ECON 710 Midterm: ECON 710 UW Madison Midterm 2012a
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Sample answers (a) nearly everyone missed this question. The references to e ciency and large samples are irrele- vant. The residual variances are unrelated to the e ciency of the estimators of (cid:12): the solutions is algebraic and elementary. Least squares, by de(cid:133)nition, minimizes the sum of squared errors. Therefore ^(cid:27)2 is smaller than any other residual variance constructed from any other estimator. ance constructed from any other estimator satisfying the restriction, including e cient minimum. Thus b(cid:27)2 (cid:20)e(cid:27)2 and b(cid:27)2 (cid:20) (cid:27)2: constrained least squares minimizes the sum of squared errors among all estimators which satisfy the restriction. Therefore e(cid:27)2 is smaller than any other residual vari- distance. Thus e(cid:27)2 (cid:20) (cid:27)2: thus the variance estimators algebraically satisfy ^(cid:27)2 (cid:20) ~(cid:27)2 (cid:20) (cid:22)(cid:27)2; which is the reverse of the assertion. (b) since. Applying standard asymptotic theory, so ie(cid:12)(cid:17) (cid:0)(cid:16)yi (cid:0) x0 ib(cid:12)(cid:17) ~ei (cid:0) ^ei = (cid:16)yi (cid:0) x0 i(cid:16)b(cid:12) (cid:0)e(cid:12)(cid:17) nxi=1 nxi=1(cid:16)b(cid:12) (cid:0)e(cid:12)(cid:17) xix0.
