estimate it. In SFA, the relationship between the inputs used and the output produced is assumed to be stochastic. Then, one advantage of the SFA model over the DEA approach is that deviations from the frontier reflect not only inefficiencies, as in the DEA approach, but also noise in the data. By addressing the stochastic nature of the benchmark, SFA produces inefficiency measures that in principle are not contaminated by the presence of the noise term.

Although SFA was initially developed to estimate productive efficiency (see, e.g., Kumbhakar and Lovell 2000; Kumbhakar, Wang and Horncastle 2015), one can view it as a tool to define benchmarks in many situations. For example, in analyzing health care delivery, the World Health Report (2000) measured and ranked the efficiency of the health systems of 191-member countries (Tandon _et al_ . 2000). There, SFA is used to model and estimate the benchmark against which each country is compared and ranked. As in the traditional production function approach, maximizing health output is considered as the objective of a health system, given health expenditure and some other control variables. Based on this notion of health production function, the study judged the health systems for the 191 countries in terms of their efficiency (derived from SFA) in turning health expenditures into health services. In general, SFA can address questions such as the following. By how much a country’s educational quality can be improved after controlling for its structural and country-specific characteristics. How much child mortality can be reduced given the resources of a country? How much a country can gain in terms of its growth rate by trade openness?

In summary, SFA estimates a benchmark as a stochastic frontier. This produces two advantages: the estimated inefficiency (gap with respect to the benchmark) is always one-sided and does not include the noise term. We now turn to a detailed description of the SFA methodology.

# **4. Methodology: Stochastic Frontier as a Benchmark**

The basic stochastic frontier model for panel data is the following: ′ ′ i 0 i i i ii where i is the indicator variable (usually in logs) for country yy = ββ + xx ββ+ zz γγ−uu _i_ at time + vv _t_ , _β0_ is a common intercept, _xit_ is the vector of structural variables; and _β_ is the associated vector of technology parameters to be yy estimated. The country-specific (time-invariant) variables are denoted by , and γ is the corresponding coefficient vector; i is a random two-sided iid noise term. Finally, ≥0 is the non-negative one-sided ii zz inefficiency term, the gap between the observed indicator and the benchmark (stochastic frontier). The ii idea of separating the vv variables from the variables is that the former vary in both uu and dimensions while the latter is time-invariant. The parameters of the model are estimated by the maximum likelihood (ML) method based on distributional assumptions on the noise and the inefficiency terms. xx zz ii tt The dependent variable ( i ) is the outcome variable being explained by the covariates ( i ). The noise term i captures the effect of unobserved factors that can affect the outcome variable both positively ∗ ′ ′ ii and negatively. Thus, the regression part of the above equation yy i 0 i xxi can be viewed and zz vv as the stochastic frontier (maximum or minimum value of the outcome variable, given the covariates, ii yy ≡ββ + xx ββ+ zz γγ+ vv depending on the problem).<sup>5</sup> This frontier defines the benchmark. To allow for the possibility that a micro

> 5 The component without the term can be viewed as the deterministic frontier. vv

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