# Bayesian Model Averaging and Prior Sensitivity in Stochastic Frontier Analysis

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

- (i)
- normal-exponential, labelled NEX, with ${\psi}_{v}$ = 2, ${\psi}_{u}$ = 1 and statistical parameters: $\beta $, ${\sigma}_{v}$, ${\sigma}_{u}$,
- (ii)
- normal-half-normal, NHN, with ${\psi}_{v}$ = 2, ${\psi}_{u}$ = 2 and statistical parameters: $\beta $, ${\sigma}_{v}$, ${\sigma}_{u}$,
- (iii)
- normal-half-GED, NHG, with ${\psi}_{v}$ = 2 and statistical parameters: $\beta $, ${\sigma}_{v}$, ${\sigma}_{u}$, ${\psi}_{u}$,
- (iv)
- GED-half-GED, GHG, with statistical parameters $\beta $, ${\sigma}_{v}$, ${\sigma}_{u}$, ${\psi}_{v}$, ${\psi}_{u}$.

^{2}I

_{k}), $p\left({\psi}_{v}\right)$ and $p\left({\psi}_{u}\right)$ are of the same form, implying that $\left({\psi}_{(.)}-1\right)~G\left(1,1\right)$—consequently, we rule out ${\psi}_{(.)}$ < 1, although this assumption might be relaxed. For $p\left({\sigma}_{v}\right)$, we follow the suggestion of Koop et al. (2000), who elicit a Gamma-type prior that is assumed to mimic the traditional Jeffrey’s prior for precision in linear models (the latter is improper, so the interpretation is of course approximate). We take the prior ${\sigma}_{v}^{-2}~G$(0.5Q, 0.5Q) with Q = 10

^{−4}and make use of it in order to maintain comparability with the aforementioned papers. Though we leave the task to formulate adequate prior for ${\sigma}_{v}$ for further research, we have checked sensitivity with respect to changes in $p\left({\sigma}_{v}\right)$ and found our results to be quite robust.

## 3. Investigation Results

#### 3.1. Prior Analysis

- ${\sigma}_{u}^{-1}~G\left(1,-\mathrm{ln}{\mathrm{r}}^{*}\right)$ based on van den Broeck et al. (1994), which we refer to as prior 1;
- ${\sigma}_{u}^{-2}~G\left(5,10{\mathrm{ln}}^{2}{\mathrm{r}}^{*}\right)$ which is based on van den Broeck et al. (1994) for a = 0 (implying a half-normal distribution on $r$; see van den Broeck et al. 1994, p. 286); we refer to it as prior 2;
- ${\sigma}_{u}^{-2}~G\left(0.5\mathrm{N},0.5\mathrm{Q}\right)$ following (Tsionas 2002; Tsionas and Kumbhakar 2014); we refer to it as prior 3.

^{−4}. Although, Q = 10

^{−6}is also sometimes used, we find that the results are virtually the same for both values and that Q = 10

^{−4}is somewhat less informative or restrictive. Thus, only results Q = 10

^{−4}are presented, with a note that they remain virtually unchanged for Q = 10

^{−6}.

#### 3.2. Posterior Analysis

- A real-life dataset, labeled A, on aggregate production from (Makieła 2014). This is a well-researched dataset covering 27 EU Member States, USA, Japan and Switzerland in 1996–2010 (450 observations). It contains information on GDP, capital stock and labor. BMA is used here to average results over the four models mentioned in Section 2 (i–iv) based on different priors (0–3).
- An artificial dataset generated under the assumption of no inefficiency in the production process, labelled B (200 observations). This allows us to explore how popular SFA models (NHN, NEX) with different prior structures react to an “efficient” process. In this context BMA is used to confront non-SF models with NHN and NEX, again under different prior settings.

#### 3.2.1. Results Based on Dataset A

- have a negligible impact on technology parameters;
- have some impact on parameters of the stochastic structure ($\sigma $’s, $\psi $’s), the intercept (especially for prior 2 and NEX model) as well as latent variables (efficiencies); i.e., changes can be observed in terms of the “average” efficiency (“av. r” in the tables) and the relative location of object-specific efficiency-scores; however, correlations between the point estimates remain very high (often above 0.95);
- have a considerable impact on p(y); especially noticeable differences are for prior 2 and NEX model;
- differences in prior specifications have the least effect on NHG and GHG models, which give consistently stable results in terms of parameters, efficiencies and p(y).

