1. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics for Economist Chap 6. Regression 1.The Relation of Two Variables 2.The Graph of Averages 3.The Regression Method 4.The Regression Effect 5.The Regression Line for x on y and y on x

2. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 2/23 INDEX 1 The Relation of Two Variables 2 The Graph of Averages 3 The Regression Method 4 The Regression Effect 5 The Regression Line for x on y and y on x

3. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 3/23 1. Relation of Two Variables Regression and SD line go through the point of averages Height (cm) Weight (kg) SD line regression line Associated with an increase of 1 SD in height there is an increase of only 0.67 SD in weight, on the average. Scatter Plot

4. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 4/23 Regression Associated with each increase of 1SD in x there is an increase of only r SDs in y, on the average.  The regression line for y on x estimates the average value for y corresponding to each value of x. r: correlation coefficient SD x r  SD y estimation average y x 1. Relation of Two Variables

5. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 5/23 INDEX 1 The Relation of Two Variables 2 The Graph of Averages 3 The Regression Method 4 The Regression Effect 5 The Regression Line for x on y and y on x

6. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 6/23 2. Graph of Averages Graph of Averages 각 점들은 각각의 키에 대하여 그 키에 해당하는 집단의 평균 몸무게를 보여준다. 점 위에 표시된 숫자는 해당집단의 크기를 나타낸다. Many points on the graph of averages are near the line. This line is regression line. regression Graph of averages

7. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 7/23 Improper case of regression line – nonlinear association When the graph of averages is nonlinear,the regression line does not reflect theexact association. Regression line 2. Graph of Averages

8. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 8/23 INDEX 1 The Relation of Two Variables 2 The Graph of Averages 3 The Regression Method 4 The Regression Effect 5 The Regression Line for x on y and y on x

9. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 9/23 3. Regression Method Guess the weight Without being told anything else about him, the best guess on the weight of one of these men is the overall average weight 63.5kg. If you are told the man ’ s height: 179.5cm then your best guess for his weight is the average for all the 179.5cm men in the study, 74.7kg. <Health and Nutrition Examination Survey> height(cm) weight (kg)

10. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 10/23 Example 1  Followings are the results of 100 students majoring in economics in a certain college. Economics 1 average = 3.00 SD X = 0.87 Statistics average = 2.80 SD Y = 0.86 r = 0.36 EX) Predict the statistics GPA of a student whose GPA of Economics I is 3.70. 3. Regression Method

11. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 11/23 Regression Analysis 1.0.8SDx above average on the Economics 1. 3.7-3.0 = 0.7 = 0.87  0.8 2. 0.36  0.8 = 0.29 0.29 SDy above average on the Statistics. 4. The predicted GPA of statistics is 2.80 + 0.25 = 3.05 Example1 3. That is, 0.29  0.86 = 0.25(GPA) above average on the Statistics. Corr. coeff SD X SD y 3. Regression Method

12. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 12/23 notice ☞ when estimating on new subjects, Foregoing data can represent the new subjects ; reasonable!! Foregoing data and the new subjects are different in groups or in the range ; you have to think about the issue carefully before applying the regression method. 3. Regression Method

13. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 13/23 In the case of [example 1] … Would be the data from the students majoring in Economics applied directly to Jane whose major is Philosophy?  What if Jane majors in Philosophy not in Economics?  What if Jane ’ s GPA of Economics I is 4.3 but 3.7?  If most GPAs of the Economics 1 (x-value) are between 2.0 and 4.0, Jane ’ s x-value(4.3) is out of the previous x-range. Would the estimated regression line based on the x- value range 2.0~4.0 be applied to Jane ’ s GPA which is over the range?  [The extrapolation is not a simple problem.] 3. Regression Method

14. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 14/23 Regression Analysis example2- predict percentile ranks  Suppose the percentile rank of one student on the Economics 1 GPA is 90%. Predict her percentile rank on statistics. 1.This student scored 1.3 SDx above average on the Economics1. 2. 0.36  1.3 = 0.47 SDy above average on the statistics. =68% z = 1.3 z = 0.47 =90% 3. The percentile rank on statistics is predicted as 68%. 3. Regression Method

15. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 15/23 INDEX 1 The Relation of Two Variables 2 The Graph of Averages 3 The Regression Method 4 The Regression Effect 5 The Regression Line for x on y and y on x

16. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 16/23 containing too many points with small x-value. Most of these points ’ y-values are above the SD line. Y-values of points are spread symmetrically along the SD line overall. Containing too many points with large x-value. Most of these points ’ y-values are below the SD line. 4. Regression Effect regression to mediocrity Regression line SD line

17. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 17/23 Regression effect Final average scores corresponding to each midterm scores SD line Regression line – summarizes each average points well 4. Regression Effect

18. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 18/23 regression fallacy  The regression effect occurs when each point representing data is spread out around the SD line.  The fallacy that the regression effect must be due to something important, not just the spread around the line ☞ Regression Fallacy 4. Regression Effect

19. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 19/23 Observed score= 140 145135 Suppose that the probability error is  5 with probability 0.5. The people scored 140 can be divided into two groups with true score 135 or 145. The model for regression effect If someone scores above average on the first test, the true score is probably a bit lower than the observed score. (observed score) = (true score) + (chance error) 4. Regression Effect

20. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 20/23 INDEX 1 The Relation of Two Variables 2 The Graph of Averages 3 The Regression Method 4 The Regression Effect 5 The Regression Line for x on y and y on x

21. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 21/23 5. Regression line for x on y and y on x Two different regression lines weight height Regression of weight on height The regression line is solid, and the SD line is dashed

22. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 22/23 exmaple3  The men whose IQ was 140 had wives whose IQ averaged 120. Look at the wives whose IQ was 120; should the average IQ of their husbands be 140? (IQ scores are scaled to have an average of about 100, and an SD of about 15, both for men and women. The correlation is about 0.50) Wife ’ s IQ Husbnad ’ s IQ 120 140 Husband ’ s IQ 140 140 ? 5. Regression line for x on y and y on x

23. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 23/23 example 3 Wife ’ s IQ Husband ’ s IQ Regression line for husband ’ s IQ on wife ’ s IQ Regression line for wife ’ s IQ on husband ’ s IQ 120 140110 There are two different regression lines; one for predicting the wife ’ s IQ from her husband ’ s IQ and the other for vice versa. 5. Regression line for x on y and y on x