ECON 303, EXAM 1 FALL 2008 DUE: WEDNESDAY, OCTOBER 8, 2008 at 12:40 PM Please show your work. There are 3 sections. Each section carries equal weight. If you have ANY questions, please ASK ME. Yes, the LC HONOR CODE APPLIES Section 1: Interpreting Information 1) A few years ago the news magazine The Economist listed some of the stranger explanations used in the past to predict presidential election outcomes. These included whether or not the hemlines of women’s skirts went up or down, stock market performances, baseball World Series wins by an American League team, etc. Thinking about this problem more seriously, you decide to analyze whether or not the presidential candidate for a certain party did better if his party controlled the house. Accordingly you collect data from 34 past presidential elections. You think of these data as comprising a population which you want to describe, rather than a sample from which you want to infer behavior of a larger population. You generate the accompanying table: Joint Distribution of Presidential Party Affiliation and Party Control of House of Representatives, 1860-1996 Democratic President ( X 0 ) Republican President ( X 1 ) Total Democratic Control of House ( Y 0 ) 0.412 Republican Control of House ( Y 1 ) 0.030 Total 0.441 0.176 0.382 0.559 0.588 0.412 1.00 (a) Interpret one of the joint probabilities and one of the marginal probabilities. (b) Compute E ( X ) . How does this differ from E ( X | Y 0) ? Explain.. (c) If you picked one of the Republican presidents at random, what is the probability that during his term the Democrats had control of the House? (d) What would the joint distribution look like under independence? Check your results by calculating the two conditional distributions and compare these to the marginal distribution. 1 2) Math SAT scores (Y) are normally distributed with a mean of 500 and a standard deviation of 100. An evening school advertises that it can improve students’ scores by roughly a third of a standard deviation, or 30 points, if they attend a course which runs over several weeks. (A similar claim is made for attending a verbal SAT course.) The statistician for a consumer protection agency suspects that the courses are not effective. She views the situation as follows: H 0 : Y 500 vs. H1 : Y 530 . (a) Sketch the two distributions under the null hypothesis and the alternative hypothesis. (b) The consumer protection agency wants to evaluate this claim by sending 50 students to attend classes. One of the students becomes sick during the course and drops out. What is the distribution of the average score of the remaining 49 students under the null, and under the alternative hypothesis? (c) Assume that after graduating from the course, the 49 participants take the SAT test and score an average of 520. Is this convincing evidence that the school has fallen short of its claim? What is the p-value for such a score under the null hypothesis? (d) What would be the critical value under the null hypothesis if the size of your test were 5%? Section 2: Some Theory Suppose the population regression function is: y 0 1 x , where ε is a random variable satisfying the Least Squares Assumptions, discussed in Stock & Watson’s section 4.4, with mean of 0 and unknown variance of 2 . You take a sample of size n and construct the OLS estimator ˆ1 . 1) Investigate the properties of the Sampling Distribution of ˆ . (That is what are its 1 probability distribution, mean, and variance?) 2) Evaluate the estimator ˆ1 for unbiasedness, efficiency, and consistency. 3) Why might it be reasonable to assume that it would be more usual for regression errors to be heteroskedastic than homoskedastic? Section 3: Regression Application 1) Sir Francis Galton, a cousin of James Darwin, examined the relationship between the height of children and their parents towards the end of the 19th century. It is from this study that the name “regression” originated. You decide to update his findings by collecting data from 110 college students, and estimate the following relationship: 2 Studˆenth = 19.6 + 0.73×Midparh, R2 = 0.45, SER = 2.0 (7.2) (0.10) where Studenth is the height of students in inches, and Midparh is the average of the parental heights. Values in parentheses are heteroskedasticity robust standard errors. (Following Galton’s methodology, both variables were adjusted so that the average female height was equal to the average male height.) (a) Interpret the estimated coefficients. (b) What is the meaning of the regression R2 ? (c) What is the prediction for the height of a child whose parents have an average height of 70.06 inches? (d) What is the interpretation of the SER here?. (e) Given the positive intercept and the fact that the slope lies between zero and one, what can you say about the height of students who have quite tall parents? Who have quite short parents? (f) Test for the statistical significance of the slope coefficient. (g) If children, on average, were expected to be of the same height as their parents, then this would imply two hypotheses, one for the slope and one for the intercept. (i) What should the null hypothesis be for the intercept? Carry out the hypothesis test at the 1% level. (ii) What should the null hypothesis be for the slope? Carry out the hypothesis test at the 5% level. (h) Can you reject the null hypothesis that there is no linear relation? (i) Construct a 95% confidence interval for a one inch increase in the average of parental height. (j) Galton was concerned about the height of the English aristocracy and referred to the above result as “regression towards mediocrity.” Can you figure out what his concern was? Why do you think that we refer to this result today as “Galton’s Fallacy?” 