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Power-law (PL) formalism is known to provide an appropriate framework for canonical modeling of nonlinear systems. We estimated three stochastically distinct models of constant elasticity of substitution (CES) class functions as non-linear inverse problem and showed that these PL related functions should have a closed form. The first model is related to an aggregator production function, the second to an aggregator utility function (the Armington) and the third to an aggregator technical transformation function. A q-generalization of K–L information divergence criterion function with

This paper proposes a new approach for modeling stochastic non-linear inverse problems. The approach is based upon Kullback-Leibler (K–L) relative entropy [

The first of these three models is the CES production stochastic function. In the case of production system, the fact that this aggregator model exhibits technical CES means that a constant percentage reduction in the quantity of one factor (e.g., labour) must be compassed by one extra unit of another factor (e.g., capital), so that total producer output remains unchanged. The second model is the constant elasticity of commercial substitution model (CECS) which aggregates domestic and imported demand for goods. According to Armington’s contribution related to merchandise demand theory [

Structurally, the CES production model displays causality relationships between supply and input factors. The CECS remains a quasi-identity utility function since it is just missing a quasi-constant variable (the indirect taxes) to constitute an identity. The CET model remains an identity equation, covariate values of which sum up to the explained value of the model, suggesting that national production aggregates two classes of goods (the locally demanded product and the export product). These two goods are supplied through a constant elasticity of technical transformation [

Thanks to the proposed non-extensive entropy approach—enabled by PL characterization of CES functions—implications on the stochastic features of these three distinct models are pointed out through robust estimation procedure. To our knowledge, this nonlinear class of functions remains analytically intractable when using traditional statistical techniques. This constitutes the main contribution of the paper.

The document is organized as follows: Section 2 presents some transitional properties linking PL and low frequency time series. Section 3 eclectically presents the PL nature of the CES function. Section 4 generalizes the Kullback-Leibleir divergence entropy to non-ergodic systems and discusses constraining problems of such models in the context of the generalized non-extensive entropy econometric model presented in Section 5. Section 6 presents some formalisms related to the parameter confidence area of the model. Finally, Section 7 presents the outputs of the model and Section 8 gives concluding remarks.

Parameters of the proposed nonlinear models were estimated on the basis of an annual time scale statistical sample. We consider that PL-related Tsallis entropy will still exist, even in the case of low frequency series, a valuable device for aggregated data modeling since the outputs provided by exponential family law, e.g., the Gibbs-Shannon entropy approach, correspond to the limiting case of Tsallis entropy when the Tsallis q-parameter equals unity. Next, perhaps more a pertinent argument for using Tsallis non-extensive entropy formalism is the existence of a number of complex phenomena involving long-range correlations, still observable when data is time scale-aggregated [

Delimiting the threshold values of the PL path towards the Gaussian distribution (or to the exponential family law) as a function of the data frequency level remains statistically difficult since each phenomenon may display its own speed of convergence—if any—towards the central limit theorem attractor. The next source of statistical concern may be related to systematic errors from statistical data collecting and processing which can generate a kind of heavy distribution. Thus, a systematic application of the Shannon-Gibbs entropy approach in the above cases—even on the basis of annual data—could be misleading and, in the best case, lead to instable solutions. Having in mind that non-extensive Tsallis entropy generalizes the exponential family law [

In the early 1960s, Arrow, Chenery, Minhas, and Solow [_{t}_{t}_{t}

or one of its generalized formulations as:

where:

and τ^{e} is a CES between factors; ε_{t} stands for the random disturbance with unknown distribution. In _{t}^{e} converging to 0 suggests perfectly substitutable factors. In the case of more than two inputs _{i}

Let us now focus on the connection between the CES class of functions and a PL. In fact, to better display that relation, let us aggregate components of Model (1) into one variable without conserving additivity. Then, we get a generic case of a PL of the form:

where in this case the endogenous variable _{t}_{t}_{t}, itself, is assumed to follow PL structure. Index _{i}

The K–L index of information divergence [

or in discrete case:

^{(0)}, under the hypothesis that

However, as is shown in our recent article (in press), the Curado-Tsallis [

seem to lead to more stable outputs. In the present work, in reverse, Escort distribution (

