To assess relative productivity, many researchers use the stochastic frontier estimation (henceforth, SFE). This method was developed independently by Aigner et al. (1977) and Meeusen and Van den Broeck (1977). SFE generates a production (or cost) frontier with a stochastic error term that consists of two components: a conventional random error (“white noise”) and a term that represents deviations from the frontier, or relative inefficiency.
In SFE, a production function, a Cobb-Douglas production function of the following form is estimated:
where the subscript i denotes the ith decision-making unit (a plant, firm, industry, or nation), Q represents output, X is a vector of inputs, β is the unknown parameter vector, and ε is an error term with two components, εi = (Vi − Ui) where Ui represents a non-negative error term to account for technical inefficiency, or failure to produce maximal output, given the set of inputs used. Vi is a symmetric error term that accounts for random effects. The standard assumption, following Aigner et al. (1977), is that the Ui and Vi have the following distributions:
That is, the inefficiency term (Ui) is assumed to have a half-normal distribution, that is, a decision-making unit is either on the frontier or below it. An important parameter in this model isthe ratio of the standard error of technical inefficiency to the standard error of statistical noise, which is bounded between 0 and 1. Note that γ = 0 under the null hypothesis of an absence of inefficiency, signifying that all of the variance can be attributed to statistical noise.
In recent years, SFE models have been developed that allow the technical inefficiency (relative productivity) term to be expressed as a function of a vector of environmental and organizational variables. For example, Reifschneider and Stevenson (1991) assume that the Ui are independently distributed as truncations