Jackknife Resampling Plan Estimates - Springer Link

Health Care Management Science 7, 173–183, 2004  2004 Kluwer Academic Publishers. Manufactured in The Netherlands.

Determinants of Health Expenditure Growth of the OECD Countries: Jackknife Resampling Plan Estimates ∗ ALBERT A. OKUNADE ∗∗ Department of Economics, Rm. 450BB (The FCoBE), The University of Memphis, Memphis, TN 38152, USA E-mail: aokunade@memphis.edu

MUSTAFA C. KARAKUS Department of Health Policy and Management, The Johns Hopkins University, Baltimore, MD 21205, USA E-mail: mkarakus@jhsph.edu

CHARLES OKEKE College of Southern Nevada, Department of Economics and Regional Studies, Las Vegas, NV 89146-1164, USA E-mail: charles_okeke@ccsn.nevada.edu

Abstract. Due to the lack of internal consistency across unit root and cointegration test methods for short time-series data, past research findings conflict on whether the OECD health expenditure data are stationary. Stationarity reasonably guarantees that the estimated OLS relationship is nonspurious. This paper departs from past investigations that applied asymptotic statistical tests of unit root to insufficient time-series lengths. Instead, data were calibrated in annual growth rates, in 5-year (1968–72, . . . , 1993–97) partitions, for maximum likelihood estimation using flexible Box–Cox transformations model and bias-reducing jackknife resampling plan for data expansion. The drivers of OECD health care spending growth are economic and institutional. Findings from the growth convergence theory affirm that health care expenditure growth accords with conditional β convergence. Statistical significance and optimal functional form models are not unique across the growth period models. Our findings exemplify the benefits of jackknife resampling plan for short data series, and caution researchers against imposing faulty functional forms and applying asymptotic statistical methods to short time-series regressions. Policy implications are discussed. Keywords: OECD health expenditure, new growth theory, Box–Cox transformation model, panel data estimates, jackknife resampling methods, health policy

1. Introduction 1.1. Past studies of health care expenditure Rising health care spending is a persistent concern of patients and government policy makers in OECD and other countries. Past studies identified major culprits and suggested cost containment policies [31]. Previous researchers mostly fitted econometric models to typically short, less than 50 aggregate time-series data length to test the unit root hypothesis (or that of joint stationarity) in health spending, per capita GDP, and hardly other economic or demographic determinants. Studies of the determinants of OECD health care spending usually employ data calibrated in levels and so, recent work has focused on how the estimation of classical time-series regression models may yield spurious results in the absence of stationary or cointegrated time-series [19,25,28]. The absence of unit root in time-series regressions strengthens one’s research confidence that the estimated relationships are non-spurious. Okunade and Suraratdecha [36] contends that applying as∗ This is the revised version of a paper earlier presented at the Annual Joint

Statistical Meetings of the American Statistical Association (Business and Economic Statistics Section) in Baltimore, MD (USA). ∗∗ Corresponding author.

ymptotic statistical hypothesis tests to short time-series or panel data and the use of disparate unit root schemes have generated conflicting conclusions on the stationarity hypothesis in health care expenditure models of the OECD countries. Past empirical studies also fitted models to samples of one, and each sample consisted of n observations from the relevant domain, a severe and familiar data collection problem in social sciences and survey research. (The famed sub-field of experimental economics, using controlled laboratory market experiments allowing replication, pioneered by Vernon Smith, the 2002 Nobel Laureate in Economics, is a landmark effort towards reducing the observational data limitations in the social sciences.) Statistical methods relying on large sample properties are expected to generate regression model estimates that are closer to true population values. The advent of fast and cheap computing algorithms makes data reuse an efficient method for inducing asymptotic results from a sample data of one, consisting of n independent time-series, and drawn from the relevant distribution. The jackknife is highly favored among resampling plans [20]. Therefore, the implications of finding conflicting unit root test results for the small samples typically used in health expenditure studies suggest the need for better methods and discussion of any policy findings. Generally, these methods

174

include (a) nonparametric methods; (b) parametric unit root hypotheses test techniques possessing desirable small sample properties; (c) the rarely used method of calibrating timeseries data in growth rates rather than in levels, and estimating a parametric regression model using maximum likelihood (ML) method; and (d) using jackknife resampling method jointly with strategy (c), to resample continuously from the finite time-series data and repeated regression estimation based on the growth convergence theory for health care expenditure. The goal is to reduce any bias in the estimated parameters by the inclusion of all the sub-sample iterations, and to draw policy inferences. 1.2. Objectives of this research Consequently, this research departs from using data calibrated in levels to test stationarity or cointegration but instead constructs models of the major drivers of health expenditure growth for OECD countries. Barro and Sala-i-Martin [4] specified the new growth model for predicting the convergence of expenditures to a steady state. Given a set of homogeneous countries (e.g., OECD), those that had a high (low) initial level of health care spending tend to raise spending growth at a slower (faster) pace such that spending across countries eventually converges (β convergence theory). The σ type growth convergence obtains if the dispersion in real per capita GDP narrows over time. The β-type convergence theory has been tested and confirmed for per-capita GDP levels and growth across high- and low-GDP countries in Africa and the OECD [6,29]. After correcting the major econometric faults of earlier research, Tsangarides [51] reported that the GDP in both the African and OECD countries tend to converge at the rate of 10%. This contrasts sharply with the 2–3% earlier reported and has implications for transitional dynamics. Newhouse [30] had conjectured that the major factors driving medical care cost increases in advanced economies tend “. . .to be common across these countries”. Consequently, Barro and Sala-i-Martin [4] tested the β-type convergence predictions for GDP growth models using health related determinants. Per capita real GDP growth depends positively on the initial endowments of human capital, e.g., educational attainments and health, inversely on the ratio of government consumption spending to GDP, and negatively on measured distortions of markets and political stability. More recently, Barros [5] econometrically modeled the determinants of health care expenditure growth of OECD countries. While finding a clear reduction in health expenditure growth share of GDP he could not explain the slowdown in the growth of health expenditures. Finding convergence, its rate, and time path have implications for differentiated tax policies and public investment planning of national health infrastructures. Therefore, the novel and inter-related goals of this research follows. First, per-capita health care expenditure growth is modeled as dependent on health system policy design variables (e.g., public integrated or not), demographic, and ma-

A.A. OKUNADE ET AL.

jor economic factors. Second, the general form of jackknife resampling is then implemented and used to re-estimate the model. The jackknife, a highly powerful sample reuse statistical methodology designed to squeeze the data dry, is used to enlarge the sample data and the model re-estimated in order to generate parameter estimates that approximate the true population parameter [45]. The current study essentially exploits the bias reducing capacity of the jackknife to generate additional samples for use in the maximum likelihood (ML) econometric model estimation to produce improved parameter estimates and related measures of precision and fit. So far, econometric models of health care cost, demand, and expenditure have rarely implemented this bias-reduction technique for data expansion [33]. Third, within the framework of growth convergence theory, a long run analysis, functional form optimality of the empirical model is tested. This is because past studies largely relied on ad hoc linear or doublelog model specification without first testing for its consistency with the observed data structures. Testing the validity of degenerate (e.g., log–log) functional forms in health expenditure growth models is accomplished by first specifying the more general, skew-correcting, Box–Cox [8] power family transformations model, and then applying the likelihood ratio (LR) test to select the optimal functional form for modeling the data (see table 2, note b) so as to reduce bias that would be induced by fitting erroneous models. The rest of this paper proceeds as follows. Section 2 looks at the study data, specification of the model, and econometric estimation. Section 3 is on the jackknife resampling methodology used to resample from the original data set for achieving improved parameter estimation. Section 4 is on empirical results and study findings, and section 5 concludes with policy implications of the major findings and suggests an agenda for future research.

2. Data and the theoretical model The data (number of OECD countries) span 1968–72 (17), 1973–77 (19), 1978–82 (21), 1983–87 (25), 1988–92 (25), and 1993–97 (25) growth rate sub-periods, respectively. Data came from CREDES-OECD 1998. The specific determinants included in the healthcare expenditure models are largely atheoretical [16,43] and are, to some extent primarily driven by data availability and consistent measurements of the relevant model variables spanning the period of study. The dependent variable is per capita health care spending growth (HCE) at constant 1990 prices in Purchasing Power Parity (or PPP) currency units. Since health system policy or institutional design variables influence health care spending levels of the OECD countries [16] they are likely to drive health care expenditure growth. The need to adjust for theory uncertainty in standard growth models [7] thus, justifies incorporating institutional design variables affecting demand, funding and supply of health care in different countries. Capturing the roles of the major health care system design variables on health expenditure growth are the 0–1 dum-

HEALTH EXPENDITURE GROWTH OF THE OECD COUNTRIES: JACKKNIFE ESTIMATES

mies for gatekeeping (GK), fee-for-service (FFS), public reimbursement (PR), capitated primary care (CA), and public integration (PI). While no country fits completely in a system design category, a country is classified (i.e., dummy variable value is 1) for the more dominant form. The use of ambulatory care, gatekeeping doctors can stem resource overuse (i.e., moral hazard) because coordinated care can reduce cost growth and raise treatment outcomes. GK countries are Australia, Canada, Denmark, Germany, Iceland, Ireland, Italy, the Netherlands, New Zealand, Norway, Portugal, Spain, and the UK. The fee-for-service reimbursement method, characteristic of systems with multiple insurers and multiple (largely private) providers, uses retrospective fee-for-service systems for ambulatory care and so tends to be more costly. FFS countries comprised Australia, Austria, Belgium, Canada, France, Germany, Greece, Ireland (until March 1989), Italy (until 1977), Japan, Luxemburg, New Zealand, Norway, Switzerland, and the US. Medical care is usually supplied under a reimbursement, contract, or integration system. These methods differ on the structure of relationship between funders of health services and care providers, and on the relative roles of patients and third party payers (e.g., insurance, public agencies) in the apportionment of health funding among the providers [31]. Under the reimbursement approach, providers get paid for services directly by patients (in turn reimbursed in part or fully by the insurer) or by insurers after patient contact and billing. Cost control incentives are fairly weak in public reimbursement (PR) countries, such as Australia, Belgium, France, Italy (until 1978), Japan, Luxembourg, Switzerland and the US. The contract method, largely a prospective agreement between third party payers (e.g., private or public insurance) and providers, grants payers greater controls of health expenditure budgets and its distribution than the reimbursement method. Consumer choice is restricted to providers under contract with funders but other suppliers receive lower reimbursement with patients responsible for higher deductibles, and hospitals are funded on per diem or case mix basis. Capitated care providers receive prospective per diem for primary health services of registered patients and primary doctors decide on referrals for specialty care. CA countries comprised Denmark, Iceland, Ireland (from March 1989), Italy (since 1978), the Netherlands, Spain (until 1983), and the UK. Finally, a public integrated system is one in which an agency, local or central government, controls funding and provision of care. Doctors are salaried and the balance of hospital funding is bulk. The benefits of a PI design, relative to contract and reimbursement structures, include contractual cost savings through internal coordination and administrative simplicity. Countries where PI system dominates are Denmark, Finland, Greece (since 1983), Iceland, Ireland, Italy (since 1979), New Zealand, Norway, Portugal (since 1978), Spain (from 1984), Sweden and UK. The use of ambulatory gatekeepers is a priori expected to reduce health system expenditure growth (i.e., ∂HCE/ ∂GK < 0) because primary care physicians must approve patients for the more expensive specialty care if the specialty

175

doctors are to be reimbursed. Systems that remunerate physicians and other health care providers on the basis of services rendered are likely to be more expensive (∂HCE/∂FFS > 0). Health systems that pay providers a fee to cover the primary care of registered patients are expected to moderate the growth of health care costs (i.e., ∂HCE/∂CA < 0) since there are more effective economic incentives to cut health care cost. The basic organization of health care systems includes public reimbursement (PR) and public integrated (PI) models. PR is the most liberal and PI is most stringent. The a priori expected impacts are ∂HCE/∂PR < 0 (public reimbursement systems tend to raise health care expenditure growth, all else equal) and ∂HCE/∂PI < 0 (public integrated health care systems are more likely to restrain HCE growth). The remaining explanatory variables in our model are the respective period’s beginning or first year level of health care expenditure divided by 1000 (LEV), the square of the level variable (i.e., LEV2 ), per capita GDP growth (GDP), growth in the relative price of health care (RELP), growth in doctor density (doctors per 1000 population, PHY), rates of growth in the population segments 65 years (POP65) and 0, ∂HCE/∂RELP > 0, ∂HCE/ ∂PHY > 0, ∂HCE/∂POP65 > 0, ∂HCE/∂POP15 > 0 and ∂HCE/∂GOV < 0. PHY captures the competitive effects of high physician density that lead to reduced market share per provider and creation of demand for therapeutically unnecessary services (i.e., supplier-induced demand) or rising consumer taste for affluence leading to rising health care spending (i.e., demand-induced supply). Theoretically, rising per capita GDP, high price of health care, high physician density, and high-consuming population segments are expected to raise health spending whereas greater government financing or provision of health care is theoretically likely to reduce the rate of expenditure growth. The general specification of the proposed theoretical model of health expenditure growth is ˙ + α4 RELP ˙ = α0 + α1 LEV + α2 LEV2 + α3 GDP ˙ HCE ˙ + α7 POP15 ˙ + α8 GOV ˙ ˙ + α6 POP65 + α5 PHY + [vector of institutional dummies] + ν, (1) where the variables and parameter estimates (α and ) of the determinants in the above model are as previously defined and the stochastic term ν, assumed well behaved, is the residual. 3. Jackknife resampling methodology Sample reuse plans are largely the bootstrap and the jackknife, but there are different designs within each scheme (e.g., one-delete, n-delete, and variable jackknife for the jackknife method, and the weighted and unweighted bootstrap for the

176

A.A. OKUNADE ET AL.

