# Directional distance functions: Optimal endogenous directions

## Author

Listed:
• Atkinson, Scott E.
• Tsionas, Mike G.

## Abstract

A substantial literature has dealt with the problem of estimating multiple-input and multiple-output production functions, where inputs and outputs can be good and bad. Numerous studies can be found in the areas of productivity analysis, industrial organization, labor economics, and health economics. While many papers have estimated the more restrictive output- and input-oriented distance functions, here we estimate a more general directional distance function. A seminal paper on directional distance functions by Chambers (1998) as well as papers by Färe et al. (1997), Chambers et al. (1998), Färe and Grosskopf (2000), Grosskopf (2003), Färe et al. (2005), and Hudgins and Primont (2007) do not address the issue of how to choose an optimal direction set. Typically the direction is arbitrarily selected to be 1 for good outputs and −1 for inputs and bad outputs. By estimating the directional distance function together with the first-order conditions for cost minimization and profit maximization using Bayesian methods, we are able to estimate optimal firm-specific directions for each input and output which are consistent with allocative and technical efficiency. We apply these methods to an electric-utility panel data set, which contains firm-specific prices and quantities of good inputs and outputs as well as the quantities of bad inputs and outputs. Estimated firm-specific directions for each input and output are quite different from those normally assumed in the literature. The computed firm-specific technical efficiency, technical change, and productivity change based on estimated optimal directions are substantially higher than those calculated using fixed directions.

## Suggested Citation

• Atkinson, Scott E. & Tsionas, Mike G., 2016. "Directional distance functions: Optimal endogenous directions," Journal of Econometrics, Elsevier, vol. 190(2), pages 301-314.
• Handle: RePEc:eee:econom:v:190:y:2016:i:2:p:301-314
DOI: 10.1016/j.jeconom.2015.06.006
as

File URL: http://www.sciencedirect.com/science/article/pii/S030440761500175X

File URL: https://libkey.io/10.1016/j.jeconom.2015.06.006?utm_source=ideas
LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
---><---

As the access to this document is restricted, you may want to search for a different version of it.

## References listed on IDEAS

as
1. Shawna Grosskopf, 2003. "Some Remarks on Productivity and its Decompositions," Journal of Productivity Analysis, Springer, vol. 20(3), pages 459-474, November.
2. R. Färe & S. Grosskopf & G. Whittaker, 2013. "Directional output distance functions: endogenous directions based on exogenous normalization constraints," Journal of Productivity Analysis, Springer, vol. 40(3), pages 267-269, December.
3. Laurits R. Christensen & Dale W. Jorgenson, 1970. "U.S. Real Product And Real Factor Input, 1929–1967," Review of Income and Wealth, International Association for Research in Income and Wealth, vol. 16(1), pages 19-50, March.
4. Yaisawarng, Suthathip & Klein, J Douglass, 1994. "The Effects of Sulfur Dioxide Controls on Productivity Change in the U.S. Electric Power Industry," The Review of Economics and Statistics, MIT Press, vol. 76(3), pages 447-460, August.
5. Mark Girolami & Ben Calderhead, 2011. "Riemann manifold Langevin and Hamiltonian Monte Carlo methods," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 73(2), pages 123-214, March.
6. Atkinson, Scott E. & Dorfman, Jeffrey H., 2005. "Bayesian measurement of productivity and efficiency in the presence of undesirable outputs: crediting electric utilities for reducing air pollution," Journal of Econometrics, Elsevier, vol. 126(2), pages 445-468, June.
7. Feng, Guohua & Serletis, Apostolos, 2014. "Undesirable outputs and a primal Divisia productivity index based on the directional output distance function," Journal of Econometrics, Elsevier, vol. 183(1), pages 135-146.
8. Mark Agee & Scott Atkinson & Thomas Crocker, 2012. "Child maturation, time-invariant, and time-varying inputs: their interaction in the production of child human capital," Journal of Productivity Analysis, Springer, vol. 38(1), pages 29-44, August.
9. R. G. Chambers & Y. Chung & R. Färe, 1998. "Profit, Directional Distance Functions, and Nerlovian Efficiency," Journal of Optimization Theory and Applications, Springer, vol. 98(2), pages 351-364, August.
10. Fare, Rolf & Grosskopf, Shawna & Noh, Dong-Woon & Weber, William, 2005. "Characteristics of a polluting technology: theory and practice," Journal of Econometrics, Elsevier, vol. 126(2), pages 469-492, June.
11. Fernandez, Carmen & Koop, Gary & Steel, Mark F.J., 2005. "Alternative efficiency measures for multiple-output production," Journal of Econometrics, Elsevier, vol. 126(2), pages 411-444, June.
12. Rolf Färe & Shawna Grosskopf, 2000. "Theory and Application of Directional Distance Functions," Journal of Productivity Analysis, Springer, vol. 13(2), pages 93-103, March.
13. Pittman, Russell W, 1983. "Multilateral Productivity Comparisons with Undesirable Outputs," Economic Journal, Royal Economic Society, vol. 93(372), pages 883-891, December.
Full references (including those not matched with items on IDEAS)

