S UPPLEMENT TO “C ROSS -S ECTIONAL AVERAGES VERSUS P RINCIPAL C OMPONENTS ”: P ROOFS AND A DDITIONAL D ETAILS Joakim Westerlund∗

Jean-Pierre Urbain

Deakin University

Maastricht University

Australia

The Netherlands

August 29, 2013

Abstract In this supplement, we (i) provide the proof of Theorem 1 in Westerlund and Urbain (2013), (ii) study the asymptotic distributions of the PC and CA factor estimators (which are discussed in Remark 6 of the main paper), (iii) provide a comparison with the residual-based PC estimator of Bai (2009a), and (iv) provide some results on the biasadjusted estimators discussed in Remark 8 of the main paper.

1 Notation ′ ) ′ can be written in matrix notation as The model for zi,t = (yi,t , xi,t

zi = FCi + ui ,

(1)

where zi = (yi , xi ) = (zi,1 , ..., zi,T )′ is T × (m + 1), F = ( F1 , ..., FT )′ is T × r, Ci = (Λ′i β + λi , Λ′i )

is r × (m + 1) and ui = (ui,1 , ..., ui,T )′ = (ηi β + ǫi , ηi ) is T × (m + 1). Alternatively, the model for zi,t can be written as the following N (m + 1)-dimensional system: zt = CFt + ut ,

(2)

′ , ..., z′ ) ′ and u = ( u ′ , ..., u ′ ) ′ are N ( m + 1) × 1, and C = ( C , ..., C ) ′ is where zt = (z1,t t N 1 N,t 1,t N,t

N (m + 1) × r. The matrix notation z = FC ′ + u

(3)

∗ Corresponding

author: Deakin University, Faculty of Business and Law, School of Accounting, Economics and Finance, Melbourne Burwood Campus, 221 Burwood Highway, VIC 3125, Australia. Telephone: +61 3 924 46973. Fax: +61 3 924 46283. E-mail address: [email protected].

1

will also be used, where z = (z1 , ..., z N ) and u = (u1 , ..., u N ) are T × N (m + 1). In what follows the representations in (1)–(3) will be used interchangeably. Many of the results can be expressed in terms of ( Fˆ PC − FH ) and ( Fˆ CA − FC). It is therefore going to be convenient to introduce some special notation to simplify such expressions. Consider the PC estimator. As in Bai (2003, page 158), if we denote by V the k × k diag-

onal matrix consisting of the first k eigenvalues of ( NT )−1 zz′ in descending order, then, by the definition of eigenvalues and eigenvectors, Fˆ PC = ( NT )−1 zz′ Fˆ PC V −1 . Thus, letting H = Q( T −1 F ′ Fˆ PC )V −1 , where Q = N −1 ∑iN=1 Ci Ci′ = N −1 C ′ C, we have dPC = Fˆ PC − FH = ( NT )−1 zz′ Fˆ PC V −1 − ( NT )−1 FC ′ CF ′ Fˆ PC V −1

= ( NT )−1 (zz′ − FC ′ CF ′ ) Fˆ PC V −1 = ( NT )−1 (uu′ + uCF ′ + FC ′ u′ ) Fˆ PC V −1 = ( NT )−1 g Fˆ PC V −1,

(4)

where g = (uu′ + uCF ′ + FC ′ u′ ) = ( g1′ , ..., g′T )′ is T × T and gt = (uut + FC ′ ut + uCFt ) is ′ T × 1. Note also that dPC = (d1PC , ..., dPC T ) , where

′ dtPC = FˆtPC − H Ft = ( NT )−1 V −1 ( Fˆ PC )′ (uut + FC ′ ut + uCFt ) = ( NT )−1 V −1 ( Fˆ PC )′ gt . (5)

In case of the CA estimator, we use zi = FCi + ui , suggesting that Fˆ CA = z = FC + u, or dCA = Fˆ CA − FC = u.

(6)

In vector notation, ′ dCA = FˆtCA − C Ft = ut . t

(7)

2 Proof of Theorem 1 Before we come to the proof of Theorem 1 we state some useful lemmas. Lemma PC1. Under Assumptions JOINT, PC1 and PC2, 1 T

T

∑ ||dtPC ||2 = O p (T −1 ) + O p ( N −1 ).

t =1

Proof of Lemma PC1. By the definition of dtPC , 1 T

T

3

T

∑ ||dtPC ||2 ≤ ||V −1 ||2 N2 T3 ∑ (||( Fˆ PC )′ uut ||2 + ||( Fˆ PC )′ FC′ ut ||2 + ||( Fˆ PC )′ uCFt ||2 ).

t =1

t =1

2

Here 1 N2 T3

T

ˆ PC ′

∑ ||( F

t =1

) uut ||

2

2 T PC ′ ˆ ∑ ∑ Fs us ut t =1 s =1 ! ! 1 T ˆ PC 2 1 T T || Fs || ∑ ∑ ( N −1 u′s ut )2 = O p (T −1 ), T s∑ T =1 t =1 s =1

=

1 N2 T3

≤

1 T

T

where the last equality holds because || FˆsPC ||2 = r, and, since N −1 u′t ut = O p (1) and N −1/2 u′s ut = O p (1) whenever t 6= s, 1 T

T

∑

T

∑ ( N −1 u′s ut )2 =

t =1 s =1

1 T

T

∑ ( N −1 u′t ut )2 + t =1

T 1 NT

T

T

∑ ∑ ( N −1/2 u′s ut )2 = O p (1) + O p (TN −1 ). t =1 s 6 = t

Also, from || Fs || = O p (1) and || N −1/2 C ′ ut || = O p (1), T

1 2 N T3

∑ ||( Fˆ PC )′ FC′ ut ||2

2 T ∑ ∑ FˆsPC Fs′ C′ ut t =1 s =1 ! ! 1 T 1 T ˆ PC 2 || Fs || ∑ ||Fs ||2 T s∑ T =1 s =1

=

1 2 N T3

≤

1 N

t =1

= Op(N

T

−1

1 T

T

∑ || N −1/2 C′ ut ||2

t =1

!

),

with N −2 T −3 ∑tT=1 ||( Fˆ PC )′ uCFt ||2 being of the same order. Hence, since ||V −1 || = O p (1), 1 T

T

∑ ||dtPC ||2 ≤ ||V −1 ||2

t =1

= O p (T

−1

3 2 N T3

T

∑ (||( Fˆ PC )′ uut ||2 + ||( Fˆ PC )′ FC′ ut ||2 + ||( Fˆ PC )′ uCFt ||2 )

t =1

) + O p ( N −1 ),

and so the proof is complete.



Lemma CA1. Under Assumptions JOINT and CA, 1 T

T

2 −1 ∑ ||dCA t || = O p ( N ).

t =1

Proof of Lemma CA1.

√ The proof of Lemma CA1 is a simple consequence of the fact that || Nut || = O p (1), as seen by writing N T

T

2 ∑ ||dCA t || ≤

t =1

1 T

T

∑ ||

t =1

√

Nut ||2 = O p (1).

 3

Lemma PC2. Under the conditions of Lemma PC1, √ √ || NT −1/2 F ′ dPC || = O p ( NT −1/2 ) + O p (1). Proof of Lemma PC2. Write

√

NT

−1/2 ′ PC

Fd

= =

√ N √ T √

T

T

1

∑ Ft (dtPC )′ = √ NT3/2 ∑ Ft gt′ Fˆ PC V −1 t =1

t =1

T

1 NT 3/2

∑ Ft (u′t u′ + u′t CF′ + Ft′ C′ u′ ) Fˆ PC V −1.

(8)

t =1

The first term on the right involves

√

T

1

∑ Ft u′t u′ Fˆ PC = NT 3/2 t =1

where 1 √ NT 3/2

√

1

T

1

T

∑ Ft u′t u′ dPC + √ NT3/2 ∑ Ft u′t u′ FH, NT 3/2 t =1

t =1

T T 1 ∑ Ft u′t u′ dPC = √ NT3/2 ∑ ∑ Ft u′t us (dsPC )′ t =1 t =1 s =1  2 1/2 !1/2 T T T 1 1 1 PC 2 ′ ||ds || ≤  ∑ √ Ft ut us  T s=1 NT t∑ T s∑ =1 =1 √ = [O p ( NT −1/2 ) + O p (1)][O p ( N −1/2 ) + O p ( T −1/2 )] √ = O p ( NT −1 ) + O p ( N −1/2 ) + O p ( T −1/2 ), T

as T −1 ∑sT=1 ||dsPC ||2 = O p ( N −1 ) + O p ( T −1 ) by Lemma PC1 and

√

T

1 NT

∑ Ft u′t us = t =1

√

NT −1/2 N −1 Fs u′s us + √

1

T

∑ Ft u′t us = O p ( NT

√

NT −1/2 ) + O p (1).

t6=s

Also,

√

1

T

Ft u′t u′ F ∑ 3/2 NT t =1

=

√

1

T

T

Ft u′t us Fs′ ∑ ∑ 3/2 NT t =1 s =1

√ T T 1 N 1 T 1 ′ ′ √ √ F u u F + = √ Ft u′t us Fs′ t t t t ∑ ∑ ∑ NT T T NT t=1 s6=t t =1 √ −1/2 √ = O p ( NT ) + O p ( T −1/2 ) = O p ( NT −1/2 ),

suggesting that, with || H || = O p (1), T T 1 1 1 ′ ′ ˆ PC ′ ′ PC √ ≤ F u u F F u u d + √ √ t t ∑ ∑ t t 3/2 3/2 NT NT 3/2 NT t =1 t =1 √ = O p ( NT −1/2 ) + O p ( N −1/2 ). 4

′ ′ F u u F t ∑ t || H || t =1 T

(9)

From ||( NT )−1/2 ∑tT=1 Ft u′t C || = O p (1), we further have T T T 1 1 Ft u′t CFs (dsPC )′ Ft u′t CF ′ dPC = √ √ ∑ ∑ ∑ 3/2 3/2 NT NT t =1 t =1 s =1 1/2  2 T T 1 1 Ft u′t C || Fs ||2  ≤  ∑ √ ∑ T s =1 NT t=1

= Op(N

−1/2

) + O p (T

−1/2

1 T

T

∑ ||dsPC ||2

s =1

!1/2

),

and 1 √ NT 3/2

1 ′ ′ F u CF F ≤ √ t ∑ t NT t =1 T

′ F u C t ∑ t ||T −1 F′ F|| = O p (1), t =1 T

giving T 1 ′ ′ ˆ PC F u CF F √ t t NT 3/2 t∑ =1 T 1 1 ′ ′ PC Ft ut CF d + √ ≤ √ NT 3/2 t∑ NT 3/2 =1

Similarly, 1 √ NT 3/2

T

∑ t =1



Ft u′t CF ′ F || H ||

= O p ( 1) .

