ABSTRACT
Let’s start by defining what is meant by a linear model. Suppose we want to model the response Y in terms of three predictors, X1, X2 and X3. One very general form for the model would be:
Y = f (X1,X2,X3)+ ε
where f is some unknown function and ε is the error in this representation. ε is additive in this instance, but could enter in some even more general form. Still, if we assume that f is a smooth, continuous function, that still leaves a very wide range of possibilities. Even with just three predictors, we typically will not have enough data to try to estimate f directly. So we usually have to assume that it has some more restricted form, perhaps linear as in:
Y = β0+β1X1+β2X2+β3X3+ ε
where βi, i = 0,1,2,3 are unknown parameters. Unfortunately this term is subject to some confusion as engineers often use the term parameter for what statisticians call the variables, Y , X1 and so on. β0 is called the intercept term.
