TECHNICAL APPENDIX FOR ASYMPTOTIC PROPERTIES OF A ...

TECHNICAL APPENDIX FOR ASYMPTOTIC PROPERTIES OF A ROBUST VARIANCE MATRIX ESTIMATOR FOR PANEL DATA WHEN T IS LARGE (TO BE OMITTED FROM PUBLICATION) CHRISTIAN HANSEN

Appendix A. Preliminaries Throughout the appendix, let kAk = [trace(A0 A)]1/2 be the Euclidean norm of a matrix A, and let Pn P PT P Pk i = i=1 , t = t=1 , and h = h=1 . Repeated use will be made of the following simple results, which are stated here for convenience. P

Lemma A.1. For matrices A and B, EkA ⊗ Bkr ≤ (EkAk2r EkBk2r )1/2 . Proof. EkA ⊗ Bkr = E{[trace(AA0 ⊗ BB 0 )]r/2 } = E{[trace(AA0 )]r/2 [trace(BB 0 )]r/2 } = E(kAkr kBkr ) ≤ (EkAk2r EkBk2r )1/2 where the equalities follow from the definition of kAk and properties of the Kronecker product and the inequality results from the Cauchy-Schwarz inequality.  Lemma A.2. Suppose {Zi,T } are independent across i for all T with E[Zi,T ] = µi,T and E|Zi,T |1+δ < ∆ < ∞ Pn p for some δ > 0 and all i, T . Then n1 i=1 (Zi,T − µi,T ) → 0 as {n, T } → ∞ jointly. Proof. The proof follows from standard arguments, cf. Chung (2001) Chapter 5. Details are given in Appendix G.  Lemma A.3. For k × 1 vectors Zi,T , suppose {Zi,T } are independent across i for all T with E[Zi,T ] = 0, Pn 0 E[Zi,T Zi,T ] = Ωi,T , and EkZi,T k2+δ < ∆ < ∞ for some δ > 0. Assume Ω = limn,T n1 i=1 Ωi,T is positive P d definite with minimum eigenvalue λmin > 0. Then √1n ni=1 Zi,T → N (0, W ) as {n, T } → ∞ jointly. Proof. The result follows from verifying the Lindeberg condition of Theorem 2 in Phillips and Moon (1999) using an argument similar to that used in the proof of Theorem 3 in Phillips and Moon (1999). Details are given in Appendix G.  The final lemma simply restates Doukhan (1994) Theorem 2 with a slight change of notation. Its proof may be found in Doukhan (1994) p. 25-30. Lemma A.4. Let {zt } be a strong mixing sequence with E[zt ] = 0, Ekzt kτ + < ∆ < ∞, and mixing coefficient α(m) of size (1−c)r r−c where c ∈ 2N, c ≥ τ , and r > c. Then there is a constant C depending only on PT τ and α(m) such that E| t=1 yt |τ ≤ CD(τ, , T ) with D(τ, , T ) defined in Doukhan (1994) and satisfying D(τ, , T ) = O(T ) if τ ≤ 2 and D(τ, , T ) = O(T τ /2 ) if τ > 2. 1

Finally note X c= 1 W x0 b i b  0 xi nT i i i 1 X 0 0 = x  i  xi nT i i i 1 X 0 − x xi (βb − β)0i xi nT i i 1 X 0 b x i (β − β)0 x0i xi − nT i i 1 X 0 + x xi (βb − β)(βb − β)0 x0i xi , nT i i

(O.1) (O.2) (O.3) (O.4)

and Vb

= =

where

1 X c ))(vec(x0 b c ))0 ] [(vec(x0i b i b 0i xi − W 0i xi − W i i b nT i X 1 X 1 X c )) 1 c ))0 (vec(x0i b i b 0i xi ))(vec(x0i b i b 0i xi ))0 − (vec(W (vec(W nT i nT i nT i

