The General Linear Model

Abstract

Consider the General Linear Model (GLM)

$$ y = Ax + \varepsilon ,\,\varepsilon \sim {\rm N}\left( {0,\sigma ^2 \Omega } \right) $$

((4.1))

where y ∈ ℜ^m is the response vector, A∈ ℜ^mx(n-1) is the exogenous data matrix with full rank, y ∈ ℜ^(n-1) is the vector of parameters to be estimated and ε ∈ ℜ^m is the noise vector normally distributed with zero mean and variance-covariance matrix σ²Ω. Without loss of generality it will be assumed that Ω ∈ ℜ^mxm is symmetric positive definite. The BLUE of x is obtained by solving the generalized linear least squares problem (GLLSP)

$$ \mathop {\arg \min }\limits_{x,u} \left\| {\left. u \right\|} \right.^2 \,subject\,to\,y = Ax + Bu, $$

((4.2))

where Ω = BB^T and $u \sim N\left( {0,\sigma ^2 I_m } \right)$. If B is the Cholesky lower triangular factor of Ω, then the solution of the GLLSP (4.2) employs as a main computational tool the Generalized QL Decomposition (GQLD):

$$ {{Q}^{T}}(y\;A) = \left( {\begin{array}{*{20}{c}} 0 \\ {{{{\tilde{R}}}^{T}}} \\ \end{array} } \right) \equiv \mathop{{\left( {\begin{array}{*{20}{c}} 0 \\ \eta \\ {{{y}_{2}}} \\ \end{array} \begin{array}{*{20}{c}} 0 \\ 0 \\ {{{R}^{T}}} \\ \end{array} } \right)\begin{array}{*{20}{c}} {m - 2} \\ 1 \\ {n - 1} \\ \end{array} }}\limits^{{1\quad n - 1}} $$

((4.3a))

and

$$ ({{Q}^{T}}B)P = L \equiv \left( {\begin{array}{*{20}{c}} {\mathop{{{{L}_{{11}}}}}\limits^{{m - n}} } & {\mathop{0}\limits^{1} } & {\mathop{0}\limits^{{n - 1}} } \\ {{{g}^{T}}} & \rho & 0 \\ {{{L}_{{21}}}} & r & {{{L}_{{22}}}} \\ \end{array} } \right)\begin{array}{*{20}{c}} {m - 2} \\ 1 \\ {n - 1} \\ \end{array} , $$

((4.3b))

where R^T,L₁₁ and L₂₂ are lower triangular non-singular matrices. The BLUE of x, say y, is obtained by solving the lower triangular system

$$ {{R}^{T}}\hat{x} = {{y}_{2}} - r(\eta /\rho ). $$

where n²/(p²(m — n + 1)) is an unbiased estimator of σ² [91, 109, 111].