Both in the pioneering work on maximum entropy econometrics by authors [

where unknown _{k}_{k}

where _{km}_{km}_{i}_{i}

where _{n}_{n}_{km}_{km}

Subject to:

For reasons of formal presentation, the criterion function _{h}_{i}

where α,

In this section, we follow the reference work of [_{j}^{2} is proposed under the entropy symbol _{j}_{q}_{q}_{i}_{q}

and:

In

with _{j}

where:

and:

As far as estimator properties are concerned, both indexes fulfill the basic Fisher-Rao-Cramer information index properties, including continuity, symmetry, maximum, and additivity [

This section presents outputs of the three computed constant elasticity class models. The first was presented in detail in Section 3 as a CES production function

Before undergoing the computation procedure, data have been dimensioned at logarithmic scale. The computations of the NCE model were carried out with the General Algebraic Modeling System (GAMS) code. Those with the NLLS technique were done in a common spread sheet (Microsoft Excel). Computations by the GMM and ML approaches were executed with special code from the open source GRETL.

Let us first show mathematical formulation of the next two CES model classes. A CECS utility function of domestic economic absorption aggregator (_{t}

where

with τ^{e} CES, δ and ε_{t} standing, respectively, for distribution parameter and random disturbances with unknown distribution.

The last model CET, the technical transformation function aggregator (_{t}

where:

and _{t}_{t}

For a clearer understanding of model outputs, let us first recall a key concept of elasticity of substitution (ES). It represents the percentage change in the ratio of consumption of two different commodities (or two factors of production) due to the variation of the marginal rate of substitution (MRS) between the two commodities (or two factors of production). Next, MRS is the rate at which a consumer (producer) is ready to give up one commodity (factor of production) in exchange for another commodity (factor of production) while maintaining the same level of utility (production). Formally, assuming free market conditions, if we let the utility over consumption of two commodities (_{1} and _{2}) be explained by

The last equality which explains

In this study, priors were initiated from NLLS outputs. As known, such priors are not deterministically fixed. They are updated according to the Bayesian information processing rule. For simulation purposes, different

As suggested in the Introduction, each of the above three models displays its particular stochastic level. Only the new presented Tsallis NCE estimator replicates these differences. This is so because, as already shown in

Using traditional nonlinear, least square methods, we have linearized the

In the case of the CESP and CECS models, the GMM computation procedure has been initialized with: A = 0.700, δ = 0.71,

The next technique applied is the ML. In the case of the production model CESP, we have initialized the model with: A = 2.2500, δ = 0.5,

Coming back to the NCE outputs, the estimated parameters reflect long-run optimal equilibrium values of the system. Taking into account the fact we are dealing with the aggregated accounts of 27 EU countries, the estimated parameters remain consistent with our expectations. For instance, in the case of the CESP production model, the estimated parameter

The model quality is likewise shown in

For the CESP production model, minimum LS errors are obtained for

To verify the Tsallis related model outputs, we have computed a classical S-K-L cross-entropy econometric model, which, as expected, has produced the same values as those obtained from Tsallis formalism for

We seem to observe a kind of invariant scale error structure at three levels, along with the q-parameter steadily evolving on a convex space towards the global minimum point of the model where

The present work has developed a new Tsallis cross-entropy econometric approach for instable, nonlinear econometric models. A large class of economic and financial models should fall into this category. We have limited our study to the three CE class functions with three distinctive stochastic forms. Only outputs produced by Tsallis formalism reflect these stochastic differences. A Super-convergence of the Tsallis entropy estimator should be owing to a strong similarity between the sample and the data generating PL system. Furthermore, we have noted a plausible presence of a fractionally integrated moving average (ARFIMA) error structure. More investigation is needed to confirm the proposed PL-based approach for econometric modeling.

The author gratefully acknowledges University of Information Technology and Management in Rzeszow (Poland) for having financed this research.

The authors declare no conflict of interest.

(

Model disturbance (CV) curve as a function of q, for [1 < q < 2.6] (model CESP).