1 X i b 0i xi ))(vec(x0i b i b 0i xi ))0 (vec(x0i b nT i 1 X = (vec(x0i i 0i xi ))(vec(x0i i 0i xi ))0 nT i 1 X − (vec(x0i i 0i xi ))(vec(x0i i (βb − β)0 x0i xi ))0 nT i 1 X − (vec(x0i i 0i xi ))(vec(x0i xi (βb − β)0i xi ))0 nT i 1 X + (vec(x0i i 0i xi ))(vec(x0i xi (βb − β)(βb − β)0 x0i xi ))0 nT i 1 X − (vec(x0i i (βb − β)0 x0i xi ))(vec(x0i i 0i xi ))0 nT i 1 X + (vec(x0i i (βb − β)0 x0i xi ))(vec(x0i i (βb − β)0 x0i xi ))0 nT i 1 X + (vec(x0i i (βb − β)0 x0i xi ))(vec(x0i xi (βb − β)0i xi ))0 nT i 1 X − (vec(x0i i (βb − β)0 x0i xi ))(vec(x0i xi (βb − β)(βb − β)0 x0i xi ))0 nT i 1 X − (vec(x0i xi (βb − β)0i xi ))(vec(x0i i 0i xi ))0 nT i 2

(V.1) (V.2) (V.3) (V.4) (V.5) (V.6) (V.7) (V.8) (V.9)

+ + − + − − +

1 X (vec(x0i xi (βb − β)0i xi ))(vec(x0i i (βb − β)0 x0i xi ))0 nT i 1 X (vec(x0i xi (βb − β)0i xi ))(vec(x0i xi (βb − β)0i xi ))0 nT i 1 X (vec(x0i xi (βb − β)0i xi ))(vec(x0i xi (βb − β)(βb − β)0 x0i xi ))0 nT i 1 X (vec(x0i xi (βb − β)(βb − β)0 x0i xi ))(vec(x0i i 0i xi ))0 nT i 1 X (vec(x0i xi (βb − β)(βb − β)0 x0i xi ))(vec(x0i i (βb − β)0 x0i xi ))0 nT i 1 X (vec(x0i xi (βb − β)(βb − β)0 x0i xi ))(vec(x0i xi (βb − β)0i xi ))0 nT i 1 X (vec(x0i xi (βb − β)(βb − β)0 x0i xi ))(vec(x0i xi (βb − β)(βb − β)0 x0i xi ))0 . nT i

(V.10) (V.11) (V.12) (V.13) (V.14) (V.15) (V.16)

Theorems 1-4 then follow by examining the properties of equations (O.1)-(O.4) and (V.1)-(V.16).

Appendix B. Proof of Theorem 1 √ p d 1 Pn 0 (i) βb − β → 0 and nT (βb − β) → Q−1 N (0, W = limn nT i=1 E[xi Ωi xi ]) follow immediately under the conditions of Theorem 1 from the Markov LLN and the Liapounov CLT. (a) (b) P P √ [ t h (E|x2ith |2+2δ )1/(2+2δ) ]2+2δ (c) (kT )2+2δ ∆ We also have that Ekx0i xi /T k2+2δ ≤ Ekxi / T k4+4δ ≤ < T 2+2δ , where T 2+2δ (a) is by the Cauchy-Schwarz inequality, (b) follows from the definition of kAk and Minkowski’s inequality, and (c) is by E|xith |4+δ < ∆. Making use of E|ith |4+δ < ∆, Ekx0i i /T k2+2δ < k 2+2δ ∆ follows similarly. c are op (n−1/2 ) by Noting that E[x0i xi ⊗ 0i xi ] = 0 by Assumption 3.a, it follows that terms (O.2)-(O.4) of W P 0 0 p p 1 1 c the Markov LLN. The Markov LLN also yields nT i xi i i xi → W , which implies W → W .