## Most related items

These are the items that most often cite the same works as this one and are cited by the same works as this one.
1. Deng, Zhongqi & Jiang, Nan & Pang, Ruizhi, 2021. "Factor-analysis-based directional distance function: The case of New Zealand hospitals," Omega, Elsevier, vol. 98(C).
2. Atkinson, Scott E. & Primont, Daniel & Tsionas, Mike G., 2018. "Statistical inference in efficient production with bad inputs and outputs using latent prices and optimal directions," Journal of Econometrics, Elsevier, vol. 204(2), pages 131-146.
3. Guohua Feng & Chuan Wang & Apostolos Serletis, 2018. "Shadow prices of $$\hbox {CO}_{2}$$ CO 2 emissions at US electric utilities: a random-coefficient, random-directional-vector directional output distance function approach," Empirical Economics, Springer, vol. 54(1), pages 231-258, February.
4. Kumbhakar, Subal C. & Tsionas, Efthymios G., 2016. "The good, the bad and the technology: Endogeneity in environmental production models," Journal of Econometrics, Elsevier, vol. 190(2), pages 315-327.
5. Emir Malikov & Raushan Bokusheva & Subal C. Kumbhakar, 2018. "A hedonic-output-index-based approach to modeling polluting technologies," Empirical Economics, Springer, vol. 54(1), pages 287-308, February.
6. Emir Malikov & Subal C. Kumbhakar & Efthymios G. Tsionas, 2015. "Bayesian Approach to Disentangling Technical and Environmental Productivity," Econometrics, MDPI, vol. 3(2), pages 1-23, June.
7. Gary Koop & Lise Tole, 2008. "What is the environmental performance of firms overseas? An empirical investigation of the global gold mining industry," Journal of Productivity Analysis, Springer, vol. 30(2), pages 129-143, October.
8. Huang, Wei & Bruemmer, Bernhard, 2017. "Balancing economic revenue and grazing pressure of livestock grazing on the Qinghai–Tibetan–Plateau," Australian Journal of Agricultural and Resource Economics, Australian Agricultural and Resource Economics Society, vol. 61(4), October.
9. Agee, Mark D. & Atkinson, Scott E. & Crocker, Thomas D. & Williams, Jonathan W., 2014. "Non-separable pollution control: Implications for a CO2 emissions cap and trade system," Resource and Energy Economics, Elsevier, vol. 36(1), pages 64-82.
10. Emir Malikov & Subal C. Kumbhakar & Mike G. Tsionas, 2016. "A Cost System Approach to the Stochastic Directional Technology Distance Function with Undesirable Outputs: The Case of us Banks in 2001–2010," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 31(7), pages 1407-1429, November.
11. Holtkamp, A.M. & Brummer, B., 2018. "Environmental efficiency of smallholder rubber production," 2018 Conference, July 28-August 2, 2018, Vancouver, British Columbia 277518, International Association of Agricultural Economists.
12. Atkinson, Scott E. & Dorfman, Jeffrey H., 2005. "Bayesian measurement of productivity and efficiency in the presence of undesirable outputs: crediting electric utilities for reducing air pollution," Journal of Econometrics, Elsevier, vol. 126(2), pages 445-468, June.
13. Feng, Guohua & Serletis, Apostolos, 2014. "Undesirable outputs and a primal Divisia productivity index based on the directional output distance function," Journal of Econometrics, Elsevier, vol. 183(1), pages 135-146.
14. Scott E. Atkinson & Kamalini Ramdas & Jonathan W. Williams, 2016. "Robust Scheduling Practices in the U.S. Airline Industry: Costs, Returns, and Inefficiencies," Management Science, INFORMS, vol. 62(11), pages 3372-3391, November.
15. Zhou, P. & Zhou, X. & Fan, L.W., 2014. "On estimating shadow prices of undesirable outputs with efficiency models: A literature review," Applied Energy, Elsevier, vol. 130(C), pages 799-806.
16. Badau, Flavius & Färe, Rolf & Gopinath, Munisamy, 2016. "Global resilience to climate change: Examining global economic and environmental performance resulting from a global carbon dioxide market," Resource and Energy Economics, Elsevier, vol. 45(C), pages 46-64.
17. Zhiqian Yu & Ning Zhu & Tomas Baležentis, 2017. "Impact of Public Education and Regional Economic Growth in China: A Shadow-Price Perspective," Sustainability, MDPI, vol. 9(8), pages 1-10, July.
18. Soledad Moya & Jordi Perramon & Anselm Constans, 2005. "IFRS Adoption in Europe: The Case of Germany," Working Papers 0501, Departament Empresa, Universitat Autònoma de Barcelona, revised Feb 2005.
19. Simar, Léopold & Vanhems, Anne, 2012. "Probabilistic characterization of directional distances and their robust versions," Journal of Econometrics, Elsevier, vol. 166(2), pages 342-354.
20. Färe, Rolf & Pasurka, Carl & Vardanyan, Michael, 2017. "On endogenizing direction vectors in parametric directional distance function-based models," European Journal of Operational Research, Elsevier, vol. 262(1), pages 361-369.

### Keywords

Optimal directions; Bayesian estimation; Directional distance function; Productivity change with goods and bads;
All these keywords.

### JEL classification:

• C11 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Bayesian Analysis: General
• C33 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Models with Panel Data; Spatio-temporal Models
• D24 - Microeconomics - - Production and Organizations - - - Production; Cost; Capital; Capital, Total Factor, and Multifactor Productivity; Capacity

## Corrections

All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:econom:v:190:y:2016:i:2:p:301-314. See general information about how to correct material in RePEc.

For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: . General contact details of provider: http://www.elsevier.com/locate/jeconom .

If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/jeconom .

Please note that corrections may take a couple of weeks to filter through the various RePEc services.

IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.