This exercise is regarding the projection method that begins with the identity (12.28).

(i) Show that E(

m1 i=1 
m2 j=1 δ1,ijk)2 = O(m1m2). [Hint: Note that, given u and v, the δ1,ijk’s are conditionally independent with E(δ1,ijk|u, v) = 0.]
(ii) Show that

(a) if i1 = i2, j1 = j2, then δ2,i1j1k and δ2,i2j2k are independent;

(b) if j1 = j2, then E(δ2,ij1kδ2,ij2k|u) = 0; and

(c) if i1 = i2, then E(δ2,i1jkδ2,i2jk|v) = 0. It follows that E(

m1 i=1 
m2 j=1 δ2,ijk)2 = O(m1m2).
(iii) Show that 
m1 i=1 
m2 j=1 ζ1,ijk = 
m1 i=1 ζ1,ik(ui ), where ζ1,ik(ui ) = 
m2 j=1 ζ1,ijk is a function of ui . Therefore, we have 
m1 i=1 
m2 j=1 ζ1,ijk = OP(m 1/2 1 m2).
(iv) Show that 
m1 i=1 
m2 j=1 ζ2,ijk = 
m2 j=1 ζ2,jk(vj ), where ζ2,jk(vj ) = 
m1 i=1 ζ2,ijk is a function of vj . Therefore, we have 
m1 i=1 
m2 j=1 ζ2,ijk = OP(m1m 1/2 2 ).