The general Gauss–Markov model, Y = Xβ + e, E(e) = 0, Cov(e) = σ <Superscript>2</Superscript> V, has been intensively studied and widely used. Most studies consider covariance matrices V that are nonsingular but we focus on the most difficult case wherein C(X), the column space of X, is not contained in C(V). This forces V to be singular. Under this condition there exist nontrivial linear functions of Q′Xβ that are known with probability 1 (perfectly) where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$${C(Q)=C(V)^\perp}$$</EquationSource> </InlineEquation>. To treat <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$${C(X) \not \subset C(V)}$$</EquationSource> </InlineEquation>, much of the existing literature obtains estimates and tests by replacing V with a pseudo-covariance matrix T = V + XUX′ for some nonnegative definite U such that <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$${C(X) \subset C(T)}$$</EquationSource> </InlineEquation>, see Christensen (Plane answers to complex questions: the theory of linear models, <CitationRef CitationID="CR2">2002</CitationRef>, Chap. 10). We find it more intuitive to first eliminate what is known about Xβ and then to adjust X while keeping V unchanged. We show that we can decompose β into the sum of two orthogonal parts, β = β <Subscript>0</Subscript> + β <Subscript>1</Subscript>, where β <Subscript>0</Subscript> is known. We also show that the unknown component of X β is <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$${X\beta_1 \equiv \tilde{X} \gamma}$$</EquationSource> </InlineEquation>, where <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$${C(\tilde{X})=C(X)\cap C(V)}$$</EquationSource> </InlineEquation>. We replace the original model with <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$${Y-X\beta_0=\tilde{X}\gamma+e}$$</EquationSource> </InlineEquation>, E(e) = 0, <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$${Cov(e)=\sigma^2V}$$</EquationSource> </InlineEquation> and perform estimation and tests under this new model for which the simplifying assumption <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$${C(\tilde{X}) \subset C(V)}$$</EquationSource> </InlineEquation> holds. This allows us to focus on the part of that parameters that are not known perfectly. We show that this method provides the usual estimates and tests. Copyright Springer-Verlag 2013
