Abstract
An adequate stochastic model for shares as dependent variables is provided by the Dirichlet distribution. The paper considers two different parameterizations which lead to linear and nonlinear Dirichlet share equations. Using an inequality for the trigamma function the global concavity of the likelihood function for the nonlinear case is shown. The same inequality is employed in proving positive definiteness of the information matrix for the linear case. Suitability of the Dirichlet specification in econometric demand systems (such as AIDS and Translog) is discussed.
Research has been financially supported by DFG
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References
Abramowitz, M., and LA. Stegun (1965). Handbook of Mathematical Functions (eighth printing). New York: Dover.
Aitchison, J. (1982). The Analysis of compositional data. Journal of the Royal Statistical Society, Ser. B 44, 139–177 (with discussion).
Chipman, J.S., and G. Tian (1989). Stochastic Specification and Maximum-Likelihood Estimation of the Linear Expenditure System. Advances in Econometrics and Modelling (edited by Baldev Raj). Dordrecht, Holland: D. Reidel Publishing Company, 129–139.
Christensen, L.R., D.W. Jorgensonand LJ. Lau(1975). Transcendental logarithmic utility functions. American Economic Review 65, 367–383.
Considine, T.J., and T. D. Mount (1984). The use of linear logit models for dynamic input demand systems. Review of Economics and Statistics 66, 434–443.
Deaton, A., and J. Muellbauer (1980). An almost ideal demand system. American Economic Review 70, 312–326.
Deprins, D., and L. Simar (1985). A note on the asymptotic relative efficiency of m.l.e. in a linear model with gamma disturbances. Journal of Econometrics 27, 383–386.
Dhrymes, P.J. (1978). Introductory Econometrics. New York: Springer-Verlag.
Graybill, F.A. (1983). Matrices with Applications in Statistics. Second Edition. Belmont: Wadsworth.
Johnson, N.L., and S. Kotz (1970). Continuous Univariate Distributions-2. Boston: Houghton Mifflin.
Johnson, N.L., and S. Kotz (1972). Distributions in Statistics: Continuous Multivariate Distributions. New York: Wiley.
Lee, L.F., and Pit, M.M. (1986). Microeconomic Demand Systems WithBinding Nonnegativity Constraints: The Dual Approach. Econometrica 54, 1237–1242.
Rao, CR. (1973). Linear statistical inference and its Applications. New York: Wiley.
Ronning G. (1986). On the curvature of the trigamma function. Journal of Computational and Applied Mathematics 15, 397–399.
Ronning, G. (1989). Maximum likelihood estimation of Dirichlet distributions. Journal of Computational Statistics and Simulation 32, 215–221.
Rothenberg, T. J. (1971). Identification in parametric models. Econometrica 39, 577–591.
Woodland, A.D. (1979). Stochastic specification and the estimation of share equations. Journal of Econometrics 10, 361–383.
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Ronning, G. (2002). Linear and Nonlinear Dirichlet Share Equations Models. In: Klein, I., Mittnik, S. (eds) Contributions to Modern Econometrics. Dynamic Modeling and Econometrics in Economics and Finance, vol 4. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3602-1_13
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DOI: https://doi.org/10.1007/978-1-4757-3602-1_13
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4419-5331-5
Online ISBN: 978-1-4757-3602-1
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