1. 732G21/732G28/732A35 Lecture 4

2. Variance-covariance matrix for the regression coefficients 2

3. Variance-covariance matrix of the model errors/residuals 3 where and

4. Multiple regression model (theoretical) 4 Multiple regression model (for a sample)

5. ANOVA table for multiple regression 5 Source of variationSSdfMS Regression p-1 Error n-p Total n-1 where J is a n * n matrix of ones

6. Example data set (car prices) 6 Y (Price in SEK)X 1 = YearX 2 = No. of kilometers driven 279000199755000 302000199845000 449900200222000 259000199753000 265000199734000 349000200050500 369000200233500...... 295000199864990 n = 59 We have collected information about cars of a certain model.

7. Scatter chart price/year 7

8. Scatter chart price/No. kilometers 8

9. Regression output from car example Regression Analysis: Price versus Year, Kilometers The regression equation is Price = - 35397446 + 17928 Year - 2.61 Kilometers Predictor Coef SE Coef T P Constant -35397446 5767248 -6.14 0.000 Year 17928 2881 6.22 0.000 Kilometers -2.6149 0.2039 -12.83 0.000 S = 30028.2 R-Sq = 90.2% R-Sq(adj) = 89.9% Analysis of Variance Source DF SS MS F P Regression 2 4.65369E+11 2.32684E+11 258.05 0.000 Residual Error 56 50494815376 901693132 Total 58 5.15864E+11 9

10. Interval estimation for multiple regression  Confidence interval: where  Prediction interval: where 10

11. Regression Analysis: Price versus Year, Kilometers The regression equation is Price = - 35397446 + 17928 Year - 2.61 Kilometers Predictor Coef SE Coef T P Constant -35397446 5767248 -6.14 0.000 Year 17928 2881 6.22 0.000 Kilometers -2.6149 0.2039 -12.83 0.000 S = 30028.2 R-Sq = 90.2% R-Sq(adj) = 89.9% Analysis of Variance Source DF SS MS F P Regression 2 4.65369E+11 2.32684E+11 258.05 0.000 Residual Error 56 50494815376 901693132 Total 58 5.15864E+11 Predicted Values for New Observations New Obs Fit SE Fit 95% CI 95% PI 1 415141 7201 (400716, 429567) (353282, 477001) Values of Predictors for New Observations New Obs Year Kilometers 1 2002 30000 11

12. Four-in-one plot of residuals for car example 12

13. Residuals plotted against predictors 13

14. Example data set (car prices) 14 Y (Price in SEK) X 1 = Year X 2 = No. of kilometers X 3 = Equipment level 279000199755000Standard 302000199845000Standard 449900200222000Luxury 259000199753000Standard 265000199734000Standard 349000200050500Luxury 369000200233500Luxury...... 295000199864990Standard

15. Scatter chart of price/equipment level 15

16. Regression Analysis: Price versus Year, Kilometers, Equipment The regression equation is Price = - 20833056 + 10618 Year - 2.08 Kilometers + 57904 Equipment Predictor Coef SE Coef T P Constant -20833056 6309217 -3.30 0.002 Year 10618 3154 3.37 0.001 Kilometers -2.0768 0.2022 -10.27 0.000 Equipment 57904 10408 5.56 0.000 S = 29269.6 R-Sq = 90.0% R-Sq(adj) = 89.4% Analysis of Variance Source DF SS MS F P Regression 3 4.22692E+11 1.40897E+11 164.46 0.000 Residual Error 55 47118909984 856707454 Total 58 4.69810E+11 Source DF Seq SS Year 1 2.82889E+11 Kilometers 1 1.13288E+11 Equipment 1 26514947048 16