Conceptual framework

The analysis that follows borrows from the analytical toolkit of microeconometrics; specifically, those tools that are used to estimate production function frontiers. According to William Greene, “the frontier production function is an extension of the familiar regression model based on the microeconomic premise that a production function … represents an ideal, the maximum output attainable given a set of

inputs.” With the setting of this ideal comes the theoretical proviso that all 13

observations will lie below it. As such, estimation of the production frontier is generally used as a means to another analytical end: the analysis of technical efficiency. Analysis of technical efficiency in the microeconomic sense refers to the degree to which producers are successful in allocating the inputs they have at their disposal to produce certain outputs in an effort to meet some specified objective.

Greene, W.H. “The Econometric Approach to Efficiency Analysis,” In The Measurement of Productive 13

This objective could be to minimise the number and level of inputs to produce a given output (input-approach), or to maximise output with a given number and level of inputs (output-approach). By means of estimating the production function a measure of efficiency emerges since what it corresponds to is the distance between the actual observation and the expected estimate of the ideal.

Figure 5.1: Output approach to measuring technical efficiency

The standards against which efficiency is measured are provided by the production

frontier, f(x) in Figure 5.1. A producer using xainputsto produce a single output of

yais inefficient to the tune of u since it is operating beneath f(x).

In the case of measuring the extent to which a state complies with or falls short of its obligation to fulfil the right to health the same notion of efficiency can be applied.

Each state can be treated as a decision making unit that produces the health basket

under the behavioural assumption that it operates to maximise attainment of the basket (output) given its ability to do so (inputs). Maximum expected attainment at different levels of ability can then be predicted and it is these expected values that set

the minimum obligation frontier. Compliance, like efficiency, can then be measured as the difference between observed attainment of the health basket and the expected level of attainment, or in other words, the level states have an unconditional obligation to provide.

The standards against which states can be measured are provided by the minimum obligation frontier in Figure 5.2. For example, it is expected that for a state with an

ability level of a1, attainment of the health basket should be at level m1; state a1 has

an unconditional obligation to fulfil m1 level of the health basket. According to the

hypothesis proposed throughout this thesis: that compliance with the right to health is a function of a state’s ability and willingness to fulfil it, provision of the health

basket at any point below the frontier, such as m-1, may signal unwillingness on the

part of the state to mobilise its maximum available resources to fulfil the right to health, which in turn signals non-compliance with the obligation to do so. The degree of non-compliance can then be measured as the distance between the observed level provided and the level set by the minimum obligation frontier, in this

example: m-1/(m-1 + m1).

Essentially, there are two main methodologies for measuring efficiency: the mathematical (non-parametric) approach, and the econometric (parametric) approach. The two techniques have both virtues and limitations in their respective

bids to envelop data and there is no prescriptive rulebook for which method is best. 14

Though the technical differences between the two methods are many, their relative advantageousness can be assessed with reference to two central methodological characteristics:

i. The econometric approach is parametric so the shape of the frontier has to be specified from the very beginning. This could make the model vulnerable to functional form misspecification. The mathematical approach, on the other hand, is non-parametric so has the advantage that no assumption has to be made as to the shape of the frontier.

ii. The econometric approach is stochastic, which allows for the model to distinguish between the effects of inefficiency and the effects of random noise. The mathematical approach, however, is deterministic and provides only a general measure of inefficiency, which is likely to hide within it random noise, and hence risk being either under or overestimated.

Deciding when one method should be chosen over the other comes down to an assessment of appropriateness to the individual dataset and the research question(s) being asked. Presently, the primary objective of the analysis is to reveal the presence and degree of non-compliance within and across states. The data involved are, by nature, highly heterogeneous, of widely differing quality, and are therefore likely to carry noise. As such, the certain advantages of a model, which allows for real non- compliance to be distinguished from random noise must outweigh the potential

See, e.g. Greene, W.H. Ibid., pp. 112-114 14

limitations posed by a risk of form misspecification. It is for these combined reasons that this study proceeds along an econometric path.

There exist a number of studies that have ventured into some form of stochastic frontier analysis in a health-output-orientated context. The first and most well known

are the studies by Evans et al., presented in the 2000 World Health Report under 15

the auspices of the WHO, which presents an assessment of the relative 16

performance of national health care delivery systems across the 191 WHO member States. Using the production function framework, two health care outcomes were used as outputs (disability adjusted life expectancy, and a composite measure of achievement across five dimensions: health, health inequality, responsiveness-level, responsiveness-distribution, and fair-financing) whilst spending on health care and the level of education were treated as inputs. The study opted for a form of the

‘fixed effects’ model based on that proposed by Schmidt and Sickles and used a 17

five-year panel dataset (1993-1997).