(10)



∑ Ft Ft′ C′ u′ Fˆ PC = O p (1). t =1 T

(11)

from which it follows that, with ||V −1 || = O p (1), T √ 1 −1/2 ′ PC ′ ′ ′ ′ ′ ′ ′ ˆ PC || NT F d || ≤ √ Ft (ut u + ut CF + Ft C u ) F ||V −1 || NT 3/2 t∑ =1 √ = O p ( NT −1/2 ) + O p (1). This completes the proof.

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Lemma CA2. Under the conditions of Lemma CA1,

√ || NT −1/2 F ′ dCA || = O p (1). Proof of Lemma CA2. The proof is completed by noting that, by the independence of Ft and ut , √ T √ −1/2 ′ CA 1 T √ ′ N ′ NT F d = √ ∑ Ft (dCA t ) = √ ∑ Ft Nu t = O p (1). T t =1 T t =1 5

(13)

 Lemma PC3. Under the conditions of Lemma PC1, ′

NT −1 (dPC )′ dPC = H Q

−1

SQ

−1

√ H + O p ( N −1/2 ) + O p ( NT −2 ) + O p ( NT −3/2 ) + O p ( T −1/2 ).

Proof of Lemma PC3. By definition, NT −1 (dPC )′ dPC

= N −1 T −3 V −1 ( Fˆ PC )′ g′ g Fˆ PC V −1 ′

′

= N −1 T −3 V −1 ( H F ′ g′ gFH + H F ′ g′ gdPC + (dPC )′ g′ gFH + (dPC )′ g′ gdPC )V −1 . (14) Consider the send term on the right-hand side. Clearly, 1 T −1 −3 ′ ′ PC ′ ′ ′ || N T F g gd || = F g g d s s NT 3 s∑ =1

≤

1 T

T

∑ || N

−1 −2 ′ ′

T

s =1

F g gs ||

2

!1/2

1 T

T

∑ s =1

||dsPC ||2

!1/2

,

(15)

where the order of the second term in the product is given by Lemma PC1. As for the first term, N − 1 T − 2 F ′ g ′ gs

= N −1 T −2 F ′ (uu′ + FC ′ u′ + uCF ′ )(uus + FC ′ us + uCFs ) = N −1 T −2 ( F ′ uu′ uus + F ′ FC ′ u′ uus + F ′ uCF ′ uus + F ′ uu′ FC ′ us + F ′ FC ′ u′ FC ′ us + F ′ uCF ′ FC ′ us + F ′ uu′ uCFs + F ′ FC ′ u′ uCFs + F ′ uCF ′ uCFs ) = I1 + ... + I9 ,

(16)

where the dependence on s in I1 , ..., I9 has been suppressed.

6

Consider I1 , which can be expanded as I1 = N −1 T −2 F ′ uu′ uus

= =

T

T

1 NT 2

t =1 k=1

1 NT 2

∑ Fk u′k us u′s us + NT2 ∑ ∑ Fk u′k ut u′t us

∑ ∑ Fk u′k ut u′t us T

1

= NT −2 N −1 Fs u′s us u′s us + 1 NT 2

+

T

t 6 = s k=1

k=1

T

T

N 1 √ 3/2 3/2 T N T

T

T

1

1

T

∑ Fk u′k us u′s us + √T NT3/2 ∑ Ft u′t ut u′t us t6=s

k6 = s

∑ ∑ Fk u′k ut u′t us t6 = s k6 = t

√ = O p ( NT −2 ) + O p ( NT −3/2 ) + O p ( T −1/2 ),

(17)

where last equality follows from direct evaluation of the individual terms. The order of the second term, for example, is obtained as √ 1 1 T T N N 1 ′ ′ ′ √ ≤ F u u u u F u u s s i,s k k ∑ ∑ ∑ s k i,k 3/2 N NT 2 k6=s T NT k6=s i=1

√ ∑ u′j,s u j,s = O p ( NT −3/2 ). j=1 N

Consider I2 . Let Σu = diag(Σu,1 , ..., Σu,N ), such that E(ut u′t ) = Σu , an N (m + 1) × N (m +

1) matrix. Note that C ′ (ut u′t − Σu )us =

N

N

∑ ∑ Ci [ui,t u′j,t − E(ui,t u′j,t )]u j,s

i=1 j=1

N

N

=

N

∑ Ci (ui,t u′i,t − Σu,i )ui,s + ∑ ∑ Ci ui,t u′j,t u j,s , i=1 j6 = i

i=1

where both terms on the right are mean zero and independent across s 6= t, suggesting 1 NT

= = +

T

∑ C′ ut u′t us t6=s

1 NT

√

T

∑ C′ (ut u′t − Σu )us +

t6=s

1 NT

√

1

T

( T − 1) ′ C Σu us NT

N

1

1

T

N

N

√ ∑ ∑ ∑ Ci ui,t u′j,t u j,s Ci (ui,t u′i,t − Σu,i )ui,s + √ ∑ ∑ NT TN T t 6 = s i=1

t 6 = s i=1 j6 = i

1 ( T − 1) N √ √ ∑ Ci Σu,i ui,s N NT i=1

= O p (( NT )−1/2 ) + O p ( T −1/2 ) + O p ( N −1/2 ) = O p ( T −1/2 ) + O p ( N −1/2 ).

7

It follows that

|| I2 || = || N −1 T −2 F ′ FC ′ u′ uus || 1 T ′ ′ ≤ ||T −1 F ′ F || C u u u t t s NT t∑ =1

1 ≤ NT −1 || T −1 F ′ F |||| N −1/2 C ′ us |||| N −1 u′s us || + || T −1 F ′ F || NT √ = O p ( NT −1 ) + O p ( T −1/2 ) + O p ( N −1/2 ).

√

∑ C′ ut u′t us t6=s T

(18)

For I3 , we use the fact that

√

1 NT

T

∑ Fk u′k us k=1

= √

T

N

1

∑ ∑ Fk ui,k u′i,s NT i=1 k=1

√ 1 N 1 N = √ Fs ui,s u′i,s + √ ∑ N T NT i=1 √ = O p ( NT −1/2 ) + O p (1),

N

T

∑ ∑ Fk ui,k u′i,s i=1 k6 = s

giving 1 1 T T 1 ′ ′ √ F u u F u C || I3 || = || N −1 T −2 F ′ uCF ′ uus || ≤ √ s t k k t NT k∑ T NT t∑ =1 =1 √ √ = T −1 [O p ( NT −1/2 ) + O p (1)] = O p ( NT −3/2 ) + O p ( T −1 ).

(19)

Also, since N −1/2 T −2 F ′ uu′ F =

√

√

1

T

T

∑ ∑ Ft u′t uk Fk′ NT 2 t =1 k=1

T T 1 1 N 1 T Ft u′t ut Ft′ + √ Ft u′t uk Fk′ ∑ ∑ T NT t=1 T NT t=1 k∑ 6=t √ √ −1 −1 −1 = O p ( NT ) + O p ( T ) = O p ( NT ),

=

we get

√ || I4 || = || N −1 T −2 F ′ uu′ FC ′ us || ≤ || N −1/2 T −2 F ′ uu′ F |||| N −1/2 C ′ us || = O p ( NT −1 ).

(20)

As for I5 , by using arguments similar to those used in the above,

|| I5 || = || N −1 T −2 F ′ FC ′ u′ FC ′ us || ≤ T −1/2 ||T −1 F ′ F ||||( NT )−1/2 C ′ u′ F |||| N −1/2 C ′ us || = O p ( T −1/2 ), with || I6 || being of the same order. 8

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For I7 ,

|| I7 || = || N −1 T −2 F ′ uu′ uCFs || 1 T T ′ ′ F u u u CF = ∑ t t k k s NT 2 t∑ =1 k=1 1 T 1 ′ ′ ≤ F u u u CF + √ t t s ∑ t t 2 NT t=1 NT 2 √ = O p ( NT −1 ) + O p ( T −1/2 ), where the last equality holds because 1 T ′ ′ F u u u CF t t t t s NT 2 t∑ =1 !1/2 √ N 1 T −1 ′ 2 ≤ || N Ft ut ut || T T t∑ =1 and 1 √ NT 2

1 T

∑ ∑ Ft u′t uk u′k CFs t =1 k6 = t  2 1/2 T T 1 1 1 ′  √  ∑ √ ∑ Ft ut uk T T k6=t NT t=1

≤

T

∑ t =1

∑ ∑ Ft u′t uk u′k CFs t =1 k6 = t T

T

|| N −1/2 u′t C ||2

!1/2

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√ || Fs || = O p ( NT −1 ),

T

T

1 T

T

∑ k6 = t

|| N −1/2 u′k C ||2

!1/2

|| Fs || = O p ( T −1/2 ).

