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First Exam: Economics 388, Econometrics Spring 2008 in R. Butler’s class YOUR NAME:________________________________________ Section I (30 points) Questions 1-10 (3 points each) Section II (40 points) Questions 11-14 (10 points each) Section III (30 points) Question 15 (30 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. show that N N i 1 i 1 ( yi y )( xi x ) ( yi y ) xi - 3. type II error - 4. conditional probability density function of y given x- 5. omitted variable bias (say left out Z in a regression of Y on X)- 6. adjusted R-square - 7. unbiased estimator- 8. homoskedasticity - 9. orthogonal projection- 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. faminc 1988 family income, $1000s ; * 4. bwght birth weight, ounces ; * 5. fatheduc father's yrs of educ ; * 6. motheduc mother's yrs of educ ; * 7. parity birth order of child ; * 10. cigs cigs smked per day while preg ; . reg bwght cigs parity faminc motheduc 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 ------------------------------------------------------------------------------ This output above is right (all correct), but now we modify a variable with STATA code (see just below) and the output below contains some errors (has been changed by Coach from what originally printed); . gen bwght2 = bwght + 2; . regress bwght2 cigs parity faminc motheduc fatheduc; Source | SS df MS -------------+-----------------------------Model | 18705.5567 5 3741.11135 Residual | 464041.135 1185 391.595895 -------------+-----------------------------Total | 482746.692 1190 405.669489 Number of obs F( 5, 1185) Prob > F R-squared Adj R-squared Root MSE = = = = = = 1191 9.55 0.0000 0.0514 0.0501 19.789 -----------------------------------------------------------------------------bwght2 | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------cigs | -2.595936 .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 31.25 0.000 109.2092 123.8394 ------------------------------------------------------------------------------ List all the errors in the output printed just above: 2 12. You are told that X is normally distributed with mean of 8 and variance of 4, and Y is normally distributed with mean of 0 and variance of 16. X and Y are independently distributed. The what are distributions for the following variables: a. W1 = 3X + 2 b. W2 = X + 4Y c. W3 = [(X-8)/2]2 + [(Y-0)/4]2 d. W4 = [( X 8) / 2]2 [(Y 0]/ 4]2 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. My son has a pyramid dice, with four sides numbered from 1 to 4. Let W be the random variable corresponding to number that's on the bottom side when the dice is rolled. If the dice is not fair, but the probability that the sides with numbers 1, 2 or 3 will occur is one sixth (for each of these events taken separately) then a. What is the expected value of the random variable W and what is the variance of W? b. If we did not know whether the dice were fair or not (i.e., that each outcome was equally probable), and decided to test for that by repeatedly (25 times) throwing the die, and got a sample average of 2.0, 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 the least squares estimator is BLUE (that is, prove the Gauss Markov theorem). 5