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E471/M504: Econometric Theory and Practice
Assignment 2 Due: Thursday 23rd September, 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. This question is a sequel to Question 1 in Assignment 1. Recall that you downloaded the file Assignment1data.csv, which contains three series: CITCRP, MARKET, and RKFREE with data from January 1978 to December 1987. Recall the regression rc,i − rf,i = β0 + β1(rm,i − rf,i) + ²i. (1) The capital asset pricing model (CAPM) suggests that β0 should be zero. Using the R chunks from our lectures, extend your program to: (a) formally test the null hypothesis H0 : β0 = 0 (in your answer, please state the null hypthesis, the alternative hypothesis, the values of βˆ0 and σˆβˆ0 , the value of the t statistic, the critical values for tests with significance levels 0.01, 0.05, and 0.1, and the outcome of the test, i.e., “reject” or “don’t reject”); (b) compute a p-value for the null hypothesis H0 : β0 = 0; (c) and to construct 90%, 95%, and 99% confidence intervals for β0 and β1. 2. This question is a sequel to Question 2 in Assignment 1. Consider the following regression model without intercept Yi = βXi + Ui, (2) where (Xi, Ui) are independent and identically distributed. In addition to the OLS estimator βˆM = 1 n ∑n i=1 XiYi 1 n ∑n i=1 X 2 i (3) consider the following alternative estimator for β: βˆM∗ = 1 n ∑n i=1 Yi 1 n ∑n i=1 Xi . (4) The goal is to compare the sampling properties of the two estimators. (a) Replace Yi by βXi + Ui and show that βˆ M ∗ can be written as βˆM∗ = β + 1 n ∑n i=1 Ui 1 n ∑n i=1 Xi . (b) Following our calculations in class (Slides 6), use the law of large numbers to show that the βˆM∗ is consistent, meaning that βˆ M ∗ p−→ β as the sample size n increases. (c) Again, following our calculations in class, use the central limit theorem to approxi- mate the distribution of βˆM∗ : βˆM∗ approx∼ N ( β, ? ) . (d) Is the new estimator βˆM∗ more or less precise than the OLS estimator?