(ii) In addition, under E|xith |8+δ < ∆ and E|ith |8+δ < ∆, Ekx0i i /T k4+4δ < k 4+4δ ∆ and Ekx0i xi /T k4+4δ < k 4+4δ ∆ follow by an argument similar to that used to show Ekx0i xi /T k2+2δ < k 2+2δ ∆. c are op (n−1/2 ), it follows that Then, using that (O.2)-(O.4) of W X √ √ d c − W )] = nT [vec( 1 nT [vec(W x0 i 0 xi − W )] + op (1) → N (0, V ) nT i i i 1 Pn 0 0 0 0 0 where V = limn nT i=1 E[(vec(xi i i xi − W ))(vec(xi i i xi − W )) ] by the Liapounov CLT. p

It is also straightforward to show that βb−β → 0, Ekx0i i /T k4+4δ < k 4+4δ ∆, and Ekx0i xi /T k4+4δ < k 4+4δ ∆ imply that terms (V.2)-(V.16) of Vb are op (1) and that Ekx0i i /T k4+4δ < k 4+4δ ∆ implies 1 X p [(vec(x0i i 0i xi ))(vec(x0i i 0i xi ))0 − E[(vec(x0i i 0i xi ))(vec(x0i i 0i xi ))0 ]] → 0. nT i 1Under

c follows. Assumption 3.b, (O.2)-(O.4) will be op (1) and consistency of W 3

p We then have Vb − V → 0. 

Appendix C. Proof of Theorem 2 (a) (b) P P √ [ t h (E|x2ith |2+2δ )1/(2+2δ) ]2+2δ (c) (kT )2+2δ ∆ (i) Ekx0i xi /T k2+2δ ≤ Ekxi / T k4+4δ ≤ < , where (a) is by the T 2+2δ T 2+2δ Cauchy-Schwarz inequality, (b) follows from the definition of kAk and Minkowski’s inequality, and (c) is by E|xith |4+δ < ∆. Making use of E|ith |4+δ < ∆, Ekx0i i /T k2+2δ < k 2+2δ ∆ follows similarly. Then Lemma P P p p p A.2 gives n1 i x0i xi /T → Q and n1 i x0i /T → 0 as {n, T } → ∞ jointly, so βb − β → 0. In addition, since P Pn d Ekx0i i /T k2+2δ < k 2+2δ ∆, we have that √1n i x0i /T → N (0, W = limn,T nT1 2 i=1 E[x0i Ωi xi ]) by Lemma √ d A.3, so n(βb − β) → Q−1 N (0, W ).

c /T : Now consider term (O.2) of W √ 1 1 X 0 1 1 X 0 vec( 2 xi xi (βb − β)0i xi ) = √ ( 2 xi i ⊗ x0i xi ) n(βb − β) = √ Op (1)Op (1) nT i n nT i n

by Lemma A.2 since Ekx0i i /T ⊗ x0i xi /T k1+δ ≤ (Ekx0i i /T k2+2δ Ekx0i xi /T k2+2δ )1/2 < C by Lemma A.1 √ and the argument in the preceding paragraph. That (O.3) is Op (1/ n) and (O.4) is Op (1/n) follow similarly. Finally, Ekx0i i 0i xi /T 2k1+δ ≤ Ekx0i i /T k2+2δ < C where the first inequality follows from the CauchyP Schwarz inequality and the second inequality is proven above. Hence, by Lemma A.2 nT1 2 i vec(x0i i 0i xi − p c /T − W → E[x0i i 0i xi ]) = op (1). It follows immediately that W 0. P √ (ii) Under Assumption 3.a, we also have that √1n ( nT1 2 i x0i i ⊗ x0i xi ) n(βb − β) = √1n op (1)Op (1) = op ( √1n ) P since E[x0i i ⊗ x0i xi ] = 0. Similarly, √1n ( nT1 2 i x0i xi ⊗ 0i xi ) = op ( √1n ). It follows from Lemma A.3 that P √ c √ d n(W /T − W ) = n( nT1 2 i vec(x0i i 0i xi − E[x0i i 0i xi ])) + op (1) → N (0, V ) where V = lim n,T

n 1 X E[(vec(x0i i 0i xi − W ))(vec(x0i i 0i xi − W ))0 ] nT 4 i=1

since Ekx0i i 0i xi /T 2 k2+δ ≤ Ekx0i i /T k4+2δ < C by an argument similar to that used to show Ekx0i i /T k2+2δ < k 2+2δ ∆ as long as E|xith |8+δ < ∆ and E|ith |8+δ < ∆. p To show Vb /T 3 − V → 0, consider

