An alternative look at the linear regression model

• Published in: Statistical papers (Berlin, Germany) Vol. 63; no. 5; pp. 1499 - 1509
• Main Authors: Baksalary, Oskar Maria, Trenkler, Götz
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.10.2022; Springer Nature B.V
• Subjects: Economic Theory/Quantitative Economics/Mathematical Methods; Economics; Finance; Insurance; Management; Mathematics and Statistics; Matrices (mathematics); Operations Research/Decision Theory; Probability Theory and Stochastic Processes; Regression analysis; Regression models; Regular Article; Statistics; Statistics for Business; Columnwise partitioned matrix; Moore–Penrose inverse; Astronomical observations; Least squares method; Experimental data processing; Estimation theory
• ISSN: 0932-5026; 1613-9798
• DOI: 10.1007/s00362-021-01280-x

An alternative look at the linear regression model is taken by proposing an original treatment of a full column rank model (design) matrix. In such a situation, the Moore–Penrose inverse of the matrix can be obtained by utilizing a particular formula which is applicable solely when a matrix to be inverted can be columnwise partitioned into two matrices of disjoint ranges. It turns out that this approach, besides simplifying derivations, provides a novel insight into some of the notions involved in the model and reduces computational costs needed to obtain sought estimators. The paper contains also a numerical example based on astronomical observations of the localization of Polaris, demonstrating usefulness of the proposed approach.