- ${\psi}_{u}$ and ${\psi}_{v}$ are close to posterior independence;
- ${\psi}_{u}$ has a more concentrated distribution than ${\psi}_{v}$;
- restriction ${\psi}_{u}$ = 2 (half-normality of $u$) is in the tail of p(${\psi}_{u}$|data) and there is virtually no posterior probability around restriction ${\psi}_{u}$ = 1 ($u$ being exponential); this reiterates our results based on p(y);
- restriction ${\psi}_{v}$ = 2 (normality of $v$) is within a region of high posterior density of p(${\psi}_{v}$|data); normality of $v$ is thus likely;
- ${\sigma}_{u}$ and ${\psi}_{u}$ are positively correlated; this is not the case for ${\sigma}_{v}$ and ${\psi}_{v}$, which do not seem to show any particular dependence;
- ${\sigma}_{u}$ and ${\sigma}_{v}$ are negatively correlated (substitutability of variances) and positioned around quite different values.

#### 3.2.2. Results Based on Dataset B

## 4. Concluding Remarks

- it allows for a formal model comparison within a broad, flexible parametric class;
- it allows for an in-depth analysis of prior sensitivity, as the numerical methods used do not impose any particular class of priors (contrary to the Gibbs sampling approach).

- it has virtually no impact on the technology parameters;
- it has some impact in terms of inference on the latent variables (i.e., the posterior efficiency estimates, especially in terms of “average” posterior mean and the relative spread of posterior means of efficiency);
- it has substantial impact on posterior model probabilities, which are crucial in BMA.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Aigner, Dennis J., C. A. Knox Lovell, and Peter Schmidt. 1977. Formulation and Estimation of Stochastic Frontier Production Function Models. Journal of Econometrics 6: 21–37. [Google Scholar] [CrossRef]
- Fernández, Carmen, Jacek Osiewalski, and Mark F. J. Steel. 1997. On the use of panel data in stochastic frontier models. Journal of Econometrics 79: 169–93. [Google Scholar] [CrossRef]
- Florens, Jean-Pierre, Léopold Simar, and Ingrid Van Keilegom. 2019. Estimation of the Boundary of a Variable Observed with Symmetric Error. Journal of the American Statistical Association 11. [Google Scholar] [CrossRef]
- Greene, William H. 2003. Simulated likelihood estimation of the normal-gamma stochastic frontier function. Journal of Productivity Analysis 19: 179–90. [Google Scholar] [CrossRef]
- Greene, William H. 2005. Fixed and random effects in stochastic frontier models. Journal of Productivity Analysis 23: 7–32. [Google Scholar] [CrossRef] [Green Version]
- Greene, William H. 2008. The Econometric Approach to Efficiency Analysis. In The Measurement of Productive Efficiency and Productivity Growth. Edited by Harold O. Fried, C. A. Knox Lovell and Shelton S. Schmidt. New York: Oxford University Press, pp. 92–250. [Google Scholar]
- Griffin, Jim E., and Mark F. J. Steel. 2004. Semiparametric Bayesian inference for stochastic frontier models. Journal of Econometrics 123: 121–52. [Google Scholar] [CrossRef]
- Griffin, Jim E., and Mark F. J. Steel. 2007. Bayesian stochastic frontier analysis using WinBUGS. Journal of Productivity Analysis 27: 163–76. [Google Scholar] [CrossRef] [Green Version]
- Griffin, Jim E., and Mark F. J. Steel. 2008. Flexible mixture modelling of stochastic frontiers. Journal of Productivity Analysis 29: 33–50. [Google Scholar] [CrossRef]
- Hajargasht, Gholamreza. 2015. Stochastic frontiers with a Rayleigh distribution. Journal of Productivity Analysis 44: 199–208. [Google Scholar] [CrossRef]
- Harvey, Andrew, and Rutger-Jan Lange. 2017. Volatility modeling with a generalized t distribution. Journal of Time Series Analysis 38: 175–90. [Google Scholar] [CrossRef] [Green Version]
- Horrace, William C., and Christopher F. Parmeter. 2018. A Laplace stochastic frontier model. Econometric Reviews 37: 260–80. [Google Scholar] [CrossRef] [Green Version]
- Koop, Gary, Jacek Osiewalski, and Mark F. J. Steel. 1997. Bayesian efficiency analysis through individual effects: Hospital cost frontiers. Journal of Econometrics 76: 77–105. [Google Scholar] [CrossRef]
- Koop, Gary, Jacek Osiewalski, and Mark F. J. Steel. 1999. The Components of Output Growth: A Stochastic Frontier Analysis. Oxford Bulletin of Economics and Statistics 61: 455–87. [Google Scholar] [CrossRef]
- Koop, Gary, Jacek Osiewalski, and Mark F. J. Steel. 2000. Modeling the Sources of Output Growth in a Panel of Countries. Journal of Business and Economic Statistics 18: 284–99. [Google Scholar]
- Kumbhakar, Subal C., Christopher F. Parmeter, and Efthymios G. Tsionas. 2013. A zero inefficiency stochastic frontier model. Journal of Econometrics 172: 66–76. [Google Scholar] [CrossRef]
- Ley, Eduardo, and Mark F. J. Steel. 2009. On the effect of prior assumptions in Bayesian model averaging with applications to growth regression. Journal of Applied Econometrics 24: 651–74. [Google Scholar] [CrossRef] [Green Version]
- Ley, Eduardo, and Mark F. J. Steel. 2012. Mixtures of g-priors for Bayesian model averaging with economic applications. Journal of Econometrics 171: 251–66. [Google Scholar] [CrossRef] [Green Version]
- Makieła, Kamil. 2014. Bayesian Stochastic Frontier Analysis of Economic Growth and Productivity Change in the EU, USA, Japan and Switzerland. Central European Journal of Economic Modelling and Econometrics 6: 193–216. [Google Scholar]
- Makieła, Kamil. 2017. Bayesian Inference and Gibbs Sampling in Generalized True Random-Effects Models. Central European Journal of Economic Modelling and Econometrics 9: 69–95. [Google Scholar]
- Makieła, Kamil, and Błażej Mazur. 2020. Stochastic Frontier Analysis with Generalized Errors: Inference, Model Comparison and Averaging. Working Paper. Available online: http://arxiv.org/abs/2003.07150 (accessed on 16 March 2020).
- Makieła, Kamil, and Jacek Osiewalski. 2018. Cost Efficiency Analysis of Electricity Distribution Sector under Model Uncertainty. The Energy Journal 39: 31–56. [Google Scholar] [CrossRef]
- Makieła, Kamil, Jerzy Marzec, and Andrzej Pisulewski. 2017. Productivity change analysis in dairy farms following Polish accession to the EU—An output growth decomposition approach. Outlook on Agriculture 46: 295–301. [Google Scholar] [CrossRef]
- Meeusen, Wim, and Julien van den Broeck. 1977. Efficiency estimation from Cobb-Douglas Production Function with Composed Error. International Economic Review 18: 435–44. [Google Scholar] [CrossRef]
- Newton, Michael A., and Adrian E. Raftery. 1994. Approximate Bayesian Inference with the Weighted Likelihood Bootstrap. Journal of the Royal Statistical Society 56: 3–48. [Google Scholar] [CrossRef]
- Osiewalski, Jacek, and Mark F. J. Steel. 1993. Una perspectiva bayesiana en selección de modelos [A Bayesian Perspective on Model Selection]. Cuadernos Economicos 55: 327–51. [Google Scholar]
- Osiewalski, Jacek, and Mark F. J. Steel. 1998. Numerical Tools for the Bayesian Analysis of Stochastic Frontier Models. Journal of Productivity Analysis 10: 103–17. [Google Scholar] [CrossRef]
- Pajor, Anna. 2017. Estimating the marginal likelihood using the arithmetic mean identity. Bayesian Analysis 12: 261–87. [Google Scholar] [CrossRef]
- Pitt, Mark M., and Lung-Fei Lee. 1981. The Measurement and Sources of Technical Inefficiency in the Indonesian Weaving Industry. Journal of Development Economics 9: 43–64. [Google Scholar] [CrossRef]
- Stacy, E. Webb. 1962. A generalization of the gamma distribution. The Annals of Mathematical Statistics 33: 1187–92. [Google Scholar] [CrossRef]
- Stead, Alexander D., Phill Wheat, and William H. Greene. 2018. Estimating Efficiency in the Presence of Extreme Outliers: A Logistic-Half Normal Stochastic Frontier Model with Application to Highway Maintenance Costs in England. In Productivity and Inequality (Springer Proceedings in Business and Economics). Edited by William H. Greene, Lynda Khalaf, Paul Makdissi, Robin C. Sickles, Michael Veall and Marcel-Cristian Voia. Cham: Springer, pp. 1–20. [Google Scholar]
- Stead, Alexander D., Phill Wheat, and William H. Greene. 2019. Robust Stochastic Frontier Analysis: A Student’s t-Half Normal Model with Application to Highway Maintenance Costs in England. Journal of Productivity Analysis 51: 21–38. [Google Scholar]
- Steel, Mark F. J. 2019. Model averaging and its use in economics. Journal of Economic Literature. [Google Scholar] [CrossRef]
- Stevenson, Rodney E. 1980. Likelihood Functions for Generalized Stochastic Frontier Estimation. Journal of Econometrics 13: 58–66. [Google Scholar] [CrossRef]
- Subbotin, Mikhail. 1923. On the law of frequency of error. Математический сбoрник 31: 296–301. [Google Scholar]
- Tchumtchoua, Sylvie, and Dipak K. Dey. 2007. Bayesian Estimation of Stochastic Frontier Models with Multivariate Skew t Error Terms. Communications. Statistics: Theory and Methods 36: 907–16. [Google Scholar]
- Theodossiou, Panayiotis, Dimitris A. Tsouknidis, and Christos S. Savva. 2020. Freight Rates in Downside and Upside Markets: Pricing of Own and Spillover Risks from Other Shipping Segments. Journal of the Royal Statistical Society. in press. [Google Scholar] [CrossRef]
- Tran, Kien C., and Mike G. Tsionas. 2016. Zero-inefficiency stochastic frontier models with varying mixing proportion: A semiparametric approach. European Journal of Operational Research 249: 1113–23. [Google Scholar] [CrossRef]
- Tsionas, Efthymios G. 2002. Stochastic Frontier Models with Random Coefficients. Journal of Applied Econometrics 17: 127–47. [Google Scholar] [CrossRef]
- Tsionas, Efthymios G. 2006. Inference in dynamic stochastic frontier models. Journal of Applied Econometrics 21: 669–76. [Google Scholar] [CrossRef]
- Tsionas, Efthymios G. 2007. Efficiency Measurement with the Weibull Stochastic Frontier. Oxford Bulletin of Economics and Statistics 69: 693–706. [Google Scholar] [CrossRef]
- Tsionas, Efthymios G., and Subal C. Kumbhakar. 2014. Firm Heterogeneity, Persistent and Transient Technical Inefficiency: A Generalized True Random-Effects model. Journal of Applied Econometrics 29: 110–32. [Google Scholar] [CrossRef]
- van den Broeck, Julien, Gary Koop, Jacek Osiewalski, and Mark F. J. Steel. 1994. Stochastic Frontier Models: A Bayesian Perspective. Journal of Econometrics 61: 273–303. [Google Scholar] [CrossRef]
- Wheat, Phill, Alexander D. Stead, and William H. Greene. 2019. Controlling for Outliers in Efficiency Analysis: A Contaminated Normal-Half Normal Stochastic Frontier Model. Working Paper. Available online: http://www.its.leeds.ac.uk/fileadmin/documents/research/bear/Allowing_for_outliers_in_stochastic_frontier_models_200318.pdf (accessed on 10 December 2019).