bootstrap). Quenouille [42] pioneered the jackknife method for reducing bias of a serial correlation estimator by breaking the sample into g groups of size h each, n = gh. More recent modifications of the jackknife include Efron [11], Tukey [53], Wu [56], and others. The two separate aspects of jackknife resampling plans are bias reduction and interval estimation. Resampling uses the full data set at hand, or a given datagenerating mechanism (e.g., coin toss) that is a model of the process of interest, to produce a new sample of simulated data and to examine the results of those samples. Wu [56] proposed a class of weighted least squares jackknife variance estimators for the least squares estimator by deleting any fixed number of data observations at a time. They are unbiased for homoskedastic residuals and the special case of one-delete jackknife is almost unbiased for heteroskedastic errors (p. 1261). The most popular one-delete jackknife is used in this study as it is particularly useful for bias and variance estimation in smooth estimators (ibid). Simon and Bruce [48] lists the steps for implementing resampling. First, construct a simulated ‘universe’ of the randomizing mechanism whose composition is similar to the universe whose behavior we investigate. Second, specify the procedure that produces a pseudo-sample, which simulates the real life sample of interest. Third, compute the probability of interest from the outcomes of the resampling plan trials. Sample reuse and the Monte Carlo (MC) methods differ; MC relies on the standard normal theory, an unnecessary assumption in resampling plan designs. One can motivate implementation of the one-delete jackknife plan by modeling the response y as y(t) = a exp(−mt) for integer values of t [27]. The tabulated data {y(0), y(1), y(2), y(3), . . . , y(L)} are a list of values {yi (0), yi (1), yi (2), . . . , yi (L)}, where i = 1, 2, . . . , n is the labeled list of these measurements (ibid). Since fluctuations in the data are correlated, the usual χ 2 distribution test is no longer valid for selecting the best fit unless modified to account for dependence. The need to raise confidence in the estimates makes the relatively robust jackknife a solid alternative for determining the propagation of error from the data to the parameters. So, given a sample of n measurements, the one-delete jackknife deletes the first measurement from the entire data set, resulting in a jackknife data set of n − 1 resampled values. Statistical analysis is then performed on the reduced sample, yielding a measured value of a parameter, say βJ1 . The first deleted observation is replaced and the next one (that is, second observation) is deleted, yielding a new measured value of the parameter, call it βJ2 . This process, repeated for each set i in the sample, generates a set of parameter values {mJi , i = 1, . . . , n}. The standard error, where m is the result of fitting the complete sample, is computed as square root of the jackknife variance formula σˆ m2

= (n − 1)

n

(mJi − m)2 /n.

(2)

i=1

The adjustment factor (n − 1)/n is required, as the jackknife variates tend to be smaller. Confidence interval estimates for

Table 1 Descriptive data (at means, in percentages, except LEV) for each growth period. 1968–72 1973–77 1978–82 1983–87 1988–92 1993–97 HCE LEV GDP RELP PHY POP65 POP15 GOV Countries

12.81 139.14 9.43 1.20 2.89 1.35 −0.56 1.09 17

14.29 222.84 10.29 0.52 4.41 1.50 −1.32 1.14 19

12.65 427.04 10.28 0.58 4.36 1.07 −1.69 1.16 21

7.34 724.68 6.52 0.65 3.03 0.88 −1.63 −0.18 25

8.55 1028.20 6.26 0.48 2.92 1.37 −1.14 1.03

4.77 1456.92 4.51 −0.54 2.07 1.28 −0.75 −0.14

25

25

the parameter m are calculated by subtracting (or adding) the product of the critical values of the t-distribution with n − 1 degrees of freedom and the bootstrap estimate of the standard error of m. The jackknife is further capable of yielding an estimate of the sampling bias since there may be estimates in which the parameter estimates drift up or down from its true value, particularly if the data sample is too small. If this is the case, the estimate m obtained from fitting the entire n data points may be higher or lower than the true value. Therefore, deletion of a measurement in one-delete jackknife is likely to amplify the bias. This effect is measured by comparing the mean of the jackknife values mJi , say mJ , with m, that of fitting the full data set. If (mJi − m) is not 0, the bias is corrected using the formula m ˜ = m − (n − 1)(mJ − m).

(3)

4. Empirical findings and discussion 4.1. Evaluating the statistical adequacy of the empirical model estimates Table 1, containing mean data values of each 5-year growth period, suggests that the OECD health spending growth rates decreased from roughly 12.8% in 1968–72 to 4.8% in 1993– 97. One observes a similar pattern for average GDP growth rates. The classic Box–Cox (CBC) model is the most flexible the data accommodated for each 5-year period model, and is compared with the linear and log-linear models using the likelihood ratio test. The hypothesis test findings (table 2) reinforce at least two conclusions. First, the response distribution, health care expenditure, is skewed and so using the standard OLS estimation method, rather than the ML technique for the CBC, is not justifiable [37]. Second, no functional form bias is imposed if the 1968–72, 1973–77 and 1978–82 data are modeled using linear-linear (untransformed data) specification but the CBC form model dominates over both the simple linear-linear or double log models for 1983– 87 and 1988–93 data. The post-1983 change in functional forms, potentially deriving from changes in health care system policies or signaling periodic divergence or convergence, cautions against fitting one uniform model across all growth


Table 2 Likelihood ratio (LR) χ 2 hypothesis test statistic results for functional forms.a 1968–72 1973–77 1978–82 1983–87 1988–92 1993–97 CBCb vs linear 1.2438 1.7878 0.7842 1.1302 2.8148∗ 3.0416∗ CBC vs log-linear 0.4222 0.0490 0.0061 6.4406∗ 12.6048∗∗∗ 6.1450∗∗ a The LR test statistic is 2[L(λ ) − L(λ )] ∼ χ 2 , where U and R are, U R df

respectively, the unrestricted and restricted maximum log likelihood function values at λU and λR . Numerically, λR = 1 for linear (untransformed) model, and λU , real-valued, is iteratively estimated jointly with the regression model parameters. The critical χ 2 (1) values at the 0.01, 0.05, and 0.10 levels are, respectively, 6.63, 3.84, and 2.71. Statistical significance at these respective levels are indicated with ∗∗∗ , ∗∗ , and ∗ . Caution that this LR test is only approximately correct, since the Box–Cox likelihood function is a quasi-likelihood and not the true likelihood function [1]. Moreover, in small samples a bootstrap version of the generalized likelihood ratio (GLR) test would be more appropriate [55]. b The Classic Box–Cox (CBC) model takes the form y (λ) = Xβ + ε, where y (λ) = [y λ−1 /λ] for λ = 0 and y ≈ log y as λ → 0, is the powertransformed response, X is the untransformed (i.e., linear) set of the independent variables, β is the vector of the regression slopes, and ε the residual. The slope parameter estimates β are jointly estimated with the optimal value of the transformation parameter λ. The CBC model is meritorious because it is capable of correcting for the skewed distribution of the response variable y, such as health expenditure in this study, and can generalize the functional form model to reduce functional form specification bias. (See, for details: Box and Cox [8], Greene [18], Okunade [37], Okunade et al. [38], and Parkin et al. [39].)

data periods. The Box–Cox power transformation to symmetric distribution with homoskedastic residuals is convex (i.e., optimal λ > 1) if the response data are left skewed and concave (i.e., optimal λ < 1) for right skewed data [10]. Since 25 OECD countries existed for the study period but 20 had all the data necessary to implement the regression models, one may ask whether the application of asymptotic distributions for inferences for 20 countries (in tables 2 and 3(a)) in the absence of data resampling plans, makes sense. Carroll and Ruppert [10] explored alternative computational methods (Fisher information, concentrated Fisher information, pseudo-model, fixed λ, M-estimation, bootstrap) for the standard errors in a transform-both-sides model, and found none to be globally superior (e.g., the pseudo-model method does not consistently estimate the standard error of ˆ and the M-estimation lacks λˆ or the covariance of λˆ with β, wide application). Published studies of OECD health spending (e.g., Parkin et al. [39], Gerdtham [15]) and research studies in business and the social sciences (e.g., Link et al. [24]) estimated Box–Cox power models and applied maximum likelihood techniques to less than 20 data observations (see, Amemiya [1], for more citations). These studies drew statistical inferences for models using the central F distribution and t tests since under the null, the non-centrality parameter of the F distribution is zero. Such uses of asymptotic tests for inferences are further justifiable on practical grounds as the theoretical power of the Box–Cox method for small sample experiments is not well developed. Amemiya [1] introduced the Box–Cox λ parameter in the log-likelihood function, but the function is penalized the greater the departure of λ from 1. Greene [18] further esti-

