Suppose that the linear model, written in matrix notation, y 5 Xb 1 u satisfies Assumptions E.1, E.2, and E.3. Partition the model as y 5 X1b1 1 X2b2 1 u, where X1 is n 3 1k1 1 12 and X2 is n 3 k2.

(i) Consider the following proposal for estimating b2. First, regress y on X1 and obtain the residuals, say, y

$

. Then, regress y

$

on X2 to get b˘ 2. Show that b˘ 2 is generally biased and show what the bias is. [You should find E1b˘ 2 0X2 in terms of b2, X2, and the residual-making matrix M1.]

(ii) As a special case, write y 5 X1b1 1 bkXk 1 u, where Xk is an n 3 1 vector on the variable xtk. Show that E1b

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k 0X2 5 a SSRk gn t51x2 tk bbk, where SSRk is the sum of squared residuals from regressing xtk on 1, xt1, xt2, p, xt, k21. How come the factor multiplying bk is never greater than one?

(iii) Suppose you know b1. Show that the regression y 2 X1b1 on X2 produces an unbiased estimator of b2 (conditional on X).