Introduction

Abstract

This book is about linear models. Linear models are models that are linear in the parameters. A typical model considered is

$$Y = X\beta + e$$

where Y is an n × 1 vector of observations, X is an n × p matrix of known constants called the design (or model) matrix, β is a p × 1 vector of unobservable parameters, and e is an n × 1 vector of unobservable random errors. Both Y and e are random vectors. We assume that E(e) = 0 and Cov (e) = σ ² I, where σ ² is some unknown parameter. (The operations E(·) and Cov(·) will be defined formally a bit later.) Our object is to explore models that can be used to predict future observable events. Much of our effort will be devoted to drawing inferences, in the form of point estimates, tests, and confidence regions, about the parameters β and σ ². In order to get tests and confidence regions, we will assume that e has an n-dimensional normal distribution with mean vector (0, 0, ..., 0)′ and covariance matrix σ ² I, i.e., e ~ N (0, σ ² I).