**Figure 4.**Natural logs of Bayes factors for normal-half-normal (NHN) and normal-exponential (NEX) under different r*.

**Figure 5.**Scatter plots between posterior estimates of r for normal-exponential (NEX) and normal-half-normal (NHN) models.

**Figure 7.**Posterior predictive density of $\epsilon +{\beta}_{0}$ for normal-exponential (NEX) and normal-half-GED (NHG) under prior 0.

Prior 0 | Prior 1 | Prior 2 | Prior 3 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

NEX | NHN | NHG | NEX | NHN | NHG | NEX | NHN | NHG | NEX | NHN | NHG | |

Me | 0.821 | 0.828 | 0.826 | 0.750 | 0.767 | 0.766 | 0.745 | 0.750 | 0.753 | 0.989 | 0.990 | 0.990 |

IQR | 0.380 | 0.335 | 0.340 | 0.488 | 0.425 | 0.434 | 0.338 | 0.267 | 0.276 | 0.025 | 0.020 | 0.020 |

Avg. | 0.723 | 0.745 | 0.741 | 0.640 | 0.661 | 0.659 | 0.698 | 0.729 | 0.725 | 0.963 | 0.968 | 0.967 |

Std. | 0.272 | 0.245 | 0.250 | 0.318 | 0.302 | 0.305 | 0.226 | 0.178 | 0.189 | 0.098 | 0.088 | 0.092 |

NEX Prior 1 | NHN Prior 2 | NHN Prior 3 | Naive BMA: E(.|y) | |||||
---|---|---|---|---|---|---|---|---|

E(.|y) | D(.|y) | E(.|y) | D(.|y) | E(.|y) | D(.|y) | ver.1 | ver.2 | |

${\beta}_{0}$ | 0.0553 | 0.0191 | 0.1227 | 0.0168 | 0.1016 | 0.0183 | 0.0870 | 0.1015 |

