)n 0.902OECD �5.57 (0.747)n 1.23 (0.078)n 0.347 (0.196)n 0.917All �4.14 (0.201)n 1.09 (0.031)n 0.268 (0.059)n 0.936

Estimated standard errors in parentheses.n Indicates significant at the 95% level.

Table 4. Health care outcome regressionsa

Constant Hexp Educ Gini Income Income2 R2

NonOECD DALE 3.04 0.187 0.017 �0.578 0.098 0.676(0.176)n (0.028)n (0.033) (0.148)n (0.037)n

�0.712 0.130 0.033 �0.703 1.046 �0.058 0.715(0.832) (0.029)n (0.313) (0.142)n (0.209)n (0.013)n

COMP 3.56 0.100 0.036 �0.426 0.064 0.769(0.101)n (0.016)n (0.191) (0.086)n (0.0214)n

1.94 0.075 0.043 �0.481 0.474 �0.025 0.784(0.495)n (0.017)n (0.019)n (0.085)n (0.124)n (0.008)

OECD DALE 3.82 �0.042 0.046 �0.102 0.0157 0.725(1.72)n (0.271) (0.244) (0.078) (0.0316)�0.261 �0.041 0.036 �0.103 �0.896 �0.046 0.760(2.25) (0.026) (0.024) (0.074) (0.477) (0.024)

COMP 3.91 0.035 0.341 �0.119 0.031 0.846(0.137)n (0.216) (0.195) (0.621) (0.025)�0.507 0.036 0.023 �0.120 0.967 �0.049 0.880(1.69) (0.195) (0.180) (0.056)n (0.358)n (0.019)n

All DALE 3.01 0.197 �0.0078 �0.397 0.105 0.714(0.162)n (0.026)n (0.029) (0.125)n (0.034)n

�0.481 �0.128 0.031 �0.625 0.980 �0.054 0.757(0.627) (0.026)n (0.028) (0.122)n (0.156)n (0.009)n

COMP 3.54 0.107 0.024 �0.338 0.068 0.823(0.093)n (0.015)n (0.017) (0.072)n (0.196)1.97 0.076 0.041 �0.440 0.462 �0.024 0.840(0.371)n (0.016)n (0.0165)n (0.072)n (0.092)n (0.006)n

Estimated standard errors in parentheses.aDALE, COMP, Expenditure, Education, and Per Capita Income are all in logs. Results are all based on the 1997 data.n Indicates significant at the 95% level.



non-OECD countries than for OECD countries.The pattern of the quadratic relationships suggeststhat the improvement in health outcomes providesgreater benefits at lower incomes than at higherones.

The basic production function analyzed here(and in HW) is of the simple form

Healthit ¼ f ðEducationit;ExpenditureitÞ þ vit � uit

¼ f ðxitÞ þ vit � uit ð19Þ

We will augment the production function with twocovariates that seem more likely to shift theproduction function than the mean of inefficiency,

zi ¼ ½Tropicsi;PopDeni;� ð20Þ

Finally, in our preferred model, we will allow fortime variation (technical change) with the yearspecific dummy variables,

t ¼ year1993; year1994; year1995; yearl1996 ð21Þ

Additional influences on health outcomes thatappear in the distribution of the inefficiency termare

hi¼ ½GEff i;Voicei;Ginii;GDPi;PubFini;OECDi�

ð22Þ

The functional form of the production modelremains to be determined. ETML specified atranslog model,

LogHealth ¼ aþ b1 log Expþ b2 log Educ

þ b11 ½12log2 Educ�

þ b22 ½12log2 Exp�

þ b12 ½log EducÞðlog ExpÞ�

þ vit � uit ð23Þ

then, in the interest of parsimony, dropped the lasttwo terms. Hollingsworth and Wildman [9], in theinterest of comparability, adopted the same func-tional form. One might question the truncation ofthe production function. In principle, the translogform is used to approximate an arbitrary under-lying function, and dropping those terms wouldsignificantly degrade the approximation. We refitthe full translog stochastic frontier model. Withall terms included, we found for COMP, first,that the scale elasticity, @ logCOMP/@ log Exp+@ logCOMP/q log Educ, was roughly 0.35, suggest-ing large diseconomies of scale. Second, thefunction is not monotonic in the inputs for all

values – marginal products were negative for manyobservations. Thus, the function would not beconcave. Therefore, a strictly orthodox interpreta-tion of the model as conforming to an optimiza-tion of output in the presence of a neoclassicalproduction function seems optimistic. A muchlooser interpretation of the relationship betweenthe health outcomes and the inputs seems moreappropriate. To maintain continuity of this strandof analysis in the literature, we have maintainedthe simpler relationship