177

mated a Box–Cox regression model of money demand with 20 data points and used competing computational methods to generate standard errors for coefficient estimates in the regression model. The implications from numerical estimates, excepting the intercept and one computational method, are fairly similar. He ascribed the differences to “small sample variation” and concluded: “there is no obvious answer to which (computational method) is right; all are. The choice is usually based on ease of computation, . . .”. Thus, it is possible for the significance level or even the sign of some regressors in the estimated model to change if some other statistical methodology is used. This possibility however, “doesn’t imply that other methods are bad” [13]. Hartigan [20], in his published comments on Efron and Tibshirani [13], and drawing further from his wealth of personal experience working with sub-sampling methods, declared that he has “some general reservations about the bootstrap”. Consequently, jackknife standard errors, competitor to the bootstrap, are computed (using equation (2) formula in section 3 of this paper) and used for determining the significance of right hand side variables (see, table 3(b)). 4.2. Discussion of the empirical regression model results Table 3(a) presents Box–Cox regression model estimates for each growth epoch data, and for the pooled 1968–97 data. The F statistic for each model suggests that the determinants collectively explain a significant portion of the total variance in health care expenditure. The adjusted R 2 s of the growth regressions range from 0.24 to 0.57. Consistent with Newhouse [30], the 50–60% unexplained variation in each growth period models may be attributable to the ‘march of science’ (i.e., technological change) or the combined effects of the minor determinants not captured in the estimated models. With the exception of the panel model, the Box–Cox λ parameter estimates differing significantly from unity across the growth models suggests that the OECD data rejected the commonly fitted linear or log–log models. Differences in the estimated λ values agree with an earlier study suggesting no single functional form dominates in OECD health spending models. Caution here that the parameter estimates (within and across tables 3(a) and 3(b)) are not directly comparable because the λ estimates fitting the observed data represent different transformations. The regression estimates indicate that the dependent variable, evaluated at its level value in the first year of each growth interval model, is consistently significant in explaining growth in health care spending. The negative LEVEL and positive LEVEL2 variable in all except the 1973–77 growth epoch means that health systems with lower initial health care expenditure per capita later grow at faster rates, ceteris paribus. The positive LEVEL and negative LEVEL2 coefficient signs for the 1973–77 growth period shows that systems with initially high HCE later grow at a slower pace. Consequently, the OECD health spending growth is consistent with the conditional β convergence theory in Barro and Sala-iMartin [4]. GDP growth on the demand side of the health care

178

A.A. OKUNADE ET AL.

Table 3 (a) Growth period-specific Box–Cox transformation regression results.a Independent variable

1968–72 period

1973–77 period

1978–82 period

1983–87 period

1988–92 period

Constant LEVEL LEVEL2 GDP RELP PHY POP65 POP15 GOV GK FFS PR CA PI

0.750 −0.410∗ 1.061∗ −0.004∗ 0.0007 0.0001 0.0001 −0.004∗ −0.001∗ 0.007 −0.010∗ 0.005 −0.014∗ −0.009∗

2.093 3.999∗ −10.230∗ −0.019 0.012∗ −0.020 −0.026 −0.007 0.002 −0.029 −0.014 0.036 0.004 −0.137∗

4.219 −3.315∗ 2.613∗ −0.025 −0.024 −0.035 −0.097∗ −0.002 −0.121∗ −0.227∗ 0.105 −0.025 0.482∗ 0.228∗

−9.940 −20.477∗ 9.628 4.164∗ 1.092 2.303∗ 0.010 1.656 −1.011 −0.118 0.996 4.883 2.488 0.051

41.449 −52.201∗ 20.793∗ 1.830∗ −1.103 2.876∗ 3.351 1.879 −1.542 21.633∗ −13.765∗ 9.690 −20.770∗ −5.107

8.100 −8.101∗ 1.673∗ 0.194 0.126∗ −0.149 0.304 0.221∗ −0.326∗ 0.804 1.375∗ 1.022 0.792 −1.025∗

0.037∗∗ −0.054∗∗∗ 0.015∗∗ 0.866∗∗∗ 0.416∗∗∗ 0.047 0.455∗∗ 0.036 −0.137 0.003 0.007 0.010∗ 0.051 0.004

Summary statistics Number of countries Adjusted R 2 F -test statistic Jarque–Bera testc Durbin–Watson testd Geary (runs) test

17 0.429 10.91∗∗∗ 0.957 1.752 −1.158

19 0.233 5.40∗∗∗ 1.609 2.218 −0.128

25 0.580 8.05∗∗∗ 0.898 2.175 0.622

25 0.393 4.99∗∗∗ 0.486 1.284 −1.015

17 0.899b 60.83∗∗∗

Box–Cox λˆ estimatee

−1.46

−0.20

21 0.440 6.63∗∗∗ 0.243 1.362 0.235

25 0.513 6.69∗∗∗ 0.792 2.339 −0.548

0.11

1.63

1.90

1993–97 period

0.58

1968–1997 panel data

2.56 1.00

(b) Jackknife resampling regression estimates of growth epoch-specific Box–Cox transformation regression results, 1968–1997.a Independent variable Constant LEVEL LEVEL2 GDP RELP PHY POP65 POP15 GOV GK FFS PR CA PI

1968–72 period 56.99 −297.4 8.73 −2.37 −0.20 0.07 −0.33 −3.10 −0.63 6.47 −7.09 3.71 −8.20 −6.14

1973–77 period 3.13 0.33 −3.60 −0.05 0.03 −0.03 −0.13 −0.09 0.001 0.26 −0.03 0.24 0.11 −0.42

Summary statistics Number of countries Adjusted R 2 F -test statistic Jarque–Bera testc Durbin–Watson testd Geary (runs) test

17 0.4940 54.67∗∗∗ 2.8719 1.6890 1.0047

19 0.3088 15.60∗∗∗ 1.0987 2.1784 0.2620

Box–Cox λ estimatee

−1.3818

−0.3374

1978–82 period 106.2 −106.13 66.01 −3.98 0.52 −2.38 −3.86 −0.80 −4.54 −6.78 −0.40 3.26 25.25 9.37 21 0.4703 0.87 0.9481 1.4319 −0.0566 0.1571

1983–87 period

1988–92 period

1993–97 period

1968–1997 panel data

−11.3 −23.19 11.39 4.47∗∗ 1.15∗∗ 2.50∗ 0.09 1.77 −1.07∗∗ −0.28 0.88 5.43 2.32 0.04

43.90∗∗∗ −60.2∗∗∗ 24.55∗∗ 2.23 −0.92 3.76 3.68 2.00∗ −1.02 25.25∗ −14.4∗∗ 11.54 −23.5∗∗ −4.70

8.376∗∗∗ −8.40∗∗∗ 1.79∗∗ 0.22 0.13∗∗∗ −0.19 0.32∗ 0.22∗∗ −0.32∗∗∗ 0.85 1.34∗∗ 0.099 0.78 −1.13∗

0.0373∗∗∗ −0.0822 0.0147∗∗∗ 0.8629∗∗∗ 0.4112∗∗∗ 0.0498∗∗ 0.4561∗∗∗ 0.0418 −0.1274∗∗∗ 0.0030 0.0071∗∗∗ 0.0101∗∗∗ 0.0520∗ 0.0042∗∗

