Until now, we have not found it necessary to assume any distributional form for the errors ". However, if we want to make any confidence intervals or perform any hypothesis tests, we will need to do this. The common assumption is that the errors are normally distributed. In practice, this is often, although not always, a reasonable assumption. We have already assumed that the errors are independent and identically distributed (i.i.d.) with mean 0 and variance #2, so we have " ~ N(0, #2I). Now since y=X!+", we have y ~ N(X!, #2I). which is a compact description of the regression model. From this we find, using the fact that linear combinations of normally distributed values are also normal, that:

3.1 Hypothesis Tests to Compare Models

Given several predictors for a response, we might wonder whether all are needed. Consider a larger model, #, and a smaller model, ', which consists of a subset of the predictors that are in ( If there is not much difference in the fit, we would prefer the smaller model on the principle that simpler explanations are preferred. On the other hand, if the fit of the larger model is appreciably better, we will prefer it. We will take ' to represent the null hypothesis and ( to represent the alternative. A geometrical view of the problem may be seen in Figure 3.1.