18.15 (Consistency of clustered standard errors.) Consider the panel data model Yit =

bXit + ai + uit, where all variables are scalars. Assume that Assumptions #1, #2, and #4 in Key Concept 10.3 hold and strengthen Assumption #3 so that Xit and uit have eight nonzero finite moments. Let M = IT -T -1II, where I is a T * 1 vector of ones. Also let Yi = (Yi1 Yi2 g YiT), Xi = (Xi1 Xi2 g XiT), ui = (ui1 ui2 g uiT), Y

i = MYi, X

i = MXi, and u 

i = Mui. For the asymptotic calculations in this problem, suppose that T is fixed and n¡ .

a. Show that the fixed effects estimator of b from Section 10.3 can be written as b n

= (gni

=1X 

i

X 

i)-1gni

=1X 

i

Y 

i.

b. Show that b n - b = (gni

=1X

iX

i)-1gni

=1X 

iui. (Hint: M is idempotent.)

c. Let QX

 = T -1E(X 

iX  i) and Qn X

 = 1 nTgni

=1gTt

=1X2 it. Show that Qn X ¡p QX

.

d. Let hi = Xiui > 2T and s2 h = var(hi). Show that 31n gni

=1hi ¡d N(0, s2h

).

e. Use your answers to

(b) through

(d) to prove Equation (10.25); that is, show that 2nT(b n -

b) ¡d N(0, s2 h>Q2 X ).

f. Let s 2 h,clustered be the infeasible clustered variance estimator, computed using the true errors instead of the residuals so that s 

2 h,clustered = 1 nTgni

=1(Xi ui)2. Show that s2 h,clustered ¡ p s2 h.

g. Let u n 1 = Y

i - b nX

i and sn 2

h, clustered = n n - 1 1 nTgni

=1(X 

i

= un i)2 [this is Equation (10.27) in matrix form]. Show that sn 2

h, clustered ¡ p s2 h.

[Hint: Use an argument like that used in Equation (17.16) to show that sn 2

h, clustered - s2 h, clustered ¡ p 0 and then use your answer to (f).]