25 0.4907 7.02∗∗∗ 0.9352 2.324 −0.5343

25 0.5856 8.62∗∗∗ 0.9272 2.219 0.4447

25 0.4160 5.58∗∗∗ 0.5308 1.4201 −0.8863

17 0.8851b 60.83∗∗∗ 1.7182 1.7517 −2.5611

1.6048

1.9324

0.5996

1.000

a The symbols ∗∗∗ , ∗∗ , and ∗ , respectively, represent statistical significance at the 0.01, 0.05 and 0.10 levels. b Buse R 2 for panel regression. c Normality test χ 2 (df = 2) = 5.99 at the 0.01 and 4.61 at the 0.05 levels, respectively. Caution that this asymptotic test can only be approximate for small

samples. d Specification test of spatial autocorrelation for geographically contiguous data. e Jointly estimated and optimal with the regression model parameter estimates.

market also fueled significant HCE growth during 1983–87 and 1988–92 epochs and the 1968–97 panel data, significantly moderated HCE growth in 1968–72 and had a neutral effect

for 1973–77 and 1978–82 growth time periods. The GDP growth elasticity of about 0.9 suggests that the income elasticity is at or around unity for health care spending. A 100 per-


cent growth of the real GDP would tend to induce a 90 percent growth in real per capita HCE signals that health care behaves like a technical necessity. RELP, the relative price of health care, raising HCE growth significantly for the 1973–77 and 1993–97 periods and the 1968–97 panel, supports an earlier finding for the US [28]. Consistent with supplier inducement, the growth in physician density, PHY, significant for 1983–87 and 1988–92 epochs, raised HCE growth. Due to the strong public finance implications for sectoral budgetary allocations, there is a rising macroeconomic policy interest in whether the rapidly burgeoning elderly population stock in most countries raises national HCE. Surprisingly, the hypothesis findings in past literature are mixed. Zweifel et al. [57], for example, finding strong support for this hypothesis in the last two years of life for Swiss age 65–95 in 1992 using private health insurance (microdata) records, suggests that one ought not to be as concerned if the highly skewed elderly health care spending is distributed over the entire country’s population lifetime. Getzen [17] critiqued this assertion on the methodological inconsistency of eliciting macroeconomic policy prescriptions for HCE from microdata structures. Seshamani and Gray [47], using a 29-year longitudinal English data set and a two-part random effects panel data model, most recently found proximity to death and not age per se as a major demographic driver of rising health care costs. They found that approaching death raises costs up to 15 years before death and that the large ten fold cost increases from 5 years before death to the last year of life dwarfs the 30% rise in costs from age 65 to 85. Nonetheless, the growth of elderly population age 65 and above in our study, POP65, consistently raised HCE growth rate insignificantly effective 1983–87. Barros [5], using a somewhat similar methodology and 1960–90 growth data partitioned in ten-year intervals, reported a negative but insignificant effect of POP65 on the health spending of 24 OECD countries. Serup-Hansen et al. [46] fairly recently reported that ageing per se and high health care costs in the last year of life matter for projecting greater health care costs in Denmark. The positive and significant POP65 effect for the 1968–97 growth period panel data mimics the recent cointegration model findings of Murthy and Okunade [28] for the US based on 1960–96 time-series levels data. The young, 15 years old and under, also appears to be high cost consumers of health care resources. This less frequently tested hypothesis is captured here using variable POP15. The effect on HCE is mixed: significantly negative for the 1968–72 period, but significantly positive during the 1993–97 period. Results of the 1968–97 panel data model indicate that POP15 is not a significant major driver of health care spending growth. 4.3. Cross country differences in health system designs: do they influence cost growth? The OECD country governments are interested in the systematic influences of health care system design characteristics on performance, such as health expenditure growth and outcomes. Our growth convergence estimates are suitable

179

for addressing system policies that target long run spending controls. The effectiveness of performance measurement systems, from the principal (e.g., patient)-agent (e.g., the clinician, third party payers) perspective, depends on factors such as the incentives for clinicians to document and make sound medical decisions [49]. Countries with very fragile primary care infrastructures have poorer health performance, and a subset of other policy design attributes distinguishes those countries with overall good health [50]. While generally known to systematically impact health expenditure levels [16], our results indicate that the health system institutional controls (GK, FFS, PR, CA, PI) influence health spending growth rates but not uniformly across the growth sub-periods. Nonetheless, GOV, systems with greater government involvements in health care provision and financing consistently moderated spending growth significantly for the 1968–72, 1978–82, and 1993–97 periods. GK, systems with gate-keeping primary care physicians, reduced HCE growth only in three consecutive growth periods: 1973–77, 1978–82 and 1983–87. When compared with the capitation (CAP) design, a feefor-service system (FFS) motivates medical practitioners to raise patient contact volume, increase the number of medical procedures performed (demand creation), and reduce the mean length of patient consultations to maximize reimbursements. Our results show that FFS successfully slowed HCE growth rates significantly during 1968–72 and 1988–92 periods – periods in which CAP also succeeded the most in reducing HCE growth. Public integration systems, PI, reduced health expenditure growth in all but the 1983–87 epoch. Finally, the potentially confounding effects of intervening policy initiatives that operate simultaneously may account for the lack of consistent effects of policy controls on HCE growth across periods. Past research using levels data indicates that institutional factors are consistently more important than when growth data are modeled. Our paper, using annual growth data, finds a number of significant effects of health system institutional controls. However, an earlier paper [5], similarly using growth data partitioned in 10-year growth intervals rather than the 5-year growth used in this paper), a limited set of policy controls, and an a priori limited functional form model, detected none of the system policy controls to be significant. 4.4. Comparison of the jackknife with standard Box–Cox regression models Resampling plans, not previously used in the OECD health care expenditure models, are useful for generating multiple samples of n − j (j 1) from one sample of n observations. Jackknife resampling methodology is therefore, applied to expand the number of samples and sensitize the regression estimates to the exclusion of specific countries and time periods from analysis. The jackknife estimates yield tighter confidence limits and greater levels of statistical significance than the standard regressions based on one original sample of n observations. The original data set for Box–Cox transforma-

180

tion, resampling and estimation averaged 20 countries with complete data for the fitted models. The absence of computational convergence difficulties in this paper during estimation suggests the fitted models are adequate or correctly parameterized [10]. Theoretically, the parameter estimates of resampling models are expected to approach those of the true population distribution. Table 3(b) presents Box–Cox regression model estimates based on the one-delete jackknife sample reuse scheme. The contrasts, across growth epochs and the panel, with the original sample data regressions in table 3(a), are most illuminating. The summary statistics within and between each of tables 3(a) and 3(b) are not directly comparable, since their functional forms differ. Moreover, the reported Buse R 2 for the panel model estimates in tables 3(a) and 3(b), differs from standard R 2 , and the standard F distribution is assumed for testing the joint significance of the right hand side variables in table 3(b). Panel data models are richer since they control for heterogeneity and omitted variables. Heterogeneity across countries can be captured using country-specific indicator variables and specific institutional or health system design indicators for country groups. Preliminary model estimates showed strong collinearity among country-specific indicators and the policy variables set. Since the latter variables are more informative and important in this study, and because the observations for resampling are limited, country-specific dummies that proxy permanent effects were excluded from final estimation. Some major differences between jackknife and standard regressions persist, however. First, the 1968–72, 1973–77, and 1978–82 growth period regressions are not significant according to the hypothesized determinants of HE growth. Second, each of the growth model’s overall F -statistic for the 1983–87, 1988–92, and 1993–97, as well as the 1968– 97 panel data show higher statistical significance. Moreover, compared with their corresponding estimates based on the original sample (table 3(a)), the jackknife growth models (table 3(b)) gained in both number of statistically significant regressors and increased precision or statistical significance levels of the parameter estimates. Fourth, slightly different functional forms (see λ estimates) characterize the original as compared with the jackknife resampling models for similar growth epochs. Fifth, and quite important for policy is the greater statistical significance of all but one of the health system’s institutional or policy variables in the jackknife panel regression model estimates (table 3, column 8). This contrasts sharply with the corresponding panel model estimates in table 3(a), a finding that should not surprise because time-series and panel models tend to differ in this line of inquiry [16]. Finally, the use of jackknife for testing data sensitivity is illustrated in table 4 reporting parameter estimates for the 1993–97 period following the deletion of a country, one at a time with replacement, from iteration. Sensitivity is indicated if there is a significant change in the parameter estimates of the model when a country is deleted (e.g., while 0.45 < λˆ 0.70, the λˆ rose to 1.02 with data for Greece omitted from estimation).