The order of I8 can be obtained from 1 NT

T

∑ C′ ut u′t C t =1

= =

√ √

T

1

C ′ (ut u′t − Σu )C + N −1 C ′ Σu C ∑ NT t =1

1 NT

√

1 NT

= O p (( NT )

T

∑

N

∑ Ci (ui,t u′i,t − Σu,i )Ci′ +

t =1 i=1

−1/2

1 NT

T

N

∑∑

N

∑ Ci ui,t u′j,t Cj′ +

t =1 i=1 j=1

1 N

N

∑ Ci Σu,i Ci′ i=1

) + O p ( 1) = O p ( 1) ,

where 1 NT

T

N

N

∑ ∑ ∑ Ci ui,t u′j,t Cj′ = t =1 i=1 j=1

1 NT

T

N

T

1

N

N

∑ ∑ Ci ui,t u′i,t Ci′ + NT ∑ ∑ ∑ Ci ui,t u′j,t Cj′ = O p (1),

t =1 i=1

t =1 i=1 j6 = i

implying 1 || I8 || = || N −1 T −2 F ′ FC ′ u′ uCFs || ≤ ||T −1 F ′ F || NT 9

′ ′ C u u C t ∑ t || Fs || = O p (1). t =1 T

(23)

It remains to consider I9 , whose order is given by 2 1 T 1 −1 ′ F u C || I9 || = || N −1 T −2 F ′ uCF ′ uCFs || = √ t t || Fs || = O p ( T ). T NT t∑ =1

(24)

Hence, by putting everything together,

√ || N −1 T −2 F ′ g′ gs || ≤ || I1 || + ... + || I9 || = O p ( NT −2 ) + O p ( NT −1 ) + O p (1),

(25)

and therefore

|| N −1 T −3 F ′ g′ gdPC || ≤

1 T

T

∑ || N −1 T −2 F′ g′ gs ||2

s =1

−2

√

!1/2

1 T

T

∑ ||dsPC ||2

s =1

!1/2

) + O p (1)][O p ( N −1/2 ) + O p ( T −1/2 )] √ = O p ( N −1/2 ) + O p ( NT −5/2 ) + O p ( NT −3/2 ) + O p ( T −1/2 ). = [O p ( NT

) + O p ( NT

−1

(26)

The order of || N −1 T −3 (dPC )′ g′ gF || is the same. Thus, since ||V −1 || = O p (1) and the order of

|| N −1 T −3 (dPC )′ g′ gdPC || is dominated by that of || N −1 T −3 F ′ g′ gdPC ||, we obtain NT −1 (dPC )′ dPC ′

′

= N −1 T −3 V −1 ( H F ′ g′ gFH + H F ′ g′ gdPC + (dPC )′ g′ gFH + (dPC )′ g′ gdPC )V −1 √ ′ = N −1 T −3 V −1 H F ′ g′ gFHV −1 + O p ( N −1/2 ) + O p ( NT −5/2 ) + O p ( NT −3/2 ) + O p ( T −1/2 ).

(27)

Consider N −1 T −3 F ′ g′ gF, which we expand in the following obvious fashion: N −1 T −3 F ′ g′ gF

= = =

1 NT 3 1 NT 3 1 NT 3

T

∑ F′ gt gt′ F t =1 T

∑ F′ (uut + FC′ ut + uCFt )(u′t u′ + u′t CF′ + Ft′ C′ u′ ) F

t =1 T

∑ ( F′ uut u′t u′ F + F′ FC′ ut u′t u′ F + F′ uCFt u′t u′ F + F′ uut u′t CF′ F

t =1

+ F ′ FC ′ ut u′t CF ′ F + F ′ uCFt u′t CF ′ F + F ′ uut Ft′ C ′ u′ F + F ′ FC ′ ut Ft′ C ′ u′ F + F ′ uCFt Ft′ C ′ u′ F ) = J1 + ... + J9 .

(28)

10

For J1 , 1 NT 3

J1 =

1 NT 3

=

1 NT 3

=

1 NT 3

+ where 1 NT 3 1 NT 3

≤

and 1 NT 3

T

∑ F′ uut u′t u′ F t =1 T T

T

∑ ∑ ∑ Fs u′s ut u′t uk Fk′

t =1 s =1 k=1 T

T

T

1

t =1

T

T

t =1 s 6 = t

T

t =1 s 6 = t k6 = t

N 1 ′ ′ ′ F u u u u F t t t ∑ t t t ≤ T2 T t =1

T

∑ || N −1 Ft u′t ut ||2 = O p ( NT −2 ),

t =1

T T ′ ′ ′ F u u u u F s t t ∑ ∑ s t t t =1 s 6 = t  2 1/2 T T 1 1 1 ′  √ F u u s t ∑ ∑ s T 3/2 T t=1 NT s6=t T

t =1 k6 = t

∑ ∑ ∑ Fs u′s ut u′t uk Fk′ ,

T

T

T

T

1

∑ Ft u′t ut u′t ut Ft′ + NT3 ∑ ∑ Fs u′s ut u′t ut Ft′ + NT3 ∑ ∑ Ft u′t ut u′t uk Fk′

T

∑∑∑ t =1 s 6 = t k6 = t



11 ≤ TT

Fs u′s ut u′t uk Fk′

1 T

T

∑ || N −1/2 u′t ut Ft′ ||2

t =1

1 ∑ √ NT t =1 T

2

T

Fs u′s ut

∑ s6=t

!1/2

= O p ( T −3/2 ),

= O p ( T −1 ).



∑tT=1 ∑kT6=t Ft u′t ut u′t uk Fk′ /NT 3 is of the same order as ∑tT=1 ∑sT6=t Fs u′s ut u′t ut Ft′ /NT 3 . Hence,

|| J1 || = O p ( NT −2 ) + O p ( T −1 ).

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For J2 , we use 1 NT 2

T

T

∑ ∑ C′ ut u′t us Fs′ = t =1 s =1

1 NT 2

T

T

1

T

∑ C′ ut u′t ut Ft′ + NT2 ∑ ∑ C′ ut u′t us Fs′ ,

t =1

t =1 s 6 = t

where 1 NT 2

= = +

T

∑ C′ ut u′t ut Ft′ t =1

1 NT 2

√ √

T

∑ C′ (ut u′t − Σu )ut Ft′ +

t =1

1 NT 3/2 1 NT 3/2

= Op(N

√ √

1 NT 1

T

N

∑∑ t =1 i=1 T

1 NT 2

Ci (ui,t u′i,t

T

∑ C′ Σu ut Ft′ t =1

− Σu,i )ui,t Ft′

N

t =1 i=1 −1/2 −3/2

√ √ ) + O p ( NT −3/2 ) = O p ( NT −3/2 ), 11

1

T

√

T 3/2 N 3/2 T

N

Ci Σu,i ui,t Ft′ ∑ ∑ NT

T

+

√

N

N

∑ ∑ ∑ Ci ui,t u′j,t u j,t Ft′ t =1 i=1 j6 = i

and, by exactly the same argument, 1 NT 2

T

T

∑ ∑ C′ ut u′t us Fs′ t =1 s 6 = t

= +

√ √

1 NT

giving

T

1

N

T

T

1 1

T

N

N

Ci ui,t u′j,t u j,s Fs′ Ci (ui,t u′i,t − Σu,i )ui,s Fs′ + ∑ ∑ ∑ ∑ ∑ ∑ ∑ T NT NT

1 NT 3/2

= Op(N Therefore, 1 NT 2

√

t =1 s 6 = t i=1

√

T

1

t =1 s 6 = t i=1 j6 = i

N

∑ ∑ Ci Σu,i ui,s Fs′ NT

−1/2 −1

T

s 6 = t i=1

√ √ ) + O p ( T −1 ) + O p ( NT −3/2 ) = O p ( T −1 ) + O p ( NT −3/2 ).

√ ′ ′ ′ C u u u F ∑ ∑ t t s s = O p (T −1 ) + O p ( NT −3/2 ). t =1 s =1 T

|| J2 || ≤ ||T

T

1 F F || NT 2

T

−1 ′

∑C t =1

′



ut u′t u′ F

√ = O p ( T −1 ) + O p ( NT −3/2 ).

(30)



√ We know from before that || N −1/2 T −2 F ′ uu′ F || = O p ( NT −1 ) and ||( NT )−1/2 F ′ uC || = O p (1). This implies 1 1 −1/2 ′ F uC || √ || J3 || ≤ √ ||( NT ) NT 2 T

T

T

∑∑ t =1 s =1



Ft u′t us Fs′

√ = O p ( NT −3/2 ).

(31)



Also, since J4 = J2′ , || J4 || is of the same order of magnitude as || J2 ||. Consider J5 , where 1 NT

T

∑ C′ ut u′t C = t =1

= = = =

T

N

N

1 NT

t =1 i=1 j=1

1 NT

∑ ∑ Ci ui,t u′i,t Ci′ + √T N √ T ∑ ∑ ∑ Ci ui,t u′j,t Cj′

1 NT 1 N 1 N

∑ ∑ ∑ Ci ui,t u′j,t Cj′ T

N

1

T

1

t =1 i=1 T

N

N

t =1 i=1 j6 = i

N

∑ ∑ Ci ui,t u′i,t Ci′ + O p (T −1/2 )

t =1 i=1 N

1

1

T

N

∑ Ci Σu,i Ci′ + √ NT √ NT ∑ ∑ Ci (ui,t u′i,t − Σu,i )Ci′ + O p (T −1/2 ) t =1 i=1

i=1 N

∑ Ci Σu,i Ci′ + O p (( NT )−1/2 ) + O p (T −1/2 )

i=1

= S + O p ( T −1/2 ),

12

suggesting that J5 = ( T −1 F ′ F )

1 NT

T

∑ C′ ut u′t C(T −1 F′ F) = (T −1 F′ F)S(T −1 F′ F) + O p (T −1/2 ).

(32)

t =1

The order of || J6 || is 1 1 T 1 ′ ′ ′ ≤ F uCF u CF F || J6 || = √ t ∑ t 3 NT t=1 T NT

2 ∑ Ft u′t C ||T −1 F′ F|| = O p (T −1 ), t =1 T

(33)

√ while that of || J7 || is O p ( NT −3/2 ), as follows from the fact that J7 = J3′ . Similarly, since J8 = J6′ , we have || J8 || = O p ( T −1 ).

