Linear Models — transformation for generalized least squares

For uncorrelated observations $(y_1, y_2, …, y_n) = Y$, we define a linear model-I. $\mathbf{Y} = \mathbf{X}\beta + \varepsilon$ such that $\mathbb{E}(\varepsilon) = 0$, $D(\varepsilon) = \sigma^2\mathbf{I}$. $\mathbb{E}(\mathbf{Y}) = \mathbf{X}\beta$ and $D(\mathbf{Y}) = \sigma^2\mathbf{I}$. We then define a more general model-II. $\mathbb{E}(\mathbf{Y}) = \mathbf{X}\beta$ and $D(\mathbf{Y}) = \sigma^2\mathbf{G}$, where $|\mathbf{G}| \ne 0$ introduces correlations among the observations such that $\mathbf{G}$ is a known matrix, $\mathbf{I}$ is a unit matrix of order $n$, $D$ is a variance-covariance matrix and $\beta$, $\sigma^2$ are unknown. Model-II can be reduced to Model-I by considering a transformation.