E471/M504: Econometric Theory and Practice I Fall 2021 Assignment 3 Due: Thursday 7th October, 2021, 3:15 pm Instructions: • Please upload an electronic copy of your answers to Canvas combining all results in one pdf/word file. If there is any handwritten part, please scan it and include it with the rest of the answers. • Please also upload your code to Canvas before the due time. The code accounts for 50% of the points of the empirical questions. • You are allowed to collaborate in groups, but required to write up answers and code independently. Direct copying will be treated as cheating. Please write the names of your collaborators at the beginning of your work, if any. • If you cannot make it to the discussion session, please let me know at least one day before the due time, and you will be randomly assigned to present your answer to some questions one day before the due time and upload your presentation video to the Discussion board on Canvas before the due time (for the “Presentation” part of the course requirement). Questions: 1. Suppose that E[U |X] = 0, E[X2] = μXX , and V[U |X] = σ2. We used the following two results in our derivations for the OLS estimator: (a) Show that E[UX ] = 0. (b) Show that V[UX ] = E[U2X2] = σ2μXX 2. Consider the following regression model Yi = βXi + Ui, β = 1, E[X2i ] = 1 4 , V(Ui|Xi) = 1 and the sample size is n = 100. We are interested in testing the null hypothesis that β = βH versus the alternative that β 6= βH using a t-statistic of the form t = βˆ − βH σˆβˆ . (a) What is the correct critical value to guarantee that the hypothesis test has a type-I error of 5%? (b) Define the type-II error of a hypothesis test. (c) Suppose an econometrician tests the “false” hypothesis that β = βH = 1.2 (the “truth” is that β = 1). What is the type-II error associated with this test? (d) Repeat the calculation for βH = 1.6, βH = 1.8, and βH = 2. (e) What is the power of a test? (f) True or False: The t-test has less power against alternative hypothesis that are far away from the null hypothesis. Hint: use your previous results to answer this question. Page 2 3. You are working for a life insurance company and are preparing for a briefing of the board. Your economic intuition tells you that the best predictor of life insurance holdings is income. You gather the relevant data (family life insurance and family income, both in thousands of dollars) and want to analyze it, running a regression lifeinsi = β0 + β1incomei + ui The data set is provided in the file Assignment3Data.csv. (a) Estimate the above regression model. What is your point estimate for β1? What is the interpretation of that estimate? (b) Provide a 95% confidence interval for the coefficient on income. (c) One of the managers suggests that the industry rule of thumb is that people buy five dollars life insurance for each additional dollar of their income. Another manager disagrees and says it could be more or less. You want to examine the difference of opinion. What null and alternative hypotheses would you use here to discriminate between these hypotheses? Test the hypothesis, using a 5% type I error and interpret the results. (d) Calculate a p-value for the test in (c). (e) The company wants to offer life insurance to low income households. The chairman asks you how much life insurance would a household with an income of 20,000 dollars buy. Calculate a point prediction. (f) Explain why it is better to consider an interval prediction rather than a point prediction. (g) Calculate 90%, 95%, and 99% prediction intervals for the life insurance holding of a family that earns 20,000 dollars. Page 3