Ekvec(x0i i 0i xi /T 2 )vec(x0i i 0i xi /T 2 )0 k1+δ ≤ Ekx0i i /T k4+4δ < C

where the first inequality is by repeated application of the Cauchy-Schwarz inequality and the second is by an argument similar to that used to show Ekx0i i /T k2+2δ < k 2+2δ ∆ which holds if E|xith |8+δ < ∆ and E|ith |8+δ < ∆. It then follows by Lemma A.2 that 1 X p [vec(x0i i 0i xi )vec(x0i i 0i xi )0 − E[vec(x0i i 0i xi )vec(x0i i 0i xi )0 ]] → 0. nT 4 i Turning to (V.2), we have vec[(vec(x0i i 0i xi /T 2 ))(vec(x0i i (βb − β)0 x0i xi /T 2 ))0 ]

= [(x0i i /T ⊗ x0i i /T ) ⊗ (x0i xi /T ⊗ 0i xi /T )]vec(βb − β). 4

Ek(x0i i /T ⊗ x0i i /T ) ⊗ (x0i xi /T ⊗ 0i xi /T )k1+δ ≤ [(Ekx0i i /T k4+4δ )3 Ekx0i xi k4+4δ ]1/4 < C where the first inequality is by repeated application of Lemma A.1 and the second inequality is by the moment conditions. P It then follows from Lemma A.2 that n1 i (x0i i /T ⊗ x0i i /T ) ⊗ (x0i xi /T ⊗ 0i xi /T ) = Op (1), so (V.2) is op (1). Using similar arguments, it is also straightforward to show that terms (V.3)-(V.16) are o p (1), and the conclusion follows.  Appendix D. Proof of Theorem 3 (b) P P (a) √ [ t h (E|x2ith |2+2δ )1/(2+2δ) ]2+2δ (c) (kT )2+2δ ∆ (i) Ekx0i xi /T k2+2δ ≤ Ekxi / T k4+4δ ≤ < , where (a) is by the T 2+2δ T 2+2δ Cauchy-Schwarz inequality, (b) by definition of kAk and Minkowski’s inequality, and (c) by E|x ith |r+δ < ∆. √ Also, Ekx0i i / T k2+2δ < C by Lemma A.4, E|it |r+δ < ∆, and the mixing condition that α is of size −3r/(r− √ Pn d 1 0 b → Q−1 N (0, W = limn,T nT 4) for r > 4. It follows by Lemmas A.2 and A.3 that nT (β−β) i=1 E[xi Ξi xi ]) as {n, T } → ∞. √ P 0 P 0 1 0 0 b b c : vec( 1 Now consider term (O.2) of W i xi xi (β − β)i xi ) = ( (nT )3/2 i xi i ⊗ xi xi ) nT (β − β) = nT √ √1 op (1)Op (1) by Lemma A.2 since E[x0 i ⊗ x0 xi ] = 0 by Assumption 3.a and Ekx0 i / T ⊗ x0 xi /T k1+δ ≤ i i i i n √ 2+2δ 0 0 2+2δ 1/2 (Ekxi i / T k Ekxi xi /T k ) < C by Lemma A.1 and the argument in the preceding paragraph. That √ √ (O.3) is op (1/ n) and (O.4) is Op (1/n) follow similarly.2 Finally, Ekx0i i 0i xi /T k1+δ k ≤ Ekx0i i / T k2+2δ < C p c −W → by the Cauchy-Schwarz inequality and the preceding argument. It then follows that W 0 by Lemma A.2.