${\beta}_{1}$ | −0.0018 | 0.0013 | −0.0005 | 0.0013 | −0.0011 | 0.0012 | −0.0011 | −0.0011 |

${\beta}_{2}$ | 0.8546 | 0.0128 | 0.8468 | 0.0144 | 0.8552 | 0.0133 | 0.8318 | 0.8552 |

${\beta}_{3}$ | 0.1125 | 0.0140 | 0.1186 | 0.0149 | 0.1117 | 0.0142 | 0.1130 | 0.1117 |

${\beta}_{4}$ | 0.0351 | 0.0193 | 0.0008 | 0.0231 | 0.0237 | 0.0213 | 0.0176 | 0.0237 |

${\beta}_{5}$ | 0.1014 | 0.0222 | 0.0615 | 0.0272 | 0.0874 | 0.0242 | 0.0796 | 0.0875 |

${\beta}_{6}$ | −0.1132 | 0.0404 | −0.0407 | 0.0494 | −0.0892 | 0.0445 | −0.0753 | −0.0893 |

${\sigma}_{u}$ | 0.1001 | 0.0131 | 0.2060 | 0.0114 | 0.1797 | 0.0140 | 0.1497 | 0.1795 |

${\sigma}_{v}$ | 0.0758 | 0.0094 | 0.0384 | 0.0090 | 0.0522 | 0.0098 | 0.0558 | 0.0522 |

$\overline{r}$ | 0.9474 | 0.0398 | 0.8916 | 0.0365 | 0.9129 | 0.0418 | 0.8991 | 0.9129 |

ln p(y) | 247.6 | 247.7 | 253.7 | |||||

1: p(M) | 0.5 | 0.5 | 1 | |||||

p(M|y) | 0.4889 | 0.5111 | 1 | |||||

2: p(M) | 0.5 | 0.5 | 1 | |||||

p(M|y) | 0.0023 | 0.9977 | 1 |

NEX | NHN | NHG | GHG | BMA | BMA | |||||
---|---|---|---|---|---|---|---|---|---|---|

E(.|y) | D(.|y) | E(.|y) | D(.|y) | E(.|y) | D(.|y) | E(.|y) | D(.|y) | All | ||

${\beta}_{0}$ | 0.0548 | 0.0191 | 0.1030 | 0.0177 | 0.1248 | 0.0207 | 0.1207 | 0.0187 | 0.1030 | 0.1161 |

${\beta}_{1}$ | −0.0019 | 0.0013 | −0.0011 | 0.0013 | −0.0007 | 0.0013 | −0.0007 | 0.0013 | −0.0011 | −0.0008 |

${\beta}_{2}$ | 0.8551 | 0.0128 | 0.8544 | 0.0134 | 0.8561 | 0.0146 | 0.8561 | 0.0149 | 0.8544 | 0.8555 |

${\beta}_{3}$ | 0.1120 | 0.0140 | 0.1123 | 0.0143 | 0.1103 | 0.0155 | 0.1103 | 0.0156 | 0.1123 | 0.1110 |

${\beta}_{4}$ | 0.0353 | 0.0193 | 0.0222 | 0.0207 | 0.0161 | 0.0225 | 0.0205 | 0.0234 | 0.0222 | 0.0193 |

${\beta}_{5}$ | 0.1013 | 0.0222 | 0.0859 | 0.0239 | 0.0748 | 0.0259 | 0.0796 | 0.0265 | 0.0859 | 0.0798 |

${\beta}_{6}$ | −0.1133 | 0.0404 | −0.0861 | 0.0436 | −0.0701 | 0.0471 | −0.0792 | 0.0489 | −0.0861 | −0.0778 |

${\sigma}_{u}$ | 0.0987 | 0.0131 | 0.1808 | 0.0133 | 0.2275 | 0.0264 | 0.2252 | 0.0251 | 0.1807 | 0.2103 |

${\sigma}_{v}$ | 0.0763 | 0.0094 | 0.0518 | 0.0098 | 0.0384 | 0.0128 | 0.0458 | 0.0137 | 0.0518 | 0.0448 |

${\psi}_{u}$ | (=1) | - | (=2) | - | 3.0169 | 0.6258 | 3.0316 | 0.6172 | 1.9996 | 26568 |

${\psi}_{v}$ | (=2) | - | (=2) | - | (=2) | - | 3.3754 | 1.4389 | 2.0000 | 22968 |

