T t t t ssr x y x x x u 3 simplifying 1 1 2 2 ˆ ˆ

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t t t t SSR X Y X X X u (3) Simplifying: 0 1 1 2 2 ˆ ˆ ˆ t t t Y T X X (4) 2 1 0 1 1 1 2 1 2 ˆ ˆ ˆ t t t t t t X Y X X X X (5) 2 2 0 2 1 1 2 2 2 ˆ ˆ ˆ t t t t t t X Y X X X X (6) If we divide the first normal equation (4) by T , we get 0 1 1 2 2 ˆ ˆ ˆ t t t Y X X ( 4 ) If we multiply equation ( 4 ) by 1 t TX we get: 1 0 1 1 1 1 2 2 1 ˆ ˆ ˆ t t t t t t t TY X TX TX X TX X thus, 2 1 0 1 1 1 2 2 1 ˆ ˆ ˆ t t t t t t TY X X TX TX X Subtracting it from the second normal equation, then the second normal equation becomes, 2 1 0 1 1 1 2 1 2 ˆ ˆ ˆ t t t t t t X Y X X X X
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ECON 301 (01) - Introduction to Econometrics I April, 2012 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: [email protected] Lecture Notes 4 thus, 2 1 0 1 1 1 2 2 1 ˆ ˆ ˆ t t t t t t TY X X TX TX X 2 2 1 1 1 1 1 2 1 2 1 2 ˆ ˆ - - - t t t t t t t t t t X Y TX Y X TX X X TX X ( 5 ) Similarly if we multiply the equation ( 4 ) by 2 t TX , and subtract it from the third normal equation, the third normal equation becomes: 2 2 2 2 1 1 2 1 2 2 2 2 ˆ ˆ - - - t t t t t t t t t t X Y TX Y X X TX X X TX ( 6 ) Recall that: 2 2 2 1 1 T T t t t t X TX x 1 1 T T t t t t t t X Y TXY x y Hence, we can rewrite the normal equations ( 5 ) and ( 6 ) as follows: 2 1 1 1 2 1 2 1 1 1 ˆ ˆ T T T t t t t t t t t x y x x x ( 5  ) 2 2 1 1 2 2 2 1 1 1 ˆ ˆ T T T t t t t t t t t x y x x x ( 6  ) A. Formula for 0 ˆ From ( 4 ), 0 1 1 2 2 ˆ ˆ ˆ t t t Y X X , we get 0 1 1 2 2 ˆ ˆ ˆ t t t Y X X (7)
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ECON 301 (01) - Introduction to Econometrics I April, 2012 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: [email protected] Lecture Notes 5 B. Formula for 1 ˆ Multiplying ( 5  ) by 2 2 1 T t t x , and ( 6  ) by 1 2 1 T t t t x x produces 2 2 2 2 1 2 1 1 2 2 1 2 2 1 1 1 1 1 1 ˆ ˆ T T T T T T t t t t t t t t t t t t t t x y x x x x x x ( 5  ) 2 2 2 1 2 1 1 2 2 2 1 2 1 1 1 1 1 ˆ ˆ T T T T T t t t t t t t t t t t t t t x y x x x x x x x ( 6  ) Subtracting ( 6  ) from ( 5  ) yields 2 2 2 2 1 2 2 1 2 1 1 2 1 1 2 1 1 1 1 1 1 1 ˆ ˆ T T T T T T T t t t t t t t t t t t t t t t t t t x y x x y x x x x x x Hence, 2 1 2 2 1 2 1 1 1 1 1 2 2 2 1 2 1 2 1 1 1 ˆ T T T T t t t t t t t t t t t T T T t t t t t t t x y x x y x x x x x x
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