(ii) In addition, under the mixing condition that α is of size −7r/(r − 8) for r > 8, E|xith |r+δ < ∆, and √ √ E|ith |r+δ < ∆, Ekx0i i / T k4+4δ < C follows by an argument similar to that used to show Ekx0i i / T k2+2δ < √ c are op (n−1/2 ), C. Then, since Ekx0i i 0i xi /T k2+δ ≤ Ekx0i i / T k4+4δ < C and using that (O.2)-(O.4) of W the second conclusion follows from Lemma A.3 since X √ √ d c − W )] = n[vec( 1 n[vec(W x0 i 0 xi − W )] + op (1) → N (0, V ) nT i i i where

V = lim n,T

.

n 1 X E[(vec(x0i i 0i xi − W ))(vec(x0i i 0i xi − W ))0 ] nT 2 i=1

p To show Vb /T − V → 0, consider

√ Ekvec(x0i i 0i xi /T )vec(x0i i 0i xi /T )0 k1+δ ≤ Ekx0i i / T k4+4δ < C

where the first inequality is by repeated application of the Cauchy-Schwarz inequality and the second is by an argument similar to that used above which holds if E|it |8+δ < ∆ < ∞, E|xith |8+δ < ∆ < ∞, and the strong mixing coefficient α is of size −7r/(r − 8) for r > 8. It then follows from Lemma A.2 that 1 X p [vec(x0i i 0i xi )vec(x0i i 0i xi )0 − E[vec(x0i i 0i xi )vec(x0i i 0i xi )0 ]] → 0. nT 2 i 2Under

c follows. Assumption 3.b, (O.2)-(O.4) will be op (1) and consistency of W 5

Turning to (V.2), we have √ 1 X vec[(vec(x0i i 0i xi /T ))(vec(x0i i ( nT (βb − β))0 x0i xi /T 3/2 ))0 ] 3/2 n i √ √ √ 1 X 0 √ = 3/2 [(xi i / T ⊗ x0i i / T ) ⊗ (x0i xi /T ⊗ 0i xi / T )]vec( nT (βb − β)). n i √ √ √ P 1 By Lemma A.2, we have that n i (x0i i / T ⊗ x0i i / T ) ⊗ (x0i xi /T ⊗ 0i xi / T ) = Op (1) since √ √ √ √ Ek(x0i i / T ⊗ x0i i / T ) ⊗ (x0i xi /T ⊗ 0i xi / T )k1+δ ≤ [(Ekx0i i / T k4+4δ )3 Ekx0i xi k4+4δ ]1/4 < C where the first inequality is by repeated application of Lemma A.1 and the second inequality is by the √ moment and mixing conditions. It then follows that (V.2) is Op (1/ n). It can similarly be shown that √ terms (V.3)-(V.16) are Op (1/ n), and the conclusion follows.  Appendix E. Proof of Theorem 4 √ d √ p d Under the hypotheses of the theorem, n(βb − β) → Q−1 N (0, W ), x0i xi /T − Qi → 0, and x0i i / T → N (0, Wi ) are immediate from a LLN and CLT for mixing sequences, cf. White (2001) Theorems 3.47 and c and b 5.20. The conclusion then follows from the definition of W i . Appendix F. Proof of Corollary 4.1 √ √ b (Rβ−r) . Under the null hypothesis, Rβ = r, so the numerator of t∗ is nT R(βb − Consider t∗ = √ nT b−1 W cQ b−1 R0 RQ P 0 −1 1 P 0 P √ d 1 ( √nT i xi i ) → RQ−1 Λ i Bi / n. From Theorem 4 and the hypotheses of the β) = R( nT i xi xi ) q P P P Corollary, the denominator of t∗ converges in distribution to RQ−1 n1 Λ( ni=1 Bi Bi0 − n1 ni=1 Bi ni=1 Bi0 )ΛQ−1 R0 . It follows from the Continuous Mapping Theorem that P √ RQ−1 Λ i Bi / n ∗ d t →q . Pn Pn Pn 1 0− 1 0 )ΛQ−1 R0 −1 Λ( RQ B B B B i i i i i=1 i=1 i=1 n n