$\overline{r}$ | 0.9475 | 0.0398 | 0.9120 | 0.0422 | 0.8891 | 0.0380 | 0.8926 | 0.0389 | 0.9121 | 0.8981 |

ln p(y) | 247.6 | 255.4 | 256.3 | 256.3 | ||||||

1: p(M) | 0.5 | 0.5 | 1 | |||||||

p(M|y) | 0.0004 | 0.9996 | 1 | |||||||

2: p(M) | 0.3636 | 0.3636 | 0.1818 | 0.0909 | 1 | |||||

p(M|y) | 0.0001 | 0.3570 | 0.4271 | 0.2158 | 1 |

NEX | NHN | NHG | GHG | BMA | BMA | |||||
---|---|---|---|---|---|---|---|---|---|---|

E(.|y) | D(.|y) | E(.|y) | D(.|y) | E(.|y) | D(.|y) | E(.|y) | D(.|y) | All | ||

${\beta}_{0}$ | 0.0553 | 0.0187 | 0.1032 | 0.0176 | 0.1239 | 0.0221 | 0.1210 | 0.0203 | 0.1032 | 0.1157 |

${\beta}_{1}$ | −0.0018 | 0.0013 | −0.0011 | 0.0013 | −0.0007 | 0.0013 | −0.0007 | 0.0013 | −0.0011 | −0.0008 |

${\beta}_{2}$ | 0.8546 | 0.0124 | 0.8544 | 0.0135 | 0.8554 | 0.0150 | 0.8567 | 0.0146 | 0.8544 | 0.8553 |

${\beta}_{3}$ | 0.1125 | 0.0139 | 0.1125 | 0.0145 | 0.1111 | 0.0159 | 0.1099 | 0.0157 | 0.1125 | 0.1113 |

${\beta}_{4}$ | 0.0351 | 0.0185 | 0.0229 | 0.0211 | 0.0172 | 0.0246 | 0.0212 | 0.0234 | 0.0229 | 0.0202 |

${\beta}_{5}$ | 0.1014 | 0.0216 | 0.0871 | 0.0246 | 0.0762 | 0.0274 | 0.0810 | 0.0261 | 0.0871 | 0.0812 |

${\beta}_{6}$ | −0.1132 | 0.0389 | −0.0880 | 0.0446 | −0.0726 | 0.0508 | −0.0813 | 0.0484 | −0.0880 | −0.0802 |

${\sigma}_{u}$ | 0.1001 | 0.0128 | 0.1816 | 0.0133 | 0.2275 | 0.0274 | 0.2259 | 0.0262 | 0.1816 | 0.2102 |

${\sigma}_{v}$ | 0.0758 | 0.0091 | 0.0514 | 0.0095 | 0.0391 | 0.0139 | 0.0453 | 0.0148 | 0.0515 | 0.0450 |

${\psi}_{u}$ | (=1) | - | (=2) | - | 3.0634 | 0.5942 | 3.0362 | 0.5978 | 1.9997 | 2.6663 |

${\psi}_{v}$ | (=2) | - | (=2) | - | (=2) | - | 3.3397 | 1.4463 | 2.0000 | 2.2870 |

$\overline{r}$ | 0.9474 | 0.0396 | 0.9115 | 0.0419 | 0.8896 | 0.0385 | 0.8924 | 0.0389 | 0.9115 | 0.8983 |

ln p(y) | 247.6 | 255.7 | 256.5 | 256.5 | ||||||

1: p(M) | 0.5 | 0.5 | 1 | |||||||

p(M|y) | 0.0003 | 0.9997 | 1 | |||||||

2: p(M) | 0.3636 | 0.3636 | 0.1818 | 0.0909 | 1 | |||||

p(M|y) | 0.0001 | 0.3677 | 0.4179 | 0.2143 | 1 |

NEX | NHN | NHG | GHG | BMA | BMA | |||||
---|---|---|---|---|---|---|---|---|---|---|

E(.|y) | D(.|y) | E(.|y) | D(.|y) | E(.|y) | D(.|y) | E(.|y) | D(.|y) | All | ||

${\beta}_{0}$ | 0.0935 | 0.0154 | 0.1227 | 0.0168 | 0.1400 | 0.0181 | 0.1379 | 0.0197 | 0.1227 | 0.1385 |

${\beta}_{1}$ | −0.0008 | 0.0014 | −0.0005 | 0.0013 | −0.0001 | 0.0013 | −0.0003 | 0.0013 | −0.0005 | −0.0002 |

