2. Marietta College Week 13 1

3. Tuesday, April 5 2 Exam 3: Monday, April 25, 12- 2:30PM

4. Leadership Q&A David Leonhardt Economics Journalist Washington Bureau The New York Times TONIGHT 7:30pm McDonough Gallery Cosponsored by McDonough Center for Leadership & Business and the Economic Roundtable of the Ohio Valley Is anyone interested in going to breakfast with him tomorrow 8 am, Lafayette Hotel

5. This is the last bonus opportunity of this semester 2 points for attending 2-5 points per question 2-10 points per summary Summaries are due before 5 pm on Friday, April 8 via an email attachment to me Total bonus points will be divided by 3 and added to your exams.

6. Return and discuss Asst 18 # 12 Page 240 a) The estimated coefficients all are in the expected direction R 2 bar seems fairly low. Always check the significance at 10 percent or better Coefficients of A, A 2 and S are significant. You can only interpret the magnitude of coefficients if they pass the t-test of significance Significance has to do with t-test Importance has to do with the absolute value of coefficient. 5

7. b) It implies that wages rise at a declining rate with respect to age and eventually fall. Does not imply perfect collinearity (non-linear correlation). c) Semilog (Ln W) is a possibility The slope coefficient represents the percentage change in wage caused by a one-unit increase in the independent variable (holding constant all the other independent variables). Since pay raises are often discussed in percentage terms, this makes sense. Phil & Yuan say, but what about the meaning of coefficient of A 2 in a semi log function? (great point) Linda says, it depends on the purpose of the study (great point) 6

8. d) It’s a good habit to ignore (except to make sure that one exists) even if it looks too large or too small. Intercept picks up the mean of the error term & and that is affected by omitted variables. e) The poor fit and the insignificant estimated coefficient of union membership are all reasons for being extremely cautious about using this regression to draw any conclusions about union membership. 7

9. Collect Asst 19 # 5 Page 234 – Including Part e (data is available online under STOCK in Chapter 7) 8

10. Imperfect Multicollinearity Problem What is it? Let’s say you estimate an regression equation, what makes you suspicious about possible mulit problem? What are the two formal tests we talked about before? 9

11. Example Income = f (wage rate, tax rate, hours of work, ….) Wage rate, tax rate and hours of work may be all highly correlated with each other Problem: simple correlation coefficient may not capture this. Sometimes 3 or more independent variables are correlated

12. Regress each independent variable (say X 1 ) on the other independent variables (X 2, X 3, X 4 ) Then calculate VIF VIF = 1 / (1- R 2 ) If VIF > 5  then X1 is highly correlated with the other independent variables Do the same for all of the other independent variables Test of Multicollinearity among 3 or more independent variables

13. Asst 20 Data set: DRUGS (Chapter 5, P 157) Estimate Equation 5.10 1.Before you run any formal tests, do you suspect an imperfect mulitcollinearity problem? Why or why not? 2.Examine the absolute values of the correlation coefficients between the independent variables included in Equation 5.10. Do you find any evidence of muliticollinerity problem? Discuss. 3.Examine the VIF of the two most suspicious independent variables in Equation 5.1 based on what your found in Section 2 above. Do you find any evidence of muliticollinerity problem? Discuss 12

14. Thursday, April 7 Exam 3: Monday, April 25, 12- 2:30PM If you asked David Leonhardt questions on Tuesday night, write it down and give it to me today. Summaries are due before 5 pm tomorrow via an email attachment. Bring laptops to class on Tuesday 13

15. Return and discuss Asst 19 # 5 Page 234 – Including Part e (data is available online under STOCK in Chapter 7) 14

16. (a) You are correct but note that the null and alternative hypotheses are not about beta hats, they are about betas. (b) It’s unusual to have a lagged variable in a cross- sectional model. BETA is lagged. 15

