Abstract
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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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.
Aït-Sahalia Y., Duarte J. (2003) Nonparametric option pricing under shape restrictions. Journal of Econometrics 116: 9–47
Barnett W. A. (1976) Maximum likelihood and iterated Aitken estimation of non-linear systems of equations. Journal of the American Statistical Association 71: 354–360
Barnett W. A. (1985) The Minflex-Laurent translog flexible functional form. Journal of Econometrics 30: 33–44
Barnett W. A. (2002) Tastes and technology: Curvature is not sufficient for regularity. Journal of Econometrics 108: 199–202
Barnett W. A., Geweke J., Wolfe M. (1991) Seminonparametric bayesian estimation of the asymptotically ideal production model. Journal of Econometrics 49(1): 5–50
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.
Barnett W. A., Pasupathy M. (2003) Regularity of the generalized quadratic production model: A counterexample. Econometric Reviews 22(2): 135–154
Chen M. H., Shao Q. M., Ibrahim J. G. (2000) Monte Carlo methods in bayesian computation. Springer, New York
Chib S., Greenberg E. (1996) Markov Chain Monte Carlo methods in econometrics. Econometric Theory 12: 409–431
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: 269–291
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.
Diewert W. E., Wales T. J. (1987) Flexible functional forms and global curvature conditions. Econometrica 55: 43–68
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. http://faculty.arts.ubc.ca/ediewert/9121.pdf.
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: 615–623
Fischer D., Fleissig A. R., Serletis A. (2001) An empirical comparison of flexible demand system functional forms. Journal of Applied Econometrics 16(1): 59–80
Fleissig A. R., Kastens T., Terrell D. (1997) Semi-nonparametric estimates of substitution elasticities. Economic Letters 54: 209–215
Fleissig A. R., Kastens T., Terrell D. (2000) Evaluating the semi-nonparametric Fourier, AIM, and neural networks cost functions. Economic Letters 68: 235–244
Gallant A. R., Golub G. H. (1984) Imposing curvature restrictions on flexible functional forms. Journal of Econometrics 26: 295–322
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 263–290
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: 107–127
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 575–590
Hildreth C. (1954) Point estimates of ordinates of concave functions. Journal of the American Statistical Association 49: 598–619
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): 649–667
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): 523–530
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): 339–346
Koop G., Osiewalski J., Steel M. F. J. (1997) Bayesian efficiency analysis through individual effects: Hospital cost frontiers. Journal of Econometrics 76: 77–105
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 409–453
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 1515–1566
Magnus J. R. (1979) Substitution between energy and non-energy inputs in the Netherlands 1950–1976. International Economics Review 20: 465–484
Mas-Colell A., Whinston M. D., Green J. R. (1995) Microeconomic theory. Oxford University Press, USA
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 2524–2558
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.
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): 709–726
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: 865–880
Racine, J. S., & Parmeter, C. F. (2008). Constrained nonparametric kernel regression estimation and inference. Working paper available at http://www1.maxwell.syr.edu/uploadedFiles/econ/kernel_cons.pdf?n=6322.
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): 331–338
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: 133–143
Simon C. P., Blume L. (1994) Mathematics for economists. W.W. Norton, New York
Terrell D. (1995) Flexibility and regularity properties of the asymptotic ideal production model. Econometric Reviews 14(1): 1–17
Terrell D. (1996) Incorporating monotonicity and concavity conditions in flexible functional forms. Journal of Applied Econometrics 11: 179–194
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.
Tripathi G. (2000) Local semiparametric efficiency bounds under shape restrictions. Econometric Theory 16: 729–739
Zellner A. (1971) An introduction to bayesian inference in econometrics. John Wiley and Sons, New York.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wolff, H., Heckelei, T. & Mittelhammer, R.C. Imposing Curvature and Monotonicity on Flexible Functional Forms: An Efficient Regional Approach. Comput Econ 36, 309–339 (2010). https://doi.org/10.1007/s10614-010-9215-1
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10614-010-9215-1
Keywords
- Nonlinear inequality constraints
- Flexible functional forms
- Metropolis-Hastings
- Accept–Reject algorithm
- Cost function
- Regularity conditions