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1 2 1 12 2 2 1 ˆ t t t t t t t x x c x why 1 2 2 2 1

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1 2 1 12 2 2 1 ˆ T t t t T t t x x c x (why?) 1 , 2. 2 2 1 1 12 1 2 1 1 ˆ ˆ T T t t t t t t SSR v x c x x (why? 2 ), and 3. 1 1 12 2 ˆ ˆ 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 y v v 1 Recall that for 0 1 t t t Y X u , 1 1 2 1 ˆ T t t t T t t x y x 2 Recall that, for 0 1 t t t Y X u , we can write 2 1 1 2 1 ˆ T t t t T t t x y R y and 2 2 1 2 1 ˆ 1 T t t T t t u R y . Then, 2 1 1 1 2 2 1 1 ˆ ˆ 1 T T t t t t t T T t t t t x y u y y , and 2 2 1 1 1 1 ˆ ˆ T T T t t t t t t t y x y u . Finally we can write: 2 2 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 0 1 t t t Y X u , from which we have 0 1 ˆ ˆ ˆ t t Y X . We know that 0 1 ˆ ˆ t t Y X , hence 0 1 ˆ ˆ t t Y X . Subtracting, we get 0 0 1 ˆ ˆ ˆ ˆ ( ) t t t t Y Y X X , or 1 ˆ ˆ t t y x . Then, the residual is as follows: 1 ˆ ˆ 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, 1 1 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 . Hence, we can conclude that, 1 ˆ measures linear influence of 1 t X on t Y after the linear influence of 2 t X on 1 t X has been eliminated. Similarly, 2 1 2 2 2 1 ˆ ˆ ˆ T t t t T t t t y v y v where 2 ˆ t v is the residual coming from the estimation of the following model: 2 20 21 1 2 t t t X c c X v . We call 1 ˆ and 2 ˆ as partial regression coefficients. Generalizing for a model with k explanatory variables, we have: 1 2 1 ˆ ˆ ˆ T t ti t i T ti t v y v where ˆ ti v is the residual in the regression of ti X on all other X’s.
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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 10 4. Simple Regression as a Special Case Recall that for a model with k explanatory variables, we have: 1 2 1 ˆ ˆ ˆ T t ti t i T ti t v y v where ˆ ti v is the residual in the regression of ti X on all other X’s. Now consider the simple regression model: 0 1 t t t Y X u This model can equivalently be written as: 0 0 1 1 t t t t Y X X u where 0 1 t X t T Now let us obtain the ˆ ti v which is the residual in the regression of ti X on all other X ’s.
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