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  • Документ
    On calculation of the pseudo-inverse econometric models matrix with a rank deficient observation matrix
    (KNUCA, 2019) Kutovyi, V.; Kutovyi, O.; Shutovskyi, O.
    The approach to estimating the parameters of linear econometric dependencies for the case of combining a number of special conditions arising in the modeling process is considered. These conditions concern the most important problems that arise in practice when implementing a number of classes of mathematical models, for the construction of which a matrix of explanatory variables is used. In most cases, the vectors that make up the matrix have a close correlation relationship. That leads to the need to perform calculations using a rank deficient matrix. There are also violations of the conditions of the Gauss-Markov theorem. For any non-degenerate square matrix X, an inverse matrix X-1 is uniquely defined such that, for random right-hand side B , the solution of the system X β = B is vector β X-1 b . If X is a degenerate or rectangular matrix, then there is no inverse to it. Moreover, in these cases, the system X β = B may be incompatible. Here it is natural to use a generalization of the concept of the inverse transformation, which is formulated in terms of the corresponding problem of minimizing the sum of squared residuals. In the same case, having a QR decomposition, one can use the formula X+ = R-1 Q1’. In addition, it is recommended for specific calculations. With an incomplete rank, the most convenient form of representation 1 X-1 follows from the expansion in characteristic numbers. If X = U ΣV with non-zero characteristic numbers, then X+ = VΣ+U’. We propose an alternative X+ calculation method, which relies on the decomposition of a rank deficient matrix into the product of two matrices of full rank.
  • Документ
    Analysis of the multicollinear econometric model parameters with a rank deficient observation matrix
    (KNUCA, 2018-03-12) Kutovy, Viktor; Katunina, Olga; Shutovsky, Oleg
    The topic of determining informative predictors, forming rational exogenous variables, substantiating the dimension and structure of predictor spaces is considered. The purpose of design and selection of characteristics is to prevent the effect of retraining, reduce the dimension in studying the processes apart from a master, build classifiers, reflect the process of dividing data into classes and determine the boundaries of solutions in limited space, as well as reasonable interpretation, provide in-depth understanding of the model and data for studying, visualization in spaces, the dimension of which is perceived by the researcher. The design predictor spaces and develop effective procedures problems for estimating the parameters of econometric models with multicollinear variables are developed. The study was made under alternative approaches to form the interdependencies models features. A mathematical toolkit is proposed for calculating the parameters of a linear econometric model in case of rank deficient observation matrix, based on the study of singular expansions. Using a singular toolkit for decomposing and analyzing the data matrix makes it possible to increase the operational efficiency and predictive quality of the procedures for estimating econometric models parameters. The mathematical approach to the construction of models of the interdependence of factors is intended to select characteristics and construct predictor spaces in the study of systems with multicollinear variables and rank deficient observation matrix.