${\beta}_{2}$ | 0.8480 | 0.0131 | 0.8468 | 0.0144 | 0.8558 | 0.0166 | 0.8568 | 0.0166 | 0.8468 | 0.8565 |

${\beta}_{3}$ | 0.1173 | 0.0139 | 0.1186 | 0.0149 | 0.1093 | 0.0179 | 0.1086 | 0.0178 | 0.1186 | 0.1088 |

${\beta}_{4}$ | 0.0099 | 0.0201 | 0.0008 | 0.0231 | 0.0101 | 0.0257 | 0.0124 | 0.0266 | 0.0008 | 0.0116 |

${\beta}_{5}$ | 0.0783 | 0.0244 | 0.0615 | 0.0272 | 0.0659 | 0.0275 | 0.0683 | 0.0279 | 0.0615 | 0.0675 |

${\beta}_{6}$ | −0.0649 | 0.0436 | −0.0407 | 0.0494 | −0.0561 | 0.0522 | −0.0608 | 0.0534 | −0.0407 | −0.0592 |

${\sigma}_{u}$ | 0.1570 | 0.0090 | 0.2060 | 0.0114 | 0.2621 | 0.0193 | 0.2581 | 0.0194 | 0.2060 | 0.2591 |

${\sigma}_{v}$ | 0.0524 | 0.0064 | 0.0384 | 0.0090 | 0.0259 | 0.0126 | 0.0304 | 0.0151 | 0.0384 | 0.0290 |

${\psi}_{u}$ | (=1) | - | (=2) | - | 3.7537 | 0.6158 | 3.6747 | 0.5804 | 2.0000 | 3.6911 |

${\psi}_{v}$ | (=2) | - | (=2) | - | (=2) | - | 2.9550 | 1.4703 | 2.0000 | 26469 |

$\overline{r}$ | 0.9237 | 0.0413 | 0.8916 | 0.0365 | 0.8710 | 0.0291 | 0.8743 | 0.0309 | 0.8916 | 0.8733 |

ln p(y) | 218.8 | 247.7 | 252.5 | 253.9 | ||||||

1: p(M) | 0.5 | 0.5 | 1 | |||||||

p(M|y) | 0.0000 | 1 | 1 | |||||||

2: p(M) | 0.3636 | 0.3636 | 0.1818 | 0.0909 | 1 | |||||

p(M|y) | 0.0000 | 0.0052 | 0.3174 | 0.6774 | 1 |

NEX | NHN | NHG | GHG | BMA | BMA | |||||
---|---|---|---|---|---|---|---|---|---|---|

E(.|y) | D(.|y) | E(.|y) | D(.|y) | E(.|y) | D(.|y) | E(.|y) | D(.|y) | All | ||

${\beta}_{0}$ | 0.0471 | 0.0280 | 0.1016 | 0.0183 | 0.1208 | 0.0237 | 0.1176 | 0.0201 | 0.1016 | 0.1117 |

${\beta}_{1}$ | −0.0019 | 0.0013 | −0.0011 | 0.0012 | −0.0010 | 0.0013 | −0.0008 | 0.0013 | −0.0011 | −0.0010 |

${\beta}_{2}$ | 0.8554 | 0.0133 | 0.8552 | 0.0133 | 0.8555 | 0.0149 | 0.8564 | 0.0137 | 0.8552 | 0.8555 |

${\beta}_{3}$ | 0.1118 | 0.0148 | 0.1117 | 0.0142 | 0.1111 | 0.0157 | 0.1103 | 0.0149 | 0.1117 | 0.1112 |

${\beta}_{4}$ | 0.0368 | 0.0198 | 0.0237 | 0.0213 | 0.0179 | 0.0249 | 0.0212 | 0.0222 | 0.0237 | 0.0211 |

${\beta}_{5}$ | 0.1012 | 0.0220 | 0.0874 | 0.0242 | 0.0769 | 0.0282 | 0.0815 | 0.0254 | 0.0875 | 0.0824 |

${\beta}_{6}$ | −0.1152 | 0.0407 | −0.0892 | 0.0445 | −0.0737 | 0.0519 | −0.0815 | 0.0465 | −0.0893 | −0.0820 |

${\sigma}_{u}$ | 0.0907 | 0.0231 | 0.1797 | 0.0140 | 0.2199 | 0.0303 | 0.2173 | 0.0272 | 0.1796 | 0.2017 |

${\sigma}_{v}$ | 0.0803 | 0.0136 | 0.0522 | 0.0098 | 0.0421 | 0.0137 | 0.0478 | 0.0145 | 0.0522 | 0.0476 |

