Multiple regression model

Chapter 4

Multiple regression analysis

Introduction

In Chapter 3 we showed how regression analysis could be used to develop an equation that would estimate the relationship between two variables. Recall that we limited our discussion to the development of a linear relationship between the two variables, or what is commonly referred to as simple linear regression. There are many important situations, however, where the underlying relationship between two variables can not be explained adequately with a straight-line relationship. In addition, the most of real world problems require the consideration of more than one independent variable in order to predict the dependent variable. In this chapter we discuss how multiple regression analysis can be used to handle such situations.

Multiple regression model

Usually a dependent variable is affected by more than one independent variable. When we include two or more independent variables in a regression model, it is called a multiple regression model.

A multiple regression model with y as a dependent variable and as independent variables is written as

where the numbers and must be estimated from sample data.

More generally, a multiple regression model with y as a dependent variable and and as independent variables is written as

(1)

where the numbers represents the constant term and are the regression coefficients of an independent variables and , respectively.

In (1) if each of the independent variables is set to 0, it follows that

Thus, is expected value of the dependent variable when every independent variable takes value 0. Frequently this interpretation does not carry practical interest and often leads to meaningless.

The interpretation of the coefficients is extremely important. For example, is expected increase in y resulting from 1 unit increase in when the values of the other independent variables remain constant. In general, is expected increase in the dependent variable resulting from a 1-unit increase in the independent variable when the values of the other independent variables remain constant.

If model (1) is estimated using sample data, which is usually the case, the estimated regression model is written as

(2)

In model (2) are the sample statistics, which are the point estimators of , respectively.

In model (1) y denotes the actual values of the dependent variable. In model (2), denote the predicted or estimated values of the dependent variable. The difference between y and gives the error of prediction.

The method of fitting multiple regression of least squares model is similar to that of fitting the linear regression model: method of least squares. That is, we choose the estimated model

that minimizes

.

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