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EC450 Homework Assignment 2 Due Date: Wednesday, February, 12, 2019 Instructions: This homework is due at the beginning of class on Wednesday, February, 12, 2020. Late submissions will NOT be accepted. This homework assignment is required and will be graded. Only the hard copy of homework submitted at the start of class on the due date will be accepted. You must show all formulas that you use and how you use them. Please show all your work in order to receive any credit. You are allowed to have discussions with your classmates. You must write down your own answer independently. If I have reason to believe that you have copied other classmate's answer, all parties involved will receive zeros. Work problems neatly and in a logical, organized manner. If you are handing in multiple pages, please staple the pages together. Problem 1. (Total: 35 points) Let Y1, Y2 and Y3 be independent, identically distributed random variables from a population with mean µ = 12 and variance σ2 = 192. Let Ȳ = 13 (Y1 + Y2 + Y3) denote the average of these three random variables. 1. (5 points) What is the expected value of Ȳ , i.e., E(Ȳ ) = ? Is Ȳ an unbiased estimator of µ? (Hint: E(aU + bV + cW ) = a · E(U) + b · E(V ) + c · E(W ). ) 2. (5 points) What is the variance of Ȳ , i.e, V ar(Ȳ ) =? (Hint: if U , V and W are independent random variables, then V ar(aU + bV + cW ) = a2V ar(U) + b2V ar(V ) + c2V ar(W ). ) 1 EC450 Applied Econometrics Spring 2020 3. (5 points) Consider a di�erent estimator of µ: W = 1 8 Y1 + 3 8 Y2 + 1 2 Y3 This is an example of a weighted average of the Yi. What is the expected value ofW , i.e., E(W ) =? Is W an unbiased estimator of µ? (Use the hint in question 1.) 4. (5 points) Find the variance of W , i.e., V ar(W ) =? (Use the hint in question 2.) 5. (5 points) Consider another estimator of µ: V = 1Y1 + 0Y2 + 0Y3 This estimator only uses Y1 and completely ignores Y2 and Y3. What is the expected value of V , i.e., E(V ) =? Is V also an unbiased estimator of µ ? 6. (5 points) Find the variance of V , i.e., V ar(V ) =? 7. (5 points) Based on your answers in previous parts, which estimator of µ do you prefer? Please explain. Problem 2. (Total: 35 points) A mortgage specialist would like to analyze the average mortgage rates for Atlanta, George. He collects data on the annual percentage rates (APR in %) for 30-year �xed loans as shown in the following table. He is willing to assume that these rates are randomly drawn from a normally distributed population. Financial Institution APR S Squared Financial 4.125 Best Possible Mortgage 4.250 Hersh Financial Group 4.250 Total Mortgage Services 4.375 Wells Fargo 4.375 Quicken Loans 4.500 Amerisave 4.750 2 EC450 Applied Econometrics Spring 2020 1. (5 points) What is the average APR among these 7 �nancial institutions? 2. (5 points) What is the standard deviation of the APR among these 7 �nancial institutions? (Please round your answer to 4 decimal places.) 3. (5 points) Let µ be population mean of mortgage rates of all �nancial institutions. State two competing hypotheses to determine whether the population mean mortgage rate exceeds 4.2 (%). 4. (5 points) Calculate the value of test statistic. (Please round your answer to 4 decimal places.) 5. (5 points) Find the p-value in this test. (Please round your answer to 4 decimal places.You can either approximate the p-value using t-table, or �nd the exact p-value using Excel or any software.) 6. (5 points) Construct a 95% con�dence interval on the population mean mortgage rate. (Please round your answer to 4 decimal places.) 7. (5 points) Can you conclude that the population mean mortgage rate exceeds 4.2% at the 10% level of signi�cance ? Please explain. Problem 3. (Total: 30 points) The Citizen Bank at Tremont St, Boston, employs two appraisers. When approving borrowers for mortgages, it is imperative that the appraisers value the same types of properties consistently. To make sure that this is the case, the bank examines six properties (in $1,000) that the appraisers had valued recently. Property Value from Appraiser 1 Value from Appraiser 2 1 235 239 2 195 190 3 264 271 4 315 310 5 435 437 6 515 525 1. (5 points) Let µD = µ1 −µ2, where µ1 is the mean value of properties from Appraiser 1, and µ2 is the mean value of properties from Appraiser 2. Specify the competing hypotheses that determine whether there is any di�erence between the values estimated by Appraiser 1 and Appraiser 2. 2. (10 points) Assuming the value di�erence is normally distributed, calculate the value of the test statistic. (Please round your answer to 4 decimal places.) 3 EC450 Applied Econometrics Spring 2020 3. (5 points) Find the p-value in this test. (Please round your answer to 4 decimal places. You can either approximate the p-value using t-table, or �nd the exact p-value using Excel or any software.) 4. (5 points) At the 5% signi�cance level, is there su�cient evidence to conclude that the appraisers are inconsistent in their estimates? Please explain. 5. (5 points) Construct a 95% con�dence interval of the mean di�erence between values estimated by Appraiser 1 and Appraiser 2, i.e., 95% con�dence interval of µD. (Please round your answer to 4 decimal places.) 4