(Consistency of clustered standard errors.) Consider the panel data model Yit = Xit + i + uit, where all variables are scalars. Assume that Assumptions

(Consistency of clustered standard errors.) Consider the panel data model Yit = βXit + αi + 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-l i€², where i is a T × 1 vector ones. Also let

and u1 = Mui. For the asymptotic calculations in this problem, suppose that T is fixed and n †’ ˆž.
a. Show that the fixed effects estimator of β from Section 10.3 can be written as

b. Show that

c. Let

d. Let

e. Use your answers to (b) through (d) to prove Equation (10.25); that is, show that

f. Let

be the infeasible clustered variance estimator computed using the true errors instead of the residuals so that

Show that

g. Let

[this is Equation (10.27) in matrix form]. Show that

nclustered .clustered= -1".,(X,1.)2 g2dlastred -, 2 111-Yi _ , and clustered-n-1-2-1 r, clustered-o?