A.A. OKUNADE ET AL.

5. Conclusions 5.1. Summary of the research objectives and findings Health care expenditure models of the OECD countries almost exclusively used data calibrated in levels. Investigating the determinants of health spending growth rates appears more interesting from cost-containment viewpoints. This is because for many reasons (e.g., technological change and insurance subsidy, population growth, resource constraints, etc.) health expenditures are expected to rise over time. However, the rate of the rise is the more important concern from cost containment policy perspectives. This research, using econometric and statistical methodologies, tested hypotheses linking per-capita health spending growth in the OECD countries to the major demographic, health system policy or design attributes, and economic factors. The Box–Cox regression models applied to the annual growth rates data partitioned at 5-year intervals (1968–72, 1973–77, . . . , 1993–97) and the 1968–97 growth panel data were separately modeled. A sample reuse method was then implemented for the Box–Cox models using growth rates data using the complete 1968–97 OECD panel dataset. LR statistic was used for testing adequacy of traditional degenerate forms against the more general Box–Cox specification. The Box–Cox, based on ML estimation method, and data resampling plans are fairly powerful tools in statistical methodology. Sample reuse technology offers potentials for highly accurate methods of inference [23]. The inferential aspect of statistical analysis includes constructing a confidence region, attaching a standard error to an estimate, conducting hypotheses tests, or selecting the appropriate regression model from among the alternatives [9,23]. Regrettably, applications of sample reuse for modeling economic phenomena, and specifically in healthcare economics and management policies, are quite rare. Consequently, this study illustrated with the estimation of an econometric growth model of factors driving the health care expenditures in OECD countries. Our results showed differential impacts for certain determinants across growth sub-periods. This study confirms that the determinants of OECD health expenditure levels in past studies do not necessarily explain health expenditure growth rates. The research results are consistent with the Barro and Sala-i-Martin [4] hypothesis that health expenditure of economies or systems (e.g., OECD countries) with similar tastes and technologies indeed converge to a steady state. The effectiveness of policy design characteristics of health systems (e.g., gate-keeping, integrated systems, fee-for-service, capitation, public integration, government provision and financing in total health spending) tends to vary across growth epochs – a finding supporting Gerdtham [15] but contrasting Barros [5]. Contrary to past research, no single functional form model (e.g., the log–log) dominates all of the 5-year growth epochs. That is, the functional form linearity or log–log assumption may yield biased results if applied, without explicit testing, to the entire data set. The one-delete jackknife data reuse


181

Table 4 Jackknife regression results, 1993–1997 growth data. Independent

Excluded country

variable

Australia

Austria

Belgium

Canada

Denmark

Finland

France

Germany

Constant LEV LEV2 GDP RELP PHY POP65 POP15 GOV GK FF PR CA PI

7.6254 −7.7685 1.5852 0.2318 0.1175 −0.1492 0.2861 0.2204 −0.3168 0.8309 1.3743 1.1804 0.7391 −0.9955

7.7669 −7.6839 1.5842 0.1929 0.1194 −0.2557 0.3436 0.2332 −0.2979 0.7993 1.2410 1.1558 0.8550 −0.9226

8.2296 −8.0201 1.6136 0.2069 0.1149 −0.1986 0.2533 0.2354 −0.3493 0.8692 1.2946 1.3756 0.7180 −1.1307

13.9910 −7.7324 1.5095 0.0577 0.1333 −1.5148 0.3713 0.0756 −0.4177 −0.8486 1.7144 −2.2038 1.7712 −2.9600

7.9764 −7.8234 1.6071 0.1671 0.1238 −0.1608 0.3025 0.2187 −0.2654 0.7779 1.3861 0.9453 0.8149 −0.9878

5.9703 −6.8085 1.4401 0.3546 0.1305 0.2612 0.2660 0.0461 −0.3685 −0.1810 1.1787 0.5391 0.1185 −0.2131

8.2938 −8.3735 1.7384 0.2163 0.1267 −0.1584 0.2888 0.2252 −0.3638 0.8479 1.3378 0.9979 0.7262 −1.0920

9.3672 −8.8141 1.7882 0.1491 0.1155 −0.2095 0.1511 0.1856 −0.3426 0.5130 1.1352 1.3213 0.9204 −0.9134

Summary statistics Box–Cox λ 0.5700 0.7266 R2 0.3711 Adjusted R 2 F statistic 4.9050 Durbin–Watson 1.2151 Runs test −1.2264 Jarque–Bera 0.4810 normality test

0.5500 0.7228 0.3624 4.8140 1.1166 −1.6469 0.5894

0.5600 0.7504 0.4258 5.5490 1.1246 −0.8059 0.6489

0.6800 0.8052 0.5521 7.6330 1.8397 0.5732 0.8694

0.5700 0.7217 0.3600 4.7890 1.1958 −1.6697 0.6564

0.6700 0.7423 0.4073 5.3180 1.4303 −0.8348 0.3592

0.5800 0.7251 0.3677 4.8690 1.3555 −1.6469 0.6742

0.5800 0.7475 0.4192 5.4640 1.5918 −0.8059 0.3923

Greece

Iceland

Ireland

Italy

5.4362 13.6840 −5.3416 −11.1440 1.0176 2.2612 0.7765 −0.1239 0.0832 0.1141 −0.6239 0.0200 0.4299 −0.2751 0.4161 0.0998 −0.0238 −0.5588 3.6640 0.4026 −0.9026 0.6558 3.3202 0.9611 −0.7890 −0.1004 −2.7391 −1.8252

7.7637 −7.7825 1.6265 0.1634 0.1311 −0.1360 0.3841 0.2543 −0.3367 0.8556 1.3123 0.9588 0.6497 −1.0675

5.5899 −5.9523 1.2942 0.0244 0.1280 0.1115 0.4855 0.1850 −0.2159 0.5765 1.2841 0.6125 1.2210 −0.4491

0.5500 0.7201 0.3563 4.7510 1.6396 −0.8348 0.3567

0.4400 0.7595 0.4469 5.8300 1.5217 −0.8059 1.1980

1.0200 0.7869 0.5100 6.8190 2.0872 0.0350 0.1810

0.5400 0.8240 0.5953 8.6450 1.3486 −0.8348 0.0624

Table 4 (Continued.) Independent

Excluded country

variable

Japan

Korea

Luxembourg

Netherlands

New Zealand

Norway

Portugal

Spain

Sweden

Switzerland

Turkey

Constant LEV LEV2 GDP RELP PHY POP65 POP15 GOV GK FF PR CA PI

8.0772 −8.1787 1.6879 0.1890 0.1349 −0.1473 0.3681 0.2378 −0.3404 0.8438 1.4192 1.1020 0.8310 −1.0438

