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ps4 - Problem Set 4 ECON 837 Prof Simon Woodcock Spring...

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Problem Set 4 ECON 837 Prof. Simon Woodcock, Spring 2006 Due: Friday Feb 17 1. [A warmup question from the 2004 Midterm] Suppose the data generating process is y = X ° + " ; where E [ " ] = 0 , E [ "" 0 ] = ° 2 I n ; and X includes an intercept term. You do not observe the data set Z = [ y X ] : Instead you observe Z 0 Z = 2 4 100 10 25 10 20 0 25 0 75 3 5 : Compute the least squares estimators ^ ° ; s 2 ; and R 2 . Is there anything to be gained by observing the full data set? 2. [A follow-up question] Find a (matrix-valued) su¢ cient statistic for ° and ° 2 in the multivariate linear regression under normality. Comment on your solution to Question 1. 3. [2004 Midterm] Suppose the data generating process is y i = x 0 i ° + " i where the errors are spherical and have mean zero. The data fall into one of two groups of equal size. In the °rst group (group 1) of n= 2 observations, x 0 i = [1 1] : In the second group (group 2), x 0 i = [1 ° 1] : Despite your knowledge of least squares, you devise a new estimator ~ ° by noting that ° y 1 = ± 1 + ± 2 + ° " 1 (1) ° y 2 = ± 1 ° ± 2 + ° " 2 (2) where ° y 1
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