1. THE MULTIPLE REGRESSION MODEL

2. MULTIPLE REGRESSION • In a multiple regression we are trying to evaluate the cumulative effects that changes to more than one independent variable (x1, x2, x3, etc.) or will have on a dependent variable (y)

3. Transformations to a Linear Model • Multiple regression can used to evaluate models like: y = 0 + 1 x1 + 2 x2 + 3 x12 + 4 x1x2+ 5 x1/x2 + 6 logx1 +  • Define • x3 = x12 • x4 = x1 x2 • x5 = x1/x2 • x6 = log x1 • Then the model becomes: y = 0 + 1 x1 + 2 x2 + 3 x3 + 4 x4 + 5 x5 + 6x6 + 

4. GENERAL FORM OF A MULTIPLE REGRESSION MODEL Since we can make substitutions similar to those just described, the general multiple regression model can be expressed as: y = 0 + 1 x1 + 2 x2 + 3 x3 + …. + k xk + 

5. THE REGRESSION APPROACH • Hypothesize a form of the model • Determine the best estimates for the ’s • Assumptions about  • Testing the strength of the model • Using the model for prediction/estimation

6. Example • It is felt that the price of a house in Laguna Hills is a function of its square footage, its lot size, and its age. • A sample of 38 recent sales in Laguna Hills is taken.

8. STEP 1: Hypothesizing a form of the model • One variable -- scatterplot • If it looks curved, hypothesize a higher order model and make transformations to a linear model • More than one variable • Simply HYPOTHESIZE – make a best judgment as the form of the model • Make appropriate substitution of variables so that the model is linear

9. Laguna Hills Model • There are three variables. • Hypothesize: y = 0 + 1x1 + 2x2 + 3x3 +

10. STEP 2: Determining the Best Estimates for the ’s • Involves complicated matrix operations but still uses the method of least squares. • Use computer (EXCEL) only • But the best values for the ’s minimizes the sum of the squared errors between the actual values of y and the predicted values for y -- i.e. They minimize SSE.

11. Note B1:D39 Must be a contiguous range Using Excel to Get the b’s Go to TOOLS/DATA ANALYSIS/REGRESSION

12. The regression equation: ŷ = 145326 +240.34591x1 +935401.9x2 – 12287.5x3

13. Since there is more than one x, we say x’s -- not just x That’s the only difference STEP 3: Assumptions For  For any given set of the x’s: •  has a normal distribution • E() = 0 Also: • Errors are independent •  does vary between different values of the x’s

14. STEP 4:Assessing the Strength of the Model • Question 1: Can we conclude that at least one of the independent variables (x’s) is useful in predicting y? • Question 2: If yes, which of the independent variables (x’s) are useful in predicting y? • Question 3:What proportion of the overall variation in y is due to the changes in the x’s? These are addressed in another module.

15. STEP 5: Use the Model for Prediction/Estimation

16. Prediction/Confidence Intervals • These are possible • but not easily with EXCEL • Other Stat packages -- MINITAB, SPSS, SAS perform these calculations.

17. Important Excel Note -- Inputting a Contiguous Range for the X’s • Suppose in this example we wished to regress Price on only Sq. Feet (column B) and Age (column D). • These are not next to each other • They must be next to each other for the regression module in Excel to work • Highlight the data in column D and click “CUT” • Click cell C1, which is where you want the data to begin, with right mouse key • Click INSERT CUT CELLS

18. Highlight cells D1:D39. • With right mouse key click Cut 3. Place cursor on cell C1. 4. With right mouse key click Insert Cut Cells.

19. Column D (Age) has been moved before column C (Land)

20. Review • Multiple regression is used when – • y is a function of more than one x • y includes terms of x raised to a power • This can be converted to a linear term • Excel (or another stat package) is used to calculate the best estimates of the ’s • The assumptions about the error term are the same •  is constant for all values of all the x’s