Sol_ps4_all - PS4 SOLUTIONS JEREMIE COHEN-SETTON ELIRA KARAJA 1 Demonstrate that =(X X)1 X y is such that it minimizes the SSR Given y = X and y = X e

Sol_ps4_all - PS4 SOLUTIONS JEREMIE COHEN-SETTON ELIRA...

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PS4 SOLUTIONS J ´ ER ´ EMIE COHEN-SETTON, ELIRA KARAJA 1. Demonstrate that ˆ β = ( X 0 X ) - 1 X 0 y ˆ β is such that it minimizes the SSR . Given y = + , and y = X ˆ β + e , we have SSR = e 0 e . e 0 e = ( y - X ˆ β ) 0 ( y - X ˆ β ) [by definition] = ( y 0 - ˆ β 0 X 0 )( y - X ˆ β ) [by ( A + B ) 0 = A 0 + B 0 and ( AB ) 0 = B 0 A 0 ] = y 0 y - y 0 X ˆ β - ˆ β 0 X 0 y + ˆ β 0 X 0 X ˆ β [by distribution] = y 0 y - 2 ˆ β 0 X 0 y + ˆ β 0 X 0 X ˆ β because y 0 X ˆ β = ˆ β 0 X 0 y since these are 1 × 1 matrices, i.e scalars Since we look for a minimum we take FOC of this expression with respect to ˆ β 0 . d ( e 0 e ) d ˆ β 0 = 0 - 2 X 0 y + 2 X 0 X ˆ β = 0 [by definition of derivative of a matrix] X 0 y = X 0 X ˆ β ( X 0 X ) - 1 X 0 y = ( X 0 X ) - 1 X 0 X ˆ β [Assuming ( X 0 X ) - 1 exists] ( X 0 X ) - 1 X 0 y = I K ˆ β [by definition of the inverse of a matrix] ( X 0 X ) - 1 X 0 y = ˆ β [by I K v = v for any K × 1 vector v ] 2. Demonstrate that R 2 = β 0 X 0 y 0 y (or that R 2 = ˆ β 0 X 0 X ˆ β y 0 y ) We can proceed in two ways. If we choose to use the theoretical decomposition of y (i.e y = + or what the book calls the PRF), we will have to use the assumption that E ( X 0 ) = 0. If we use the empirical decomposition of y (i.e. y = X ˆ β + e or what the book calls the SRF) we will use the algebraic result that we get from the normal equations: X 0 e = 0.
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