Computational Economics

, Volume 36, Issue 4, pp 309–339 | Cite as

Imposing Curvature and Monotonicity on Flexible Functional Forms: An Efficient Regional Approach

  • Hendrik WolffEmail author
  • Thomas Heckelei
  • Ron C. Mittelhammer


In many areas of economic analysis, economic theory restricts the shape of functions. Examples are the monotonicity and curvature conditions that apply to utility, profit, and cost functions. Here we extend upon a currently available estimation method (Terrell, J Appl Econometr 11:179–194, 1996) for imposing regularity regionally on a connected subset of the regressor space. Our method offers important advantages by imposing theoretical consistency not only locally, at a given evaluation point but also within the whole empirically relevant region of the domain associated with the function being estimated. The method also provides benefits through higher flexibility, which generally leads to a better model fit to the sample data. Specific contributions of this paper are (a) to increase the computational speed, (b) to provide regularity preserving point estimates, and (c) to illustrate the benefits of this revised regional approach via numerical simulation results.


Nonlinear inequality constraints Flexible functional forms Metropolis-Hastings Accept–Reject algorithm Cost function Regularity conditions 

JEL Classification

C51 D21 C11 


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  1. Adkins, L. C., Rickman, D. S., & Hameed, A. (2002). Bayesian estimation of regional production for CGE modeling. Paper presented at the fourteenth international conference on input–output techniques October 10–15, 2002, Montréal, Canada.Google Scholar
  2. Aït-Sahalia Y., Duarte J. (2003) Nonparametric option pricing under shape restrictions. Journal of Econometrics 116: 9–47CrossRefGoogle Scholar
  3. Barnett W. A. (1976) Maximum likelihood and iterated Aitken estimation of non-linear systems of equations. Journal of the American Statistical Association 71: 354–360CrossRefGoogle Scholar
  4. Barnett W. A. (1985) The Minflex-Laurent translog flexible functional form. Journal of Econometrics 30: 33–44CrossRefGoogle Scholar
  5. Barnett W. A. (2002) Tastes and technology: Curvature is not sufficient for regularity. Journal of Econometrics 108: 199–202CrossRefGoogle Scholar
  6. Barnett W. A., Geweke J., Wolfe M. (1991) Seminonparametric bayesian estimation of the asymptotically ideal production model. Journal of Econometrics 49(1): 5–50CrossRefGoogle Scholar
  7. Barnett, W. A., Kirova, M., & Pasupathy, M. (1995). Estimating policy-invariant deep parameters in the financial sector when risk and growth matter. Journal of Money, Credit Bank, 27(4), part 2, 1402–1430.Google Scholar
  8. Barnett W. A., Pasupathy M. (2003) Regularity of the generalized quadratic production model: A counterexample. Econometric Reviews 22(2): 135–154CrossRefGoogle Scholar
  9. Chen M. H., Shao Q. M., Ibrahim J. G. (2000) Monte Carlo methods in bayesian computation. Springer, New YorkGoogle Scholar
  10. Chib S., Greenberg E. (1996) Markov Chain Monte Carlo methods in econometrics. Econometric Theory 12: 409–431CrossRefGoogle Scholar
  11. Chua C. L., Griffiths W. E., O’Donnell C. J. (2001) Bayesian model averaging in consumer demand systems with inequality constraints. Canadian Journal of Agricultural Economics 49: 269–291CrossRefGoogle Scholar
  12. Cuesta, R. A., O’Donnell, C. J., Coelli, T. J., & Singh, S. (2001). Imposing curvature conditions on a production frontier: With applications to Indian dairy processing plants. CEPA Working Papers, No. 2/2001, ISBN 1 86389 749 6. Armidale: School of Economics, University of New England.Google Scholar
  13. Diewert W. E., Wales T. J. (1987) Flexible functional forms and global curvature conditions. Econometrica 55: 43–68CrossRefGoogle Scholar
  14. Diewert, W. E., & Wales, T. J. (1991). Multiproduct cost functions and subadditivity tests: A critique of the Evans and Heckman research on the U.S. Bell Systems. Discussions Paper 91-21 at the Department of Economics, University of British Columbia, Vancouver.
  15. Evans D. S., Heckman J. J. (1984) A Test for Subadditivity of the Cost Function with an Application to the Bell System. American Economic Review 74: 615–623Google Scholar
  16. Fischer D., Fleissig A. R., Serletis A. (2001) An empirical comparison of flexible demand system functional forms. Journal of Applied Econometrics 16(1): 59–80CrossRefGoogle Scholar
  17. Fleissig A. R., Kastens T., Terrell D. (1997) Semi-nonparametric estimates of substitution elasticities. Economic Letters 54: 209–215CrossRefGoogle Scholar
  18. Fleissig A. R., Kastens T., Terrell D. (2000) Evaluating the semi-nonparametric Fourier, AIM, and neural networks cost functions. Economic Letters 68: 235–244CrossRefGoogle Scholar
  19. Gallant A. R., Golub G. H. (1984) Imposing curvature restrictions on flexible functional forms. Journal of Econometrics 26: 295–322CrossRefGoogle Scholar
  20. Griffiths W. E. (2003) Bayesian inference in the seemingly unrelated regressions model. In: Giles D. E. A. (eds) Computer-aided econometrics. Marcel Dekker, New York, pp 263–290Google Scholar
