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Unformatted text preview: ECON 4261: Introduction to Econometrics Fall 2009 Problem Set 2 – Answer Key Exercise 1 Suppose you model a variable Y depending on a nonstochastic variable X , according to the relation Y i = βX i + u i . The errors are iid with mean zero and variance σ 2 . Considering estimating β by the slope of a line between the origin and one out of the n plotted observations ( X i ,Y i ) . (a) Is this estimator unbiased? Yes. Note that in our case b β = slope between (0 , 0) and ( X i ,Y i ) = Y i X i . Then E ( b β ) = E Y i X i = E βX i + u i X i = E βX i X i + E u i X i = E ( β )  {z } β + 1 X i E ( u i )  {z } =0 = β. (b) Find the variance of this estimator. Again, by construction var ( b β ) = var Y i X i = var βX i + u i X i = var β + u i X i = var u i X i = 1 X 2 i var ( u i ) = σ 2 X 2 i . 1 (c) Which of the observations would you choose to calculate the estimate? Justify your answer. Looking at the answer in (b), we would pick the observation with the greatest X i , since this would minimize the variance of the estimator. Exercise 2 You are given 30 pairs of observations ( X i ,Y i ) which are to be represented by the following model Y i = β + β 1 X i + u i , and the errors are iid with mean zero and variance σ 2 . Suppose you know that R 2 = 0 . 25, ∑ n i =1 ( X i X ) 2 = 49 and ∑ n i =1 ( Y i Y ) 2 = 49. Calculate the OLS estimate for β 1 (assume is positive). From class we got the following formula: R 2 = ( b β 1 ) 2 ∑ n i =1 ( X i X ) 2 ∑ n i =1 ( Y i Y ) 2 . 25 = ( b β 1 ) 2 (49) (49) . 25 = ( b β 1 ) 2 ⇒ b β 1 = 0 . 5 Exercise 3 ( Regression without any regressor ) Suppose you are given the model Y i = β 1 + u i . Use OLS to find the estimator of....
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 Fall '08
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 Econometrics, Linear Regression, Regression Analysis, Variance, Yi, Xi Yi − Y Xi

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