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First Exam: Economics 388, Econometrics Winter 2008 in R. Butler’s class YOUR NAME:________________________________________ Section I (30 points) Questions 1-10 (3 points each) Section II (50 points) Questions 11-13 (15 points each); Question 14 (5 points) Section III (20 points) Question 15 (20 points) Section I. Define or explain the following terms (3 points each) 1. If Y ~ N ( , ) then for the square matrix of constants, A, AY ~ N (??,???) 2. formula for the F-statistic (based on SSR)- 3. type II error - 4. conditional probability density function of y given x- 5. plim or probability limit- 6. adjusted R-square - 7. unbiased estimator- 8. heteroskedasticity - 9. probability significance values (i.e., ‘p-values’)- 10. Var(w) where w is a nx1 vector of random variables- 1 II. Some Concepts 11. For the following Stata Output, indicate what a) the statistic is (formula) and b) what it indicates for the following circled statistics * 1. * 4. * 5. * 6. * 7. * 10. faminc 1988 family income, $1000s ; bwght birth weight, ounces ; fatheduc father's yrs of educ ; motheduc mother's yrs of educ ; parity birth order of child ; cigs cigs smked per day while preg ; Source | SS df MS Number of obs -------------+-----------------------------F( 5, 1185) Model | 18705.5567 5 3741.11135 Prob > F Residual | 464041.135 1185 391.595895 R-squared -------------+-----------------------------Adj R-squared Total | 482746.692 1190 405.669489 Root MSE = = = = = = 1191 9.55 0.0000 0.0387 0.0347 19.789 -----------------------------------------------------------------------------bwght | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------cigs | -.5959362 .1103479 -5.40 0.000 -.8124352 -.3794373 parity | 1.787603 .6594055 2.71 0.007 .4938709 3.081336 faminc | .0560414 .0365616 1.53 0.126 -.0156913 .1277742 motheduc | -.3704503 .3198551 -1.16 0.247 -.9979957 .2570951 fatheduc | .4723944 .2826433 1.67 0.095 -.0821426 1.026931 _cons | 114.5243 3.728453 30.72 0.000 107.2092 121.8394 -----------------------------------------------------------------------------. test (cigs=0) (parity=0); F( 2, 1185) = 17.95 Prob > F = 0.0000 I. II. III. IV. 2 12. Indicate whether the following statement is True, False or Uncertain and explain why. You are graded only on the explanation for your answer. “For the population linear regression model, y X , the sample predicted value of y and the residual are independently distributed (uncorrelated). That is, yˆ X̂ and ̂ are uncorrelated.” 13. For the simpliest regression (one slope variable, no intercept in the model), we have yi xi i , and the following picture for our particular sample, where length of the y-vector is 8 as indicated, and the length of the x vector is 5. If the angle between the x vector and the y-vector is 45 degrees, than a) what is the OLS estimate, ˆ , and b) what will be the residual sum of squares? (Warning, the picture is deliberately NOT drawn to scale, so do the math—one hindrance, one help: the three angles of a triangle sum to 180 degrees, and that the square root of 5 is 2.236) Y: length= 8 45 X: length=5 3 14. A regular dice (cube with, respectively, the numbers 1 through 6 on each side) may have been tampered with, so that 6 comes up on half the throws on average, and the numbers 1, 2, 3, and 4 each come up only one twelfth of the time. But you’re not sure if the die were really tampered with. a) Indicate what is the expected, average outcome (mean or average of numbers that appear in repeated trials) if the dice has been tampered with as described above? b) Indicate what is the expected, average outcome if the dice has NOT been tampered with, with each side equally likely to come up? c) Indicate what z-score (t-test) formula you would use to test to the hypothesis that the die had not been tampered with (the null hypothesis) if you had experimental outcomes (i.e., the actual means from 25 tosses of the die). (if you need to use a standard deviation for your test, use the standard deviation under the null hypothesis in your calculation). d) Is this a one tail or two tail test of these hypotheses? Draw a picture indicating where the critical region would lie for your test statistic. e) If I throw the die repeatedly (25 times), and got a sample average of 4.5, would I likely reject the null hypothesis at the 5 percent level (guess the best you can about statistical significance, drawing upon your extensive knowledge of the empirical rule for normal distributions)? 4 15. Under the model assumptions, prove that s2 is an unbiased estimator of 2 for the OLS regression model, using all the necessary assumptions employed in the proof in class or in the book, and proving it in the general case using matrix algebra. Do your proof carefully, spelling out fully all your steps. 5