Define δ = (RQ−1 ΛΛQ−1 R0 )1/2 , so

P

√ B1,i / n t →U = q P Pn n δ2 1 Pn 0 0 i=1 B1,i B1,i − n i=1 B1,i i=1 B1,i ) n( δ

∗ d

=q

i

e1,n B

. 2 −B e2 ) B1,i 1,n e1,n ∼ N (0, 1), that P B 2 − B e 2 ∼ χ2 , and that P B 2 − B e 2 and It is straightforward to show that B 1,n n−1 1,n i 1,i i 1,i e B1,n are independent, from which it follows that  1/2  1/2 e1,n n B n qP U= ∼ tn−1 . n−1 n−1 2 −B e 2 )/(n − 1) ( i B1,i 1,n 1 n(

P

i

The result for F ∗ is obtained through a similar argument.  6

Appendix G. Proof of Lemma A.2 and Lemma A.3 Proof of Lemma A.2 Instead of proving Lemma A.2 directly, we prove the following: If {Z i,T } are independent across i for all T with E[Zi,T ] = 0 and E|Zi,T |1+δ < ∆ < ∞ for some δ > 0 and all i, T , P p 1 i Zi,T → 0 as {n, T } → ∞ jointly. Lemma A.2 then follows immediately from an argument similar to n that used to prove White (2001) Corollary 3.9. P P Define Yi,T = Zi,T 1(|Zi,T | ≤ n). Then V ar( i Yi,T /n) = i V ar(Yi,T /n) by independence of Zi,T across R P P i. Now i V ar(Yi,T /n) ≤ E(Yi,T /n)2 = i |Z|≤n (Z 2 /n2 )dFi,T (Z) where Fi,T is the distribution function of Zi,T . Z 2 /n2 ≤ Z 1+δ /n1+δ for |Z| ≤ n implies XZ XZ XZ (Z 2 /n2 )dFi,T (Z) ≤ (Z 1+δ /n1+δ )dFi,T (Z) ≤ (Z 1+δ /n1+δ )dFi,T (Z) < ∆/nδ |Z|≤n

i

i

|Z|≤n

i

where the last inequality results from E|Zi,T |1+δ < ∆. It follows that X V ar( Yi,T /n) < ∆/nδ → 0

(1)

i

as {n, T } → ∞ jointly. Now consider |E

XZ 1X Yi,T | = | (Z/n)dFi,T (Z)| n i |Z|≤n i XZ XZ = | (Z/n)dFi,T (Z) − i

= | ≤

XZ i

XZ i

i

(Z/n)dFi,T (Z)| |Z|>n

(Z/n)dFi,T (Z)| |Z|>n

(|Z|/n)dFi,T (Z) |Z|>n

by the Triangle inequality and Jensen’s inequality. For |Z| > n, |Z|/n ≤ |Z|1+δ /n1+δ , so XZ XZ X Z |Z|1+δ |Z|1+δ |Z| dFi,T (Z) ≤ dF (Z) ≤ dFi,T (Z) ≤ ∆/nδ → 0, i,T 1+δ 1+δ n n n |Z|>n |Z|>n i i i

(2)

which yields |E

1X Yi,T | → 0. n i

(3)

By Chebyshev’s inequality and (1), X X lim P (| (Yi,T − E[Yi,T ])/n| ≥ ) ≤ lim V ar( Yi,T /n)/2 = 0, n,T

so

1 n

P

i

Yi,T −

1 n

P

i

n,T

i

p

E[Yi,T ] → 0, which implies, using (3), that 7

i

1 n

P

i

p

Yi,T → 0.

