8 c formula for 2 ˆ multiplying 5 by 1 2 1 t t t t x

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(8) C. Formula for 2 ˆ Multiplying ( 5  ) by 1 2 1 T t t t x x , 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 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 ( 5  ) 2 2 2 2 2 1 1 1 2 1 2 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 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 2 1 1 1 2 2 2 1 2 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 2 1 1 1 2 1 1 1 1 2 2 2 2 2 1 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 (9) 3. Interpreting the Multiple Regression Coefficients Algebraically 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 2 1 2 1 1 1 ˆ T T t t t t t x y x 2 2 1 T t t x 2 1 2 1 1 2 2 1 2 2 1 2 1 1 T T t t t t t t T t t T T t t t t x y x x x x x 2 2 1 T t t x 2 1 2 1 2 2 1 T t t t T t t 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 7 1 2 1 1 2 2 1 1 2 1 1 1 2 2 1 1 1 2 2 1 1 2 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 t t t t x x x y x y x x x x x x x Denoting 1 2 1 12 2 2 1 ˆ T t t t T t t x x c x , we have: 1 12 2 1 1 1 2 1 12 1 2 1 1 ˆ ˆ ˆ T T t t t t t t T T t t t t t x y c x y x c x x Hence, 1 12 2 1 1 2 1 12 1 2 1 1 ˆ ˆ ˆ T t t t t T T t t t t t y x c x x c x x Now consider the following regression model: 1 10 12 2 1 t t t X c c X v where 1 t v is the disturbance term. Hence we can write that:
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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 8 1.
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