Chapter 1 Further properties of the linear sufficiency in the partitioned linear model Augustyn Markiewicz and Simo Puntanen Abstract A linear statistic Fy, where F is an f × n matrix, is called linearly sufficient for estimable parametric function Kβ under the model M = {y, Xβ , V}, if there exists a matrix A such that AFy is the BLUE for Kβ . In this paper we consider some particular aspects of the linear sufficiency in the partitioned linear model where X =(X 1 : X 2 ) with β being partitioned accordingly. We provide new results and new insightful proofs for some known facts, using the properties of relevant covariance matrices and their expressions via certain orthogonal projectors. Particular attention will be paid to the situation under which adding new regressors (in X 2 ) does not affect the linear sufficiency of Fy. Key words: Best linear unbiased estimator, generalized inverse, linear model, lin- ear sufficiency, orthogonal projector, L¨ owner ordering, transformed linear model. 1.1 Introduction In this paper we consider the partitioned linear model y = X 1 β 1 + X 2 β 2 + ε , or shortly denoted M 12 = {y, Xβ , V} = {y, X 1 β 1 + X 2 β 2 , V} , (1.1) where we may drop off the subscripts from M 12 if the partitioning is not essential in the context. In (1.1), y is an n-dimensional observable response variable, and ε is an unobservable random error with a known covariance matrix cov(ε )= V = Augustyn Markiewicz Department of Mathematical and Statistical Methods, Pozna ´ n University of Life Sciences, Wojska Polskiego 28, PL-60637 Pozna´ n, Poland, e-mail: amark@up.poznan.pl Simo Puntanen Faculty of Natural Sciences, FI-33014 University of Tampere, Finland, e-mail: simo.puntanen@uta.fi 1 This is the accepted manuscript of the article, which has been published in Matrices, Statistics and Big Data : Selected Contributions from IWMS 2016. 2019, 1-22. https://doi.org/10.1007/978-3-030-17519-1_1