1 BA 275 Quantitative Business Methods Residual Analysis Multiple Linear Regression Adjusted R-squared Prediction Dummy Variables Agenda

2 Final Examination Wednesday, 3/22/06 from 7:30 – 9:20 a.m. Room: Withycombe 109 Check the seating chart when you arrive. You must sit in the assigned seat to receive your own exam booklet. Topics: Comprehensive Regression analysis accounts for 50%. Need a calculator that works and a good night sleep. Formulas and tables will be provided. Check over the weekend for additional office hours during the finals week.

3 Simple Linear Regression Model population sample True effect of X on Y Estimated effect of X on Y Key questions: 1. Does X have any effect on Y? 2. If yes, how large is the effect? 3. Given X, what is the estimated Y?

4 Residual Analysis (p.26) The three conditions required for the validity of the regression analysis are: the error variable is normally distributed. the error variance is constant for all values of x. the errors are independent of each other. How can we diagnose violations of these conditions?

5 Residual Analysis (p.26) Examining the residuals (or standardized residuals), help detect violations of the required conditions. Residual = actual Y – estimated Y We do not have  (random error), but we can calculate residuals from the sample.

6 Residuals, Standardized Residuals, and Studentized Residuals

7 The random error  is normally distributed (p.26)

8 The error variance   is constant for all values of X and estimated Y (p.26) Constant spread !

9 Constant Variance When the requirement of a constant variance is violated we have a condition of heteroscedasticity. Diagnose heteroscedasticity by plotting the residual against the predicted y The spread increases with y ^ y ^ Residual

10 The errors are independent of each other (p.26) Do not want to see any pattern.

Time Residual Time Note the runs of positive residuals, replaced by runs of negative residuals Note the oscillating behavior of the residuals around zero. 00 Non Independence of Error Variables

12 Multiple Regression Model (p.30)

13 Correlations (p.31)

14 Fitted Model (p.32) Multiple Regression Analysis Dependent variable: Price Standard T Parameter Estimate Error Statistic P-Value CONSTANT Age Bidder ? ? ? ? H 0 :  AGE = 0 H a :  AGE ≠ 0 H 0 :  BIDDER = 0 H a :  BIDDER ≠ 0 Q: Effect of AGE? Q: Effect of BIDDER? Degrees of freedom = ?

15 Fitted Model (p.32) Multiple Regression Analysis Dependent variable: Price Standard T Parameter Estimate Error Statistic P-Value CONSTANT Age Bidder Fitted Model: Estimated price = AGE BIDDER

16 Analysis of Variance (p.32) ? ?

17 Model Selection (p.33)

18 Using the Model (p.33) What is the total variation of auction prices? How much has been explained by the model? If there are 10 bidders and the age of the clock is 100 years old, what is the expected auction price? If AGE is held fixed and the number of bidders increases from 10 to 11, how much does PRICE increase?

19 Prediction and Confidence Intervals Statgraphics demo Fitted Model: Estimated price = AGE BIDDER