7.9154 −7.7730 1.5936 0.2423 0.1395 −0.1068 0.2924 0.2228 −0.2163 0.6819 1.1002 0.7665 0.7341 −1.2791

8.0467 −8.0566 1.6712 0.1765 0.1235 −0.1254 0.3000 0.2105 −0.3119 0.7708 1.3971 0.9080 0.8161 −0.9671

8.3456 −8.2183 1.6963 0.1823 0.1280 −0.1529 0.2997 0.2252 −0.3353 0.8126 1.3389 0.9963 0.8309 −1.1208

9.7377 −10.3100 2.1118 0.2458 0.1556 −0.2347 0.4004 0.3094 −0.3834 1.1395 2.2734 1.1404 1.0416 −0.9231

7.3791 −7.5558 1.5527 0.2154 0.1069 −0.1506 0.2586 0.2227 −0.2648 0.8792 1.5902 0.9509 0.7180 −0.7702

7.4147 −7.8038 1.6191 0.2120 0.1240 −0.0491 0.3134 0.2028 −0.3022 0.5014 1.6979 0.7871 1.0856 −0.8954

7.8844 −7.9682 1.6423 0.2066 0.1208 −0.1471 0.2902 0.2378 −0.3155 0.7470 1.4607 0.9839 0.8965 −1.0360

5.0198 −6.6739 1.3794 0.3112 0.0865 −0.0684 0.2168 −0.1245 −0.2887 1.3679 1.5422 1.4465 0.7003 −0.8835

6.9541 −8.5553 1.8093 0.3698 0.1726 −0.2784 0.7360 0.4343 −0.3923 1.3938 1.8015 0.9903 1.0049 −1.1680

7.9640 −7.9322 1.6420 0.1587 0.1933 −0.1086 0.3124 0.2136 −0.3272 0.7251 1.2904 0.9126 0.8267 −1.0056

Summary statistics Box–Cox λ 0.5800 0.5300 0.5800 0.5800 0.7000 0.5800 0.6000 0.5700 0.7193 0.7039 0.7219 0.7210 0.7736 0.7163 0.7223 0.7209 R2 0.3544 0.3189 0.3604 0.3584 0.4792 0.3474 0.3613 0.3580 Adjusted R 2 F statistic 4.7310 4.3880 4.7920 4.7720 6.3070 4.6600 4.8020 4.7680 Durbin–Watson 1.2801 1.4201 1.2915 1.3267 1.3488 1.5028 1.2946 1.2731 Runs test −0.8059 −0.8348 −1.6697 −0.8059 −0.7165 −0.7165 −0.8348 −0.8348 Jarque–Bera 0.3983 0.5632 0.4640 0.5194 0.3382 0.3164 0.5344 0.6438 normality test

plan, a highly powerful statistical methodology rarely used for modeling economic phenomena, was used for resampling the original data and to sensitize the regression estimates to the exclusion of specific countries and time periods. The

UK

USA

9.4352 11.5440 −8.9785 −16.7760 1.8339 4.9026 0.1291 0.2969 0.1324 0.2303 −0.1770 −0.0420 0.2541 0.6400 0.2369 0.3975 −0.2976 −0.4914 0.7191 1.5493 1.5224 1.6960 0.9869 1.7200 1.1759 0.9907 −1.0571 −0.5986

0.4900 0.5900 0.5600 0.6200 0.7000 0.7691 0.7934 0.7190 0.7306 0.8093 0.4688 0.5249 0.3536 0.3803 0.5613 6.1480 7.0920 4.7230 5.0060 7.8330 1.5571 1.5306 1.3530 1.3108 1.5474 0.0000 −1.6469 −0.8348 −0.8059 −1.1464 0.4783 0.8730 0.5347 0.3119 0.8257

Box–Cox models fitted to the larger jackknifed data, yielded tighter confidence limits and higher statistical significance levels, when compared with the usual model fitted to the original sample data. Our GLS panel estimates, showing the GDP

182

elasticity of HCE of around 0.6 suggests that health care behaves like a necessity, tends to reconcile aggregate panel with cross-sectional data findings. 5.2. Suggestions for future research Suggestions for further research include expanding the data set to include the latest data points, and the application of jackknife methods other than the one-delete or the other resampling plans including the bootstrap in models of health expenditure growth. The use of jackknife for testing sensitivity of an omitted country data in model estimation, particularly the outliers, ought to be investigated to adjust for their presence in an econometric model estimation to improve precision of the parameter estimates. Third, while health care spending growth is correlated with policy design variables, the causality direction is uncertain. One may account for this lack of knowledge using policy lags, a procedure not entirely free from reverse causality [22] and an exercise for future research. Fourth, researchers interested in projecting future health care costs are better off shifting demographic paradigms from measurements based on simple population aging to that of ‘time to death’. Fifth, econometric studies of health expenditure assume homogeneous technology across countries, a potentially erroneous assumption [44]. In effect, just as nations differ in their initial level of health spending, augmenting the growth model using each nation’s starting level value of an aggregate medical technology stock is a testable proposition. Finally, modeling the demand for good health through health care spending is an incomplete story, as it needs to accommodate other wide-ranging factors, such as the social environment and activities that augment health status but which fall outside of the formal health care sector. One example is the WHO (World Health Organization) country-specific policy goals under “Health for All”, and they include health promotion and disease prevention [31,32] outside of the formal health care system. These measures are yet to be incorporated in models of health care spending growth as potential decelerators. Finally, future research might test the health expenditure growth convergence thesis for non-OECD nation groups at similar development stages (e.g., middle- and low-income, and the newly industrialized countries). This is because internally consistent nation clusters worldwide (e.g., the newly industrializing countries (NICs), the transition economies) habitually experiment with the health system designs, policies, and institutions of the OECD nations in attempting to also contain their own health care expenditure growth.

Acknowledgements The authors thank the conferees, Ulf-G. Gerdtham, George E. Relyea, the Editors-in-Chief and three anonymous referees of Health Care Management Science for productive comments. They also acknowledge partial funding for this research through The Wang Center for International Business

A.A. OKUNADE ET AL.