LogHealth ¼ aþ b1 log Expþ b2 log Educ

þ b11 ½ðlog2 EducÞ=2� þ � � � þ vit � uit

¼ aþ x0

itbþ � � � þ vit � uit ð24Þ

Random and fixed effects estimates of

inefficiency

Tables 5(a) and 5(b) present estimated productionfunctions based on the simple panel data specifica-tions, ETML’s [4,5] (Cornwell et al.’s [3]) fixedeffects model, the Pitt and Lee [24] random effectsmodel, and the true fixed and random effectsmodels.m None of these has any built in accom-modation for cross country heterogeneity in theproduction function or the inefficiency. Theestimated models differ somewhat, though thedifferences in the estimates of the structuralparameters might be misleading – the estimatedinefficiency terms (estimates of ui) are quite similarfor the two pairs of specifications, fixed/randomeffects and true fixed/true random effects. Theestimated random effects models in Table 5(a) areconsistent with Gravelle et al.’s [6,7] observation,that there is little within group variation. Thevariance decomposition is dominated by ui; theestimates of l ¼ su=sv are 25.47 for DALE and30.09 for COMP. Both of these are quite large bycommon standards. These values fall to 3.26 and15.44 in the true random effects models for DALEand COMP, respectively. The relatively smallervalues of 5.526 for DALE and 6.136 for COMPfor the true fixed effects models appear consistentwith this as well (though there is no counterpart inthe ETML fixed effects model). The estimatessuggest that these models in Table 5(b) are movingsome of the variation out of ui. This would beconsistent with purging the invariant ui of sometime invariant heterogeneity. (We note, however,

W.Greene968


that the crucial difference might be in the means,rather than the variances. This is addressed belowand in Table 6.)

Estimated inefficiencies are computed using themethods discussed earlier. In all cases, to simplifycomparisons, we have used the direct estimate ofinefficiency, #uui rather than expð� #uuiÞ. For the fixedeffects estimator, this is simply max(ai) – ai. Therandom effects estimator is computed usingthe Jondrow et al. [12] estimator in (7), as are theestimates for the true fixed and random effectsmodels in Table 5(b). Tables 6(a) and 6(b) presentanalyses of these effects. As evidenced in the plotsin Figure 1, the correspondence between the twosets of estimates, for both health outcomes isstriking for both modeling approaches. The simple

correlations between the pairs of estimates isalmost 1 for the time invariant estimates, and0.9515 and 0.8276 for DALE and COMP for thetime varying estimates based on the 1997 observa-tions.n The random effects estimators also repro-duce the rankings of the fixed effects estimatorwhich, in turn, gives largely the same results asETML obtained with their normalized version.(France remains fourth in the DALE results andfirst in the COMP results, for example.) Thisdegree of correspondence between these twoestimators has been observed elsewhere (e.g.,[34]). This finding suggests that the impact of thespecific distributional assumption of the stochasticfrontier model is not so severe as suggested earlier.The correspondence is similar between the true

Table 5. Estimated frontier production models

Constant Expenditure Education Education2

(a) Estimated frontier production models with time invariant inefficiencya

DALE Fixed effects 0.00884 0.0629 0.0435 s=0.109(0.00305)n (0.0363) (0.0285) R2=0.998

Random effects 3.96 0.0152 0.0970 0.0130 su=0.2771(0.0159)n (0.00234)n (0.0175)n (0.0149) sv=0.0109

l=25.47COMP Fixed effects 0.00654 0.0495 0.0455 s=0.0066

(0.00185)n (0.0221)n (0.0171)n R2=0.999

Random effects 4.24 0.0105 0.0617 0.0411 su=0.1989(0.0101)n (0.0011)n (0.0101)n (0.0925)n sv=0.00661

l=30.09

(b) Estimated frontier production models with time varying inefficiencya

DALE True fixed effects 0.0665 0.2896 �0.1147 l=5.526(0.00156)n (0.0102)n (0.0069)n s=0.2106

su=0.2072sv=0.0375

True random effects 3.632 0.04099 0.0172 0.1015 l=3.26, s=0.0194(0.0044)n (0.0007)n (0.0039)n (0.0042)n su=0.01858

sv=0.0057sw=0.1760

COMP True fixed effects 0.0684 0.1301 �0.0311 l=6.136, s=0.1224(0.00086)n (0.0053)n (0.0037)n su=0.1207

sv=0.0197True random effects 3.551 0.0698 0.4762 �0.2559 l=15.44, s=0.0781

(0.0121)n (0.0025)n (0.0185)n (0.0185)n su=0.07093sv=0.00459sw=0.0773

All variables in logarithms. Estimated standard errors in parentheses.n Indicates significant at the 95% level.aBased on 140 countries observed in all 5 years and Algeria observed in four. STi=704.



random and true fixed effects values, though thecorrelation is somewhat reduced. Also to beexpected, though somewhat disconcerting is thevirtual lack of correlation between the CSS fixedand true fixed effects estimates, and between thePitt and Lee [24] and the true random effectsestimators. Not only are both the means andvariances of the estimated inefficiencies much lowerin the true effects models, but the full distributionover countries seems to have changed as well. But,again, this is consistent with the purging of sometime invariant heterogeneity from the time invar-iant estimates in the CSS and Pitt and Lee [24]estimates. Figure 2 illustrates this for the compar-ison of the CSS fixed to the corresponding 1997estimates from the true fixed effects model.

