ECON 710 Midterm: ECON 710 UW Madison Midterm 2008a
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1. (a) this is easiest solved using matrix notation. Write the model as y = x1(cid:12)1 + x2(cid:12)2e and. 1: by the properties the short regression as y = x1 ^(cid:12)1 + ^e: let m1 = i (cid:0) x1 (x 0 of least-squares and the fact that m1x1 = 0; Thus since m1 is idempotent (n (cid:0) k1) s2 = (x2(cid:12)2 + e)0 m1m1 (x2(cid:12)2 + e) = (x2(cid:12)2 + e)0 m1 (x2(cid:12)2 + e) Since x2 and m1 are functions of x; and e (e j x) = 0 (n (cid:0) k1) e(cid:0)s2 j x(cid:1) = e(cid:0)e0m1e j x(cid:1) + (cid:12) 0. 2m1x2(cid:12)2: the second equality since e (ee0 j x) = i(cid:27)2 and tr [m1] = rank(m1) = n(cid:0) k1 imply that. E(cid:0)e0m1e j x(cid:1) = tr(cid:2)m1e(cid:0)ee0 j x(cid:1)(cid:3) = tr [m1] (cid:27)2 = (n (cid:0) k1) (cid:27)2. 1: we learned in class that (cid:12) 0. 1 n e0m1e = since 1 n x 0.
