On calculation of the pseudo-inverse econometric models matrix with a rank deficient observation matrix

Abstract

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 1 X  is uniquely defined such that, for random right-hand sideB , the solution of the system XB  is vector 1 Xb  . If X is a degenerate or rectangular matrix, then there is no inverse to it. Moreover, in these cases, the system XB  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 1 1 X R Q   . In addition, it is recommended for specific calculations. With an incomplete rank, the most convenient form of representation 1 X  follows from the expansion in characteristic numbers. If X U V with non-zero characteristic   
numbers, then X V 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.