Finally, consider J9 , which can be evaluated in the following fashion: 1 T 11 T ′ ′ ′ ′ || J9 || = ≤ F uCF F C u F ||( NT )−1/2 F ′ uC ||2 || Ft ||2 = O p ( T −1 ). t ∑ t NT 3 t=1 T T t∑ =1

(34)

Hence, by adding the results, N −1 T −3 F ′ g′ gF

= J1 + ... + J9 √ = ( T −1 F ′ F )S( T −1 F ′ F ) + O p ( NT −2 ) + O p ( NT −3/2 ) + O p ( T −1/2 ),

(35)

which in turn implies ′

NT −1 (dPC )′ dPC = N −1 T −3 V −1 H F ′ g′ gFHV −1 + O p ( N −1/2 ) + O p ( NT −5/2 ) √ + O p ( NT −3/2 ) + O p ( T −1/2 ) ′

= V −1 H ( T −1 F ′ F )S( T −1 F ′ F ) HV −1 + O p ( N −1/2 ) + O p ( NT −2 ) √ + O p ( NT −3/2 ) + O p ( T −1/2 ) (36) According to Lemma PC2, || T −1 F ′ dPC || = O p ( T −1 ) + O p (( NT )−1/2 ), from which it follows that H = Q( T −1 F ′ Fˆ PC )V −1 = Q( T −1 F ′ FH )V −1 + Q( T −1 F ′ dPC )V −1

= QT −1 F ′ FHV −1 + O p ( T −1 ) + O p (( NT )−1/2 ), or, since Q is invertible, T −1 F ′ FHV −1 = Q

−1

H + O p ( T −1 ) + O p (( NT )−1/2 ),

13

with which we obtain −1

′

−1

NT −1 (dPC )′ dPC = H Q SQ H + O p ( N −1/2 ) + O p ( NT −2 ) √ + O p ( NT −3/2 ) + O p ( T −1/2 ),

(37)

as was to be shown.



Lemma CA3. Under the conditions of Lemma CA1, NT −1 (dCA )′ dCA = Σu + O p ( T −1/2 ).

Proof of Lemma CA3. A direct calculation reveals that NT −1 (dCA )′ dCA

N T

= = =

T

CA ∑ dCA t ( dt ) = t =1

N T

T

∑ ut u′t = t =1

N

T

1 NT

1 1 ∑ ∑ ui,t u′i,t + √ T N √ T t =1 i=1

1 NT

∑ ∑ ui,t u′i,t + O p (T −1/2 )

1 NT

T

N

T

N

N

∑ ∑ ∑ ui,t u′j,t t =1 i=1 j=1 N

∑ ∑ ∑ ui,t u′j,t t =1 i=1 j6 = i

N

T

t =1 i=1

= Σu + √

1 NT

= Σu + O p ( T

√

1

T

N

(ui,t u′i,t − Σu,i ) + O p ( T −1/2 ) ∑ ∑ NT

−1/2

t =1 i=1

),

(38)

as was to be shown.



Lemma PC4. Under the conditions of Lemma PC1, 1 T

N

∑ ηi′ dPC = i=1

1 N

N

∑ Ση,i ( β, Im )Ci′ Q

−1

√ H + O p ( NT −3/2 ) + O p ( N −1/2 ) + O p ( T −1/2 ).

i=1

Proof of Lemma PC4. The proof of this lemma is very similar to that of Lemma PC2. We begin by noting that 1 T

N

∑ ηi′ dPC = i=1

1 NT 2

N

T

∑ ∑ ηi,t (u′t u′ + u′t CF′ + Ft′ C′ u′ ) Fˆ PC V −1. i=1 t =1

Ignoring V −1 , the first term on the right is 1 NT 2

N

∑

T

∑ ηi,t u′t u′ Fˆ PC =

i=1 t =1

1 NT 2

N

∑

T

∑ ηi,t u′t u′ dPC +

i=1 t =1

14

1 NT 2

N

T

∑ ∑ ηi,t u′t u′ FH. i=1 t =1

(39)

Clearly, 1 NT

T

N

∑ ∑ ηi,t u′t us

1 NT

=

i=1 t 6 = s

T

N

∑∑

T

N

1

N

T

i=1 t 6 = s

i=1 j6 = i t 6 = s

= O p ( 1) + O p ( T − 1 ) = O p ( 1) ,

and therefore 1 NT

N

∑ ∑ ηi,t u′i,t ui,s + NT ∑ ∑ ∑ ηi,t u′i,t u j,s

√

1 N 1 N √ ∑ ηi,s ( N −1 u′s us ) + T NT N i=1 √ = O p ( NT −1 ) + O p (1).

ηi,t u′t us

=

i=1 t =1

N

T

∑ ∑ ηi,t u′t us i=1 t 6 = s

Thus, 1 NT 2

1 N T T ∑ ∑ ηi,t u′t u′ dPC = NT2 ∑ ∑ ∑ ηi,t u′t us (dsPC )′ i=1 t =1 i=1 t =1 s =1  2 1/2 !1/2 T N T T 1 1 1 ′ PC 2 ≤  ∑ ||ds || ∑ ηi,t ut us  T s=1 NT i∑ T s∑ =1 =1 t =1 √ = [O p ( NT −1 ) + O p (1)][O p ( N −1/2 ) + O p ( T −1/2 )] √ = O p ( NT −3/2 ) + O p ( N −1/2 ) + O p ( T −1/2 ). N

T

Moreover, 1 NT 2

= = = +

N

T

∑ ∑ ηi,t u′t u′ F i=1 t =1

1 NT 2 1 NT 2

T

N

∑ ∑ ∑ ηi,t u′t us Fs′ i=1 t =1 s =1 N T

∑

∑ ηi,t u′t ut Ft′ +

i=1 t =1

1

N

1 √

T 3/2 N T

√

T

1 NT

√

1

T

1 NT 2

ηi,t u′i,t ui,t Ft′

∑∑ i=1 t =1 N

T

N

T

T

∑ ∑ ∑ ηi,t u′t us Fs′ i=1 t =1 s 6 = t

+

√

N

1

N

√

T 3/2 N 3/2 T

T

1 1

N

T

∑ ∑ ∑ ηi,t u′j,t u j,t Ft′ i=1 j6 = i t =1 N

N

T

T

ηi,t u′i,t ui,s Fs′ + ηi,t u′j,t u j,s Fs′ ∑ ∑ ∑ ∑ ∑ ∑ ∑ 3/2 T NT NT i=1 t =1 s 6 = t

i=1 j6 = i t =1 s 6 = t

√

= O p ( T −3/2 ) + O p ( NT −3/2 ) + O p (( NT )−1/2 ) + O p ( T −1 ) √ = O p ( NT −3/2 ) + O p (( NT )−1/2 ) + O p ( T −1 ),

from which we deduce that 1 N T 1 N T 1 N T ′ ′ ˆ PC ′ ′ PC ′ ′ ≤ + η u u F η u u d η u u F i,t i,t i,t ∑ t ∑ t ∑ t || H || NT 2 i∑ NT 2 i∑ NT 2 i∑ =1 t =1 =1 t =1 =1 t =1 √ = O p ( NT −3/2 ) + O p ( N −1/2 ) + O p ( T −1/2 ). (40) 15

Let us now consider the second term in the expansion of T −1 ∑iN=1 ηi′ dPC . By the definition H, we have H = T −1 QF ′ Fˆ PC V −1 , or, since Q is invertible, T −1 F ′ Fˆ PC V −1 = Q

−1

H. This

implies 1 NT

N

∑

T

N

1 NT

∑ ηi,t u′t CT −1 F′ Fˆ PC V −1 =

i=1 t =1

∑

T

∑ ηi,t u′t CQ

−1

1 NT

H=

i=1 t =1

N

T

N

∑ ∑ ∑ ηi,t u′j,t Cj′ Q

−1

H.

i=1 t =1 j=1

′ β, η ′ )] = ( Σ β, Σ ) = Σ ( β, I ), suggesting that Here, E(ηi,t u′i,t ) = E[ηi,t (ǫi,t + ηi,t m η,i η,i η,i i,t

1 NT

N

T

N

∑ ∑ ∑ ηi,t u′j,t Cj′ i=1 t =1 j=1

= = + = =

1 NT 1 N

N

∑

T

∑ ηi,t u′i,t Ci′ +

i=1 t =1

N

1 N

N

N

T

∑ ∑ ∑ ηi,t u′j,t Cj′ i=1 t =1 j6 = i

1

N

1

T

∑ Ση,i ( β, Im )Ci′ + √ NT √ NT ∑ ∑ [ηi,t u′i,t − Ση,i ( β, Im )]Ci′ i=1 t =1

i=1

1 1 √ √ TN T 1 N

1 NT

N

T

N

∑ ∑ ∑ ηi,t u′j,t Cj′ i=1 t =1 j6 = i

N

∑ Ση,i ( β, Im )Ci′ + O p (( NT )−1/2 ) + O p (T −1/2 ) i=1 N

∑ Ση,i ( β, Im )Ci′ + O p (T −1/2 ).

i=1

Therefore, since || H || = O p (1), 1 NT 2

N

T

∑ ∑ ηi,t u′t CF′ Fˆ PC V −1

=

i=1 t =1

=

1 NT 1 N

N

T

N

∑ ∑ ∑ ηi,t u′j,t Cj′ Q

−1

H

i=1 t =1 j=1

N

∑ Ση,i ( β, Im )Ci′ Q

−1

H + O p ( T −1/2 ).

(41)

i=1

One term remains, the order of which is given by 1 N T ′ ′ ′ ˆ PC η F C u F i,t ∑ ∑ t NT 2 i=1 t=1 1 N T 1 N T ′ ′ ′ ′ ′ ′ PC || η F C u F η F C u d ≤ H || + ∑ i,t t ∑ i,t t NT 2 i∑ NT 2 i∑ =1 t =1 =1 t =1

= O p (( NT )−1/2 ) + O p ( T −1 ),

as follows from noting 1 N T 1 1 ′ ′ ′ = η F C u F √ i,t ∑ ∑ t 2 NT i=1 t=1 T NT

′ η F i,t ∑ ∑ t ||( NT )−1/2 C′ u′ F|| = O p (T −1 ), i=1 t =1 N

16

T

(42)

and 1 N T ′ ′ ′ PC η F C u d i,t ∑ t NT 2 i∑ =1 t =1 1 N T T ′ ′ PC ′ = η F C u ( d ) ∑ ∑ i,t t s s NT 2 i∑ =1 t =1 s =1  2 1/2 N T 1  1 T 1 ′ −1/2 ′ √ η F ( N C u ) ≤ √ s  i,t t ∑ ∑ ∑ T T NT i=1 t=1 s =1

1 T

T

∑ s =1

||dsPC ||2

!1/2

= O p (( NT )−1/2 ) + O p ( T −1 ).

