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

Abstract

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 GaussNewton algorithm are applied and completed by the least square methods approaches. Due to conceptual nature of such methods, denitely estimated functions are dierent 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 dierent 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 GibbsShannon 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 condence 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 dierent from the ceteris paribus interpretation related to the classical econometrical modeling. The approach seems to produce very ecient parameters in comparison to those obtained from the classical iterative nonlinear method which will be presented, too.

Non-Extensive Entropy Econometrics: New Statistical Features of Constant Elasticity of Substitution-Related Models

“…In the same study, authors reveal interesting aspects when q-generalized cross-entropy is associated with constraining information. Authors [15] suggest the q-generalized cross-entropy, defined later by Equation (11), to be more consistent with the expectations and the constraints form proposed by [17] known as q-averages or escort distribution:…”

“…Less adequate outputs owing to such empirical approach have led a large community of economists to question the rationale of some economic modeling techniques like the applied general equilibrium approach. In this Section, we develop the proposed entropy formalism using a classical CES production model (CESP) explaining the gross domestic product ( t VA ) by two classical factors (e.g., [14,15]): Labor ( t L ) and capital ( t K ). The next two CES model class cases will be presented in Section 7.…”

“…Both in the pioneering work on maximum entropy econometrics by authors [19] and in the work of author [15], the moment-constraints of an entropy econometric linear model for ergodic systems have been reparameterized [20]. We need to transform variables of the constraining generalized linear model into weight-probabilities over a point support space defining each of the original random parameters.…”

“…as a discrete random variable defined over a compact support space [15,19] and 2 < M < ∞ possible outcomes, it is then possible to express B k as:…”

Greenhouse Emission Forecast as an Inverse Stochastic Problem: A Cross-Entropy Econometrics Approach

“…Traditionally, as far as Bayesian and relative entropy formulations are concerned, it may be worth mentioning that both models will lead to similar results when and only when the true expected error related to the data generating system is equal to zero within a support space symmetric around zero. Similarly, Gibbs-Shannon cross-entropy and Tsallis non-extensive cross-entropy outputs will correspond when errors embodied in the model are not correlated and then evolve within the Gaussian basin of attraction [e.g., [21][22][23]. Furthermore, the magnitudes of parameter or error bounds will affect the level of estimated parameter shrinkage towards the prior distribution.…”