For simple regression model case the regression of ti

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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
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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 11 Here we are given that 0 1 t X t T , then: 0 1 0 1 ( ) t t X v 1 t t X v 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 1 1 ˆ ˆ t t t v X X , or equivalently 1 1 ˆ t t 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 v y v reduces to 1 1 2 1 ˆ T t t t T t t x y 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 . The overlap of X with Y , the blue area, represents the variation that Y and X have common in the sense that this variation in Y can be explained by X via an OLS regression. The blue area reflects information employed by the estimating procedure in estimating the slope coefficient x , the larger this area, the more information is used to form the estimate and thus the smaller is its variance. Now consider Figure 2, in which a Ballantine for a case of two explanatory variables, X and Z , is portrayed (i.e., now Y is determined by both X and Z ) 4 Note that this part heavily based on Kennedy, P. (2003) A Guide to Econometrics, 5th edition, pp.53-56. The Ballentine was named by its originators Cohen and Cohen (1975), after a brand of US beer whose logo resembles Figure 2.
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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 13 Figure 2 Interpreting Multiple Regression with the Ballantine In general, the X and Z circles will overlap, reflecting some collinearity between the two, this is shown in Figure 1 by the red+orange area. If Y were regressed on X alone, information in the blue+red area would be used to estimate x .
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