Econ220B_prob1

# Econ220B_prob1 - 1 Econ 220B Winter 2012 James Hamilton...

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–1– Econ 220B, Winter 2012 James Hamilton Problem Set 1 Due Thursday, Jan 19 1.) Let X be a ( T × k ) matrix whose columns are linearly independent, and let M = I T X ( X 0 X ) 1 X 0 . Show that M is symmetric and idempotent. Calculate the eigenvalues and rank of M , and show that it is positive semide f nite. 2.) Consider a regression of y t on x t where the f rst element of x t is a constant term. The R 2 or coe cient of determination is de f ned as R 2 =1 P T t =1 ( y t x 0 t b ) 2 P T t =1 ( y t y ) 2 where b is the OLS regression coe cient and y is the sample mean. Show that 0 R 2 1 . 3.) Let P be a nonsingular symmetric ( k 1 × k 1 ) matrix, Q a nonsingular symmetric ( k 2 × k 2 ) matrix, and R an arbitrary ( k 1 × k 2 ) matrix,. Verify the following formula for the inverse of a partitioned matrix: · PR R 0 Q ¸ 1 = · W WRQ 1 Q 1 R 0 W ( Q 1 + Q 1 R 0 WRQ 1 ) ¸ for W =( P RQ 1 R 0 ) 1 . 4.) Consider a regression of

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## This note was uploaded on 03/02/2012 for the course ECON 220b taught by Professor Hamilton,j during the Spring '08 term at UCSD.

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Econ220B_prob1 - 1 Econ 220B Winter 2012 James Hamilton...

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