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17.5 Weighted Least Squares 745

17.9 Prove Equation (17.16) under Assumptions #1 and #2 of Key Concept 17.1
plus the assumption that Xi and ui have eight moments.

17.10 Let un be an estimator of the parameter u, where un might be biased. Show
that if E3(un - u)24 ¡ 0 as n ¡ ∞ (that is, the mean squared error
of un tends to zero), then un ¡p u. [Hint: Use Equation (17.43) with
W = un - u.4

17.11 Suppose that X and Y are distributed bivariate normal with density given
in Equation (17.38).

a. Show that the density of Y given X = x can be written as

fY͉X = x(y) = 1 exp c - 1y - mY͉X 2
sY͉X 22p 2a sY͉X b d

where sYX = 2sY2 (1 - r2XY) and mY͉X = mY - (sXY>sX2 )(x - mX).
[Hint: Use the definition of the conditional probability density

fY 0 X = x(y) = 3gX, Y(x, y)4 > 3fX(x)4, where gX,Y is the joint density of X
and Y, and ƒX is the marginal density of X.]

b. Use the result in (a) to show that Y 0 X = x N(mY 0 X, sY2 0 X).

c. Use the result in (b) to show that E(Y 0 X = x) = a + bx for suitably

chosen constants a and b.

17.12 a. Suppose that u N(0, s2u). Show that E(eu) = e12su2
b. Suppose that the conditional distribution of u given X = x is

N(0, a + bx2), where a and b are positive constants. Show that

E(eu 0 X = x) = e12(a + bx2).

17.13 Consider the heterogeneous regression model Yi = b0i + b1iXi + ui, where
b0i and b1i are random variables that differ from one observation to the next.

Suppose that E(ui 0 Xi) = 0 and (b0i, b1i) are distributed independently of Xi.

a. Let bn1OLS denote ¡thpe OEL(Sb1e)s,tiwmhaetroerEo(fbb11) given in Equation (17.2).
Show that bnO1 LS is the average value of b1i
in

the population. [Hint: See Equation (13.10).]

b. Suppose that var(ui 0 Xi) = u0 + u1X2i , where u0 and u1 are known posi-
tDivoeecsobnn1WsLtaSn¡ts.p LeEt b(nbW1 1L)S?
denote the weighted least squares estimator.
Explain.

17.14 Suppose that Yi, i = 1, 2, c, n, are i.i.d. with E(Yi) = m, var(Yi) = s2,
and finite fourth moment. Show the following: