Chapter 15 Multiple Regression I The Multiple Regression

Chapter 15 Multiple Regression

I. The Multiple Regression Model Multiple regression is the study of how a dependent variable y is related to two or more independent variables. Ex. The sales of a chain of small auto parts stores (y) might be modeled as a function of a town’s population (x 1), their average income (x 2) and the number of competitors in each town (x 3).

A. The Multiple Regression Model The theoretical model looks much like before, except there a total of k independent variables.

B. Multiple Regression Equation Take the expected value and get the regression equation. Once again, if we had the true parameters, we wouldn’t have to estimate anything. That’s not the case in the world, so we’ll have to take a sample and estimate each coefficient with least squares.

Estimates A simple random sample is used to compute sample statistics b 0, b 1, b 2, …, bk.

II. Least Squares Method The procedure is the same, it’s just that the minimization process becomes more involved. A statistical package becomes quite handy.

Interpretation of Coefficients A coefficient estimate is interpreted in the same way as before, with one modification. Before, b 1 represented the slope of a line, or rate of change in y when x was changed by one unit. Now, b 1 represents the change in y when x 1 is changed by one unit, and all other variables are held constant. If you’re familiar with calculus, this is like taking a partial derivative.

Example Look at #4 in the text. A shoe store is modeling sales with two variables. Y= sales ($1000’s) x 1= inventory investment ($1000’s) x 2= advertising expenditures ($1000’s)

Interpretations b 1=10, means that a $1000 increase in inventory investment increases sales by $10, 000 if all other variables are held constant. b 2= 8, means that a $1000 increase in advertising expenditures increases sales by $8000 if all other variables are held constant.