WISE International Masters

WISE International Masters ECONOMETRICS Instructor: Brett Graham INSTRUCTIONS TO STUDENTS 1 The time allowed for this examination paper is 2 hours. 2 This examination paper contains 32 questions. You are REQUIRED to answer ALL questions. No marks will be deducted for wrong answers. 3 For each multiple choice question there is one and ONLY ONE suitable answer. 4 All numerical answers should be rounded to 3 decimal places. I will accept every answer to within 0.002 of the correct answer. All probabilities should be expressed in decimal form. 5 This examination paper contains 18 pages including this instruction sheet, an answer sheet for the first 30 questions, an answer sheet for question 31, an answer sheet for question 32 and a blank page at the end of the exam. 6 This is a closed-book examination. You are allowed to bring one handwritten 105mm by 75mm piece of paper to the exam. You are also allowed to use a financial calculator. 7 You are required to return all examination materials at the end of the examination. 8 Where required, please use the following critical values, Tail-end Probabilities of the Normal Distribution z 0.994 1.405 1.751 2.054 2.326 2.576 Pr(Z z) (%) 16 8 4 2 1 0.5 5% Significance Level Critical Values of the χ 2 m Distribution Degrees of Freedom (m) 1 2 3 4 5 6 7 8 9 Critical Value 3.84 5.99 7.81 9.49 11.07 12.59 14.07 15.51 16.92

Use the following information to answer the next 3 questions (3-??): Let X be a discrete random variable with the following probability distribution. x Pr(X = x) 1 0.2 2 0.3 3 0.3 4 0.2 1. What is the value of the cumulative distribution function of X when x = 2? Thus, F X (2) = 0.5 0 if x < 1 0.2 if 1 x < 2 F(x) = 0.5 if 2 x < 3 0.8 if 3 x < 4 1 if 4 x 2. What is the mean of X. E(X) = 2.5. 3. What is the variance of X. var(x) = 1.05. Page 2

Use the following information to answer the next 2 questions (4-5): Let X and Z be independently distributed random variables where X N( 8, 8) and Z N(5, 9), and let Y = 4X 2 4Z 3. 4. What is the E(Y )? E(X 2 ) = var(x) + [E(X)] 2 = 8 + [ 8] 2 = 72 E(Z 2 ) = var(z) + [E(Z)] 2 = 9 + [5] 2 = 34 0 = E[(Z µ Z ) 3 ] = E[Z 3 ] 3µ Z E[Z 2 ] + 3µ 2 ZE[Z] µ 3 Z = E[Z 3 ] 3µ Z (var(z) + µ 2 Z) + 2µ 3 Z = E[Z 3 ] 3µ Z var(z) µ 3 Z E[Z 3 ] = 3µ Z var(z) + µ 3 Z = 260 E(Y ) = E(4X 2 4Z 3 ) = 4E(X 2 ) 4E(Z 3 ) = 288 (1040) = 752 5. What is the E(Y X = 6)? E(Y X = 6) = E(4 6 2 4Z 3 X = 6) = 4 36 4E(Z 3 ) = 896. Page 3

6. The random variables Y i, i = 1,..., n are i.i.d. and each has a Bernoulli distribution with p = 0.6. Let Ȳ denote the sample mean. The sample size is n = 200. Using the Central Limit Theorem, find the value that exactly 2% of sample means are larger than, i.e. find Ȳ 0.02 where Pr(Ȳ > Ȳ0.02) = 0.02. Using the CLT, ( ) Ȳ p Pr > 2.054 = 0.02 Pr(Ȳ > Ȳ0.02) = 0.02. p(1 p)/n where Ȳ0.02 = p + 2.054 p(1 p)/n = 0.671. 7. A pizza delivery store is considering offering customers a discount on any delivery that takes more than half an hour. They will only offer the discount if less than 10% of current deliveries take more than half an hour. From a random sample of 293 current deliveries, 27 take more than half an hour. What is the value of the test statistic? This is a question about population proportion. The null hypothesis is that p = 0.1. ˆp p 0.092 0.1 z = = = 0.448. p(1 p)/n 0.1(0.9)/293 Page 4