This completes the proof.



Lemma CA4. Under the conditions of Lemma CA1, 1 T

N

∑ ηi′ dCA = i=1

1 N

N

∑ Ση,i ( β, Im ) + O p (T −1/2 ). i=1

Proof of Lemma CA4. By using the fact that E(ηi,t u′i,t ) = Ση,i ( β, Im ) (see Proof of Lemma PC4), we obtain 1 T

N

∑ ηi′ dCA

=

i=1

= = + =

1 T

i=1 t =1

1 NT 1 N

T

N

∑ ∑ ηi,t u′t = N

T

N

∑ ∑ ∑ ηi,t u′j,t i=1 t =1 j=1 N

1

T

N

∑ ∑ ηi,t u′i,t + NT ∑ ∑ ∑ ηi,t u′j,t

i=1 t =1

i=1 t =1 j6 = i

N

∑ Ση,i ( β, Im ) + √

i=1

1 1 √ √ TN T 1 N

T

N

1 NT

N

T

1 NT

√

1 NT

N

T

∑ ∑ [ηi,t u′i,t − Ση,i ( β, Im )]

i=1 t =1

N

∑ ∑ ∑ ηi,t u′j,t i=1 t =1 j6 = i

N

∑ Ση,i ( β, Im ) + O p (T −1/2 ),

(43)

i=1

as required for the proof.



Lemma PC5. Under the conditions of Lemma PC1, 2 (1, 0)Ci′ Q NT −1 ǫi′ dPC = σǫ,i

−1

√ H + O p ( N −1/2 ) + O p ( NT −1 ) + O p ( T −1/2 ) + O p ( NT −3/2 ).

Proof of Lemma PC5. 17

This proof is very similar to Proof of Lemma PC4. We have NT −1 ǫi′ dPC =

1 T2

T

∑ ǫi,t (u′t u′ + u′t CF′ + Ft′ C′ u′ ) Fˆ PC V −1.

(44)

t =1

Here 1 T2

T

∑ ǫi,t u′t u′ Fˆ PC = t =1

1 T2

T

∑ ǫi,t u′t u′ dPC + t =1

T

1 T2

∑ ǫi,t u′t u′ FH, t =1

with 1 T ′ ′ PC ǫ u u d i,t t NT 2 t∑ =1 1 T T ′ PC ′ = ǫ u u ( d ) s i,t ∑ ∑ t s NT 2 t=1 s=1   2 1/2 T T 1 1 ′  ≤ ∑ ǫi,t ut us  T s∑ NT =1 t =1

1 T

T

∑ s =1

||dsPC ||2

!1/2

= [O p ( N −1 ) + O p ( T −1 ) + O p (( NT )−1/2 )][O p ( N −1/2 ) + O p ( T −1/2 )]

= O p ( N −3/2 ) + O p ( N −1/2 T −1 ) + O p ( N −1 T −1/2 ) + O p ( T −3/2 ),

(45)

where we have made use of the fact that 1 NT

T

∑ ǫi,t u′t us

1 NT

=

t =1

1 1 NT

=

T

1

T

N

∑ ǫi,t u′i,t ui,s + NT ∑ ∑ ǫi,t u′j,t u j,s j6 = i t =1

t =1 T

N

1 1

1

1

N

t =1

= Op(N

−1

) + O p (T

−1

j6 = i

) + O p (( NT )

−1/2

j6 = i t6 = s

).

By using the same steps as in the proof of Lemma PC4, we can further show that 1 NT 2

= +

T

∑ ǫi,t u′t u′ F t =1

1 1 √ 3/2 NT T 1 √

1

N T T 3/2

T

∑ ǫi,t u′i,t ui,t Ft′ + t =1 T

∑

T

1 √

1

N

T 3/2 N T

∑ ǫi,t u′i,t ui,s Fs′ + √

t =1 s 6 = t

T

∑ ǫi,t u′i,t ui,s + T N ∑ ǫi,s u′j,s u j,s + √ NT √ NT ∑ ∑ ǫi,t u′j,t u j,s

1 NT

T

∑ ∑ ǫi,t u′j,t u j,t Ft′ j6 = i t =1

√

1

N

T

T

ǫi,t u′j,t u j,s Fs′ ∑ ∑ ∑ NT j6 = i t =1 s 6 = t

= O p ( N −1 T −3/2 ) + O p ( T −3/2 ) + O p ( N −1 T −1/2 ) + O p ( N −1/2 T −1 ) = O p ( T −3/2 ) + O p ( N −1 T −1/2 ) + O p ( N −1/2 T −1 ),

18

and so we obtain 1 N T 1 ′ ′ ˆ PC F η u u ≤ 2 ∑ ∑ i,t t 2 T i=1 t =1 T

T

N

∑∑ i=1 t =1

1 T2



ηi,t u′t u′ dPC +

√



N

T

∑∑ i=1 t =1



ηi,t u′t u′ F || H ||

= O p ( N −1/2 ) + O p ( NT −1 ) + O p ( T −1/2 ) + O p ( NT −3/2 ).

(46)

Again, in analogy to the proof of Lemma PC4, 1 NT

T

t =1

N

T

1 NT

∑ ǫi,t u′t CT −1 F′ Fˆ PC V −1 =

∑ ∑ ǫi,t u′j,t Cj′ Q

−1

H,

t =1 j=1

′ β, η ′ )] = ( σ2 , 0). Moreover, since where E(ǫi,t u′i,t ) = E[ǫi,t (ǫi,t + ηi,t i,t ǫ,i " !′ !# 1 T N 1 T N ′ ′ √ ∑ ∑ ǫi,t u j,t √ ∑ ∑ ǫi,t u j,t E T t =1 j6 = i T t =1 j6 = i

1 T

=

1 T

=

1 T

=

T

T

N

N

∑ ∑ ∑ ∑ E(u j,t ǫi,t ǫi,s u′k,s ) t =1 s =1 k6 = i j6 = i T

∑

N

N

∑ ∑ E(u j,t ǫi,t ǫi,t u′k,t ) +

t =1 k6 = i j6 = i T

1

N

N

t 6 = s s =1 k6 = i j6 = i

∑ σǫ,i2 E(u j,t u′j,t ) + T ∑ ∑ ∑

t =1

T

∑ ∑ ∑ ∑ E(u j,t ǫi,t ǫi,s u′k,s ) N

N

T

T

1 T

t=1 k6 = i j6 = i,j6 = k

2 2 σǫ,i E(u j,t u′k,t ) = σǫ,i Σu,j ,

′ which is bounded, we have || T −1/2 ∑tT=1 ∑ N j6 = i ǫi,t u j,t || = O p (1). Multiplication by a constant

′ ′ matrix does not affect this result. Hence, || T −1/2 ∑tT=1 ∑ N j6 = i ǫi,t u j,t C j || is O p (1) too. The impli-

cation is that 1 NT

=

T

N

∑ ∑ ǫi,t u′j,t Cj′ t =1 j=1

1 NT

T

t =1

t =1 j6 = i

2 = N −1 σǫ,i (1, 0)Ci′ +

=

N

T

1

∑ ǫi,t u′i,t Ci′ + NT ∑ ∑ ǫi,t u′j,t Cj′

2 (1, 0)Ci′ N −1 σǫ,i

1 1 √ √ N T T

+ Op(N

T

∑ [ǫi,t u′i,t − σǫ,i2 (1, 0)]Ci′ +

t =1

−1 −1/2

T

1 1 √ √ N T T

T

N

∑ ∑ ǫi,t u′j,t Cj′ t =1 j6 = i

),

giving 1 T2

T

∑ ǫi,t u′t CF′ Fˆ PC V −1 = t =1

1 T

T

N

∑ ∑ ǫi,t u′j,t Cj′ Q

−1

2 H = σǫ,i (1, 0)Ci′ Q

−1

H + O p ( T −1/2 ). (47)

t =1 j=1

Finally, since 1 1 T 1 ′ ′ ′ √ = ǫ F C u F √ i,t t T NT 2 t∑ NT =1

′ ǫ F ∑ i,t t ||( NT )−1/2 C′ u′ F|| = O p ( N −1/2 T −1 ), t =1 T

19

and 1 T ′ ′ ′ PC ǫ F C u d i,t t NT 2 t∑ =1 1 T T ′ ′ PC ′ ǫ F C u ( d ) = s i,t ∑ ∑ t s NT 2 t=1 s=1  2 1/2 T T 1 1 1 ′ −1/2 ′ ≤ √ C us )  √ ∑ ǫi,t Ft ( N ∑ T NT T t =1 s =1

1 T

T

∑ s =1

||dsPC ||2

!1/2

= O p ( N −1 T −1/2 ) + O p ( N −1/2 T −1 ),

we have 1 2 T

1 ′ ′ ′ ∑ ǫi,t Ft C u F || H || + T2 t =1 √ = O p ( T −1/2 ) + O p ( NT −1 ).

1 ∑ ǫi,t Ft′ C′ u′ Fˆ PC ≤ T2 t =1 T

T

∑ ǫi,t Ft′ C′ u′ dPC t =1 T

(48)

This establishes the required result.