The lower panels of Tables 6(a) and 6(b)presents second step analyses of the estimated

inefficiencies for each health measure for the twomodeling approaches. They suggest that incomeand the distribution of income are both significantin explaining variation in efficiency when theinefficiency is assumed to be time invariant. Sinceui is in proportional terms, the absolute magni-tudes of the coefficients give the proportionalimpacts. It appears that the most importantdeterminant is the distribution of income, withlarger values of the Gini coefficient (less equalincome distribution) having a major negativeimpact on both health outcomes. (Increases in uiimply lower efficiency.) The second largest deter-minant is per capita income, which works in theexpected direction – higher income is associatedwith more efficient delivery of health care andachievement of higher life expectancy. (Theseresults are not interpretable as direct impacts on

Table 6. Analysis of technical inefficiencies

DALE COMP

Fixed effects Random effects Fixed effects Random effects

(a) Analysis of estimated time invariant technical inefficienciesMean 0.2200 0.2089 0.1741 0.1647Std dev. 0.1862 0.1822 0.1181 0.1116Correlation 0.9989 0.9988

Second step regression resultsConstant 1.14 (0.156)n 1.075 (0.155)n 0.6965 (0.084)n 0.597 (0.0858)n

Gini 0.528 (0.139)n 0.546 (0.139)n 0.3084 (0.077)n 0.299 (0.0767)n

Voice �0.0072 (0.0190) �0.0089 (0.0189) �0.0134 (0.010) �0.0125 (0.0104)GEFF 0.0352 (0.0230) 0.0365 (0.0229) �0.0038 (0.0127) �0.00451 (0.012)LogIncome �0.150 (0.0168)n �0.144 (0.0167)n �0.0779 (0.009)n �0.0707 (0.009)n

Public finance 0.0009 (0.0062) 0.0008 (0.0006) �0.0002 (0.0003) 0.0002 (0.0003)OECD 0.0704 (0.0398) 0.0691 (0.0396) 0.0341 (0.0220) 0.0334 (0.0190)R2 0.5599 0.5447 0.6671 0.6292

(b) Analysis of estimated 1997 observations on time varying technical inefficienciesMean 0.0797 0.0159 0.0462 0.0384Std dev. 0.0187 0.0171 0.0146 0.0190Correlation 0.9515 0.8276

Second step regression resultsConstant 0.1193 (0.222)n 0.0432 (0.0205)n 0.0665 (0.018)n 0.0975 (0.0207)n

Gini 0.0142 (0.0203) 0.0159 (0.0187) 0.0068 (0.016) 0.0319 (0.0189)Voice 0.0006 (0.0027) �0.0001 (0.0025) 0.0008 (0.0022) 0.0005 (0.0026)GEFF 0.0017 (0.0033) 0.0009 (0.0031) �0.00004 (0.002) 0.0016 (0.0031)LogIncome �0.0054 (0.0024)n �0.0043 (0.0022) �0.0024 (0.0019) �0.0084 (0.0022)n

Public finance �0.00014 (0.001) �0.0001 (0.0001) �0.00001 (0.0001) �0.0002 (0.0008)OECD �0.0005 (0.0058) �0.00001 (0.005) �0.002 (0.0046) 0.0002 (0.0054)R2 0.0729 0.657 0.0482 0.6371

Estimated standard errors in parentheses.n Indicates statistical significance at 95% level.

W.Greene970


the health outcomes.) Surprisingly, OECD statusis associated with less efficient production. Note,though that per capita income is already in theequation, so whatever effect is at work in thispersistent result is net of the impact of per capitaincome. Finally, as expected, this second step hasvery little explanatory power in the regressionsinvolving the inefficiencies from the true randomand fixed effects models.

The fixed effects estimators provide no methodof accommodating time invariant indicators ofheterogeneity in the model, but most of the othersdiscussed do. The true random effects (randomconstant term) model, however, becomes highlyunstable in these data when any other covariatesare added to the model – this occurs because thereis insufficient within group variation to estimatethis type of model. Thus, from this point forward,we will focus on the random effects models. Thetrue fixed and random effects approaches still seem

promising, but analysis of them is held for furtherresearch.

