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Article Excerpt
Introduction
In the stochastic production frontier approach, the output of each production unit is bounded above by a frontier that is stochastic in the sense that its placement is allowed to vary randomly across production units.
From an economic point of view, this technique permits production units to be technically efficient relative to their own frontier.
Variation of the frontier among production units captures the effects of external factors, like unexpected favourable and unfavourable shocks beyond the control of the production units, along with errors of observation and measurement on output as well as combined effects of unspecified input variables in the production function. These random effects are usually described by a normal random variable.
The different inefficiencies across the production units are captured by an error component, assumed to be one-sided distributed, allowing observed output of any producer to fall below its maximum output, which is determined by the stochastic production frontier.
The homoscedasticity assumption is usually made both for the random effects and for the inefficiency component, which is described by using an appropriate one-sided random variable, such as half normal, exponential, truncated normal (Stevenson 1980) and gamma distributions (Greene 1990).
Although homoscedasticity is one of the standard assumptions of the classical regression model, there are several reasons why the variance of the disturbance term may be variable or heteroscedastic (Valavanis 1959). For example, heteroscedasticity can be produced by sampling strategies based on clustering or if there are subpopulation differences or other interaction effects.
Moreover, heteroscedasticity occurs in a number of applications, both in cross-section and time-series data, although the problem of heteroscedasticity is likely to be more common in cross-sectional data, where the scale of the dependent variable and the explanatory power of the model tend to vary across observations.
To this respect, Prais and Houthakker (1955), when analysing household spending patterns, found that there is a greater variation in expenditure on certain commodity groups among high-income households than among low ones.
While it is well known that, in the presence of heteroscedasticity, the ordinary least squares (OLS) estimator is unbiased, consistent and asymptotically normally distributed, but inefficient, the consequences of the existence of heteroscedasticity in stochastic frontier models have been examined only recently.
Caudill and Ford (1993), by using a Monte Carlo experiment, showed that heteroscedasticity in the one-sided error, relative to a Cobb-Douglas stochastic production frontier, leads to overestimation of the intercept and underestimation of the slope coefficients. Bojanic et al. (1998) found that heteroscedasticity in the normal random variable introduces substantial biases into Maximum Likelihood (ML) estimator of the parameters in frontier models.
Guermat and Hadri (1999), again by means of Monte Carlo methods experimented in a half normal stochastic production frontier, found that, when heteroscedasticity exists, correcting it leads to a substantial improvement of the statistical properties of estimators and to improved efficiency measures.
To explore the effects of the presence of heteroscedasticity, Caudill et al. (1995) developed and estimated a heteroscedastic frontier cost function using data from commercial banks. Their results showed dramatic changes in the parameter estimates and in the efficiency measures when accounting for heteroscedasticity in the estimation process.
Hadri (1999) suggested the estimation of a stochastic frontier cost function model as well, assuming the presence of heteroscedasticity in both the above two random components. By applying the model to the same data as those used by Caudill et al. (1995), he found that production unit efficiency measures are extremely sensitive to the proposed correction for heteroscedasticity.
Recently, Hadri et al. (2003) considered heteroscedasticity in both random variables in a stochastic production frontier model, which also estimates technical inefficiency effects, by applying the Battese and Coelli (1995) model, using panel data on English farms.
It is worth emphasizing that, if in the above studies, involving fields such as commercial banks and farms, the homoscedasticity hypothesis has been relaxed, in the field of human capital formation in the university system, which is the object of this study, homoscedasticity is even more difficult to be justified as the formation process depends heavily on people's skills.
Ferrari and Laureti (2004) modelled the human capital production process in the university system, as one in which a production unit produces itself. So the student is considered as the basic production unit (1) who, by using factors provided by the faculty (e.g. professors, equipment, seats in lecture halls and so on), as well as the student's own effort and capabilities, produces him/herself as a graduate. The model was specified under the homoscedasticity hypothesis obtaining consistent and reliable findings. In particular, by estimating the stochastic frontier for each faculty separately following the concept of meta-frontier function, the analysis shown that the joint effects of the selected explanatory variables on the inefficiencies of production are significant, although the individual effects of one or more of the variables could not be statistically significant. There was evidence that student characteristics could not be neglected without introducing a bias in the estimation of production function for educational production process.
Nevertheless, potential problems in analysing data on university formation processes can arise if there are a great number of exogenous variables, represented by the students' characteristics, e.g. prior educational qualifications and household social and economic background, which strongly affect the results. Since these exogenous factors vary from student to student, characterizing the "environment" in which formation takes place, the homoscedasticity hypothesis is no longer reasonably sustainable.
In fact, it seems inadequate to assume that the variability of the student efficiency behaviour in carrying out the formation process is the same for each student.
Therefore, in this paper, where a great number of exogenous variables are...
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