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

Info icon This preview shows pages 8–11. Sign up to view the full content.

View Full Document Right Arrow Icon
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 .
Image of page 8

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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.
Image of page 9
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.
Image of page 10

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 11
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern