ECON301_Handout_05_1213_02

# 2 1 1 1 2 2 2 1 2 1 2 1 1 1 1 1 1 1 ˆˆ t t t t t t

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2 1 1 1 2 2 2 1 2 1 2 1 1 1 1 1 1 1 ˆˆ T T T T T T T tt 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 22 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 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 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   2 12 11 1 ˆ TT t x y x   2 2 1 T t t x 2 1 2 2 2 1 t t t t T t t x y x x x xx 2 2 1 T t t x 2 1 2 2 1 T t T t t 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 12 1 2 11 2 1 1 2 1 1 1 2 2 2 1 ˆ T TT tt t T t t T t t t t T t t xx x y x y x x x x x   Denoting 1 12 2 2 1 ˆ T t T t t c x , we have: 1 12 2 1 2 1 12 1 2 ˆ ˆ ˆ t t t x y c x y x c x x Hence,   1 12 2 1 1 2 1 12 1 2 ˆ ˆ ˆ 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:
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. 12 1 12 2 2 1 ˆ T tt t T t t xx c x (why?) 1 , 2. 22 1 1 12 1 2 11 ˆˆ TT t t t t SSR v x c x x  (why? 2 ), and 3. 1 1 12 2 t t t v x c x  (why? 3 ) Therefore, 1 1 1 2 1 1 ˆ ˆ ˆ T t t t T t t yv v 1 Recall that for 01 t t t Y X u  , 1 1 2 1 ˆ T t T t t xy x 2 Recall that, for t t t Y X u , we can write 2 1 1 2 1 ˆ T t T t t R y and 2 2 1 2 1 ˆ 1 T t t T t t u R y  . Then, 2 1 ˆ ˆ 1 t t t x y u yy  , and 1 1 1 1 ˆ ˆ T T T t t t t t t t y x y u . Finally we can write: 1 1 1 1 ˆ ˆ T T T t t t t t t t u y x y 3 Consider the simple model t t t Y X u , from which we have ˆ YX . We know that , hence . Subtracting, we get 0 0 1 ˆ ˆ ˆ ˆ () t t t t Y Y X X , or 1 ˆ ˆ yx . Then, the residual is as follows: 1 ˆ t t t t t u y y y 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 9 Thus, 1 ˆ is the estimated parameter of the following regression model: 0 1 1 ˆ t t Y e e v errorterm In other words, 11 ˆ ˆ e . Note that here, 1 ˆ t v is the residual coming from the estimation of the following model: 1 10 12 2 1 t t t X c c X v
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2 1 1 1 2 2 2 1 2 1 2 1 1 1 1 1 1 1 ˆˆ T T T T T T T tt t...

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