ECON301_Handout_05_1213_02

The overlap of x with y the blue area represents the

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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 . 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.
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The overlap of X with Y the blue area represents the...

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