2) The news-magazine The Economist regularly publishes data on the so called Big Mac index and exchange rates between countries. The data for 30 countries from the April 29, 2000 issue is listed below: 3 Price of Actual Exchange Rate Country Currency Big Mac per U.S. dollar Indonesia Rupiah 14,500 7,945 Italy Lira 4,500 2,088 South Korea Won 3,000 1,108 Chile Peso 1,260 514 Spain Peseta 375 179 Hungary Forint 339 279 Japan Yen 294 106 Taiwan Dollar 70 30.6 Thailand Baht 55 38.0 Czech Rep. Crown 54.37 39.1 Russia Ruble 39.50 28.5 Denmark Crown 24.75 8.04 Sweden Crown 24.0 8.84 Mexico Peso 20.9 9.41 France Franc 18.5 7.07 Israel Shekel 14.5 4.05 China Yuan 9.90 8.28 South Africa Rand 9.0 6.72 Switzerland Franc 5.90 1.70 Poland Zloty 5.50 4.30 Germany Mark 4.99 2.11 Malaysia Dollar 4.52 3.80 New Zealand Dollar 3.40 2.01 Singapore Dollar 3.20 1.70 Brazil Real 2.95 1.79 Canada Dollar 2.85 1.47 Australia Dollar 2.59 1.68 Argentina Peso 2.50 1.00 Britain Pound 1.90 0.63 United States Dollar 2.51 The concept of purchasing power parity or PPP (“the idea that similar foreign and domestic goods … should have the same price in terms of the same currency,” Abel, A. and B. Bernanke, Macroeconomics, 4th edition, Boston: Addison Wesley, 476) suggests that the ratio of the Big Mac priced in the local currency to the U.S. dollar price should equal the exchange rate between the two countries. 4 a) Enter the data into STATA. They are located in the STATA data file, bmi.dta, in the Exam1 Materials folder on the Econ 303 webdisk cite. They are also there as a text file and as a MSWord file. Calculate the predicted exchange rate per U.S. dollar by dividing the price of a Big Mac in local currency by the U.S. price of a Big Mac ($2.51). b) Run a regression of the actual exchange rate on the predicted exchange rate and report your regression results. Do not just include the STATA output. Rather report the regression results as they are reported in the text: the estimated regression equation, appropriate standard errors, R-squared, and sample size. c) If purchasing power parity held, what would you expect the slope of the regression to be, and what would you expect the intercept of the regression to be? Write each expectation in terms of a null hypothesis and an alternative hypothesis. d) Plot the actual exchange rate against the predicted exchange rate. Include the 45 degree line in your graph. Which observations might cause the slope and the intercept to differ from zero and one? e) Using a 5% significance level, what is your decision regarding the null hypotheses given in part (c)? Why? Are you concerned with the fact that you are testing the two hypothesis sequentially when they are supposed to hold simultaneously? f) What assumptions had to be made for your choice of standard error? 3) The cost of attending your college has once again gone up. Although you have been told that education is investment in human capital, which carries a return of roughly 10% a year, you (and your parents) are not pleased. One of the college’s administrators does not make the situation better by telling you that you pay more because the reputation of your institution is better than that of others. To investigate this hypothesis, you collect data randomly for 100 national universities and liberal arts colleges from the 2000-2001 U.S. News and World Report annual rankings. Next you perform the following regression Coˆst = 7,311.17 + 3,985.20 Reputation – 0.20 Size (2,058.63) (664.58) (0.13) + 8,406.79 Dpriv – 416.38 Dlibart – 2,376.51 Dreligion (2,154.85) (1,121.92) (1,007.86) R2=0.72, SER = 3,773.35 where Cost is Tuition, Fees, Room and Board in dollars, Reputation is the index used in U.S. News and World Report (based on a survey of university presidents and chief academic officers), which ranges from 1 (“marginal”) to 5 (“distinguished”), Size is the number of undergraduate students, and Dpriv, Dlibart, and Dreligion are binary variables 5 indicating whether the institution is private, a liberal arts college, and has a religious affiliation. The numbers in parentheses are heteroskedasticity-robust standard errors. (a) Interpret the results. Do the coefficients have the expected sign? Indicate whether or not the coefficients are significantly different from zero. (a) What is the forecasted cost for a liberal arts college, which has no religious affiliation, a size of 2,000 students and a reputation level of 4.5? (All liberal arts colleges are private.) (b) To save money, you are willing to switch from a private university to a public university, which has a ranking of 0.5 less and 10,000 more students. What is the effect on your cost? Is it substantial? (c) Eliminating the Size and Dlibart variables from your regression, the estimation regression becomes Coˆst = 5,450.35 + 3,538.84 Reputation + 10,935.70 Dpriv – 2,783.31 Dreligion; (1,772.35) (590.49) (875.51) (1,180.57) R2=0.72, SER = 3,792.68 Why do you think that the effect of attending a private institution has increased now? 6