VIEWS: 5 PAGES: 16 CATEGORY: Real Estate POSTED ON: 10/3/2010
Econometrics A Midterm Exam Date: Thursday, April 30, 2009 1. The exam is closed book and closed notes. 2. No calculators are allowed. 3. Do not discuss the exam with other students, in particular students in later sections. It is in your best interest to follow this rule, as helping them by telling them what is on the exam will hurt your own grade. 4. Do not take this exam with you. 5. Any students caught cheating will fail the course. The Dean of Students will be notiﬁed as well. 6. Good luck! 1 1. Let (X, Y ) be a random vector. (a) (12 points) Suppose that E[X|Y ] = 10, E[Y ] = 5, and Var[X|Y ] = 20 · Y . i. What is E[X]? Provide a proof of your answer. ii. What is E[X 2 |Y ]? Provide a proof of your answer. iii. What is Var[X]? Provide a proof of your answer. iv. What is Cov[X, Y ]? Provide a proof of your answer. v. Is X mean independent of Y ? vi. Is X uncorrelated with Y ? 2 (b) (8 points) Suppose further (i.e., in addition to the assumptions made in part (a) above) that E[Y 2 |X] = 30, Var[Y |X] = 5, and Y ≤ 0 always. i. What is E[Y |X]? Provide a proof of your answer. ii. What is Var[Y ]? Provide a proof of your answer. iii. Is Y mean independent of X? iv. Do you have enough information to determine whether X is independent of Y ? Explain brieﬂy. 3 2. Let (H1 , W1 ), . . . , (Hn , Wn ) be an i.i.d. sample for (H, W ), where (H, W ) is the height (in meters) and weight (in kilograms) of an individual in the population. Suppose H ≤ 10 and W ≥ 1 always. (a) (2 points) Is it reasonable to assume that either H or W is normally dis- tributed? Explain brieﬂy. (b) (8 points) The Body Mass Index (BM I) of a person with height H (in meters) and weight W (in kilograms) is deﬁned to be H BM I = . W2 Propose a consistent and unbiased estimator for the expected value of BM I. Justify your answer by proving both statements. 4 (c) People with a BM I of less than 24 are considered healthy. You wish to test the null hypothesis that the expected value of BM I is less than or equal to 24 at signiﬁcance level α versus the alternative that it is greater than 24. i. (2 points) Formally state the null and alternative hypotheses. ii. (6 points) How would you perform your test? In particular, write down your test statistic, your critical value, and the rule you would use to determine whether or not to reject the null hypothesis. iii. (4 points) State in words the deﬁnition of a p-value. What is the p-value for your test? 5 3. Suppose Y = β0 + β1 X + U , where X is a binary random variable and 0 < P {X = 1} < 1. (a) Suppose, for the time being, that you, the researcher, wish to interpret this regression as a model of the best linear predictor of Y given X. i. (3 points) Interpret U and β1 . Be as precise as possible. ii. (3 points) Is U uncorrelated with X? Explain brieﬂy. iii. (3 points) Express β0 and β1 in terms of conditional expectations. 6 iv. (3 points) What is E[Y |X]? Explain brieﬂy. v. (6 points) Let (X1 , Y1 ), . . . , (Yn , Xn ) be an i.i.d. sample from (Y, X). You estimate the above equation using OLS. Suppose E[Y 4 ] < ∞. Is ˆ ˆ β1 unbiased for β1 ? Is β1 consistent for β1 ? If not, express its limit in probability in terms of β1 and features of the joint distribution of X and U . Explain brieﬂy. 7 (b) Now suppose that you wish to interpret this regression as a causal model of the eﬀect of X on Y . i. (3 points) Interpret U and β1 . Be as precise as possible. ii. (3 points) Is U uncorrelated with X? Explain brieﬂy. 8 iii. (3 points) Is β1 equal to your answer to part iii. of part (a) above? Explain brieﬂy. iv. (6 points) Let (X1 , Y1 ), . . . , (Yn , Xn ) be an i.i.d. sample from (Y, X). You estimate the above equation using OLS. Suppose E[Y 4 ] < ∞. Is ˆ ˆ β1 unbiased for β1 ? Is β1 consistent for β1 ? If not, express its limit in probability in terms of β1 and features of the joint distribution of X and U . How do your answers compare with your answers to part v. of part (a) above? Explain brieﬂy. 9 4. Let log(Y ) = β0 + β1 X + U , where Y = yearly salary (measured in dollars) X = experience (measured in years) . You, the researcher, wish to interpret this regression as a causal model of the eﬀect of X on Y . (a) (2 points) How would you interpret β1 ? (b) (2 points) How would you interpret U ? Do you believe U is likely to be correlated with X? Explain brieﬂy. (c) (3 points) Is it sensible to assume that E[U ] = 0? Explain brieﬂy. 10 (d) Let (Y1 , X1 ), . . . , (Yn , Xn ) be an i.i.d. sample from (Y, X). Suppose 0 < Var[X] < ∞, E[X 4 ] < ∞ and E[(log(Y ))4 ] < ∞. Suppose further that X and U are uncorrelated and that E[U ] = 0. You estimate this model and ﬁnd that ˆ ˆ β1 = .3 and s.e.(β1 ) = .1 . You also ﬁnd that the R2 = .09. i. (6 points) Suppose you had instead measured experience in months in- ˆ ˆ stead of years. How would β1 , s.e.(β1 ) and R2 change? ii. (4 points) A fellow researcher is concerned about the causal interpreta- tion of β1 because the R2 is so low. How do you respond? 11 iii. (4 points) Construct a (two-sided) conﬁdence interval for β1 at the 5% signiﬁcance level. (Hint: If Z ∼ N (0, 1), then P {Z ≤ 1.96} = .975.) iv. (4 points) What coverage property does the interval in part (iv) satisfy? How would you use this interval to test the null hypothesis that β1 = 1 versus the alternative that β1 = 1 at the 5% signiﬁcance level? 12 (scratch work) 13 (scratch work) 14 (scratch work) 15 (scratch work) 16