and Economic Research (CIBER) at University of Memphis. The usual caveat applies, however. References [1] T. Amemiya, Advanced Econometrics (Harvard University Press, Cambridge, MA, 1985). [2] A.C. Atkinson, Plots, Transformations and Regression. An Introduction to Graphical Methods of Diagnostic Regression Analysis, Oxford Statistical Science Series, Vol. 1 (Oxford University Press, Oxford, UK, 1985). [3] S.E. Atkinson and P.W. Wilson, The bias of bootstrapped versus conventional standard errors in the general linear and SUR models, Econometric Theory 8 (1992) 258–275. [4] R.J. Barro and X. Sala-i-Martin, Economic Growth (McGraw-Hill, Inc., New York, 1995). [5] P.P. Barros, The black box of health care expenditure growth determinants, Health Economics 7 (1998) 533–544. [6] W.J. Baumol, Productivity growth, convergence, and welfare: What the long run data show, American Economic Review 76 (1986) 1072–1085. [7] W.A. Brock and S.N. Durlauf, Growth empirics and reality, World Bank Economic Review 15 (2001) 229–272. [8] G.E.P. Box and D.R. Cox, An analysis of transformations, Journal of the Royal Statistical Society Series B 26 (1964) 211–243. [9] R. Butler and E.D. Rothman, Predictive intervals based on reuse of the sample, Journal of the American Statistical Association, Theory and Methods Section 75 (December 1980) 881–889. [10] R.J. Carroll and D. Ruppert, Transformations and Weighting in Regression Analysis (Chapman and Hall, New York, 1988). [11] B. Efron, The Jackknife, the Bootstrap and Other Resampling Plans, CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 38, Sponsored by the Conference Board of The Mathematical Sciences, Supported by National Science Foundation (Printed for the Society for Industrial and Applied Mathematics by J.W. Arrowsmith Ltd., Bristol, England, 1982). [12] B. Efron and R. Tibshirani, Bootstrap methods for standard errors, confidence intervals, and other measures of statistical accuracy, Statistical Science 1 (1986) 54–77. [13] B. Efron and R. Tibshirani, Rejoinder, Statistical Science 1 (1986) 77– 79. [14] B. Efron and R. Tibshirani, An Introduction to the Bootstrap (Chapman and Hall, New York, 1993). [15] U.-G. Gerdtham, Pooling international health expenditure data, Health Economics 1 (1992) 217–231. [16] U.-G. Gerdtham and B. Jonsson, International comparisons of health expenditure: Theory, data and econometric analysis, in: Handbook of Health Economics, eds. A.J. Culyer and J.P. Newhouse (North-Holland, Amsterdam, 2000). [17] T. Getzen, Aging and health care expenditures: A comment on Zweifel, Felder and Meiers, Health Economics 10 (2001) 175–177. [18] W.H. Greene, Econometric Analysis, 3rd Ed. (Prentice Hall, New York, 1993). [19] P. Hansen and A. King, The determinants of health care expenditure: A cointegration approach, Journal of Health Economics 15 (1996) 127– 137. [20] J.A. Hartigan, Comment, Statistical Science 1 (1986) 75–77. [21] H. Herwartz and B. Theilen, The determinants of health care expenditure: Testing pooling restrictions in small samples, Health Economics 12 (2003), in press. [22] P. Klenow, Comment on “It’s not factor accumulation: Stylized facts and growth models”, World Bank Economic Review 15 (2001) 221– 224. [23] C. Leger, D. Politis and J. Romano, Bootstrap technology and applications, Technometrics 34 (November 1992) 378–398. [24] A.N. Link, T.G. Seaks and S. Woodberry, Firm size and R&D spending: Testing for functional forms, Southern Economic Journal 54 (1988) 1027–1032.


[25] G. MacDonald and S. Hopkins, Unit root properties of OECD health care expenditure and GDP data, Health Economics 11 (2002) 371–376. [26] R.G. Miller, The jackknife – A review, Biometrika 61 (1974) 1–15. [27] R. Moore, Physics 6730 jackknife error estimates; http://www.physics. utah.edu/∼detar/phycs6730/handouts/jackknife/jackknife/node1.htm (accessed October 16, 2003). [28] N.R.V. Murthy and A.A. Okunade, Managed care, deficit financing, and aggregate health care expenditure in the US: A cointegration analysis, Health Care Management Science 3 (2000) 279–285. [29] N.R.V. Murthy and I.S. Chen, The empirics of economic growth for OECD countries: Some new findings, Economics Letters 55 (1997) 425–429. [30] J.P. Newhouse, Medical care costs: How much welfare loss? Journal of Economic Perspectives 6 (1992) 3–21. [31] OECD, New Directions in Health Policy, Health Policy Studies, No. 7 (Organization for Economic Cooperation and Development, Paris, 1995). [32] OECD, The Reform of Health Care: A Comparative Analysis of Seven OECD Countries, Health Policy Studies, No. 2 (OECD, Paris, France, 1992). [33] A. O’Hagan and J. Stevens, Assessing and comparing costs: How robust are the bootstrap and methods based on asymptotic normality? Health Economics 12 (2003) 33–49. [34] A.A. Okunade and M.C. Karakus, Unit root and cointegration: Timeseries versus panel estimates for international health expenditure models, Applied Economics 33 (2001) 1131–1137. [35] A.A. Okunade and V.N.R. Murthy, Technology as a ‘major driver’ of health care costs: A cointegration analysis of the Newhouse conjecture, Journal of Health Economics 21 (2002) 147–159. [36] A.A. Okunade and C. Suraratdecha, Health care expenditure inertia in the OECD countries: A heterogeneous analysis, Health Care Management Science 3 (2000) 31–42. [37] A.A. Okunade, Box–Cox vs. Modulus power family of response transformations, Atlantic Economic Journal 19 (1991) 57. [38] A.A. Okunade, H. Haryanto and D.B. Means, Testing the unbiasedness hypothesis of foreign exchange rates and the analysis of transformations, Review of Quantitative Finance and Accounting 6 (1996) 39–46. [39] D. Parkin, A. McGuire and B. Yule, Aggregate health expenditures and national income: Is health care a luxury good? Journal of Health Economics 6 (1987) 109–127. [40] E.A. Peden and M.S. Freeland, Insurance effects of US medical spending, 1960–1993, Health Economics 7 (1998) 671–687.

183

[41] M.H. Quenouille, Approximate tests of correlation in time-series, Journal of the Royal Statistical Society Series B 11 (1949) 68–84. [42] M.H. Quenouille, Notes on bias in estimation, Biometrika 43 (1956) 353–360. [43] J. Roberts, Sensitivity of elasticity estimates for OECD health care spending: Analysis of a dynamic heterogeneous data field, Health Economics 8 (1999) 459–472. [44] P. Romer, Comment on “It’s not factor accumulation: Stylized facts and growth models”, World Bank Economic Review 15 (2001) 225–227. [45] S.K. Sapra, A jackknife maximum likelihood estimator for the probit model, Applied Economics Letters 9 (2002) 73–74. [46] N. Serup-Hansen, J. Wickstrøm and I.S. Kristiansen, Future health care costs – Do health care costs during the last year of life matter? Health Policy 62 (2002) 161–172. [47] M. Seshamani and A.M. Gray, A longitudinal study of the effects of age and time to death on hospital costs, Journal of Health Economics (2004), in press. [48] J.L. Simon and P. Bruce, Resampling: A tool for everyday statistical work, CHANCE. New Directions for Statistics and Computing 4 (1991) 22–32. [49] P.C. Smith, Measuring health system performance, European Journal of Health Economics 3 (2002) 145–148. [50] B. Starfield and L. Shi, Policy relevant determinants of health: An international perspective, Health Policy 60 (2002) 201–218. [51] F.S. Swed and C. Eisenhart, Tables for testing randomness of grouping in a sequence of alternatives, Annals of Mathematical Statistics 14 (1943). [52] C.G. Tsangarides, On cross-country growth and convergence: Evidence from African and OECD countries, Journal of African Economies 10 (2002) 355–389. [53] J.W. Tukey, Comparison of some ratio estimators, Journal of the American Statistical Society 60 (1965) 294–307. [54] M. Veall, Bootstrapping the process of model selection: An econometric example, Journal of Applied Econometrics 7 (1992) 93–99. [55] H. White, Estimation, Inference, and Specification Analysis (Cambridge University Press, Cambridge, 1994). [56] C.F.J. Wu, Jackknife, bootstrap and other resampling methods in regression analysis, Annals of Statistics 14 (1986) 1261–1295. [57] P. Zweifel, S. Felder and M. Meiers, Ageing of population and health care expenditure: A red herring? Health Economics 8 (1999) 485–496.