A Stochastic Non-Homogeneous Constant Elasticity of Substitution Production Function as an Inverse Problem: A Non-Extensive Entropy Estimation Approach

S. Bwanakare University of Information Technology and Management in Rzeszów, Rzeszów, Poland

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The document proposes a new entropy-based approach for estimating the parameters of nonlinear and complex models, i.e. those whose no transformation renders linear in parameters. Presently, for estimating such class of functions, various iterative technics like the Gauss-Newton algorithm are applied and completed by the least square methods approaches. Due to conceptual nature of such methods, definitely estimated functions are different from the original nonlinear one and the estimated values of parameters are in most of cases far from the true values. The proposed approach, being related to the statistical theory of information, is very different from those so far applied for that class of functions. To apply the approach, we select a stochastic non-homogeneous constant elasticity of substitution aggregated production function of the 27 EU countries which we estimate maximizing a non-extensive entropy model under consistency restrictions related to the constant elasticity of substitution model plus regular normality conditions. The procedure might be seen as an attempt to generalize the recent works (e.g. Golan et al. 1996) on entropy econometrics in the case of ergodic systems, related to the Gibbs-Shannon maximum entropy principle. Since this nonlinear constant elasticity of substitution estimated model contains four parameters in one equation and statistical observations are limited to twelve years, we have to deal with an inverse problem and the statistical distribution law of the data generating system is unknown. Because of the above reasons, our approach moves away from the normal Gaussian hypothesis to the more general Levy instable time (or space) processes characterized by long memory, complex correlation and by a convergence, in relative long range, to the attraction basin of the central theorem limit. In such a case, fractal properties may eventually exist and the q non extensive parameter could give us useful information. Thus, as already suggested, we will propose to solve for a stochastic inverse problem through the generalized minimum entropy divergence under the constant elasticity of substitution model and other normalization factor restrictions. At the end, an inferential confidence interval for parameters is proposed. The output parameters from entropy formalism represent the long-run state of the system in equilibrium, and so, their interpretation is slightly different from the "ceteris paribus" interpretation related to the classical econometrical modeling. The approach seems to produce very efficient parameters in comparison to those obtained from the classical iterative nonlinear method which will be presented, too.

DOI: 10.12693/APhysPolA.123.502 PACS numbers: 89.65.Gh, 89.70.cf