Welcome to Econ 420 Applied Regression Analysis

1. Welcome to Econ 420 Applied Regression Analysis Study Guide Week Seven

2. Answer Key to Assignment 5 Question 1- Part One • Step 1 • H0: B1 = B2 = B3 = 0 • HA: At least one of these B’s is not zero • Step 2: • Level of significance = 1% • Degrees of Freedom in Numerator = k = 3 • Degrees of Freedom in Denominator = n – k – 1 = 30 – 3 – 1 = 26 • Critical F, Fc, = 4.64 (pg 319)

3. Step 3: • Run regression and find F-statistic = 40.82042 • Step 4: • Because our F-statistic, 40.82 > 4.64, the null hypothesis is rejected at the 1% significance level; it is 99% likely that at least one of these B’s is not zero.

4. Question 1- Part Two • The estimated slope coefficient for income, is 0.022756. • SE = 0.005516 • Degrees of Freedom = n – k – 1 = 30 – 3 – 1 = 26 • tc = 2.056 (pg. 313)

5. The 95% confidence interval for the coefficient on income is B^1 – tc• SE (B^1) < B1 < B^1 + tc• SE (B^1), • The 95% confidence interval is 0.0114 < B1 < 0.0340. • There is 95% chance that the true value of B1 is in the above range.

6. 2. #17, Page 63 • a. Adjusted R2 = 1 – (1 – 0.7) • (9/5) = 0.46 • b. Adjusted R2 = 1 – (1 – 0.7) • (19/15) = 0.62 • c. Adjusted R2 = 1 – (1 – 0.7) • (99/95) = 0.69 • d. With the same R2, when the sample number goes up, adjusted R2 will increase. The implication here is that when you add more observations to your sample, the degrees of freedom goes up, and therefore the goodness of fit will increase. • e. When the sample size is increased, R2 may increase, decrease or even stay the same. It depends on how well the new observations fit the regression line.

7. 3. #4, PP 81-82 • a and b. Use the following formula to calculate the real values.

8. Percentage change • Is equal to (new value- old value) divided by the old value.

10. The percent change in real tax collections tends to be much smaller than that of the nominal tax collections. This shows the importance of adjusting for inflation (see part c). • c. If you didn’t adjust for inflation, the regression process would think tax collections increased a lot more than they did. Any regression results from a model that includes the nominal (unadjusted) tax collections are likely to be misleading.

11. 4. #5, Page 82 • The model is a tautology, or it is very close to being a tautology. The right hand side simply adds up all the people who have left the nursing home for various reasons. The true value for each of the slope coefficients will always be 1. For example, if one more person leaves the nursing home to live with relatives, EXIT will always increase by 1, so the true value of B3 is 1. This is true for all the slope coefficients.

12. 5. #6, Page 83 • a. HOUSE_EXP = 7 + 0.00017 INCOME • b. HOUSE_EXP = 7,000 + 170 INCOME • c. HOUSE_EXP = 7 + 0.17 INCOME • d. HOUSE_EXP = 0.7 + 0.17 INCOME • e. “b” is the easiest to interpret. You can say that if someone has an additional 1,000 in income, on average, they will spend $170 more on housing that year. • f. A measure of the price of housing, and the number of people in the household are two possible answers.

13. Chapter 5 • This week we will cover up to Page 94: Section 5-2 Interaction variables

14. Some elementary rules of partial differentiation • Y = 2X1 + 3 X1X2 + 5 X33 • dY/dX1 measures change in Y as a result of one unit change in X1 assuming X2 and X3 are constant • dY/dX1= 2 +3X2 • dY/dX2 = 3X1 • dY/dX3 = 15X32

15. Intercept Dummies • Theory 1: Men’s earnings is ,in general, higher than women’s earnings

16. Graph of earnings versus experience Earnings Male Female Years of work

17. How would a dummy variable capture this? • Intercept dummy • Earnings = B0 + B1 (gender) + B2 (years of work) + error • Where gender is dummy variable that takes a value of 1 if the observation is a male and 0 otherwise.

18. So you add one more variable to your data set. Suppose you have 5 observations in your data set, then it will look like this

19. Testing the theory • You estimate your model as usual and get • Earnings^ = 1000+ 200 (gender) + 500(years of work) • Then you do a one sided t-test of significance on the coefficient of gender • Ho: B1 ≤0 • Ha: B1>0 • If you reject Ho, then you have found significant evidence that men, in general earn more than women

20. How much more? • If your observation is a male • Earnings^ = 1000+ 200 (1) + 500(years of work) • Earnings^ = 1200+ 500(years of work) • If your observation is female • Earnings^ = 1000+ 200 (0) + 500(years of work) • Earnings^ = 1000+ 500(years of work)

21. Graph of earnings versus experience Earnings Male 1200 Female 1000 Years of work

22. Slope Dummies • Theory 2: Men’ earnings grow at a higher rate than women’s earnings

23. Graph of earnings versus experience Earnings Male Female Years of work

24. How would a dummy variable capture this? • Slope dummy • Earnings = B0 + B1 (years of work) + B2 (years of work) *( gender) + error • Where gender is dummy variable that takes a value of 1 if the observation is a male and 0 otherwise.

25. Suppose you have 5 observations in your data set, you will create a new variable (genwork). Genwork is gender times years of work. your data set will look like this

26. Testing the theory • You estimate your model as usual and get • Earnings^ = 1000+ 500(years of work) +70(genwork) • Then you do a one sided t-test of significance on the coefficient of genwork • Ho: B2 ≤0 • Ha: B2>0 • If you reject Ho, then you have found significant evidence that men’s earnings grow at a higher rate with years of experience.

27. How much more? • If your observation is a male • Earnings^ = 1000+ 500(years of work) + 70 (years of work)* (1) • Earnings^ = 1200+ 570(years of work) • If your observation is female • Earnings^ = 1000+ 500(years of work) + 70 (years of work) * (0) • Earnings^ = 1000+ 500(years of work)

28. Graph of earnings versus experience Earnings Male slope = 570 Female slope = 500 Years of work

29. What if • The theory suggested that not only, in general, men’s salaries are higher than women’s salaries but men also receive a higher rate of increases in their salaries compared to women over time. • Then you are better off to estimate the model twice: once for male observations and once for female observations as the slope and the intercept must be allowed to vary across genders.

30. Assignment 6 (20 points)Due: before 10 PM on Friday, October 12) • Suppose the theory suggests that advertising for sun blocks is more effective in summer than any other time of the year • Formulate the model • What type of a data set will you use: time series or cross sectional? • Set up a hypothesis to test the theory

31. 2. Suppose we estimate a regression equation that sets the crime rate as a function of a state’s per capita income and the number of police officers in each state per 10,000 population. The estimated coefficient of per capita income happens to be positive. We suspect that the estimated coefficient of per capita income is biased positively because we have an omitted variable. Which of the following omitted variables is more likely to have caused the bias in our estimated coefficient of income and why? • Number of college educated individuals per 1000 population • Percentage of population living in poverty • State’s unemployment rate • Percentage of population who lives in urban areas.