Lemma CA5. Under the conditions of Lemma CA1, 2 (1, 0) + O p ( T −1/2 ). NT −1 ǫi′ dCA = σǫ,i

Proof of Lemma CA5. 2 ′ ′ The proof is implied by || T −1/2 ∑tT=1 ∑ N j6 = i ǫi,t u j,t || = O p (1) and E ( ǫi,t u i,t ) = σǫ,i (1, 0) (see

Proof of Lemma PC4), as is clear from NT −1 ǫi′ dCA =

1 T

T

∑ ǫi,t u′t = t =1

1 T

T

∑

N

∑ ǫi,t u′j,t =

t =1 j=1

1 1 2 (1, 0) + √ √ = σǫ,i T T

1 T

T

∑ ǫi,t u′i,t + t =1

T

1 T

1

T

N

∑ ∑ ǫi,t u′j,t t =1 j6 = i

1

T

N

∑ [ǫi,t u′i,t − σǫ,i2 (1, 0)] + √T √T ∑ ∑ ǫi,t u′j,t

t =1

t =1 j6 = i

2 = σǫ,i (1, 0) + O p ( T −1/2 ).

(49)

 Proof of Theorem 1. We begin by considering the PC estimator. Since H is r × k with rk( H ) = r ≤ k, the r × r ′

−

′

′

−

matrix HH is nonsingular. We may therefore define H = H ( HH )−1 , such that HH = Ir . The equation for yi can now be written as −

−

yi = xi β + Fˆ PC H λi − dPC H λi + ǫi ,

(50) 20

where dPC = Fˆ PC − FH is as before. The PC estimator of β is given by ! −1 N N βˆ ( Fˆ PC ) = ∑ xi′ M ˆ PC xi ∑ xi′ M ˆ PC yi , F

i=1

F

i=1

suggesting that

√

NT ( βˆ ( F

ˆ PC

) − β) =

1 NT

N

∑ i=1

xi′ M Fˆ PC xi

! −1

√

N

1

xi′ M Fˆ ∑ NT

PC

i=1

−

(ǫi − dPC H λi ).

We begin by considering the second term in the numerator. Clearly, M Fˆ PC dPC H

(51) −

=

−

M Fˆ PC ( Fˆ PC − FH ) H = − M Fˆ PC F, and therefore

−√

1 NT

N

−

∑ xi′ MFˆ PC dPC H λi =

√

=

√

i=1

N

1 NT 1

∑ xi′ MFˆ

PC

Fλi

i=1 N

∑ Λi F′ MFˆ PC Fλi + √

NT i=1 = K1 + K2 .

1 NT

N

∑ ηi′ MFˆ

PC

Fλi

i=1

(52)

Consider K1 . From ′

H F ′ M Fˆ PC FH = (dPC )′ M Fˆ PC dPC = (dPC )′ M FH dPC − (dPC )′ ( M FH − M Fˆ PC )dPC , we obtain K1 =

= = +

√ √ √ √

1

N

∑ Λi F′ MFˆ NT 1

i=1 N

PC

∑ Λi ( H NT

− ′

1

∑ Λi ( H NT

− ′

1

− ′

i=1 N i=1 N

∑ Λi ( H NT

Fλi ′

−

) H F ′ M Fˆ PC FHH λi −

) (dPC )′ M FH dPC H λi

i=1

−

) (dPC )′ ( M FH − M Fˆ PC )dPC H λi .

(53)

From the definitions of M FH and M Fˆ PC , M FH − M Fˆ PC

′ = dPC (( Fˆ PC )′ Fˆ PC )−1 (dPC )′ + dPC (( Fˆ PC )′ Fˆ PC )−1 H F ′ ′ ′ + FH (( Fˆ PC )′ Fˆ PC )−1 (dPC )′ + FH [(( Fˆ PC )′ Fˆ PC )−1 − ( H F ′ FH )−1 ] H F ′ ,

suggesting that

(dPC )′ ( M FH − M Fˆ PC )dPC

′ = (dPC )′ dPC (( Fˆ PC )′ Fˆ PC )−1 (dPC )′ dPC + (dPC )′ dPC (( Fˆ PC )′ Fˆ PC )−1 H F ′ dPC

+ (dPC )′ FH (( Fˆ PC )′ Fˆ PC )−1 (dPC )′ dPC ′

′

+ (dPC )′ FH [(( Fˆ PC )′ Fˆ PC )−1 − ( H F ′ FH )−1 ] H F ′ dPC . 21

(54)

Write ′ ′ ′ (( Fˆ PC )′ Fˆ PC )−1 − ( H F ′ FH )−1 = (( Fˆ PC )′ Fˆ PC )−1 [( Fˆ PC )′ Fˆ PC − H F ′ FH ]( H F ′ FH )−1 ′

′

= (( Fˆ PC )′ Fˆ PC )−1 [(dPC )′ Fˆ PC + H F ′ dPC ]( H F ′ FH )−1 . By Lemmas PC1 and PC2,

√ √ √ || NT −1/2 (dPC )′ Fˆ PC || ≤ NT || T −1 (dPC )′ dPC || + || NT −1/2 (dPC )′ F |||| H || √ √ = O p ( NT −1/2 ) + O p ( TN −1/2 ) + O p (1),

(55)

which in turn implies ′ T ||(( Fˆ PC )′ Fˆ PC )−1 − ( H F ′ FH )−1 ||

′ ′ ≤ ||( T −1 ( Fˆ PC )′ Fˆ PC )−1 ||T −1 ||(dPC )′ Fˆ PC + H F ′ dPC ||||( T −1 H F ′ FH )−1 ||

= O p ( T −1 ) + O p ( N −1 ) + O p (( NT )−1/2 ),

(56)

It follows that

||T −1 (dPC )′ ( M FH − M Fˆ PC )dPC || ≤ ||T −1 (dPC )′ dPC ||2 ||( T −1 ( Fˆ PC )′ Fˆ PC )−1 || + 2|| H ||||T −1 (dPC )′ dPC || ||( T −1 ( Fˆ PC )′ Fˆ PC )−1 || ||T −1 F ′ dPC ||

′ + || H ||2 ||T −1 (dPC )′ F ||2 T ||(( Fˆ PC )′ Fˆ PC )−1 − ( H F ′ FH )−1 ||

= [O p ( T −1 ) + O p ( N −1 )]2 + [O p ( T −1 ) + O p ( N −1 )][O p ( T −1 ) + O p (( NT )−1/2 )] + [O p ( T −1 ) + O p ( N −1 ) + O p (( NT )−1/2 )][O p ( T −1 ) + O p (( NT )−1/2 )]2 = O p ( T −2 ) + O p ( N −2 ) + O p ( N −3/2 T −1/2 ) + O p ( N −1/2 T −3/2 ) + O p (( NT )−1 ). Hence, 1 √ NT

≤

√

PC ′ PC H ) ( d ) ( M H λ Λ ( − M ) d PC ˆ i ∑ i FH F i=1 N

− ′

(57)

−

−

NT || H ||2 || T −1 (dPC )′ ( M FH − M Fˆ PC )dPC ||

1 N

N

∑ ||Λi ||||λi ||

i=1

√ √ = O p ( NT −3/2 ) + O p ( TN −3/2 ) + O p ( N −1 ) + O p ( T −1 ) + O p (( NT )−1/2 ).

22

(58)

′

′

Moreover, since (dPC )′ M FH dPC = (dPC )′ dPC − (dPC )′ FH ( H F ′ FH )−1 H F ′ dPC with 1 N − ′ PC ′ ′ ′ ′ − Λi ( H ) (d ) FH ( H F FH )−1 H F ′ dPC H λi √ ∑ NT i=1

√ 1 N − ′ || H ||2 || H ||2 || NT −1/2 (dPC )′ F ||2 ||( T −1 H F ′ FH )−1 || ∑ ||Λi || ||λi || N i=1 NT √ √ = ( NT )−1/2 [O p ( NT −1/2 ) + O p (1)]2 = O p ( NT −3/2 ) + O p (( NT )−1/2 ), √

≤

1

we obtain

√

K1 =

√

+

N

1

∑ Λi ( H NT i=1 N

NT

1 N

√ T 1 √ NN

=

− ′

+ Op(N

∑ Λi ( H

i=1 N

∑ Λi ( H

−

) (dPC )′ M FH dPC H λi − ′ −1

)T

−

(dPC )′ ( M FH − M Fˆ PC )dPC H λi

√ √ − ) NT −1 (dPC )′ dPC H λi + O p ( NT −3/2 ) + O p ( TN −3/2 )

− ′

i=1 −1

) + O p ( T −1 ) + O p (( NT )−1/2 ).

Application of Lemma PC3 now yields K1 =

√

√ √ TN −1/2 B1PC + O p ( NT −3/2 ) + O p ( TN −1 ) + O p ( N −1/2 ) + O p ( T −1 ),

where B1PC = N −1 ∑iN=1 Λi Q

−1

SQ

−1

(59)

λi −

Next, consider K2 . By using M FH Fλi = M FH FHH λi = 0, and then substitution for

( M FH − M Fˆ PC ), K2 =

√

NT

= −√ = −√ −

√

−

√

N

1 1 1

PC

Fλi

i=1 N

NT

∑ ηi′ ( MFH − MFˆ

PC

) Fλi

i=1 N

1

N

′

∑ ηi′ dPC (( Fˆ PC )′ Fˆ PC )−1 (dPC )′ Fλi − √ NT ∑ ηi′ dPC (( Fˆ PC )′ Fˆ PC )−1 H F′ Fλi NT

1 NT 1

∑ ηi′ MFˆ

i=1 i=1 N ηi′ FH (( Fˆ PC )′ Fˆ PC )−1 (dPC )′ Fλi i=1 N ′ ′ ηi′ FH [(( Fˆ PC )′ Fˆ PC )−1 − ( H F ′ FH )−1 ] H F ′ Fλi i=1

∑ ∑

NT = −K21 − ... − K24 .