  21. Griffiths W. E., O’Donnell C. J., Tan-Cruz A. (2000) Imposing regularity conditions on a system of cost and factor share equations. Australian Journal of Agricultural and Resource Economics 44: 107–127CrossRefGoogle Scholar
  22. Griffiths W. E., Skeels C. L., Chotikapanich D. (2002) Sample size requirements for estimation in SUR Models. In: Ullah A., Chaturvedi A., Wan A. (eds) Handbook of Applied Econometrics and Statistical Inference. Marcel Dekker, New York, pp 575–590Google Scholar
  23. Hildreth C. (1954) Point estimates of ordinates of concave functions. Journal of the American Statistical Association 49: 598–619CrossRefGoogle Scholar
  24. Ivaldi M., Ladoux N., Ossard H., Simioni M. (1996) Comparing Fourier and translog specifications of multiproduct technology: Evidence from an incomplete panel of French farmers. Journal of applied econometrics 11(6): 649–667CrossRefGoogle Scholar
  25. Kleit A. N., Terrell D. (2001) Measuring potential efficiency gains from deregulation of electricity generation: A bayesian approach. The Review of Economics and Statistics 83(3): 523–530CrossRefGoogle Scholar
  26. Koop G., Osiewalski J., Steel M. F. J. (1994) Bayesian efficiency analysis with a flexible form: The AIM cost function. Journal of Business and Economic Statistics 12(3): 339–346CrossRefGoogle Scholar
  27. Koop G., Osiewalski J., Steel M. F. J. (1997) Bayesian efficiency analysis through individual effects: Hospital cost frontiers. Journal of Econometrics 76: 77–105CrossRefGoogle Scholar
  28. Lau L. J. (1978) Testing and imposing monotonicity, convexity, and quasi-convexity constraints. In: Fuss M., McFadden D. (eds) Production economics: A dual approach to theory and applications (volume 1). North-Holland, Amsterdam, pp 409–453Google Scholar
  29. Lau L. J. (1986) Functional forms in econometric model building. In: Griliches Z., Intriligator M. D. (eds) Chapter 26 in handbook of econometrics. Elsevier Science, Amsterdam, North Holland, pp 1515–1566Google Scholar
  30. Magnus J. R. (1979) Substitution between energy and non-energy inputs in the Netherlands 1950–1976. International Economics Review 20: 465–484CrossRefGoogle Scholar
  31. Mas-Colell A., Whinston M. D., Green J. R. (1995) Microeconomic theory. Oxford University Press, USAGoogle Scholar
  32. Matzkin R. L. (1994) Restrictions of economic theory in nonparametric methods. In: Engle R. F., McFadden D. L. (eds) Handbook of Econometrics. North-Holland Pub Co, Amsterdam, pp 2524–2558Google Scholar
  33. O’Donnell, C. J., & Coelli, T. (2003). A bayesian approach to imposing curvature on distance functions. Paper presented at the Australasian Meeting of the Econometric Society, Sydney 2003.Google Scholar
  34. O’Donnell C. J., Rambaldi A. N., Doran H. E. (2001) Estimating economic relationships subject to firm- and time-varying equality and inequality constraints. Journal of Applied Econometrics 16(6): 709–726CrossRefGoogle Scholar
  35. O’Donnell C. J., Shumway C. R., Ball V. E. (1999) Input demands and inefficiency in U.S. agriculture. American Journal of Agricultural Economics 81: 865–880CrossRefGoogle Scholar
  36. Racine, J. S., & Parmeter, C. F. (2008). Constrained nonparametric kernel regression estimation and inference. Working paper available at
  37. Ryan D. L., Wales T. J. (1998) A simple method for imposing local curvature in some flexible consumer demand systems. Journal of Business and Economic Statistics 16(3): 331–338CrossRefGoogle Scholar
  38. Salvanes K. G., Tjøtta S. (1998) A Note on the Importance of Testing for Regularities for Estimated Flexible Functional Forms. Journal of Productivity Analysis 9: 133–143CrossRefGoogle Scholar
  39. Simon C. P., Blume L. (1994) Mathematics for economists. W.W. Norton, New YorkGoogle Scholar
  40. Terrell D. (1995) Flexibility and regularity properties of the asymptotic ideal production model. Econometric Reviews 14(1): 1–17CrossRefGoogle Scholar
  41. Terrell D. (1996) Incorporating monotonicity and concavity conditions in flexible functional forms. Journal of Applied Econometrics 11: 179–194CrossRefGoogle Scholar
  42. Terrell, D., & Dashti, I. (1997). Imposing monotonocity and concavity restrictions on stochastic frontiers. Working Paper, Department of Economics, Louisiana State University, E. J. Ourso College of Business Administration.Google Scholar
  43. Tripathi G. (2000) Local semiparametric efficiency bounds under shape restrictions. Econometric Theory 16: 729–739CrossRefGoogle Scholar
  44. Zellner A. (1971) An introduction to bayesian inference in econometrics. John Wiley and Sons, New York.Google Scholar

Copyright information

© Springer Science+Business Media, LLC. 2010

Authors and Affiliations

  • Hendrik Wolff
    • 1
    Email author
  • Thomas Heckelei
    • 2
  • Ron C. Mittelhammer
    • 3
  1. 1.Department of EconomicsUniversity of WashingtonSeattleUSA
  2. 2.Institute for Food and Resource EconomicsUniversity of BonnBonnGermany
  3. 3.School of Economic SciencesWashington State UniversityPullmanUSA

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