Though recognised for their innovation in, and evolution of, the measurement of inefficiency in health care delivery on a macro scale, the authors of the WHO study soon found themselves at the centre of critical attention. For example, Williams

questioned the normative content of the study, calling it “dangerously opaque;” 18

Gravelle and colleagues re-ran the analysis using different definitions of efficiency and different estimation methods and suggested the WHO’s ranking and efficiency

Evans D., A. Tandon, C. Murray, and J. Lauer “The Comparative Efficiency of National Health 15

Systems in Producing Health: An Analysis of 191 Countries” (2000a); and Evans D., A. Tandon, C. Murray, and J. Lauer. “Measuring Overall Health System Performance for 191 Countries.” (2000b)

WHO. The World Health Report 2000 - Health Systems: Improving Performance. 2000 16

Schmidt, P., and R. Sickles. “Production Frontiers with Panel Data.” (1984) 17

Williams, A. “Science or Marketing at WHO? A Commentary on ‘World Health 2000.’” (2001) p. 99 18

scores lacked robustness; Hollingsworth and Wildman found fault in the authors’ 19 choice to use the fixed effects method and presented evidence to suggest more

flexible panel data techniques would do the job better; and Greene criticised the 20

study’s failure to accommodate a procedure for distinguishing real inefficiency from

cross-country heterogeneity. 21

To confront the deficiencies of the WHO study in a direct way, several of the aforementioned critics reanalysed the WHO data, many of which produced

substantially differing results. Although the aims and the questions asked of the 22

WHO study — and its descendants — are very different from those under investigation here, both are subject to similar methodological issues. Unlike these previous empirical works, however, the theory upon which this study is based has much more solid and clearly defined foundations. Deciding how output and input measures are determined and how possible forms of heterogeneity should enter the model here is, therefore, much less of an arbitrary, solely statistical, job.

With its theoretical foundations in hand, this study aims to assess the extent to which states comply with their obligation to fulfil the right to health by estimating a frontier production function by panel data stochastic frontier models using a longitudinal Gravelle, H., R. Jacobs, A.M. Jones, and A. Street. “Comparing the Efficiency of National Health 19

Systems: A Sensitivity Analysis of the WHO Approach.” (2002)

Hollingsworth, J., and B. Wildman. “The Efficiency of Health Production: Re-estimating the WHO 20

Panel Data Using Parametric and Nonparametric Approaches to Provide Additional Information.” (2003)

Greene, W.H. Supra n. 5, pp. 216-250; and Greene, W.H. “Distinguishing Between Heterogeneity 21

and Inefficiency: Stochastic Frontier Analysis of the World Health Organization’s Panel Data on National Health Care Systems.” (2004)

For example, mean inefficiencies for the two output measures, DALE and COMP were reported in 22

the Evans et al. study as 0.220 and 0.174, respectively. This is compared to Greene’s truncation model, which produced estimates of 0.196 and 0.153 (Greene, W.H. Ibid., p. 974) The study by Gravelle et al. presented inefficiency score correlations, between their estimates and those produced in the WHO study, ranging between 0.597 and 0.998 (Gravelle, H., R. Jacobs, A.M. Jones, and A. Street. Supra n. 19, p. 16.)

dataset made up of health, financial, and geographical data. To this end, it is assumed that each state operates in order to maximise attainment of the health basket and produces that health basket according to the following basic stochastic production

function: 23

(1)

i = 1, … N and t = 1, … T where N is the state, T is the year, 𝑦it denotes the output

(Health Basket Attainment) and 𝐱’it is the set of inputs (financial and human

resources). Since the production function under consideration is stochastic, the error

term has two components: 𝒗it is the random component, which represents the

stochastic noise in the production function, and 𝒖i in this case represents the non-

compliance component, which measures the distance between 𝑦it and the frontier.

Estimation of 𝒖i is the central focus of the analysis that follows.

With estimation of the basic stochastic model come several assumptions with respect

to the combined error term, 𝓔i = 𝒗it - 𝒖i:

1. It is assumed that 𝒗it has a zero mean and is normally distributed.

2. 𝒖i is constrained to always be non-negative; 𝒖i ≥ 0, and is assumed to be

distributed independently of 𝒗it and of the regressors.

3. Since in this case the panel is short, 𝒖i is assumed to be time invariant.

The production function itself can take various functional forms. However, the most commonly used (and the only two to be mentioned here) are the Cobb-Douglas

Aigner, D., K. Lovell, and P. Schmidt. “Formulation and Estimation of Stochastic Frontier Models.” 23

(1977); Meeusen, W., and J. van den Broeck. “Efficiency Estimation from Cobb-Douglas Production Functions with Composed Error.” (1977)

yit =α +X

/

(CD) and translog specifications. The CD production function is the simplest and has the advantage of being easy to estimate and interpret since it requires the estimation of few parameters. There is, however, a price to pay for such simplicity: it assumes elasticities of factor substitution are constant, which may be rather unrealistic particularly for this analysis since different states are likely to have different ‘production’ elasticities and the elasticity of substitution between inputs is

likely to ≠ 1. As a result, many analysts have opted for more general, flexible

functional forms, which relax the restrictions on elasticities of substitution. In this regard, the translog production model is used most often. For the same reasons, this study will proceed using a non-separated version of the flexible translog form where, in line with the theoretical assumption that output is optimised, the production function has two properties: the marginal products of inputs are positive and the function is concave, i.e. as inputs increase output also increases but at a decreasing rate. 24