Since (dtPC )′ (( Fˆ PC )′ Fˆ PC )−1 dsPC and Ft′ (( Fˆ PC )′ Fˆ PC )−1 d′ Fλi are just scalars, the orders of K21

23

and K23 can be inferred as follows: 1 N ′ PC ˆ PC ′ ˆ PC −1 PC ′ η d (( F ) F ) ( d ) Fλ ||K21 || = √ i i NT i∑ =1 1 T N T = √ ηi,t (dtPC )′ (( Fˆ PC )′ Fˆ PC )−1 ∑ dsPC Fs′ λi ∑ ∑ NT i=1 t=1 s =1 1 N T T PC ′ PC ′ ˆ PC −1 PC ′ ˆ ( d ) (( F ) F ) d η F λ = √ ∑ t s ∑ i,t s i NT t∑ =1 s =1 i=1  T T T √ 1 1 1 ≤ T ||( T −1 ( Fˆ PC )′ Fˆ PC )−1 || ∑ ||dtPC ||2  2 ∑ ∑ √ T t =1 T t =1 s =1 N √ = O p ( TN −1 ) + O p ( T −1/2 ),

2 1/2 ′  η F λ ∑ i,t s i i=1 N

and 1 ||K23 || = √ NT 1 = √ NT 1 = √ NT

N

∑ i=1



ηi′ FH (( Fˆ PC )′ Fˆ PC )−1 (dPC )′ Fλi

∑ ∑ ηi,t Ft′ H (( Fˆ PC )′ Fˆ PC )−1 (dPC )′ Fλi i=1 t =1 N T PC ′ PC − 1 PC ′ ′ ˆ ˆ λ η H (( F ) F ) ( d ) F F ∑ i i,t ∑ t t =1 i=1 !1/2 √ 1 1 T ≤ √ ||| H || || Ft ||2 ||( T −1 ( Fˆ PC )′ Fˆ PC )−1 || || NT −1/2 (dPC )′ F || ∑ T t =1 N  2 1/2 N T 1 1 ×  ∑ √ ∑ λi ηi,t  T t =1 N i=1 √ = O p ( T −1/2 ) + O p ( TN −1 ) + O p ( N −1/2 ). N

T

′ ′ Similarly, since Ft′ H [(( Fˆ PC )′ Fˆ PC )−1 − ( H F ′ FH )−1 ] H F ′ Fλi is a scalar, 1 N ′ ′ PC ′ ˆ PC −1 ′ −1 −1 ′ ′ ˆ F ) − ( ||K24 || = √ [(( F ) H F FH ) ] T H F Fλ FHT η i i NT i∑ =1 1 N T ′ ′ ′ ′ PC ′ ˆ PC −1 −1 ′ ˆ = √ Ft H [(( F ) F ) − ( H F FH ) ] H F F ∑ λi ηi,t NT t∑ =1 i=1 !1/2 T √ 1 ′ T ||(( Fˆ PC )′ Fˆ PC )−1 − ( H F ′ FH )−1 || || T −1 F ′ F || || Ft ||2 ≤ T || H ||2 ∑ T t =1  2 1/2 N T 1 1 ×  ∑ √ ∑ λi ηi,t  T t =1 N i=1 √ = O p ( T −1/2 ) + O p ( TN −1 ) + O p ( N −1/2 ).

24

′

K22 can be expanded as follows, by adding and subtracting ( T −1 H F ′ FH )−1 :

√

K22 =

√

=

√

=

√

+

N

1

′

ηi′ dPC (( Fˆ PC )′ Fˆ PC )−1 H F ′ Fλi ∑ NT i=1 N

1 NT 1

′

∑ ηi′ dPC (T −1 ( Fˆ PC )′ Fˆ PC )−1 T −1 H F′ FHH

i=1 N

∑ ηi′ dPC H NT

−

1 N

λi

λi

i=1 N

NT

−

′

′

∑ T −1 ηi′ dPC [(T −1 ( Fˆ PC )′ Fˆ PC )−1 − (T −1 H F′ FH )−1 ]T −1 H F′ Fλi ,

i=1

where the summand in last term on the right is ′ ′ − T −1 ||ηi′ dPC [( T −1 ( Fˆ PC )′ Fˆ PC )−1 − ( T −1 H F ′ FH )−1 ] T −1 H F ′ FHH λi || !1/2 !1/2 T 1 1 T ′ ||( T −1 ( Fˆ PC )′ Fˆ PC )−1 − ( T −1 H F ′ FH )−1 || ||ηi,t ||2 ||dtPC ||2 ≤ ∑ ∑ T t =1 T t =1 ′

−

× ||T −1 H F ′ FH |||| H ||||λi || = [O p ( T −1/2 ) + O p ( N −1/2 )][O p ( T −1 ) + O p ( N −1 ) + O p (( NT )−1/2 )] = O p ( T −3/2 ) + O p ( T −1/2 N −1 ) + O p ( N −1/2 T −1 ) + O p ( N −3/2 ). √ √ The order of the second term in K22 is NT times this, which is O p ( NT −1 ) + O p ( N −1/2 ) + √ O p ( T −1/2 ) + O p ( TN −1 ). The first term is, via Lemma PC4, √

N

1

∑ ηi′ dPC H NT

−

λi

i=1

√ T 1 N ′ PC − √ ∑ ηi d H λ i N T i=1 √ √ T 1 N −1 √ Ση,i ( β, Im )Ci′ Q λi + O p ( T −1 ) + O p ( TN −1 ) + O p ( N −1/2 ) ∑ N N i=1

= =

By using this and ( β, Im )Ci′ = ( β, Im )(Λ′i β + λi , Λ′i )′ = βλ′i + ( ββ′ + Im )Λi , we obtain K22 =

√

√ √ TN −1/2 B2PC + O p ( NT −1 ) + O p ( N −1/2 ) + O p ( T −1/2 ) + O p ( TN −1 ),

where B2PC = N −1 ∑iN=1 Ση,i [ βλ′i + ( ββ′ + Im )Λi ] Q

−√

1

N

∑ xi′ MFˆ NT i=1

PC

−1

(60)

λi . Thus, by adding the results,

−

dPC H λi = K1 + K2

√

√ TN −1/2 ( B1PC − B2PC ) + O p ( NT −1 ) + O p ( N −1/2 ) √ (61) + O p ( T −1/2 ) + O p ( TN −1 ). =

25

Next, consider ( NT )−1/2 ∑iN=1 xi′ M Fˆ PC ǫi , the first term in the numerator of

√

NT ( βˆ ( Fˆ PC ) −

β). Clearly,

√

1 NT

N

∑ xi′ MFˆ PC ǫi = √

i=1

N

1 NT

∑ xi′ MFH ǫi − √

i=1

1

N

xi′ ( M FH − M Fˆ ∑ NT

PC

) ǫi ,

(62)

i=1

where

√

1

N

∑ xi′ ( MFH − MFˆ

NT

i=1

=

√

+ +

√ √

1

N

1

N

PC

) ǫi

1 xi′ dPC (( Fˆ PC )′ Fˆ PC )−1 (dPC )′ ǫi + √ ∑ NT i=1 NT NT 1

N

′

∑ xi′ dPC (( Fˆ PC )′ Fˆ PC )−1 H F′ ǫi i=1

∑ xi′ FH (( Fˆ PC )′ Fˆ PC )−1(dPC )′ ǫi i=1 N

′

′

xi′ FH [(( Fˆ PC )′ Fˆ PC )−1 − ( H F ′ FH )−1 ] H F ′ ǫi ∑ NT i=1

= L1 + ... + L4 .

(63)

The order of L1 , ..., L4 can be obtained by using the same steps as when analyzing K2 . For L1 , we use the fact that xi,t = Λi Ft + ηi,t , giving 1 N 1 1 N √ ∑ xi,t ǫi,s ≤ √ ∑ Λi ǫi,s || Ft || + √ N i=1 N N i=1 which in turn implies 1 || L1 || = √ NT 1 = √ NT

N

∑ i=1

η ǫ ∑ i,t i,s = O p (1), i=1 N



xi′ dPC (( Fˆ PC )′ Fˆ PC )−1 (dPC )′ ǫi

∑ ∑ (dtPC )′ (( Fˆ PC )′ Fˆ PC )−1 dsPC ∑ xi,t ǫis t =1 s =1 i=1  T T √ 1 T 1 1 T ∑ ||dtPC ||2 ||( T −1 ( Fˆ PC )′ Fˆ PC )−1 ||  2 ∑ ∑ √ ≤ T t =1 T t =1 s =1 N √ = O p ( T −1/2 ) + O p ( TN −1 ). T

N

T

26

2 1/2 ∑ xi,t ǫis  i=1 N

(64)

Similarly, 1 || L4 || = √ NT

√

≤

N

∑

xi′ FH [(( Fˆ PC )′ Fˆ PC )−1

i=1

N || H ||

2

1 N

N

∑ i=1

||T −1 xi′ F ||2

′ ′

− ( H F FH )

!1/2

−1

] H F ǫi ′ ′

′

T ||(( Fˆ PC )′ Fˆ PC )−1 − ( H F ′ FH )−1 ||

!1/2 1 N −1/2 ′ 2 × ||T F ǫi || N i∑ =1 √ = O p ( NT −1 ) + O p ( N −1/2 ) + O p ( T −1/2 ).

