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Stochastic Frontier Analysis

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Productivity and Efficiency Analysis

Abstract

Distance, revenue, cost and profit functions can always be written in the form of regression models with unobserved error terms representing statistical noise and different types of inefficiency. In practice, the noise components are almost always assumed to be random variables (i.e., stochastic). The associated frontiers are known as stochastic frontiers. This chapter explains how to estimate and draw inferences concerning the unknown parameters in so-called stochastic frontier models (SFMs). It then explains how the estimated parameters can be used to predict levels of efficiency and analyse productivity change. The focus is on maximum likelihood estimators and predictors.

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Notes

  1. 1.

    The dependent variable in the Aigner et al. (1977) model is an output, not the logarithm of an output. The dependent variable in the Battese and Corra (1977) model is a value, not the logarithm of a quantity. If there is no statistical noise and the dependent variable is either an output or a value, then \(u_{it}\) can no longer be interpreted as an output-oriented technical inefficiency effect.

  2. 2.

    The error term in (7.19) is a mean-corrected inefficiency effect, whereas the error term in (8.20) is a mean-corrected noise and inefficiency effect.

  3. 3.

    The n-th moment of the OLS residuals is \(s_n=\sum _t^T \sum _i^{I_t}\hat{e}_{it}^n/\sum _t^TI_t\) where \(\hat{e}_{it}\) denotes the it-th residual and \(I_t\) is the number of firms in the dataset in period t.

  4. 4.

    Elsewhere, these estimators are sometimes referred to as corrected ordinary least squares (COLS) estimators; see, for example, [Horrace and Schmidt 1996, p. 260] . In this book, the term COLS is reserved for LS estimators for the parameters in deterministic frontier models.

  5. 5.

    See, for example, [Coelli 1995, p. 250].

  6. 6.

    Schmidt and Lin (1984) describe their test as a ‘test of the existence of a frontier’.

  7. 7.

    See, for example, Stevenson (1980) , Battese and Coelli (1995) and Salas-Velasco (2018).

  8. 8.

    These equations follow from (A.48) and (A.51) in Appendix A.7. If the data are cross-section data and \(\mu =0\), then (8.28) reduces to equation (2) in Jondrow et al. (1982) .

  9. 9.

    Coelli (1995) attributes this result to Gouriéroux et al. (1982) . Those authors derive their results in the context of a linear regression model with a normally distributed error term. It is not obvious that their results carry over to the case of a (composite) error term that is not normally distributed.

  10. 10.

    In the present context, Bayes’s theorem actually says that \(p(\theta | X, y) = p(y | X, \theta )p(\theta |X)/p(y|X)\). However, it is notationally convenient, and common practice, to suppress the X.

  11. 11.

    One exception is the marginal likelihood given by \(p(y)= \int p(y|\theta )p(\theta )~d\theta \).

  12. 12.

    Primal and dual indices are computed using distance, revenue and cost functions. SFMs are underpinned by the assumption that these functions exist. However, their functional forms are generally unknown. Moreover, the variables in these functions are often unobserved and/or measured with error.

  13. 13.

    For a more complete list of assumptions, see Sect. 4.7.5.

  14. 14.

    Olson et al. (1980) refer to these estimators as corrected ordinary least squares (COLS) estimators. In this book, the term COLS is reserved for LS estimators for the parameters in deterministic frontier models.

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  1. University of Queensland, Brisbane, QLD, Australia

    Christopher J. O’Donnell

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O’Donnell, C.J. (2018). Stochastic Frontier Analysis. In: Productivity and Efficiency Analysis. Springer, Singapore. https://doi.org/10.1007/978-981-13-2984-5_8

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  • DOI: https://doi.org/10.1007/978-981-13-2984-5_8

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