## Distributional Theory

Although the motivations of the linear least squares model and Gauss-Markov model were different, they both led in the same direction usingminimal assumptions. From a statistical viewpoint, the Gauss-Markov model employed only moment assumptions on the errors: E(e) = 0, Cov (e) = σ 2I. The goal of this chapter is to extend the assumptions on the errors e to specifying its joint distribution, so that in Chapter 6 we can then look for best estimators. The distributional assumptions will also permit construction of hypothesis tests and conﬁdence regions. Most of this exposition, of course, will follow the traditional route using the normal

distribution. In applications, we often observe a process whose deviations from the mean can be thought of as the sum of a large number of independent random effects, with no few of them dominating. Consequently, the central limit theorem’s conclusion makes the assumption that the errors are normally distributed to be themost reasonable distribution to assume.