Linear Combinations of Parameters

Now we will focus on hypothesis testing of the regression parameters. Instead of solving several different types of inferential problems, e.g. involving a single parameter, involving two parameters or in some cases involving a linear combination of parameters we will develop a general theory for doing inference on linear combinations of parameters. Each of the cases described in the previous sentence can then be derived as a special case of the general theory.

For example if we want to predict a future value, at $x=5$ based on a simple linear model $$y_i=\beta_0+\beta_1 x+\epsilon_i \implies \hat y_i=\hat \beta_0+ \hat \beta_1 x$$ we are interested in the linear combination: $$\beta_0+5\beta_1,$$

which can be written as: $$(1\quad 5) \left( \begin{array}{c} \beta_0 \\ \beta_1 \end{array} \right).$$ We start by considering how to form linear functions of the parameters in a linear model.