Economics 105: Statistics

1. Economics 105: Statistics GH 22 & 23 due Thursday, 17th GH 24 (last one! Please rejoice silently.) due Thur 24th Unit 3 Review will be due Tuesday, 29th (I’ll hand it out Thur 24th). It’ll cover what we get through. I’ve decided NOT to squeeze in the individual RAP presentations. I will move that % of your grade to RAP & Review 3.

2. Quadratic Regression Model Quadratic models may be considered when the scatter diagram takes on one of the following shapes: Y Y Y Y X1 X1 X1 X1 β1 < 0 β1 > 0 β1 < 0 β1 > 0 β2 > 0 β2 > 0 β2 < 0 β2 < 0 β1 = the coefficient of the linear term β2 = the coefficient of the squared term

3. Testing the Overall Model • Estimate the model to obtain the sample regression equation: • The “whole model” F-test H0: β1 = β2 = β3 = … = β15 = 0 H1: at least 1 βi ≠ 0 • F-test statistic =

4. Testing the Overall Model Critical value = 2.082= F.INV(0.99,15,430-15-1) p-value = 0 = 1-F.DIST(120.145,15,430-15-1,1)

5. Testing for Significance of just a Quadratic Term • t-test

6. Example

7. Coefficient of Determination for Multiple Regression • Reports the proportion of total variation in Y explained by all X variables taken together • Consider this model

8. Multiple Coefficient of Determination (continued) 52.1% of the variation in pie sales is explained by the variation in price and advertising

9. Adjusted R2 • R2 never decreases when a new X variable is added to the model • disadvantage when comparing models • What is the net effect of adding a new variable? • We lose a degree of freedom when a new X variable is added • Did the new X variable add enough explanatory power to offset the loss of one degree of freedom?

10. Adjusted R2 (continued) • Penalizes excessive use of unimportant variables • Smaller than R2and can increase, decrease, or stay same • Useful in comparing among models, but don’t rely too heavily on it – use theory and statistical signif

11. Adjusted R2 (continued) 44.2% of the variation in pie sales is explained by the variation in price and advertising, taking into account the sample size and number of independent variables

12. Log Functional Forms • Linear-Log • Log-linear • Log-log • Log of a variable means interpretation is a percentage change in the variable • (don’t forget Mark’s pet peeve)

13. Log Functional Forms • Here’s why: ln(x+x) – ln(x) = • calculus: • Numerically: ln(1.01) = .00995 = .01 • ln(1.10) = .0953 = .10 (sort of)

14. Linear-Log Functional Form

15. Linear-Log Functional Form

16. Log-Linear Functional Form

17. Log-Linear Functional Form

18. Log-Log Functional Form

19. Log-Log Functional Form

20. Examples

21. Examples

22. Examples

23. Examples

24. Dummy Variables • A dummy variable is a categorical explanatory variable with two levels: • yes or no, on or off, male or female • coded as 0’s and 1’s • Regression intercepts are different if the variable is significant • Assumes equal slopes for other explanatory variables (i.e., equal marginal effects!) • “Dummy Variable Trap” • If more than two categories, the number of dummy variables included is (number of categories - 1)

25. Dummy Variable Example (with 2 categories) • E[ GPA | EconMajor = 1] = ? • E[ GPA | EconMajor = 0] = ? • Take the difference to interpret EconMajor

26. Dummy Variable Example (More than 2 categories) • Model the effect of class year on GPA, controlling for study hours