Use the following information to answer the next 5 questions (8-12): The weekly spending habits of 620 randomly chosen males and 620 randomly chosen females is recorded. Let µ m denote the male population average of weekly spending and µ w denote the female population average of weekly spending. Let X m and X w denote their sample counterparts. Let σ m denote the male population standard deviation of weekly spending and σ w denote the female population standard deviation of weekly spending. Let s m and s w denote their sample counterparts. In the survey X m = 54, X w = 54.7, s m = 12.9, s w = 11.7. 8. You are interested in the competing hypotheses: H 0 : µ m µ w = 1 vs. H 1 : µ m µ w 1. Suppose that you decide to reject H 0 if X m X w 1 > 1. In what region does the size of this test lie if σ m = σ w = 12? a. (0, 0.01) b. (0.01, 0.02) c. (0.02, 0.04) d. (0.04, 0.08) e. (0.08, 1) The size of the test is the probability of rejecting the null when the null is true. ( ) Pr( X m X 1 w 1 > 1) µ m µ w = 1) = Pr z > 122 /620 + 12 2 /620 = Pr( z > 1.467) Since z 0.04 = 1.751 > 1.467 > 1.405 = z 0.08, so 0.08 < α < 0.16. 9. Using the sample information, what is the test statistic associated with H 0 : µ m µ w = 1 vs. H 1 : µ m µ w 1. t = X m X w (µ m µ w ) s 2 m /n m + s 2 w/n w = 54 54.7 1 12.92 /620 + 11.7 2 /620 = 2.431. 2.431 is also an acceptable answer. Page 5

10. Calculate the lower confidence limit of a confidence interval for µ m µ w with 98% coverage probability. Using the critical value z 0.01 = 2.326 LCL = 54 54.7 z 0.01 12.92 /620 + 11.7 2 /620 = 2.327. 0.927 is also an acceptable answer. 11. Suppose that the survey is carried out 6 times, using independently selected people in each sample. For each of these 6 surveys, a 92% confidence interval for µ m µ w is constructed. What is the probability that the true value of µ m µ w is contained in all 6 of these confidence intervals? Pr(µ m µ w ) [LCL, UCL] = 0.92 6 = 0.606. 12. Suppose that the survey is carried out 6 times, using independently selected people in each sample. For each of these surveys, a 92 % confidence interval for µ m µ w is constructed. How many of these confidence intervals do you expect to contain the true value of µ m µ w? 5.52 Page 6

Use the following information to answer the next 5 questions (13-17): Suppose that a random sample of 300 twenty-year-old men is selected from a population and that these men s height and weight are recorded. A regression of weight on height yields W eight = -14.823 + 49.860 Height, R 2 = 0.416, SER = 5.892, (6.031) (3.434) where W eight is measured in kilograms and Height is measured in meters. 13. John, who is 1.85 meters tall and weighs 66 kilograms is one of the randomly sampled men. What is the regression s weight prediction for John? W eight (Height=1.85) = 14.823 + 49.86(1.85) = 77.418 kilograms. 14. What is the residual associated with John? û John = 66 77.418 = 11.418 kilograms. 15. A man has a late growth spurt and grows 2.76 centimeters over the course of a year. What is the regression s prediction for the increase in this man s weight? W eight ( Height=0.0276) = 49.86(0.0276) = 1.376 kilograms. Page 7

