ECON301_Handout_05_1213_02

# 5 whenever x and z are orthogonal to one another have

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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. The yellow area represents variation in Y attributable to the error (disturbance) term, and thus the magnitude of the yellow area represents the magnitude of 2 , the variance of the error term. This implies, for example, that if, in the context of Figure 2, Y had been regressed on only X, omitting Z, 2 would be estimated by the yellow+green area, an overestimate.
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5 Whenever X and Z are orthogonal to one another have zero...

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