Generalized least squares

In statistics, generalized least squares (GLS) is a technique for estimating the unknown parameters in a linear regression model. GLS can be used to perform linear regression when there is a certain degree of correlation between the residuals in a regression model. In these cases, ordinary least squares and weighted least squares can be statistically inefficient, or even give misleading inferences. GLS was first described by Alexander Aitken in 1934.

In a typical linear regression model we observe data ${\displaystyle \{y_{i},x_{ij}\}_{i=1,\dots ,n,j=2,\dots ,k}}$ on n statistical units. The response values are placed in a vector ${\displaystyle \mathbf {y} =\left(y_{1},\dots ,y_{n}\right)^{\mathtt {T}}}$, and the predictor values are placed in the design matrix ${\displaystyle \mathbf {X} =\left(\mathbf {x} _{1}^{\mathtt {T}},\dots ,\mathbf {x} _{n}^{\mathtt {T}}\right)^{\mathtt {T}}}$, where ${\displaystyle \mathbf {x} _{i}=\left(1,x_{2i},\dots ,x_{ki}\right)}$ is a vector of the k predictor variables (plus a constant) for the ith unit. The model assumes that the conditional mean of ${\displaystyle \mathbf {y} }$ given ${\displaystyle \mathbf {X} }$ is a linear function of ${\displaystyle \mathbf {X} }$, whereas the conditional variance of the error term given ${\displaystyle \mathbf {X} }$ is a known nonsingular matrix ${\displaystyle \mathbf {\Omega } }$. This is usually written as

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