Incorporating measured heterogeneity in the

production function and in the estimates of

inefficiency

We now consider expanding the functional form toaccommodate the time effects and the othermeasured covariates. The augmented form of theproduction function will be at least

yit ¼ aþ x0

itbþ z0

icþ t0hþ vit � ui

where zi=(TROPICS, POPDEN) – these twovariables seem more appropriate as impacts onthe level of production than as drivers ofinefficiency. The placement of the remainingheterogeneity indicators, hi=(GINI, Income,

0.00

1.00

0.80

0.60

0.40

0.20

0.00

1.00 0.150

0.100

0.050

0.000

0.0500

0.0750

0.1000

0.1250

0.1500

0.1750

0.2000

0.80

0.60

0.40

0.20

0.00

0.400.20 0.60 0.80 1.00

0.00

0.000 0.025 0.050 0.075 0.100 0.125

0.000 0.025 0.050 0.075 0.100 0.1250.400.20 0.60 0.80 1.00

Estimated Inefficiencies: DALE Fixed verus Random Effects

Estimated Inefficiencies: COMP Fixed verus Random Effects

Estimated Inefficiencies: DALE True FE verus True RE

Estimated Inefficiencies: COMP True FE verus True RE

UIDALEPL

UICOMPPL

UIDTRURE

UIDTRURE

UIF

ED

ALE

UID

TR

UF

E

UIF

EC

OM

P

UIC

TR

UF

E

Figure 1. Estimated fixed and random effects inefficiencies (based on the 1997 observations for the time varying estimates)



OECD, Voice, GEFF, PUBFIN) remains to beestablished. There is no clearly defined theory thatdictates how these should enter the model, so weapproached the issue as follows: First, a practicalresult, when the GINI variable appeared in theproduction model, with or without the otherelements of hi, the estimator produced wildlyerratic estimates of the distribution of ui – theestimate of su in Table 5 of 0.2771 for DALEjumps to over 1.0, and the coefficient l diverges toover 1000. The behavior of COMP is similar.When the truncation model is specified as thealternative, with

mi ¼ d0 þ d1 GINIi

the model behavior is quite reasonable, and shiftsmarginally from the results in Table 5. Weconcluded on this (admittedly nonstatistical) basisthat GINIi belonged more appropriately in themean of the inefficiency distribution. The log ofper capita income seems likewise to be morenaturally associated with the efficiency of produc-tion than production, itself. Per capita expenditurealready in the production function, will capturesome of the effect of higher incomes in any event.Thus, our departure point is

mi ¼ d0 þ d1 GINIi þ d2 LogIncome

That leaves the set of indicators,

hi¼ GEff i;Voicei;PubFini;OECDi

Though these would fit naturally in mi, they couldarguably be drivers of production instead (or aswell).

We have tested this specification issue asfollows: We have two competing models. Model(P) places the indicators in the production func-tion:

ðPÞ yit ¼ aþ x0

itbþ z0

icþ t0hþ h

0

i/þ vit � ui

mi ¼ d0 þ d1 GINIi þ d2 LogIncomei

while model (E) places them in the mean of theheterogeneity distribution;

yit ¼ aþ x0

itbþ z0

icþ t0hþ vit � ui

ðEÞ mi ¼ d0 þ d1 GINIi þ d2 LogIncomei þ h0

id3:

These models are not nested, so no simple test(such as an F) test can be used to distinguish themstatistically. We have used Vuong’s [35] test fornonnested models, instead. The two competingmodels were estimated by maximum likelihood.Let Li(P) be the contribution of country i to thelog likelihood function assuming specification Pand let Li(E) be the counterpart for the othermodel. Then, let qi=Li(P)� Li(E). The statistic isffiffiffi

np

%qq=sqwhere %qq and sq are the sample mean andstandard deviation. Vuong showed that thisstatistic has a limiting standard normal distribu-tion. The test is directional. Values in excess of+1.96 (assuming a 95% significance level) favormodel (P) while values less than �1.96 favor model(E). The intermediate values are (unfortunately)inconclusive. The estimated values of the Vuongstatistic were 3.29 for DALE and 1.78 for COMP.These favor placing the indicators of heterogeneityin the production model – thus model (P) is ourpreferred specification.

Table 7 presents estimates of the random effectstruncated normal stochastic frontier in which thetime invariant covariates GINI and LogIncomeappear in the underlying mean of Ui. The modelalso allows for technical change (the time shifts inthe production function) and for some countryeffects in the production function (tropical loca-tion and population density and the other compo-nents discussed above). This is the heterogeneousform, with E[Ui]=mi=d0+hi

0d.o The pattern inthese results is similar to the preceding outcome.Once again, per capita income and the distributionof income appear to be significant determinants ofthe mean level of inefficiency, again in the expecteddirection.

The incorporation of cross country heterogene-ity in the model has also produced the expectedresult with respect to the variation in inefficiency.

