HW10 Solution - Econ 3190 HW10 Solution 1. E ^...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Page1 / 4

HW10 Solution - Econ 3190 HW10 Solution 1. E ^...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online