An Introduction to Econometric Theory

10The Gaussian Regression Model

10.1 Testing Hypotheses

In this chapter, a key extra assumption is added to the classical regression model (CRM) specified in Section 7.2. Like the CRM itself, this is not necessarily a realistic assumption as it stands, but it often holds as an approximation and yields powerful results – not less than a complete theory of statistical inference. The Gaussian classical regression model (GCRM) shares assumptions GCRM(i)–(iii) with CRM(i)–(iii) and adds the following

Assumption GCRM(iv):

means that the disturbances are jointly normally distributed, with density function

If is any fixed conformable matrix and a fixed conformable vector, GCRM(iv) and the linearity property of the normal distribution implies that

In particular, since by (7.2) and (7.1)

(10.1)

and

it follows ...