What happens on the other hand if y is regressed on x

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What happens, on the other hand, if Y is regressed on X and Z together? In the multiple regression of Y on X and Z together, the OLS estimator uses the information in the blue area to estimate x , disregarding the information in the red area . The information in the blue area corresponds to variation in Y that matches up uniquely with variation in X; using this information should therefore produce an unbiased estimate of x . The information in the red area is not used since it reflects variation in Y that is determined by variation in both X and Z, the relative contributions of which are not a priori known. In the blue area, for example, variation in Y is all due to variation in X, so matching up this variation in Y with variation in X should allow accurate estimation
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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 14 of x . But in the red area, matching up these variations will be misleading because not all variation in Y is due to variation in X. Notice that regression Y on X and Z together creates unbiased estimates of x and z , whereas regressing Y on X and Z separately creates biased estimates of x and z since this latter method uses the red area (In fact this is the “ omitted variable case ”). But notice also that, because the former method discards the red area, it uses less information to produce its slope coefficient estimates and thus these estimates will have larger variances. 5 Whenever X and Z are orthogonal to one another (have zero collinearity) they do not overlap as in Figure 2 and the red area disappears. Because there is no red area in this case, regressing Y on X alone or on Z produces the same estimates of x and z , as if Y were regressed on X and Z together. Whenever X and Z are higly collinear and therefore overlap a lot, the blue and green areas become very small, implying that when Y is regressed on X and Z together very little information is used to estimate x and z . This causes the variances of these estimates to be very large. Thus, the impact of multicollinearity is to raise the variances of the OLS estimates. Perfect collinearity causes the X and Z circles to overlap completely, the blue and green area disappear and estimation is impossible. Multicollinearity will be discussed in detail later in this semester. In Figure 1 the blue area represents the variation in Y explained by X . Thus R 2 is given as the ratio of the blue area to the entire Y circle. In Figure 2 the blue+red+green area represents the variation 5 As is invariably the case in econometrics, the price of obtaining unbiased estimates is higher variances.
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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 15 in Y explained by X and Z together. Thus, the R 2 resulting from the multiple regression is given by the ratio of the blue+red+green area to the entire Y circle.
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