17. 16 Part C Should we include EARN in the set of our independent variables 1.Does the theory call for its inclusion? Yes, but a version of it is in dependent variable  exclude EARN 2.Is the estimated coefficient of EARN significant in the right direction? No  exclude EARN 3.As you include EARN, does the adjusted R squared goes up? Yes  include EARN 4.As you include EARN, do the other variables’ coefficients change significantly? Change somewhat! 5.As you include EARN, do AIC and SC go down? No  exclude EARN

18. (d) The functional form is a semilog left, which is appropriate both on a theoretical basis and also because two of the independent variables are expressed as percentages. 17

19. (e) EARN, DIV, and Beta all can be negative, can’t take their log. 18

20. Return and discuss Asst 20 Data set: DRUGS (Chapter 5, P 157) Estimate Equation 5.10 1.Before you run any formal tests, do you suspect an imperfect mulitcollinearity problem? Why or why not? 2.Examine the absolute values of the correlation coefficients between the independent variables included in Equation 5.10. Do you find any evidence of muliticollinerity problem? Discuss. 3.Examine the VIF of the two most suspicious independent variables in Equation 5.1 based on what your found in Section 2 above. Do you find any evidence of muliticollinerity problem? Discuss 19

21. Dependent Variable: P Method: Least Squares Sample: 1 32 Included observations: 32 VariableCoefficientStd. Errort-StatisticProb. C38.221316.3873045.9839510.0000 GDPN1.4336800.2143956.6871080.0000 CVN-0.5947320.223947-2.6556790.0133 PP7.3113306.1230841.1940600.2432 DPC-15.628646.932635-2.2543590.0328 IPC-11.384567.159258-1.5901870.1239 R-squared0.811223 Adjusted R-squared0.774920 20 Do we suspect multicollinearity problem? What should we look for? R bar squared is high but we have two insignificant variables

22. Correlation Matrix GDPNCVNPPDPCIPC GDPN 1 0.86 0.21 0.17-0.05 CVN 10.12 0.31 0.06 PP 1-0.13-0.21 DPC1 0.38 IPC 1 21 Why are the diagonal values all 1? Why did I eliminate the values in bottom half of the table? Is multicollinearity a problem?

23. Dependent Variable: GDPN Method: Least Squares Sample: 1 32 Included observations: 32 VariableCoefficientStd. Errort-StatisticProb. C12.686485.1877122.4454860.0213 CVN0.9010370.1016958.8601670.0000 PP4.3432015.4324260.7994960.4310 DPC-4.1125126.172508-0.6662630.5109 IPC-3.8838866.382854-0.6084870.5479 R-squared0.766913 22 VIF = 1/ (1-0.77) VIF = 4.34 VIF<5  no serious multicollinearity problem

24. Dependent Variable: CVN Method: Least Squares Sample: 1 32 Included observations: 32 VariableCoefficientStd. Errort-StatisticProb. C-4.6395975.415845-0.8566710.3992 DPC8.4028895.7339101.4654730.1543 GDPN0.8258060.0932048.8601670.0000 IPC2.6624346.1309640.4342600.6676 PP-1.3339225.255631-0.2538080.8016 R-squared0.775957 23 VIF = 1/ (1-0.78) VIF = 4.54 VIF<5  no serious multicollinearity problem

25. 1.If your main goal is to use the equation for forecasting and you don’t want to do specific t- test on each estimated coefficient then do nothing. ◦ This is because multicollinearity does not affect the predictive power of your equation. 2.If it seems that you have a redundant variable, drop it. ◦ Examples ◦ You don’t need both real and nominal interest rates in your model ◦ You don’t need both nominal and real GDP in your model Remedies for Multicollinearity

26. 3. If all variables need to stay in the equation, transform the multicollinear variables  Example:  Number of domestic cars sold = B 0 + B 1 average price of domestic cars + B 2 average price of foreign cars +…..+ є  Problems: Prices of domestic and foreign cars are highly correlated  Solution:  Number of domestic cars sold = B 0 + B 1 the ratio of average price of domestic cars to the average price of foreign cars +…..+ є 4. Increase the sample size or choose a different random sample Remedies for Multicollinearity

