“When, Where, and How” of Efficiency Estimation: Improved Procedures for Stochastic Frontier Modeling

ABSTRACT

The issues of functional form, distributions of the error components, and endogeneity are for the most part still open in stochastic frontier models. The same is true when it comes to imposition of restrictions of monotonicity and curvature, making efficiency estimation an elusive goal. In this article, we attempt to consider these problems simultaneously and offer practical solutions to the problems raised by Stone and addressed by Badunenko, Henderson and Kumbhakar. We provide major extensions to smoothly mixing regressions and fractional polynomial approximations for both the functional form of the frontier and the structure of inefficiency. Endogeneity is handled, simultaneously, using copulas. We provide detailed computational experiments and an application to U.S. banks. To explore the posteriors of the new models we rely heavily on sequential Monte Carlo techniques.

KEYWORDS:

• Bayesian inference
• Efficiency estimation
• Fractional polynomial approximations
• Sequential Monte Carlo
• Smoothly mixing regressions
• Stochastic frontiers

Acknowledgments

The author thanks two anonymous referees for their useful comments on earlier versions of the article. The usual disclaimer applies.

Notes

1 Here, s1 to s4 refer to different scenarios in their simulations. Suppose σ²_v and σ²_u denote the variances of two sided error term (v) and one-sided error term (u), respectively, and λ = σ_u/σ_v. “In scenario s1 (σ_v = σ_u = 0.01, λ = 1.0), both terms are relatively small. In other words, the data are measured with relatively little error and the units are relatively efficient. In scenario s2 (σ_v = 0.01 and σ_u = 0.05,  λ = 5.0), the data have relatively little noise, but the units under consideration are relatively inefficient. In scenario s3 (σ_v = 0.05 and σ_u = 0.01, λ = 0.2), the data are relatively noisy and the firms are relatively efficient. The fourth scenario s4 (σ_v = σ_u = 0.05, λ = 1.0) is redundant as λ = 1.0 as in s1. However, we show this case to emphasize that the results of the experiment depend upon the ratio of σ_u to σ_v and not their absolute values.”

2 Many other distributions have been used in the literature like the exponential, the gamma, the Weibull etc.

3 For a review, see Kumbhakar and Lovell (2003).

4 Fractional polynomials are also used by Sauerbrei and Royston (1999).

5 The restrictions involve no loss of generality as the components in parentheses can be switched without changing the value of the function.

6 Clearly, flexibility in modeling the distributions of the two error components also accounts for endogeneity, which is an additional advantage of smoothly mixing regressions.

7 Relative to other copulas, the Gaussian copula is generally robust for most application and has many desirable properties (Danaher and Smith 2011).

8 The data are available at http://qed.econ.queensu.ca/jae/datasets/malikov001/.

9 For model VII, we used a full comparison between FPA for the cost function, the two scale parameters and the weights in bases $\frac{1}{k},k=1,\dots ,10$. This involved a choice among 10,000 different models so we give only the final best choice.

10 Here, w_it stands for the vector of variables that affect the ACD specification. We remind that most elements of w_it are already in logs.