ECON301_Handout_05_1213_02

# Hence we can conclude that 1 ˆ measures linear

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. 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 yv 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 vy 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 vy 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: 01 t t t Y X u  This model can equivalently be written as: 0 0 1 1 tt 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. For simple regression model case, the regression of ti X on all other X ’s is as follows: 1 0 1 0 t t t X X v 
ECON 301 (01) - Introduction to Econometrics I April, 2012 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: [email protected] Lecture Notes 11 Here we are given that 0 1 t X t T    , then: 0 1 0 1 () tt Xv  1  where is a constant. In this case 1 ˆ ˆ t X and we can show that the OLS estimator for (denoted by ˆ ) is equal to 1 X , that is 1 ˆ X . Thus, for this model, the residual is 11 ˆ ˆ t t t v X X  , or equivalently ˆ v X X . Notice that, here ˆ t v is equal to t x (mean deviation form of X 1 ). We see that in simple regression model, the general formula 1 2 1 ˆ ˆ ˆ T t ti t i T ti t vy v reduces to 1 1 2 1 ˆ T t T t t xy x which is nothing but the expression that we have obtained before for the OLS estimator of the slope term in simple regression model.

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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 12 5. Interpreting Multiple Regression with the Ballantine Venn Diagram Consider the following Venn diagram called the Ballantine . 4 Suppose the CLR model applies, with Y determined by X and an error term. Figure 1 Ballantine Venn diagram In the figure above, the circle Y represents variation in the dependent variable, and the circle X represents variation in the independent variable X .
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Hence we can conclude that 1 ˆ measures linear influence...

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