Re-framing the linear model

The most general form of a linear model is $$\begin{eqnarray*} \boldsymbol{Y} &=& \boldsymbol{X}\boldsymbol{\beta} + \boldsymbol{\epsilon} \mbox{ or}\\ E(\boldsymbol{Y}|\mathbf{X}) &=& \boldsymbol{X}\boldsymbol{\beta}\\ \end{eqnarray*}$$

The full probability model for the response $\bf{Y}$ can be written as

The most general form of a linear model is

$$\boldsymbol{Y} \sim N(\boldsymbol{X \beta},\Sigma)$$ where $\Sigma$ is an $n \times n$ covariance matrix. This suggests that

$$\begin{eqnarray*} E(\boldsymbol{Y}|\boldsymbol{X})&=&\boldsymbol{X \beta}\\ \\ \Sigma &=& \sigma^2 I\\ \\ &=& \left( \begin{array}{cccc} \sigma^2 & 0 & \ldots & 0 \\ 0 & \sigma^2 & \ldots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & \ldots & 0 & \sigma^2 \\ \end{array} \right)\\ \end{eqnarray*}$$

Using this probability model, we can find the maximum likelihood estimates of the model parameters $\boldsymbol{\beta}$ using the five steps listed above.