HW10 Solution

# HW10 Solution - Econ 3190 HW10 Solution 1 E ^& = E& n...

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Unformatted text preview: Econ 3190 HW10 Solution 1. E ^ & = E & n i =1 c i X i = & n i =1 c i EX i = & & n i =1 c i E ^ & = & , if and only if & n i =1 c i = 1 To &nd the best unbiased estimator, we have to &nd value of c i that solves the minMSE (^ & ) problem. Since we are looking for the best estimator among the class of unbiased estimator, the problem is to &nd the value of c i that solves the min V ar (^ & ) problem. So the problem is min V ar (^ & ) = min V ar (& n i =1 c i X i ) s:t: & n i =1 c i = 1 We can use the Lagrangian function to solve this minimization problem. L = V ar (& n i =1 c i X i ) + ¡ (1 & & n i =1 c i ) = & n i =1 c 2 i V arX i + ¡ (1 & & n i =1 c i ) = & n i =1 c 2 i ¢ 2 i + ¡ (1 & & n i =1 c i ) = ¢ 2 & n i =1 c 2 i i + ¡ (1 & & n i =1 c i ) We need to &nd the F.O.C. which c i and ¡ need to satisify to minimize V ar (^ & ) . @L @c 1 = ¢ 2 2 c 1 1 & ¡ @L @c 2 = ¢ 2 2 c 2 2 & ¡ : : @L @c n = ¢ 2 2 c n n & ¡ & n i =1 c i = 1 1 = ) c 1 = & 2 ¡ 2 c 2 = 2 & 2 ¡ 2 : : c n = n& 2 ¡ 2 & n i =1 c i = 1 Plug the &rst n equations into the last one, & n i =1...
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HW10 Solution - Econ 3190 HW10 Solution 1 E ^& = E& n...

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