Multiple regression analysis is the study of how a dependent variable y is related to two or more independent variables. In the general case, we will use p to denote the number of independent variables.

**1. Regression Model and Regression Equation**

The concepts of a regression model and a regression equation introduced in the preceding chapter are applicable in the multiple regression case. The equation that describes how the dependent variable y is related to the independent variables x_{1}, x_{2}, . . . , x_{p} and an error term is called the multiple regression model. We begin with the assumption that the multiple regression model takes the following form.

In the multiple regression model, β_{0}, β_{1}, β_{2}, … , β_{p} are the parameters and the error term e (the Greek letter epsilon) is a random variable. A close examination of this model reveals that y is a linear function of x_{1}, x_{2}, . . . , x_{p} (the β_{0} + β_{1}x_{1} + ^2 + . . . + βpxp part) plus the error term ∈. The error term accounts for the variability in y that cannot be explained by the linear effect of the p independent variables.

In Section 15.4 we will discuss the assumptions for the multiple regression model and ∈. One of the assumptions is that the mean or expected value of e is zero. A consequence of this assumption is that the mean or expected value of y, denoted E( y), is equal to β_{0} + β_{1 }x_{1} + β_{2}x_{2} + . . . + β_{p}x_{p}. The equation that describes how the mean value of y is related to x_{1}, x_{2}, . . . , x_{p} is called the multiple regression equation.

**2. Estimated Multiple Regression Equation**

If the values of β_{0}, β_{1}, β_{2}, … , β_{p} were known, equation (15.2) could be used to compute the mean value of y at given values of x_{1}, x_{2}, . . . , x_{p}. Unfortunately, these parameter values will not, in general, be known and must be estimated from sample data. A simple random sample is used to compute sample statistics β_{0}, β_{1}, β_{2},…, β_{p} that are used as the point estimators of the parameters β_{0}, βi, β_{2}, …, β_{p}. These sample statistics provide the following estimated multiple regression equation.

The estimation process for multiple regression is shown in Figure 15.1.

Source: Anderson David R., Sweeney Dennis J., Williams Thomas A. (2019), *Statistics for Business & Economics*, Cengage Learning; 14th edition.

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30 Aug 2021