# Multiple Regression Analysis

4.0 Introduction

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

Yi = a + в2Х2і + в3Х3і + •• + вКXKi + ui І = 1, 2,…,п (4.1)

where Yi denotes the i-th observation on the dependent variable Y, in this case the sales of this product. Xki denotes the i-th observation on the independent variable Xk for k = 2,…,K in this case, own price, the competitor’s price and advertising expenditures. a is the intercept and в2,в3,…, вК are the (K — 1) slope coefficients. The ui’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 Xk for k = 2,…,K, is uncorrelated with the ui’s with the property that E7=1(Xki — Xk)2/n where Xk = E7=1 Xki/n has a finite probability limit which is different from zero.

Section 4.2 derives the OLS normal equations of this multiple regression model and discovers that an additional assumption is needed for these equations to yield a unique solution.