# CHAPTER 3

# MULTIPLE LINEAR REGRESSION

## 3.1 INTRODUCTION

In this chapter the general multiple linear regression model is presented. The presentation serves as a review of the standard results on regression analysis. The standard theoretical results are given without mathematical derivations, but illustrated by numerical examples. Readers interested in mathematical derivations are referred to the bibliographic notes at the end of Chapter 2, where a number of books that contain a formal development of multiple linear regression theory is given.

## 3.2 DESCRIPTION OF THE DATA AND MODEL

The data consist of *n* observations on a dependent or response variable *Y* and *p* predictor or explanatory variables, *X*_{1}, *X*_{2},…, *X _{p}*. The observations are usually represented as in Table 3.1. The relationship between

*Y*and

*X*

_{1},

*X*

_{2},…,

*X*is formulated as a linear model

_{p}where *β*_{0}, *β*_{1}, *β*_{2},…, *β _{p}* are constants referred to as the model

*partial*regression coefficients (or simply as the

*regression coefficients*) and

*ε*is a random disturbance or error. It is assumed that for any set of fixed values of

*X*

_{1},

*X*

_{2},…,

*X*that fall within the range of the data, the linear equation (3.1) provides an acceptable approximation of the true relationship between

_{p}*Y*and the

*X*'s (

*Y*is approximately a linear function of the

*X*'s, and

*ε*measures the discrepancy in that approximation). In particular,

*ε*contains no systematic information for determining ...

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