Finally, consider P (|

1X 1X 1X Zi,T − Yi,T | ≥ ) = P (| (1 − 1(|Zi,T | ≤ n))Zi,T | ≥ ) n n n i i i 1X ≤ E| (1 − 1(|Zi,T | ≤ n))Zi,T |/ n i

by the Markov inequality. E| n1 inequality, and

P

i (1 − 1(|Zi,T |

≤ n))Zi,T | ≤

1X E|(1 − 1(|Zi,T | ≤ n))Zi,T | = n i =

1X [ n i XZ i

by (2). It then follows that 

1 n

P

i

Zi,T −

1 n

P

i

Z

1 n

P

i

E|(1 − 1(|Zi,T | ≤ n))Zi,T | by the Triangle

|Z|dFi,T (Z) −

|Z|>n

Z

|Z|≤n

|Z|dFi,T (Z)

|Z| dFi,T (Z) → 0 n

p

Yi,T → 0 which, with

1 n

P

p

i

Yi,T → 0, implies

1 n

P

i

p

Zi,T → 0.

P −1/2 Proof of Lemma A.3 Define ξi,n,T = Ωn,T Zi,T where Ωn,T = i Ωi,T . By the Cramer-Wold device, P P d d k 0 i ξi,n,T → N (0, Ik ) as {n, T } → ∞ jointly if ∀ c ∈ R with kck = 1, c i ξi,n,T → N (0, 1) as {n, T } → ∞ P P d n n jointly. Then, by Ω = limn,T n1 i=1 Ωi,T > 0, √1n i=1 Zi,T → N (0, Ω) as {n, T } → ∞ jointly. To establish c0

P

i ξi,n,T

d

→ N (0, 1), it is sufficient to verify lim n,T

X

2 2 E[ξi,n,T 1(|ξi,n,T > )] = 0.

(4)

i

For a given  > 0 and c ∈ Rk with kck = 1, c0

X

0 0 E[ξi,n,T ξi,n,T 1(|c0 ξi,n,T ξi,n,T c| > )]c

i

−1/2

= c0 Ωn,T

X

−1/2

−1/2

−1/2

0 0 E[Zi,T Zi,T 1(|c0 Ωn,T Zi,T Zi,T Ωn,T c| > )]Ωn,T c.

i

Considering first the indicator function, we have −1/2

−1/2

−1/2

−1/2

0 0 Ωn,T k > ) 1(|c0 Ωn,T Zi,T Zi,T Ωn,T c| > ) ≤ 1(kckkΩn,T Zi,T Zi,T 2 ≤ 1(λmax (Ω−1 n,T )kZi,T k > )

= 1(kZi,T k2 > λmin (Ωn,T )) 8

Then −1/2

c0 Ωn,T

X

−1/2

−1/2

−1/2

0 0 E[Zi,T Zi,T 1(|c0 Ωn,T Zi,T Zi,T Ωn,T c| > )]Ωn,T c

i

−1/2

≤ kck2 kΩn,T ≤

X i

λmax (Ω−1 n,T )k

X i

−1/2

−1/2

−1/2

0 0 E[Zi,T Zi,T 1(|c0 Ωn,T Zi,T Zi,T Ωn,T c| > )]Ωn,T k 0 E[Zi,T Zi,T 1(kZi,T k2 > λmin (Ωn,T ))]k

X 1 ≤ E[kZi,T k2 1(kZi,T k2 > λmin (Ωn,T ))] λmin (Ωn,T ) i ≤

X EkZi,T k2+δ 1 λmin (Ωn,T ) i (λmin (Ωn,T ))δ

≤

δ [nλ

n∆ P

1 min ( n

i

Ωi,T

=

)]1+δ

∆ δ nδ λ1+δ min

→0

as {n, T } → ∞ jointly, and it follows that (4) → 0 as {n, T } → ∞ jointly. References Chung, K. L. (2001): A Course in Probability Theory. San Diego, California: Academic Press, third edn. Doukhan, P. (1994): Mixing: Properties and Examples, vol. 85 of Lecture Notes in Statistics (Springer-Verlag). New York: Springer-Verlag. Phillips, P. C. B., and H. R. Moon (1999): “Linear Regression Limit Theory for Nonstationary Panel Data,” Econometrica, 67(5), 1057–1111. White, H. (2001): Asymptotic Theory for Econometricians. San Diego: Academic Press, revised edn.

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