ECON301_Handout_05_1213_02

# T t t t x y x x x x 5 2 2 2 1 1 2 2 2 ˆ ˆ ˆ t t t

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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 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     22 1 1 1 1 1 2 1 2 1 2 ˆˆ - - - tt 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 1 1 2 1 2 2 2 2 - - - t t t t t t t t X Y TX Y X X TX X X TX ( 6 ) Recall that: 2 2 2 11 TT X TX x   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 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 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 Y X X , we get 0 1 1 2 2 ˆ ˆ ˆ t Y X X  (7)
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 12 1 T tt t xx produces 2 2 2 2 1 2 1 1 2 2 1 2 2 1 1 1 1 1 1 ˆˆ 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 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 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 22 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 x y x x y x x x x x x   (8) C. Formula for 2 ˆ Multiplying ( 5 ) by 1 T t , and ( 6 ) by 2 1 1 T t t x produces 2 2 1 1 2 1 1 1 2 2 1 2 1 1 1 1 1 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   ( 5  ) 2 2 2 2 2 1 1 1 2 1 2 2 1 1 1 1 1 1 1 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   ( 6 ) Subtracting ( 5 ) from ( 6 ) yields

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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 6 2 2 2 2
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t t t t X Y X X X X 5 2 2 2 1 1 2 2 2 ˆ ˆ ˆ t t t t t t...

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