Types of Linear Models

The model equation for a linear model, i.e., y = Xβ + e, has two distinct terms on the right-hand side: Xβ and e. Because Xβ = E(y), we refer to any additional assumptions made about Xβ (i.e., additional to X being nonrandom and known and β being nonrandom and unknown) as the model’s mean structure, and to any additional assumptions made about e [i.e., additional to E(e) = 0] as the model’s error structure Linear model error structure of . In this chapter, we give a brief overview of various linear models that are distinguishable from one another on the basis of these two types of structure, and we describe some terminology associated with each type. We also describe prediction-extended versions of these models, i.e., models for the mean structure and error structure of the joint distribution of y and a related, though unobservable, random vector u. Throughout the remainder of the book, many aspects of linear model theory will be exemplified by the models introduced here.