if we choose M1 >
h i 1/(2+δ) −1/(2+δ) ∗ w 2+δ max E Rt u(R ˙ jt ) .
1≤j≤JT
Then, by letting be arbitrarily small, we can show that ΠT 4 = OP (Ξ T ).
(B.12)
Z jt (xjk ) − E Z jt (xjk ) ≤ C0 MT ,
(B.13)
Var Z jt (xjk ) ≤ C0 h
(B.14)
On the other hand, note that
and
where C0 is a positive constant. By (B.13), (B.14) and Theorem 1.3(2) in Bosq (1998) with p = [(M2 MT Ξ T /4)−1 ] which tends to infinity by (4.7), we have ! T 1 X Z jt (xjk ) − E Z ij (xjk ) > M2 Ξ T P(Î T 3 > M2 Ξ T ) = P max max 1≤j≤JT 1≤k≤Nj T h t=1 JT X −qM22 Ξ 2T p = Nj 4 exp + 22 [1 + 4C0 MT /(M2 hΞ T )] qÎł 0 4C M M Ξ /h + 16C /(ph)) 0 2 T 0 T j=1 ≤ C
JT X
Nj exp {−M2 log T } + T MT2 Îł p0 = o(1),
j=1
where M2 is chosen sufficiently large and q = T /(2p). Hence we have ΠT 3 = OP (Ξ T ).
(B.15)
In view of (B.10)–(B.12) and (B.15), we have shown (B.8), completing the proof of Proposition 4.1. Proof of Theorem 4.2. Recall that |
|
b a∗ = (b a1 , . . . , b aJ−1 ) , a∗0 = (a10 , . . . , aJ−1,0 ) ,
| w w | w w , Rt (w) = (R1t , . . . , RJt ) , Rt∗ (w) = R1t , . . . , RJ−1,t | | w w w w bJ−1,t bt (w) = R b1t bJt bt∗ (w) = R b1t R ,...,R , R ,...,R .
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