Evaluating parameters of econometric models with linear limitations and a rank deficient observation matrix

Анотація

There considered the approach of the linear econometric dependences parameters estimating for the case of combining a set of special conditions arising in the simulation process. These conditions address the most important problems met in practice when implementing a series of classes of mathematical models, for the construction of which the matrix of explanatory variables can be used. In most cases the vectors that make up the matrix have a close correlation relationship; this leads to the need of performing calculations using a rank deficient matrix. There are also take place violations of the Gauss-Markov theorem condition. The list of above mentioned special conditions is augmented by the additional model parameters constraints. Cobb-Douglas's production function and the Solow model are known economic problems of this type. In this research the need to impose additional constraints on the model parameters is extended to a wider range of tasks. In general, the economic formulation of the problem with the specified features is presented.

Known ways to solve these tasks are discussed. The authors’ approach proposed takes into account the whole spectrum of these features. This approach is based on the application of pseudorandom matrices and the use of singular matrix decomposition. The use of proposed mathematical tools makes it possible to improve the quality of estimating model parameters while using real economic processes data. The analytical definition is found for the parameter evaluating vector of a linear econometric model with all the above mentioned features. Analysis of the used definition provides determination of the conditions that the matrix
must satisfy; this describes additional model parameters constraints. The term was also obtained to estimate the variance of a linear econometric model parameters vector.

The results obtained can be used in machine learning systems in the implementation of problems of econometric dependencies or discriminant models.