University Of ChicagoECONOMICS 21020Problem Set 2 Partial Solutions.

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ECON 210: Problem Set 2 Simple Linear Regression Due April 18th, 2018 at the beginning of class 1 (30 points) Stock and Watson, Exercises 4.2 (10 points) and 5.2 (20 points). Skip the parts about ... SER in 4.2c and 5.2e. 2 (20 points) Consider the regression Y = α + βX + . Suppose you know that α = 0 and have iid draws (X1; Y1); : : : ; (XN; YN) from (X; Y ). 1. Write down a formula for β in terms of population moments. Also, derive the least squares estimator for β by solving the sample minimization problem min β NXi =1 2 i 2. Prove that s2 = 1 N−1 PN i=1 ^i2 is a consistent estimator for V ar() 3. How would you test H0 : β = 0 vs. HA : β 6= 0 at the 5% level (give formulas for the test statistic and all its components)? 4. What would the least squares estimator for β be if you were instead sure that α = 7? Solution: 1. β = cov(X;Y ) var(X) is fine. Should solve for β^ = PPXXiYi2i from the minimization 2. Full credit only for noting that N1−1 = N1 NN−1, that the second term approaches 1, and so the proof from class works with one additional CMT 3. As in class. Full credit only if an estimator for ^ σβ is given 4. β^ = P XPi(XYii2−7) 3 (25 points) Consider the regression R = α + βMkt +  where R is the monthly return on a share of Microsoft and Mkt is the monthly return on the S&P 500 minus the risk free rate 1. Assume that the covariance of R and Mkt is positive. What is the expected sign of β and why? Also, what is the interpretation of β (warning for those who know what the CAPM is: this regression is not the [Show More]

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