1. A sample of 40 observations provide the following statistics: sx = 100 sy= 100 and Sxy = 2500 a. What is the sample covariance? What Information does this give us about how closely x and y are correlated? b. What is the sample correlation coefficient, r? c. Interpret the sign of the sample correlation coefficient d. Interpret the magnitude of the sample correlation coefficient 2. Now suppose instead you were given sx = 50 sy= 50 and Sxy = 2500. (All other information is same as question 1) a. What is the sample correlation coefficient now? b. Interpret the sign of the sample correlation coefficient c. Interpret the magnitude of the sample correlation coefficient. 3. For the problem let Y i be output by a firm in an oligopolistic competitive sector (so supply is upward sloping) and X i be the profit maximizing price for firm i. xx y Yi Observation Yi Xi x 1 5 15 2 15 9 3 25 12 a. Fill out the table above b. Find 1 and 0. c. Present your findings in the form: Y i = ^ 0 + ^ 1 X i d. What is the distance between the sample regression line and the observed data points called? Calculate this for each observation. Also calculate the sum. 4. In their classic growth paper, Mankiw Romer Weil find that a regression which contains national investment, population growth, depreciation, technological growth, and years of schooling on (the dependent variable) income per working age person in 1985 (GDP per capita, 1985) has an adjusted R2 of .78 for non-oil exporting countries and .24 for OECD (developed) countries. What is presented below is a simplification of their results: Sample Non-oil OECD N 98 22 Investment .69(.13) .28 (.39) Combined effect of -1.73 (.41) -1.07 (.75) population growth, technology growth, and depreciation Years of schooling .66 (.07) .76 (.29) R .78 .24 R 2 S.e.e .51 .33 a. According to the table above, which set of countries does the model fit better? Why? b. Which of the variables are statistically different than zero at the 95% confidence level? c. What is the interpretation of the value of the adjusted R2 (be specific), including which variables it takes into account? 5. Suppose you have the following estimated model with a sample size of 27: Y i = 5 + 5.7 X i and thestandard error 1 for (se1) is 1. a. Interpret the coefficient of ^ 1 b. Is ^ 1 statistically different from 0 at the 90% level? c. Is ^ 1 statistically different from 10 at the 90% level? d. Is ^ 1 statistically different from 2.5 at the 90% level? e. Is ^ 1 statistically different from 5 at the 99% level? f. Give the 99% confidence interval for ^ 1 g. Give the 95% confidence interval if our estimate for ^ 1was the same, but we had sampled 1000 people 6. Suppose we observe a plot of residuals squared on the y axis and values of x on the horizontal axis. We then run a regression on this data (i.e. we run the model I 2 = 0 + 1 X i ). Suppose the estimated value for ^ 1 is 50 with a standard error of 2. Doing a t-test against the null of ^ 1 =0 would give a t-score of 25, which is significant at any reasonable level. a. What does this test tell us? b. What does this mean about the estimates of the original model (the regression of y on x?) Are the estimates unbiased? Are the estimated efficient? c. How would you correct this problem? 7. Larry thinks he is pretty smart. He recently ran a regression of the form: feeti = 0 + inches1 + 2weight + 3gender + 4age + i Johnny got an R2 of 1! Wow! Oddly, he finds that only 1 is statistically significant, and nothing else is. a. What is the interpretation of R2 = 1 b. Is Larry actually really smart? If not, what mistake has he made? c. How can Johnny fix the mistake he made (assuming he made one)