Multiple Regression Analysis

Abstract

So far we have considered only one regressor X besides the constant in the regression equation. Economic relationships usually include more than one regressor. For example, a demand equation for a product will usually include real price of that product in addition to real income as well as real price of a competitive product and the advertising expenditures on this product. In this case Y _i = α + β ₂ X}₂ i β ₃ X ₃ i + β _itK X _itKi + u _i i = 1,2,...,n where Y _i denotes the i-th observation on the dependent variable Y, in this case the sales of this product. X _ki denotes the i-th observation on the independent variable X _k for k = 2,...,K in this case, own price, the competitor’s price and advertising expenditures. α is the intercept and β ₂,β ₃,...,β _K are the (K − 1) slope coefficients. The u _i’s satisfy the classical assumptions 1–4 given in Chapter 3. Assumption 4 is modified to include all the X’s appearing in the regression, i.e., every X _k for k = 2,...,K, is uncorrelated with the u _i’s with the property that

$$ Y_i = \alpha + \beta _2 X_{2i} + \beta _3 X_{3i } + \beta _K X_{Ki } + u_{i } i = 1, 2,...,n $$

(1)

where has a finite probability limit which is different from zero