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Distributional Forms in Stochastic Frontier Analysis

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Abstract

This chapter is a survey of developments in stochastic frontier modelling. The literature on stochastic frontiers has grown substantially in the 42 years since the seminal work by Aigner et al. (J. Econom 6 (1): 21–37, 1977). There exist many surveys of this literature that cover a broad range of contribution pathways in the field. In this chapter, we present a review of key developments in distributional specifications in stochastic frontier models, with a particular emphasis on innovations that address practical issues identified by practitioners.

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Notes

  1. 1.

    In the SF literature, ‘truncated normal’ refers specifically to the left truncation at zero of a normal distribution with mean \(\mu\) and variance \(\sigma_{u}^{2}\).

  2. 2.

    Lee and Lee (2014) focus on the upper bound on inefficiency in the normal-uniform model and appear to have been unaware of the model’s earlier introduction by Li (1996), who was motivated by the skewness issue.

  3. 3.

    Such ‘tolerance’ does not necessarily reflect the technical competence or experience of regulators per se. It could reflect the perceived limitations on the robustness of the analysis (e.g. data quality), which necessitates a risk averse efficiency finding from a regulatory review.

  4. 4.

    If we view the normal-half normal model as a skew-normal regression model in which we expect (but do not restrict) the skewness parameter \(\sigma_{u} /\sigma_{v}\) to be positive, then we view the test for the presence of inefficiency as a one-tailed test of the H0 that \(\sigma_{u} \le 0\), or equivalently that \(\sigma_{u} /\sigma_{v} = 0\), rather than as a test involving a boundary issue. Comparing the case of one inequality constraint in Gouriéroux et al. (1982) to Case 5 in Self and Liang (1987), we see the same result.

  5. 5.

    However, since Schmidt and Sickles (1984), cross-sectional models have been proposed, such as those of Kumbhakar et al. (1991), Huang and Liu (1994), and Battese and Coelli (1995), that allow for dependence between inefficiency and frontier variables. These are discussed in Sect. 4.

  6. 6.

    N being the total number of observations, so that N = IT in the case of a balanced panel.

  7. 7.

    Clearly, this model is far from parsimonious, since g(t) includes 2I parameters. In fact, the authors apply a simpler model, \(g\left( t \right) = \exp \left[ {\lambda_{1i} \left( {t - \lambda_{2} } \right)} \right]\) after failing to reject \(H_{0} : \lambda_{2i} = \lambda_{2}\).

  8. 8.

    The authors instead estimate a system of T equations via the seemingly unrelated regressions (SUR) model proposed by Zellner (1962). However, this approach offers no way of predicting observation-specific efficiencies.

  9. 9.

    However, in the context of a log-linear model, the estimate of the intercept will be biased in either case.

  10. 10.

    Despite this, Tsionas (2002) does interpret the models as incorporating technological heterogeneity.

  11. 11.

    Note that these proposals are very similar to those of Kumbhakar (1991) and Heshmati and Kumbhakar (1994), the difference being the interpretation of ai and \(\alpha_{t}\) as picking up inefficiency effects, rather than unobserved heterogeneity.

  12. 12.

    The univariate skew normal distribution is a special case of the closed skew-normal distribution. To see that \(\varepsilon_{it}\) is the sum of two closed skew-normal random variables, therefore, consider that \(v_{it}^{*} = v_{it} + w_{it}\). and \(a_{i}^{*} = a_{i} + w_{i}\) both follow skew-normal distributions. For details on the closed skew-normal distribution, see González-Farías et al. (2004).

  13. 13.

    Note that the authors in fact proposed a deterministic frontier model in which \(E\left( {u_{i} |z_{i} } \right) = \exp \left( {z_{i} \delta } \right),\) but if we interpret the random error as vi rather than a component of ui, we have an SF model with a deterministic ui.

  14. 14.

    Note, however, that since the (post-truncation) variance of the truncated normal distribution is a function of the pre-truncation mean, the Kumbhakar et al. (1991), Huang and Liu (1994), and Battese and Coelli (1995) model also implies heteroskedasticity in ui.

  15. 15.

    Note the two similar but subtly different parameterisations, \(\sigma_{ui} = \exp \left( {z_{i} \gamma } \right)\) and \(\sigma_{ui}^{2} = \exp \left( {z_{i} \gamma } \right)\).

  16. 16.

    Note the similarity of the issues here to those around ‘confidence intervals’ and prediction intervals for \(E\left( {u_{i} |\varepsilon_{i} } \right)\), discussed by Wheat et al. (2014).

  17. 17.

    However, the authors’ discussion overstates the simplicity of marginal effects in this case, since it focuses on \(\partial \ln \hat{u}_{i} /\partial z_{li}\), which is \(\eta_{l}\) regardless of the distribution of \(u_{i}^{*}\) (or indeed the choice of predictor). However, \(\partial \hat{u}_{i} /\partial z_{li}\) is more complex, and as previously noted, the translation into efficiency space via \(\partial \exp \left( { - \hat{u}_{i} } \right)/\partial z_{li}\) adds additional complexity.

  18. 18.

    For an explanation of shrinkage in the context of the predictor \(E\left( {u_{i} |\varepsilon_{i} } \right)\), see Wang and Schmidt (2009).

  19. 19.

    Holding \(\beta\) constant.

  20. 20.

    In keeping with previous terminology, ‘truncated’ (without further qualification) refers specifically to the left truncation at zero of a distribution with mean \(\mu\), and ‘half’ refers to the special case where \(\mu = 0\). Note that truncating the Laplace distribution thus yields the exponential distribution whenever \(\mu \le 0\) due to the memorylessness property of the exponential distribution.

  21. 21.

    Or more specifically, the scale contaminated normal distribution.

  22. 22.

    As discussed in Sect. 3, see Case 5 in Self and Liang (1987).

  23. 23.

    Again, as an exception to this, dependency between error components may be introduced via ‘environmental’ variables influencing the parameters of their distributions as discussed in Sect. 5.

  24. 24.

    Schmidt and Lovell (1980) fold, rather than truncate.

  25. 25.

    Kumbhakar et al. (2009), using panel data, also include a lagged regime membership (i.e. technology choice) dummy in their selection equation.

  26. 26.

    The authors actually use ui to denote the noise term and vi and wi for the one-sided errors. In the interest of consistency and to avoid confusion, we use vi to refer to the noise term and ui and wi for the one-sided errors.

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Stead, A.D., Wheat, P., Greene, W.H. (2019). Distributional Forms in Stochastic Frontier Analysis. In: ten Raa, T., Greene, W. (eds) The Palgrave Handbook of Economic Performance Analysis. Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-030-23727-1_8

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