0.00 0.16 0.32 0.48 0.64 0.80

UIFEDALE

0.200

0.160

0.120

0.080

0.040

0.000

UID

TR

UF

E

Estimated Inefficiencies: DALE, True FE vs. Fixed Effects

Figure 2. Estimated inefficiencies for DALE, True vs ETML

fixed effects

W.Greene972


In Table 5, in the simple random effects model, thestandard deviation of the underlying distributionof Ui, su, is estimated as 0.2771 and 0.1989 forDALE and COMP, respectively. In the expandedmodel in Table 7, the counterparts are 0.1628 and

0.0806. A large proportion of the variation in‘inefficiency’ appears to be explainable as hetero-geneity in the mean, instead. The estimate of theresidual variation, sv, is almost unchanged forboth DALE and COMP.

Table 7. Estimated heterogeneous truncated normal stochastic frontiers

DALE COMP

Truncation Batt./Coelli Truncation Batt./Coelli

Production function Constant 4.04 3.91 4.26 4.22(0.0319)n (0.0236)n (0.0209)n (0.0197)n

Exp 0.00826 0.00791 0.00609 0.00600(0.00306)n (0.00215)n (0.00146)n (0.00145)n

Educ 0.0484 0.241 0.0420 0.0916(0.0231)n (0.0143)n (0.0113)n (0.0105)n

Educ2 �0.0146 �0.109 0.00498 �0.0224(0.0140)n (0.0116)n (0.00873) (0.00863)n

Year 1993 0.0300 0.00350 0.0178 0.0113(0.0134)n (0.0138) (0.0102) (0.0113)n

Year 1994 0.0320 0.00761 0.0191 0.0134(0.0147)n (0.0156) (0.0101) (0.0129)

Year 1995 0.0339 0.0120 0.0205 0.0156(0.0147)n (0.0142) (0.0104)n (0.0129)

Year 1996 0.0360 0.0166 0.0200 0.0179(0.0183)n (0.0180) (0.0120) (0.0144)

Tropics �0.00979 �0.0237 �0.00591 0.00424(0.0136) (0.0125)n (0.0155) (0.0142)n

Pop. density 0.00557 �0.00309 0.00585 0.00503(0.00275)n (0.00235) (0.00192)n (0.00197)

Voice 0.0202 0.0184 0.0164 0.0189(0.00858)n (0.00683)n (0.0092) (0.00893)n

GEFF 0.00478 0.0143 0.0137 0.00926(0.00733) (0.00775) (0.0078) (0.00742)

Public finance �0.0000896 0.000270 0.000117 0.000226(0.000302) (0.000246) (0.000229) (0.000257)

OECD 0.0383 0.0216 0.0237 0.0204(0.0117)n (0.0131) (0.00941)n (0.0122)

Mean inefficiency Constant 12.80 1.92 12.24 0.949(1.64)n (0.435)n (1.50)n (0.116)n

Gini 8.41 1.51 7.36 0.610(1.78)n (0.456)n (1.41)n (0.117)n

GDP Per capita �1.98 �0.309 �1.68 �0.130(0.240)n (0.0731)n (0.176)n (0.0150)n

Distributions of u and v su 0.162n 0.197n 0.0806n 0.0786n

sv 0.0107n 0.00952n 0.00645n 0.00636l 16.79n 20.72 12.51 12.36Z �0.0190n �0.00699n

Log likelihood 1806.976 1861.687 2150.807 2158.697

Estimated standard errors in parentheses.n Indicates statistical significance at 95% level.aSample is all 141 countries. SiTi=704.



The second set of estimates in each groupingin Table 7 are for Battese and Coelli’s[26,27] extension of the random effects model.This form of the model incorporates sometime variation in the inefficiency; the genericspecification is

Uit ¼ exp½�Zðt� TÞ�jUi j ð25Þ

where t is the current year (1993–1997) and T is theterminal year (1997). Note that the Pitt and Lee[24] model is nested within this one, so we can testfor the presence of this form of evolution ofinefficiency. The log likelihoods are given inTable 7 for this purpose. For both variables, thehypothesis that Z equals zero (that is, the hypoth-esis of the Pitt and Lee [24] model) is rejected. In

practical terms, however, the scale factor in (25)brings only very minor change in the year to yearestimates of ui. The underlying random componentof the inefficiency remains time invariant. Theeffect of this on the overall nature of theinefficiency is a bit ambiguous.