27. Asst 21: Due Tuesday in class Use the data set FISH in Chapter 8 (P 274) to run the following regression equation: F = f (PF, PB, Yd, P, N) 1)Conduct all 3 tests of imperfect multicollinearity problem and report your results. 2)If you find an evidence for imperfect multicollinearity problem, suggest and implement a reasonable solution. 26

28. 27 Chapter 9 (Autocorrelation or Serial Correlation) Suppose we are using time series data to estimate consumption (C) as a function of income (Y) and other factors C t = β 1 + β 2 Y t +…..+ є t – Where t = (1, 2, 3, ….T) – This means that C 1 = β 1 + β 2 Y 1 +…. + є 1, and C 2 = β 1 + β 2 Y 2 +…. + є 2 ….. …… C T = β 1 + β 2 Y T +…. + є T …… One of the classical assumptions regarding the error terms is – No correlation among the error terms in the theoretical equation If this assumption is violated then there is a problem of pure serial correlation (autocorrelation).

29. 28 First Order Pure Autocorrelation є 2 = ρ є 1 + u 2 – That is, the error term in period 2 depends on the error term in period 1 – Where, u 2 is a normally distributed error with the mean of zero and constant variance

30. 29 Second Order Pure Autocorrelation є 3 = ρ 1 є 1 + ρ 2 є 2 + u 3 – That is, the error term in period 3 depends on the error term in period 1 and the error term in period 2. – Where, u 3 is a normally distributed error with the mean of zero and constant variance

31. 30 Higher Order Pure Autocorrelation є t = ρ 1 є t-1 + ρ 2 є t-2 + ρ 3 є t-3 + ….. + u t – That is, the error term in period t depends on the error term in period t-1, the error term in period t- 2, and the error term in period t-3,…etc. – Where, u t is a normally distributed error with the mean of zero and constant variance

32. When the true (theoretical) regression line does not have an autocorrelation problem but our estimated equation does. Why? 1.Specification error 2.Wrong functional form 3.Data error 31 What is the Impure Serial Correlation?

33. 32 Types of Serial Correlation 1.Positive Errors form a pattern A positive error is usually followed by another positive error A negative error is usually followed by another negative error More common

34. 33 Example of positive autocorrelation

35. 34 Types of Serial Correlation 2. Negative A positive error is usually followed by a negative error or visa-versa Less common

36. 35 Example of negative autocorrelation

37. 36 EViews allows you to see the residuals’ graph After you estimate the regression equation Click on View on your regression output Click on Actual, Fitted, Residual Table

38. ActualFittedResidualResidual Plot 23.5200 27.0183-3.49829|*. |. | 25.9500 28.4120-2.46201| *. |. | 25.9400 28.6804-2.74042| *. |. | 27.2200 28.8564-1.63643|.* |. | 27.8200 29.5686-1.74857|.* |. | 29.7700 30.3005-0.53051|. * |. | 32.0800 30.4455 1.63447|. | *. | 32.6200 31.9419 0.67814|. | *. | 32.8800 32.3974 0.48264|. | *. | 34.9000 32.2590 2.64105|. |. * | 36.8800 33.3502 3.52980|. |. *| 36.7400 35.0438 1.69615|. | *. | 38.4900 35.3550 3.13501|. |. * | 37.0100 33.6742 3.33580|. |. * | 36.9300 37.2539-0.32386|. *|. | 36.7000 38.0231-1.32312|. * |. | 39.8400 37.4913 2.34866|. |.* | 40.7100 39.4460 1.26398|. | *. | 43.1000 42.2249 0.87505|. | *. | 46.6400 44.7560 1.88398|. | *. | 46.9100 48.2692-1.35924|. * |. | 48.4500 50.1672-1.71724|.* |. | ……… 37 What type of serial correlation may we have? Negative residuals seem to be followed by other negative residuals  suspect positive autocorrelation