# Bayesian Analysis. of Stochastic. Frontier Models

Gary Koop and Mark F. J. Steel

Stochastic frontier models are commonly used in the empirical study of firm1 efficiency and productivity. The seminal papers in the field are Aigner, Lovell, and Schmidt (1977) and Meeusen and van den Broeck (1977), while a recent survey is provided in Bauer (1990). The ideas underlying this class of models can be demonstrated using a simple production model2 where output of firm i, Yi, is produced using a vector of inputs, Xi, (i = 1 … N). The best practice technology for turning inputs into output depends on a vector of unknown parameters, p, and is given by:

Y = f (Xi; P). (24.1)

This so-called production frontier captures the maximum amount of output that can be obtained from a given level of inputs. In practice, actual output of a firm may fall below the maximum possible. The deviation of actual from maximum output is a measure of inefficiency and is the focus of interest in many applications. Formally, equation (24.1) can be extended to:

Yi = f (Xi; p )Ti, (24.2)

where 0 < Ti < 1 is a measure of firm-specific efficiency and Ti = 1 indicates firm i is fully efficient.

In this chapter we will discuss Bayesian inference in such models. We will draw on our previous work in the area: van den Broeck, Koop, Osiewalski, and Steel (1994), Koop, Osiewalski, and Steel (1994, 1997, 1999, 2000) and Koop, Steel, and Osiewalski (1995), whereas theoretical foundations can be found in Fernandez, Osiewalski, and Steel (1997). It is worthwhile to digress briefly as to why we think these models are worthy of serious study. Efficiency measurement is very important in many areas of economics3 and, hence, worthy of study in and of itself. However, stochastic frontier models are also close to other classes of models and can be used to illustrate ideas relating to the linear and nonlinear regression models; models for panel data, variance components, random coefficients, and, generally, models with unobserved heterogeneity. Thus, stochastic frontier models can be used to illustrate Bayesian methods in many areas of econometrics. To justify our adoption of the Bayesian paradigm, the reader is referred to our work in the area. Suffice it to note here that the competitors to the Bayesian approach advocated here are the classical econometric stochastic frontier approach (see Bauer, 1990 for a survey) and the deterministic or nonparametric data envelopment analysis (DEA) approach (see, e. g., Fare, Grosskopf, and Lovell, 1994). Each of the three approaches has strengths and weaknesses, some of which will be noted in this chapter.

This chapter is intended to be reasonably self-contained. However, we do assume that the reader has a basic knowledge of Bayesian methods as applied to the linear regression model (e. g. Judge, Griffiths, Hill, Lutkepohl, and Lee, 1985, ch. 4 or Poirier, 1995, pp. 288-309 and 524-50). Furthermore, we assume some knowledge of simulation methods. Koop (1994, pp. 12-26) provides a simple survey of some of these methods. Osiewalski and Steel (1998) focuses on simulation methods in the context of stochastic frontier models. Casella and George

(1992) and Chib and Greenberg (1995) are good expository sources for Gibbs sampling and Metropolis-Hastings algorithms, respectively. Geweke (1999) is a complete survey of both Bayesian methods and computation.

The remainder of the chapter is organized as follows. The second section considers the stochastic frontier model with cross-sectional data beginning with a simple loglinear model then considering a nonlinear extension and one where explanatory variables enter the efficiency distribution. The third section discusses the issues raised by the availability of panel data.

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