We have included in Table 8 descriptivestatistics for the inefficiency estimates and com-parisons to the fixed effects estimates produced byETML. Figure 3 shows the correspondencegraphically for DALE using the Pitt and Lee [24]random effects model. Virtually the same patternarises for the Battese and Coelli [26,27] model, andfor COMP using either model as well.p In fact, thestatistical results in Table 7 fail to reveal somerather substantial adjustments. For many

Table 8. Estimates of technical inefficiency from heterogeneous random effects models

DALE COMP

Truncation B&C Truncation B&C

Estimated inefficiency(comparison to fixedeffects estimates)

Corr. with FE 0.978 0.984 0.962 0.971

Rank corr. 0.932 0.945 0.962 0.968Mean RE FE RE FE RE FE RE FEOECD 0.0607 0.0792 0.0639 0.0792 0.0554 0.0558 0.0381 0.0297NonOECD 0.231 0.258 0.221 0.258 0.181 0.206 0.173 0.206All 0.196 0.220 0.187 0.220 0.153 0.174 0.144 0.174

Std dev. RE FE RE FE RE FE RE FEOECD 0.0345 0.0474 0.0317 0.0473 0.0293 0.0400 0.119 0.112NonOECD 0.214 0.192 0.202 0.192 0.119 0.112 0.0297 0.0400All 0.204 0.186 0.191 0.186 0.119 0.118 0.120 0.118

150

150

120

120

90

90

60

60

30

300

0

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0.80

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UIDTRUNC

UIF

ED

ALE

RANKTRNC

RA

NK

ET

ML

Country Rank, Heterogeneous RE Model vs. Fixed Effects Estimated Inefficiencies FE vs. Heterogeneous RE

Figure 3. Country ranks and inefficiencies, DALE, random effects model

W.Greene974


countries, this expansion of the model appears tobe fine tuning, but for a large number of others,quite substantial differences emerge.

The results in Table 7 suggest that the efficiencyof production is significantly affected by both thedistribution of income and the level of per capitaincome, this persistent result has been hinted at inprevious analyses. We do find considerable evi-dence of the result in our estimates. The highlysignificant estimate of Z in the estimates of (25)(based on the likelihood ratio tests) suggest thatthe inefficiency varies somewhat over time as well.However, the plot in Figure 4 of the estimatedvalues country by country reveals that this modelextension actually adds relatively little actualvariation in the estimates. The fixed effectsestimate is the same in every year, so the verticalgrouping reveals the time variation in the Batteseand Coelli [26,27] estimates. Overall, 99.86% ofthe variation in the B&C estimates is betweencountries. Nearly 100% of the within groupvariation is accounted for by the 25 least efficientcountries.

Figures 5(a) and 5(b) display the estimates of uifrom this model, once again in comparison to theinitial estimates in ETML. We have segregated theestimates by OECD and non-OECD countries inthe figures and in the descriptive statistics inTable 8. The dotted ovals in the figures contain allthe OECD observations. This partition of thesample reveals a major aspect to all of the results.In the figures, the OECD countries are packed

0.80

0.64

0.48

0.32

0.32 0.48 0.64 0.80

0.16

0.16

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0.00

UIFEDALE

UID

ALE

BC

Battese/Coelli Inefficiencies vs. Fixed Effects Estimates

Figure 4. Battese/Coelli time varying inefficiency estimates

0.80

0.64

0.48

0.32

0.36 0.54 0.72 0.90

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UIFECOMP

UIT

_DA

LEU

IT_C

OM

P

DALE: Fixed Effects vs. Heterogeneous Random Effects

COMP: Fixed Effects vs. Heterogeneous Random Effects

OECD = 1 OECD = 2

0.80

0.64

0.48

0.32

0.16

0.00

OECD = 1 OECD = 2

(a)

(b)

Figure 5. (a) DALE, fixed effects vs heterogeneous random

effects, 1997 values, (b) COMP, fixed effects vs heterogeneous

random effects, 1997 values



tightly in the lower left corner of the plots. Fromthe descriptive statistics, as borne out by thefigures, we now see that nearly all of the variationin the estimated inefficiencies in both ETML’sfixed effects estimates, and these, our finalproposed alternatives, is attributable to the non-OECD countries in the sample.

It can be seen in the figures that this modelbrings some rather large changes in the results. Toexplore this further, we have plotted the countryranks for the DALE estimates in Figure 6. (Thefigure for COMP is similar.) The results havechanged considerably. Table 9 shows the effect ofthe respecified model on the ranks of the top 25countries in the original fixed effects form. Thetable suggests that the rankings change consider-ably with this change in the model. Recalling thatin the WHR, the rankings, not the absolute levelsof inefficiency, were the primary object of estima-tion, we conclude that in the terms of the study,the model specification is clearly an importantissue.

Conclusions

The WHO data analyzed here include most of theworld’s countries and embody nearly its entirepopulation. The original studies by Evans et al.