16. Suppose that instead of measuring weight and height in kilograms and meters, these variables are measured in grams and centimeters. What is the regression estimate of the coefficient on Height from this new gram-centimeter regression? The units of the coefficient ˆβ 1 is kilograms per meter. Note that 1 m = 100 cm and 1 kg = 1, 000 g. Hence, if weight was measured in grams and height in centimeters then the units of the coefficient ˆβ 1 would be grams per centimeter and ˆβ 1 = 49.86(1000/100) = 498.6. 17. Suppose that instead of measuring weight and height in kilograms and meters, these variables are measured in grams and centimeters. What is the regression SER from this new gramcentimeter regression? The units of SER is kilograms. Note that 1 m = 100 cm and 1 kg = 1, 000 g. Hence, if weight was measured in grams and height in centimeters then the units of SER would be grams and SER = 5.892(1000) = 5892. 18. For the simple regression model Y i = β 0 + β 1 X i + u i you have been given the following data: 600 600 600 Y i = 1286.2; X i = 2471.5; 600 600 X i Y i = 5744.1; Xi 2 = 11018.5; Calculate the regression slope and the intercept. Y 2 i = 63319.3. X = 2471.5/600 = 4.12 and Ȳ = 1286.2/600 = 2.14. ˆβ 1 = 600 X iy i n XȲ 600 X2 i n X = 2 5744.1 600 4.12 2.14 11018.5 600 4.12 2 = 0.532. ˆβ 0 = 2.14 ( 0.532 4.12) = 0.048 Page 8

19. You believe that the time since an artist s death has a large impact on the price of their paintings. Using a random sample of oil painting sales (in thousands of dollars) and time from the artist s death to the sale of the painting (in years) you generate the following partial regression output. Coefficient Standard Error Intercept 2 0.2 T ime 11.8 1.9 You want to test the hypothesis that an additional ten years from the time of death to the sale of a painting will increase the price of the painting by at least $100,000. What is the value of the test statistic? The null is H 0 : 10β 1 = 100, which is equivalent to H 0 : β 1 = 10. t = ˆβ 1 β 1,0 S.E.( ˆβ 1 ) = 11.8 10 1.9 = 0.947. Use the following information to answer the next 3 questions (20-22): A random sample contains n R = 120 individuals who live in a rural area and n U = 180 individuals who live in an urban area. The sample mean of years of education of those individuals from a rural area (ȲR = 1 nr n R Y R,i ) is 8.6 years, and the sample standard deviation of individuals from a rural area, s R, is 1.8 years. The corresponding values for those individuals from an urban area are ȲU = 11 years and s U = 4.3 years. Let Urban denote an indicator variable that is equal to 1 for individuals from an urban area and 0 otherwise, and suppose that all 300 observations are used to estimate the regression line Ŷ = β 0 + β 1 Urban. 20. What is the OLS estimate of β 1? ˆβ1 = ȲU ȲR = 2.4. 21. What is the standard error of the OLS estimate of β 1? S.E( ˆβ 1 ) = S.E(ȲU ȲR) = s 2 U + s2 R 1.8 2 = n U n R 120 + 4.32 180 = 0.36 years. Page 9

22. What is the value of the test statistic to test if individuals from rural and urban areas have different levels of education? t = 2.4/0.36 = 6.664 23. When a variable, which is a determinant of the dependent variable, is omitted from a linear regression model then a. the error term is heteroskedastic. b. the error term is homoskedastic. c. the OLS estimator of the coefficient of the variable of interest is biased if the omitted variable is correlated with the variable of interest. d. this has no effect on the estimator of the coefficient of the variable of interest because the omitted variable is excluded. e. this will always bias the OLS estimator of the coefficient of the variable of interest. 24. In the multiple regression model Y i = β 0 + β 1 X 1i + β 2 X 2i +... + β k X ki + u i, i = 1,..., n, the OLS estimators are obtained by minimizing a. n (Y i b 0 b 1 X 1i... b k X ki ). b. n Y i b 0 b 1 X 1i... b k X ki. c. n (Y i b 0 b 1 X 1i... b k X ki ) 2. d. n (Y i b 0 b 1 X 1i... b k X ki u i ) 2. e. n (Y i b 0 b 1 X 1i ). 25. In the multiple regression model, the adjusted R 2, R 2 a. cannot be negative. b. will never be greater than the regression R 2. c. equals the square of the correlation coefficient r. d. cannot decrease when an additional explanatory variable is added. Page 10

