1. Econ 3790: Business and Economics Statistics Instructor: Yogesh Uppal yuppal@ysu.edu

2. The equation that describes how the dependent variable y is related to the independent variables x 1, x 2,... x p and an error term is called the multiple regression model. Chapter 15: Multiple Regression Model y =  0 +  1 x 1 +  2 x 2 +... +  p x p +  where:  0,  1,  2,...,  p are the parameters, and  is a random variable called the error term

3. A simple random sample is used to compute sample statistics b 0, b 1, b 2,..., b p that are used as the point estimators of the parameters  0,  1,  2,...,  p. Estimated Multiple Regression Equation ^ y = b 0 + b 1 x 1 + b 2 x 2 +... + b p x p The estimated multiple regression equation is:

4. Interpreting the Coefficients In multiple regression analysis, we interpret each In multiple regression analysis, we interpret each regression coefficient as follows: regression coefficient as follows: b i represents an estimate of the change in y b i represents an estimate of the change in y corresponding to a 1-unit increase in x i when all corresponding to a 1-unit increase in x i when all other independent variables are held constant. other independent variables are held constant.

5. Example: Car Sales Suppose we believe that number of cars sold ( y ) is Suppose we believe that number of cars sold ( y ) is not only related to the number of ads ( x 1 ), but also to the minimum down payment required at the ( x 2 ). The regression model can be given by: Multiple Regression Model where y = number of cars sold y = number of cars sold x 1 = number of ads x 1 = number of ads x 2 = minimum down payment required (‘000) x 2 = minimum down payment required (‘000) y =  0 +  1 x 1 +  2 x 2 + 

6. Estimated Regression Equation y = 14.4 + 3.7 * x1 – 25* x2 n Interpretation? n Estimated values of y? n Error? n Prediction?

7. Multiple Coefficient of Determination n Relationship Among SST, SSR, SSE where: SST = total sum of squares SST = total sum of squares SSR = sum of squares due to regression SSR = sum of squares due to regression SSE = sum of squares due to error SSE = sum of squares due to error SST = SSR + SSE

8. Multiple Coefficient of Determination R 2 = 84.63/89.2 =.949 Adjusted Multiple Coefficient of Determination Standard Error of Estimate R 2 = SSR/SST

9. Testing for Significance: t Test Hypotheses Rejection Rule Test Statistics Reject H 0 if p -value <  or if t t   where t  is based on a t distribution with n - p - 1 degrees of freedom.

10. Example: Testing for significance of coefficients Hypotheses Rejection Rule For  =.05 and d.f. = ?, t.025 = Test Statistics

11. Testing for Significance of Regression: F Test Hypotheses Rejection Rule Test Statistics H 0 :  1 =  2 =... =  p = 0 H 0 :  1 =  2 =... =  p = 0 H a : One or more of the parameters H a : One or more of the parameters is not equal to zero. is not equal to zero. F = MSR/MSE Reject H 0 if p -value F   where F  is based on an F distribution with p d.f. in the numerator and n - p - 1 d.f. in the denominator.

12. The years of experience, score on the aptitude The years of experience, score on the aptitude test, and corresponding annual salary ($1000s) for a sample of 20 programmers is shown on the next slide. Example 2: Programmer Salary Survey Multiple Regression Model A software firm collected data for a sample A software firm collected data for a sample of 20 computer programmers. A suggestion was made that regression analysis could be used to determine if salary was related to the years of experience and the score on the firm’s programmer aptitude test.

13. 471581001669210568463378100868286847580839188737581748779947089 244323.734.335.83822.223.13033 3826.636.231.6293430.133.928.230 Exper.ScoreScoreExper.SalarySalary Multiple Regression Model

14. Suppose we believe that salary ( y ) is Suppose we believe that salary ( y ) is related to the years of experience ( x 1 ) and the score on the programmer aptitude test ( x 2 ) by the following regression model: Multiple Regression Model where y = annual salary ($1000) y = annual salary ($1000) x 1 = years of experience x 1 = years of experience x 2 = score on programmer aptitude test x 2 = score on programmer aptitude test y =  0 +  1 x 1 +  2 x 2 + 

15. Solving for  0,  1 and  2 :

16. Anova Table Source of Variation Sum of Squares Degrees of Freedom Mean Square F-statistic Regression500.34…………..………. Error……..……. Total599.8……..

17. Estimated Regression Equation SALARY = 3.174 + 1.404(EXPER) + 0.251(SCORE) b 1 = 1.404 implies that salary is expected to increase by $1,404 for each additional year of experience (when the variable score on programmer attitude test is held constant). b 1 = 1.404 implies that salary is expected to increase by $1,404 for each additional year of experience (when the variable score on programmer attitude test is held constant). b2 = 0.251 implies that salary is expected to increase by $251 for each additional point scored on the programmer aptitude test (when the variable years of experience is heldconstant).

18. Prediction Suppose Bob had an experience of 4 years and had a score of 78 on the aptitude test. What would you estimate (or expect) his score to be? = 3.174 + 1.404*(4) + 0.251(78) = 28.358 = 28.358 Bob’s estimated salary is $28,358. Bob’s estimated salary is $28,358.

19. Error Bob’s actual salary is $24000. How much error we made in estimating his salary based on his experience and score? So, we shall overestimate Bob’s salary.

20. Multiple Coefficient of Determination n Relationship Among SST, SSR, SSE where: SST = total sum of squares SST = total sum of squares SSR = sum of squares due to regression SSR = sum of squares due to regression SSE = sum of squares due to error SSE = sum of squares due to error SST = SSR + SSE

21. Multiple Coefficient of Determination R 2 = 500.3285/599.7855 =.83418 R 2 = SSR/SST Adjusted Multiple Coefficient of Determination

22. Testing for Significance: t Test Hypotheses Rejection Rule Test Statistics Reject H 0 if p -value <  or if t t   where t  is based on a t distribution with n - p - 1 degrees of freedom.

23. Example Hypotheses Rejection Rule For  =.05 and d.f. = 17, t.025 = 2.11 Reject H 0 if p -value 2.11 Test Statistics Since t=7.07 > t 0.025 =2.11, we reject H 0.

24. Testing for Significance of Regression: F Test Hypotheses Rejection Rule Test Statistics H 0 :  1 =  2 =... =  p = 0 H 0 :  1 =  2 =... =  p = 0 H a : One or more of the parameters H a : One or more of the parameters is not equal to zero. is not equal to zero. F = MSR/MSE Reject H 0 if p -value F   where F  is based on an F distribution with p d.f. in the numerator and n - p - 1 d.f. in the denominator.

25. Example Hypotheses H 0 :  1 =  2 = 0 H 0 :  1 =  2 = 0 H a : One or both of the parameters H a : One or both of the parameters is not equal to zero. is not equal to zero. Rejection Rule For  =.05 and d.f. = 2, 17; F.05 = 3.59 Reject H 0 if p -value 3.59 Test Statistics F = MSR/MSE = 250.17/5.86 = 42.8 = 250.17/5.86 = 42.8 F = 42.8 > F 0.05 = 3.59, so we can reject H 0.