00

40

40

80

80

120

120

160

160

200

200

RANKTRND

RA

NK

ET

ML

OECD = 1 OECD = 2

Country Ranks, Fixed Effects and Heterogeneous RE

Figure 6. Country ranks: fixed effects and heterogeneous

random effects, DALE

Table 9. Country ranks for the top 25 Countries in ETML sample

DALE COMP

Rank Country New rank Country New rank

1 Malta 33 France 162 Oman 8 Italy 83 Italy 23 San Marino 194 France 5 Andorra 145 San Marino 29 Malta 106 Spain 7 Singapore 87 Andorra 32 Spain 28 Jamaica 2 Oman 909 Japan 1 Austria 910 Greece 3 Japan 411 Monaco 25 Norway 1212 Saudi Arabia 101 Portugal 3113 Singapore 6 Monaco 3914 Portugal 38 Greece 115 Austria 35 Iceland 2016 Norway 22 Netherlands 1117 United Arab Emirates 95 Luxembourg 3618 Netherlands 17 Ireland 3319 Sweden 11 United Kingdom 2220 Costa Rica 9 Colombia 2921 Cyprus 20 Switzerland 2422 Chile 4 Belgium 1723 United Kingdom 13 Cyprus 2124 Iceland 15 Sweden 325 Switzerland 18 Saudi Arabia 128

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were an innovative, large scale effort to comparethese countries on the effectiveness of delivery ofhealth care using two measures, disability adjustedlife expectancy (DALE) and a recently developedcomposite measure of health care delivery(COMP). The authors used a panel data techniquebased on the fixed effects regression model toobtain both quantitative measures of effectivenessand simple rankings for each country. Subsequentresearchers have analyzed these same data with awide range of agreement or disagreement with themethodologies and results. This study has con-tinued that analysis, with several extensions. First,the model used here is considerably more generalthan those used previously. Second, we haveincorporated measured indicators of cross countryheterogeneity in the estimates. Third, we haveproduced an alternative set of individual countryspecific efficiency measures and country ranks. Ourresults differ substantially from those obtained bythe authors of the original studies.

This reanalysis of the WHO data was motivatedby several considerations.

* Both the fixed effects approach used earlier, andseveral others that have appeared in receivedstudies make no distinction between technicalinefficiency and cross country heterogeneity.Thus, some of what was reported as measuredinefficiency should instead be treated as hetero-geneity. Certainly, decomposing a two partunmeasured effect is a difficult exercise. Theseveral models proposed here do so to varyingdegrees. We find, in our preferred specification,that making this distinction brings a substantialchange in the estimated results

* The fixed effects model used in Evans et al. [4,5]does not allow the analyst to make use ofmeasured effects that capture some of the crosscountry heterogeneity in the data. The modelsproposed here can accommodate these vari-ables. We find including these measured in-dicators, such as per capita income and ameasure of income distribution, produce notice-able changes in the estimated results.

In formulating the production function, weconfirm what earlier researchers have found withrespect to the income elasticity of health expendi-ture. Our estimates range from 1.08 to 1.23, withmuch larger values for the wealthier OECDcountries than for the remaining countries in the

sample. A number of earlier studies have examinedthe relationship between income (measured by percapita GDP) and health outcomes, and found aweak to nonexistent support. We find a persistent,significant impact of income on inefficiency. Like-wise, the distribution of income has been suggestedas an important influence, but prior results weresomewhat inconclusive. As in the case of income,itself, we find significant explanatory power in thedistribution of income. It should be noted that inboth these cases, and throughout this study, thedistinction between OECD and non-OECD coun-tries, which explains much of the variation in thesemeasures, also explains much of the variation inhealth outcomes and in the efficiency of delivery ofthem.

Using the random effects model as the platform,we found that comparing a basic model with noheterogeneity (Tables 5 and 6) to one withheterogeneity in both production and the meanof the inefficiency (Tables 7 and 8), the estimatedunderlying standard deviation in the distributionof ui, su, falls from 0.2771 to 0.1620 for DALE andfrom 0.1989 to 0.0806 for COMP. The samplemeans of the estimated inefficiencies also fallslightly, from 0.2088 to 0.1960 for DALE andfrom 0.1647 to 0.1530 for COMP. These areconsistent with the original conjecture, thatunaccounted for heterogeneity was indeed show-ing up as inefficiency in the original model. Thisdoes not imply, however, that the rankings of thecountries would be changed. But, as seen in Figure5, the rankings do change, considerably.

It should be emphasized, the measures andrankings in the tables pertain to the efficiency ofdelivery of health outcomes, not to absolute levelsof those outcomes. Our results do not comment onthe levels of health attainment in countriescontained in this data set.

Acknowledgements

Elements of this paper have been presented at theconference on ‘Current Developments in Productivityand Efficiency Measurement’, University of Georgia,October 25–26, 2002 and the Second Hellenic Workshopon Productivity and Efficiency Measurement in PatrasGreece, May 30, 2003. It has also benefited fromcomments at the North American Productivity Work-shop at Union College, June, 2002, the Asian Con-ference on Efficiency and Productivity in Taipei in July,2002 and from discussions at The University of



Leicester, York University and Binghamton Universityand ongoing conversations with Mike Tsionas, Subal,Kumbhakar and Knox Lovell.

