1. Steps in Regression Analysis (1) Choose the dependent and independent variables (2) Examine the scatterplots and the correlation matrix Check for any high correlations between independent variables (if any, suspect M/C) (3) Run the regression (4) Deal with multicolinearity, remove a variable if necessary (5) Check for significance of coefficients (t-test) (6) Remove insignificant variables, starting with the one with the highest p-value (or smallest t-test)

2. Possible Problems with Regression Line Multicolinearity (M/C) A linear relationship between two independent variables Why is it a problem? One of the independent variables becomes redundant When should one suspect M/C? When you expect a linear relationship between two independent variables When the correlation between two independent variables is higher than 0.7 When the signs of (significant) variables are distorted

3. Remove the one with highest p-value (or smallest absolute t-test--it’s the same thing). Remove the one with highest p-value. neither is significant one is significant leave both, unless you believe there is a very strong linear relationship btw them orif the signs are distorted. both are significant Two independent variables are highly correlated (positively or negatively). Dealing with Multicolinearity

4. The F-test Objective: To provide a global test of the regression equation H 0 : B 1 = B 2 = B 3 = B 4 =... =B k = 0 H A : At least one B j  0 The test: If H 0 is true: Interpretation: ratio of explained to unexplained variation

5. F P The F-test If the null hypothesis were true, this would be the distribution of F.

6. (1) Choose dependent and independent variables (use contextual knowledge) (2) Study scatterplots and correlation matrix (look for M/C). (3) Run the regression. (4) Deal with multicolinearity (decision tree). (5) Begin the model building phase. If F-test is not significant, then do not continue, or return to step 1. (6) Backward elimination, using t-tests and p-values. A Step-by-Step Approach to Model Building

7. Case: Firing Experts Three experts forecasting Yen:$ Which one to fire? How to approach this problem

8. Correlation Matrix of Firing Experts TomTatsuyaBernardActual Tom1.000.77-0.080.75 Tatsuya0.771.000.280.76 Bernard-0.080.281.000.46 Actual0.750.760.461.00

9. Best Solution

10. R 2 and adjusted-R 2 Adjusted-R 2 is used to compare models of different sizes. In contrast to R 2, it can go up or down when a variable is taken out of the model.

11. No outliers

12. Effect of an Outlier

14. (1) Choose dependent and independent variables (use contextual knowledge) (2) Study scatterplots and correlation matrix (look for M/C). (3) Run the regression. (4) Deal with multicolinearity (decision tree). (5) Begin the model building phase. If F-test is not significant, then do not continue, or return to step 1. (6) Backward elimination, using t-tests and p-values. (7) Deal with outliers. If data is removed, go back to Step 2. (8) Check regression assumptions. (9) Monitor the model over time. A Step-by-Step Approach to Model Building

15. Confidence Intervals for B j 100(1 -  )% interval

16. Confidence Intervals for a Forecast Formulas for simple regression (1 independent variable) s e : This is the standard deviation of regression. s f : This is the standard error of a forecast. 100(1 -  )% Interval for forecast