${\psi}_{u}$ | (=1) | - | (=2) | - | 2.9212 | 0.6239 | 2.8674 | 0.5878 | 1.9987 | 2.5061 |

${\psi}_{v}$ | (=2) | - | (=2) | - | (=2) | - | 3.3447 | 1.4549 | 2.0000 | 2.2504 |

$\overline{r}$ | 0.9505 | 0.0394 | 0.9129 | 0.0418 | 0.8940 | 0.0397 | 0.8963 | 0.0000 | 0.9129 | 0.9027 |

ln p(y) | 247.0 | 253.7 | 254.2 | 254.2 | ||||||

1: p(M) | 0.5 | 0.5 | 1 | |||||||

p(M|y) | 0.0013 | 0.9987 | 1 | |||||||

2: p(M) | 0.3636 | 0.3636 | 0.1818 | 0.0909 | 1 | |||||

p(M|y) | 0.0005 | 0.4386 | 0.3746 | 0.1862 | 1 |

Real | Non-SF | Prior 0 | Prior 1 | Prior 2 | Prior 3 | BMA | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

val. | NEX | NHN | NEX | NHN | NEX | NHN | NEX | NHN | Prior0 | Prior1 | Prior2 | Prior3 | ||

E(.|y) | E(.|y) | E(.|y) | E(.|y) | E(.|y) | E(.|y) | E(.|y) | E(.|y) | E(.|y) | E(.|y) | E(.|y) | E(.|y) | E(.|y) | ||

${\beta}_{0}$ | 0 | 0.000 | 0.045 | 0.092 | 0.090 | 0.118 | 0.187 | 0.209 | 0.024 | 0.020 | 0.020 | 0.020 | 0.000 | 0.011 |

${\beta}_{1}$ | 0.6 | 0.609 | 0.607 | 0.609 | 0.608 | 0.609 | 0.606 | 0.607 | 0.609 | 0.609 | 0.609 | 0.609 | 0.609 | 0.609 |

${\beta}_{2}$ | 0.4 | 0.393 | 0.393 | 0.392 | 0.393 | 0.393 | 0.393 | 0.394 | 0.393 | 0.393 | 0.393 | 0.393 | 0.393 | 0.393 |

${\sigma}_{u}$ | 0.000 | 0.045 | 0.116 | 0.092 | 0.149 | 0.216 | 0.267 | 0.023 | 0.025 | 0.024 | 0.025 | 0.001 | 0.013 | |

${\sigma}_{v}$ | 0.2 | 0.202 | 0.194 | 0.186 | 0.183 | 0.180 | 0.148 | 0.145 | 0.199 | 0.200 | 0.199 | 0.198 | 0.202 | 0.201 |

E($\overline{r}$|y) | 1 | 1 | 0.900 | 0.841 | 0.775 | 0.787 | 0.612 | 0.633 | 0.958 | 0.972 | 0.962 | 0.961 | 0.999 | 0.982 |

D($\overline{r}$|y) | - | - | 0.120 | 0.121 | 0.129 | 0.113 | 0.093 | 0.084 | 0.073 | 0.047 | ||||

ln p(y) | 0.581 | −0.533 | −0.193 | −1.853 | −0.434 | −16.026 | −4.644 | 0.623 | 0.667 | |||||

p(M) | 0.500 | 0.250 | 0.250 | 0.250 | 0.250 | 0.250 | 0.250 | 0.250 | 0.250 | |||||

p(M|y) | 0.717 | 0.118 | 0.165 | 1 | ||||||||||

p(M|y) | 0.816 | 0.036 | 0.148 | 1 | ||||||||||

p(M|y) | 0.997 | 0.000 | 0.003 | 1 | ||||||||||

p(M|y) | 0.484 | 0.252 | 0.264 | 1 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Makieła, K.; Mazur, B.
Bayesian Model Averaging and Prior Sensitivity in Stochastic Frontier Analysis. *Econometrics* **2020**, *8*, 13.
https://doi.org/10.3390/econometrics8020013

**AMA Style**

Makieła K, Mazur B.
Bayesian Model Averaging and Prior Sensitivity in Stochastic Frontier Analysis. *Econometrics*. 2020; 8(2):13.
https://doi.org/10.3390/econometrics8020013

**Chicago/Turabian Style**

Makieła, Kamil, and Błażej Mazur.
2020. "Bayesian Model Averaging and Prior Sensitivity in Stochastic Frontier Analysis" *Econometrics* 8, no. 2: 13.
https://doi.org/10.3390/econometrics8020013