Notes

a. The participants were, in addition to Evanset al., William Greene of NYU, SubalKumbhakar, University of Binghamton,Knox Lovell, University of Georgia, KaliappaKalirajan, ANU, Marijn Berhoeven, IMF,Paul Wilson, University of Texas, ChristopherTong, Hong Kong Baptist University andPhilip Grossman, St. Cloud State University.

b. We have focused on parametric (stochasticfrontier) and semiparametric (fixed effectslinear regression) models. Nonparametricmethods such as data envelopment analysis(DEA) are not considered here (for commen-tary, see, e.g. [9]).

c. Other distributional assumptions have beensuggested, such as the normal-exponential [10]and the normal-gamma [15,16]. These exten-sions occasionally bring noticeable changes inthe results. But they are tangential for presentpurposes.

d. The only received application of a type of truefixed effects model in the frontiers literature isPolachek and Yoon’s [17] study of laborsupply.

e. The major practical obstacle to use of thefixed effects approach in nonlinear modelssuch as this one is the difficulty of computingthe possibly hundreds or thousands of dummyvariable coefficients. Unlike the linear regres-sion model, the stochastic frontier modelcannot be transformed to eliminate thedummy variables from the estimator. It iswidely held that this renders the modelunusable if the number of individuals (N) isvery large (see, e.g., [18]), but see Greene [19]for analysis of the solution to this computa-tional problem and Greene [22] for anapplication involving direct computation ofsome extremely large models. The methodolo-gical shortcoming of the fixed effects approachis the so called incidental parameters problem.See [20] and Greene (2004) for discussion.This issue as it relates to the stochastic frontiermodel remains completely unresolved; theonly received evidence on it is the MonteCarlo study in Greene (2004).

f. In principle, one might estimate ui in arandom effects linear regression, either basedon a full normality assumption, or as theMMSE projection, using #uui ¼ E½ui j %eei� ¼ r%eeiwhere r ¼ s2u=ðs

2u þ s2vÞ. However, this is still

unsigned, and a second transformation to #uueffi¼ max½ #uui� � #uui would still be needed. Theusefulness of this estimator seems dubious.We are not aware of any applications orinvestigations.

g. This extension of the model introduces asubtle complication. The ‘heterogeneity’ in-troduced into uit by this formulation influ-ences both the mean and variance ofinefficiency, because although Zit represents ascaling of the inefficiency, this scaling (thestandard deviation) enters the mean as well, ascan be seen in (15)–(16). More generally,stochastic frontier models that seek to incor-porate heteroscedasticity in the distribution ofuit actually, because of this result, also buildheterogeneity in the mean in to the model atthe same time. Unlike the linear regressionmodel, in the stochastic frontier model, theconditional mean of the part of the distur-bance that is of interest is a function of boththe underlying mean and the underlyingvariance.

h. The assistance of researchers at WHO, espe-cially David Evans and Ajay Tandon isgratefully acknowledged. The data set usedhere is an expanded version of the data used inETML [4,5]. The earlier papers did not use thetime invariant measures, per capita GDP,income distribution, etc.

i. One country, Algeria, was only observed fourtimes.

j. Australia, Austria, Belgium, Canada, CzechRepublic, Denmark, Finland, France, Ger-many, Greece, Hungary, Iceland, Ireland,Italy, Japan, Korea, Luxembourg, Mexico,The Netherlands, New Zealand, Norway,Poland, Portugal, Slovak Republic, Spain,Sweden, Switzerland, Turkey, United King-dom, United States.

k. Certainly other country specific covariatescould explain the heterogeneity in this dataset. Those mentioned were contained in thelarger data set. Observers might differ on thebest set of variables. We leave it for ongoingresearch to fill out the model with an optimaltreatment of the unobserved cross countryheterogeneity.

W.Greene978


l. The year 1997 dummy variable is the omittedcategory.

m. To avoid fitting panel models with ‘groups’ ofone observation, the models to follow arebased on the 140 countries with five years ofdata plus Algeria, with four.

n. Using the country means of the inefficienciesproduces largely the same results.

o. The strikingly large constant and coefficienton GINI in the first and third columns do notindicate extremely large means, as they areoffset by the coefficient on the log of GDP.The means of GINI and logGDP are 0.34 and8.4, respectively, so it can be seen that theseeffects are largely offsetting in E[Ui].

p. For the Battese and Coelli formulation, wedid the comparison based on the 1997estimate of ui. There is a small amount of timevariation in the estimates from this model, butthe patterns are essentially unchanged.

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