Multiple Regression Analysis

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 $$Y_i = \alpha + \beta_2 X_{2i} + \beta_3 X_{3i} + .. + \beta_K X_{Ki} + u_i \quad i= 1,2, \ldots, n$$ (4.1) 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 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 X k for k = 2, ’ , K, is uncorrelated with the ui’s with the property that \(\sum\nolimits^{n}_{i=1} (X_{ki} - \bar{X}_k)^2/n \ \hbox{where} \ \bar{X}_k = \sum\nolimits^{n}_{i=1} X_{ki}/n\) has a finite probability limit which is different from zero.