Question

Type or pas
2. Let the population regression model between a dependent variable y and an independent variable is given by y= Bo+ B1 x x+ u Suppose that E(u|x) = E(u) = 0 and V(ux) = o2. Based on a random sample ((y, ) i = 1,2,...n) of size n such that (xi- )2>0, let Bo and B be the OLS estimates of Bo and Bi respectively. Answer the following questions

Answer 1

We can write Bo = xi = E(Bo+B1X; + u:))} = Bo + B28 +ū. Since E(u;) = 0, we have E(ū) = 0 so that E(BolX1,.., Xn) = Bo + B12 + 0 = Bo + Blk. ...........(* Thus B1 = 0 implies E(BolX1...,xn) = Bo. However, if it is given that E(Bolx1,..,xn) = Bo, we get from (*) that B x = 0, which implies either B1 = 0 or z = 0. Hence E(Bolx1,..,xn) = Bo does not always mean B1 = 0.

C) Since, Bo = * = E(Bo+B1x; + u;) = Bo + B17 +ū, we get Var (Polx...,xn) = Var(a) = Now Bo =ý-Bit, B1 == {(x; – 1)(): – 7) /2(xi – 5)2 = {(x; – t)y:/(x; – 5)2. Then we can write Bo = 5 - B2B = Ebiyi, b;=<: - * } . Then Bo = £b;(B + Bixi + uz) = Bo£b; + B12b;x; + £b;4; so that 25(Xi - 1) (Xi - 7272 Varlſo lx...kn) = 0?? = 0°:{n nece = 351 + LEC - 5*} 12(x; -72 [3(xi – 7)212 (362333 = Var (Bolx1,.., Xn) +503Var (Bolx1,.., Xn) as x2 > 0,2(x; – )2 > 0. 02 0 + Hence the proof.

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