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 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.