Bivariate kernel density estimates between CV and q, for [1.0 < q < 7/3] (CESP model).

Aggregated data (chain-linked volumes at 2005 exchange rates) for models (1000 billion euro).

Year | 1999 | 2000 | 2001 | 2002 | 2003 | 2004 | 2005 | 2006 | 2007 | 2008 | 2009 | 2010 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

VA | 7.82 | 8.136 | 8.427 | 8.578 | 8.978 | 9.294 | 9.76 | 10.288 | 10.288 | 9.777 | 10.149 | 10.412 |

K | 3.287 | 3.42 | 3.55 | 3.641 | 3.86 | 4.02 | 4.265 | 4.528 | 4.476 | 4.105 | 4.336 | 4.444 |

L | 4.427 | 4.606 | 4.76 | 4.819 | 4.99 | 5.149 | 5.374 | 5.63 | 5.683 | 5.554 | 5.691 | 5.834 |

MO | 8.691 | 9.026 | 9.212 | 9.325 | 9.442 | 9.689 | 9.877 | 10.211 | 10.550 | 10.621 | 10.162 | 10.359 |

DMO | 5.719 | 5.693 | 5.751 | 5.792 | 5.846 | 5.813 | 5.773 | 5.711 | 5.795 | 5.797 | 5.924 | 5.693 |

Export | 2.971 | 3.333 | 3.461 | 3.533 | 3.596 | 3.876 | 4.104 | 4.500 | 4.754 | 4.825 | 4.238 | 4.666 |

Imports | 2.938 | 3.272 | 3.361 | 3.414 | 3.526 | 3.796 | 4.029 | 4.411 | 4.672 | 4.727 | 4.153 | 4.551 |

Outputs from the Tsallis NCE: Dependent var: C(t), MO(t), VA(t)-.

Exogenousvar: | ||||||
---|---|---|---|---|---|---|

CESP(L(t), K(t)) | 1.866 | 0.163 | 0.001 | 1.e+00 | 0.99 | 0.006 |

CET(DO(t),Ex(t)) | 2.000 | 0.5 | −1.E00 | 0.999 | 4.271E–7 | |

CECS(C(t),M(t)) | 2.021 | 0.504 | −1.E00 | 0.999 | 2.705E–5 |

_{q}

Outputs from the NLLS, CESP, CET, CECS models; dependent var: C(t), MO(t), VA(t).

Exogenous var | ^{2} | ||||
---|---|---|---|---|---|

CESP(L(t), K(t)) | 1.995 | 0.282 | 3.046 | 0.993 | |

Parameters T-value | 48.89 | 6.61 | 1.49 | 6.61 | 0.88 |

CET(DO(t),Ex(t)) | 2.008 | 0.497 | −0.954 | ||

Parameters T-value | 558.12 | 1551.4 | −292.5 | 0.999 | |

CECS(C(t),M(t)) | 2.147 | 0.477 | −0.532 | ||

Parameters T-value | 6.257 | 13.073 | −1.554 | 0.83 |

Outputs from GMM, models CESP, CET, CECS.

CESP | CET | CECS | ||||
---|---|---|---|---|---|---|

Estimate | Std. Error | Estimate | Std. Error | Estimate | Std. Error | |

A | 1.672 | 0.007 | 1.647 | 0.003 | 1.679 | 0.024 |

δ | 0.54 | 0.079 | 0.511 | 0.001 | 0.524 | 0.022 |

−1.53 | 1.922 | −2.344 | 0.026 | −1.852 | 0.279 | |

0.679 | 0.004 |

Outputs from ML—CESP, CET, CECS models.

CESP | CET | CECS | ||||
---|---|---|---|---|---|---|

Estimate | Std. Error | Estimate | Std. Error | Estimate | Std. Error | |

2.97 | 9.66 | 0.933 | 0.002 | 0.945 | 0 | |

δ | 0.813 | 10.569 | 0.241 | 0.008 | 0.303 | 0.002 |

1.018 | 34.114 | −0.349 | 0.003 | −0.293 | 0.001 | |

1.145 | 16.903 | |||||

Akaikecriterion | −26.46 | −267347.1 | −77504.51 |