26. Perfect multicollinearity in the multiple regression model a. is normal as many economic variables are perfectly correlated. b. implies that the OLS estimators are no longer BLUE. c. implies that the OLS estimators cannot be computed. d. implies that the OLS estimators are normally distributed. e. implies that the OLS estimators are unbiased. 27. You regress Y on X 1 and X 2. To test the joint hypothesis that β 1 = β 2 = 0, you reject the null if either t 1 > 2.576 or t 2 > 2.576 (or both), where t 1 = ˆβ 1 S.E.( ˆβ and t 1 ) 2 = ˆβ 2 S.E.( ˆβ. 2 ) Assuming that ˆβ 1 and ˆβ 2 are independent, what is the significance level of this test? 0.01 0.01 + 0.01 (1 0.01) + (1 0.01) 0.01 = 0.02. 28. For a hypothesis test with a single restriction (q = 1), against a two-tailed alternative hypothesis, the F -statistic a. has a critical value of 1.96. b. has a critical value of 3. c. is the square of the t-statistic. d. is the square root of the t-statistic. e. will be negative. Page 11

Use the following information to answer the next 2 questions (29-30): Consider the following regression model: Y i = β 0 + β 1 X 1i + β 2 X 2i + u i, i = 1..., n. Suppose a sample of n = 90 households has the sample means and sample covariances below for a dependent variable, Y, and two regressors, X 1 and X 2 : Sample Covariances Sample Means Y X 1 X 2 Y 6 2 7 7 X 1 8 6-5.2 X 2 5 9 29. What are the dimensions of ( X T X ), i.e., how many rows and how many columns does this matrix have? X has 90 rows and 3 columns. X T has 3 columns and 90 rows. Thus, ( X T X ) has 3 rows and 3 columns. 30. What is the value of ( X T X ), i.e., the value of the first row and first column of ( X T X ). 1,1 X T X 1,1 = n 1 = n. Page 12

Long Answers 31. Consider the sample mean Ȳ from a sample of i.i.d. observations drawn from a distribution (5) with a finite fourth moment. What is E(Ȳ 2 )? Recall that var(y ) = E(Y 2 ) E(Y ) 2. Hence, E(Y 2 ) = σy 2 + µ2 Y. Also, since Y i and Y j are independent E(Y i Y j ) = E(Y i )E(Y j ) = µ 2 Y. Hence, E(Ȳ 2 ) = E ( 1 n ) 2 n Y i = 1 n E(Y n 2 i ) 2 + 1 n 2 n = 1 n (σ2 Y + µ 2 Y ) + n 1 n µ2 Y = 1 n σ2 Y + µ 2 Y. n E(Y i Y j ) Alternatively, use the fact that for any random variable X, E(X 2 ) = var(x) + (E(X)) 2. In this particular case the random variable is Ȳ : j i E(Ȳ 2 ) = var(ȳ ) + (E(Ȳ )2 = σ2 Y n + µ2 Y. where the second equality uses the fact that the observations are i.i.d. Page 13

32. Show that (5) n (Y i Ȳ )(X i X) n n (X i X) = (Y ix i ) nȳ X 2 n (X2 i ) n X. 2 n (Y i Ȳ )(X i X) n n (X i X) = (Y ix i Y i X Ȳ X i + Ȳ X) 2 n (X2 i 2X X i + X 2 ) n = (Y ix i ) X n (Y i) Ȳ n (X i) + n (Ȳ X) n (X2 i ) 2 X n (X i) + n ( X 2 ) n = (Y ix i ) n X 1 n n (Y i) nȳ 1 n n (X i) + nȳ X n (X2 i ) 2n X 1 n n (X i) + n X 2 = = n (Y ix i ) n XȲ nȳ X + nȳ X n (X2 i ) 2n X X + n X 2 n (Y ix i ) nȳ X n (X2 i ) n X 2 Page 14

Answer Sheet Econometrics Mid-term Exam Name: Question Points Answer 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 1 10 1 11 1 12 1 13 1 14 1 Question Points Answer 18 1 19 1 20 1 21 1 22 1 23 1 24 1 25 1 26 1 27 1 28 1 29 1 30 1 Total: 13 15 1 16 1 17 1 Total: 17

Answer Sheet Econometrics Mid-term Exam Name: Question 31 (5 points) Page 16

Answer Sheet Econometrics Mid-term Exam Name: Question 32 (5 points) Page 17