(65)

′

Consider L2 . Adding and subtracting ( H F ′ FH )−1 give L2 =

= +

√ √ √

N

1

′

∑ xi′ dPC (( Fˆ PC )′ Fˆ PC )−1 H F′ ǫi NT i=1 N

1

′

′

xi′ dPC ( H F ′ FH )−1 H F ′ ǫi ∑ NT i=1 N

1

′

′

xi′ dPC [(( Fˆ PC )′ Fˆ PC )−1 − ( H F ′ FH )−1 ] H F ′ ǫi ∑ NT i=1

= L21 + L22 , where 1 || L22 || = √ NT 1 = √ NT

=

× = =

N

∑

xi′ dPC [(( Fˆ PC )′ Fˆ PC )−1

i=1

′ ′

− ( H F FH )

−1

] H F ǫi ′ ′

∑ ∑ ∑ xi,t (dtPC )′ [(( Fˆ PC )′ Fˆ PC )−1 − ( H F FH )−1 ] H Fs ǫi,s i=1 t =1 s =1 !1/2 √ 1 T ′ T || H || ||dtPC ||2 T ||(( Fˆ PC )′ Fˆ PC )−1 − ( H F ′ FH )−1 || ∑ T t =1 2 1/2 !1/2  T T T N 1 1 1 2  √ || Fs || ǫi,s xi,t  ∑ 2 ∑ ∑ T s∑ T N i=1 t =1 s =1 =1 √ T [O p ( T −1/2 ) + O p ( N −1/2 )][O p ( T −1 ) + O p ( N −1 ) + O p (( NT )−1/2 )] √ O p ( T −1 ) + O p ( N −1 ) + O p (( NT )−1/2 ) + O p ( TN −3/2 ). N

T

T

′ ′

′

Also, from xi = FΛ′i + ηi , L21 =

=

√ √

1

N

′

′

∑ xi′ dPC ( H F′ FH )−1 H F′ ǫi NT 1 NT

i=1 N

′

′

∑ Λi F′ dPC ( H F′ FH )−1 H F′ ǫi + √

i=1

27

1

N

′

′

ηi′ dPC ( H F ′ FH )−1 H F ′ ǫi . ∑ NT i=1

Lemma PC4 implies || NT −1 ηi′ dPC || = O p (1), from which it follows that 1 N ′ ′ ηi′ dPC ( H F ′ FH )−1 H F ′ ǫi √ ∑ NT i=1 1 1 √ NN

≤

N

′

∑ || NT −1 ηi′ dPC ||||(T −1 H F′ FH )−1 |||| H ||||T −1/2 F′ ǫi || = O p ( N −1/2 ),

i=1

and by further use of Lemma PC1, 1 N ′ ′ Λi F ′ dPC ( H F ′ FH )−1 H F ′ ǫi √ ∑ NT i=1 1 1 √ TN

≤

= T

−1/2

N

∑ ||Λi ||||

√

i=1

′

NT −1/2 F ′ dPC ||||( T −1 H F ′ FH )−1 |||| H |||| T −1/2 F ′ ǫi ||

√ √ [O p ( NT −1/2 ) + O p (1)] = O p ( NT −1 ) + O p ( T −1/2 ).

Consequently, 1 || L21 || ≤ √ NT

N

′ PC

∑ Λi F d i=1

′ ′

( H F FH )

√

−1

1 H F ǫi + √ NT ′ ′

= O p ( N −1/2 ) + O p ( NT −1 ) + O p ( T −1/2 ),

N

∑ i=1



′ ′ ηi′ dPC ( H F ′ FH )−1 H F ′ ǫi



leading to the following result for || L2 ||:

√ √ || L2 || ≤ || L21 || + || L22 || = O p ( N −1/2 ) + O p ( NT −1 ) + O p ( T −1/2 ) + O p ( TN −3/2 ). ′

(66)

Consider L3 . As when evaluating L2 , we begin by adding and subtracting ( H F ′ FH )−1 ; L3 =

= +

√ √ √

1 NT 1

N

∑ xi′ FH (( Fˆ PC )′ Fˆ PC )−1(dPC )′ ǫi i=1 N

′

∑ xi′ FH ( H F′ FH )−1 (dPC )′ ǫi NT 1

i=1 N

′

∑ xi′ FH [(( Fˆ PC )′ Fˆ PC )−1 − ( H F′ FH )−1 ](dPC )′ ǫi , NT i=1

28

where, in analogy to || L22 ||, 1 N ′ xi′ FH [(( Fˆ PC )′ Fˆ PC )−1 − ( H F ′ FH )−1 ](dPC )′ ǫi √ ∑ NT i=1 1 N T T ′ ′ ′ PC ′ ˆ PC −1 −1 PC ˆ x F H [(( F ) F ) − ( H F FH ) ] d ǫ = √ i,s ∑ ∑ i,t t s NT i∑ =1 t =1 s =1 !1/2 √ 1 T 1 ′ 2 T ≤ || Ft || || H ||T ||(( Fˆ PC )′ Fˆ PC )−1 − ( H F ′ FH )−1 || ∑ T t =1 T   2 1/2 N T T 1 1 ×  2 ∑ ∑ √ ∑ xi,t ǫi,s  T t =1 s =1 N i=1 √ = O p ( T −1 ) + O p ( N −1 ) + O p (( NT )−1/2 ) + O p ( TN −3/2 ). −

′

1

N

T

∑ s =1

||dsPC ||2

!1/2

By substituting for xi , and then use of Lemma PC5, ( H )′ H = Ir and (1, 0)(Λ′i β + λi , Λ′i )′ = 2 ( β′ Λ + λ ′ ), the first term in L can be written σǫ,i 3 i i

√

1

N

′

xi′ FH ( H F ′ FH )−1 (dPC )′ ǫi ∑ NT i=1

= = + =

√

1

N

′

∑ Λi F′ FH ( H F′ FH )−1 (dPC )′ ǫi + √

NT i=1 √ T 1 N − √ Λi ( H )′ NT −1 (dPC )′ ǫi ∑ N N i=1 1 1 √ NN √ T 1 √ NN

+ Op(N

N

′

∑ ηi′ FH ( H F′ FH )−1 (dPC )′ ǫi NT i=1

′

∑ T −1/2 ηi′ FH (T −1 H F′ FH )−1 NT −1 (dPC )′ ǫi i=1 N

∑ σǫ,i2 Λi Q

−1

√ (Λ′i β + λi ) + O p ( TN −1 ) + O p ( T −1/2 )

i=1 −1/2

√ ) + O p ( NT −1 ).

2 Λ Q Let B3PC = N −1 ∑iN=1 σǫ,i i

−1

(Λ′i β + λi ). Direct substitution into the expression for L3 now

yields L3 =

√

√ √ TN −1/2 B3PC + O p ( TN −1 ) + O p ( T −1/2 ) + O p ( N −1/2 ) + O p ( NT −1 ).

(67)

The results for L1 , ..., L4 imply

√

1

N

∑ xi′ ( MFH − MFˆ NT

PC

) ǫi

i=1

= L1 + ... + L4 √ √ √ = TN −1/2 B3PC + O p ( TN −1 ) + O p ( T −1/2 ) + O p ( N −1/2 ) + O p ( NT −1 ). 29

(68)

which in turn implies

√

1 NT

N

∑ xi′ MFˆ PC ǫi =

i=1

√

N

1

∑ xi′ MFH ǫi − NT

+ O p (T

i=1 −1/2

√

√ TN −1/2 B3PC + O p ( TN −1 )

√ ) + O p ( N −1/2 ) + O p ( NT −1 )

(69)

2 I By using the cross-section independence of ǫi , the serial independence of ηi,t , E(ǫi ǫi′ ) = σǫ,i T

′ ) = Σ , we can sow that and E(ηi,t ηi,t η,i " ! N 1 ′ ′ √ E ∑ ηi′ FH ( H F′ FH )−1 H F′ ǫi NT i=1

= = = = = =

N

1 NT

N

′

′

′

′

′

′

√

N

1 NT

′ ′

′ ′

′

′

∑ ηi′ FH ( H F FH )−1 H F ǫi i=1

!′ #

∑ ∑ E[ηi′ FH ( H F′ FH )−1 H F′ E(ǫi ǫ′j |ηi , ηj , F) FH ( H F′ FH )−1 H F′ ηj ]

i=1 j=1 N

1 NT

′

′

∑ σǫ,i2 E[ηi′ FH ( H F′ FH )−1 H F′ FH ( H F′ FH )−1 H F′ ηi ] i=1 N

1 NT

∑ σǫ,i2 E[ηi′ FH ( H F′ FH )−1 H F′ ηi ]

i=1 N

1 NT

T

T

′

′

∑ σǫ,i2 ∑ ∑ E[Ft′ H ( H F′ FH )−1 H Fs E(ηi,t ηi,s′ |F)]

1 NT

i=1 N

t =1 s =1 T

i=1 N

t =1

′

′

∑ σǫ,i2 ∑ E[Ft′ H ( H F′ FH )−1 H Ft Ση,i ]

1 k TN

∑ σǫ,i2 Ση,i = O p (T −1 ),

i=1

where last equality holds because T

′

′

∑ Ft′ H ( H F′ FH )−1 H Ft

T

=

t =1

′

′

∑ tr( Ft′ H ( H F′ FH )−1 H Ft ) t =1 T

= tr

′

′ ′

∑ H Ft Ft′ H ( H F FH )−1 t =1

!

= tr( Ik ) = k.

Thus, since the variance is O p ( T −1 ), we have 1 N ′ ′ −1/2 ′ −1 ′ ′ ). ( H F FH ) H F ǫ FH η √ i = O p ( T i NT i∑ =1

(70)

This result, together with the fact that M FH xi = M FH ηi , implies

√

1

N

∑ xi′ MFH ǫi NT

=

i=1

= =

√ √ √

1

N

∑ ηi′ MFH ǫi NT 1

i=1 N

1

N

′

′

∑ ηi′ ǫi − √ NT ∑ ηi′ FH ( H F′ FH )−1 H F′ ǫi NT 1

i=1 N

i=1

ηi′ ǫi + O p ( T −1/2 ). ∑ NT i=1

30

(71)

Moreover, since the forth-order moments of ηi,t and ǫi,t are bounded by assumption, by a central limit law for iid variates,

√

N

1

ηi′ ǫi ∼ N (0, W ) ∑ NT

(72)

i=1

2 Σ . Thus, as N, T → ∞, where ∼ signifies asymptotic equivalence and W = N −1 ∑iN=1 σǫ,i η,i √ −1 √ −1 provided that NT = o(1) and TN = o(1),

√

N

1

∑ xi′ MFˆ PC ǫi =

NT

i=1

√

1

N

∑ ηi′ ǫi − NT

√

√ TN −1/2 B3PC + O p ( TN −1 )

i=1 −1/2

√ ) + O p ( N −1/2 ) + O p ( NT −1 ) √ ∼ N (0, W ) − TN −1/2 B3PC . + O p (T

(73)

Let BPC = B1PC − B2PC − B3PC . The above results suggest the following limit for the √ numerator of NT ( βˆ ( Fˆ PC ) − β):

√

N

1

( xi′ M Fˆ ∑ NT

=

PC

i=1

√

N

1

−

ǫi − xi′ M Fˆ PC dPC H λi )

ηi′ ǫi + ∑ NT

√

√ √ TN −1/2 BPC + O p ( NT −1 ) + O p ( TN −1 )

i=1 −1/2

) + O p ( T −1/2 ) √ ∼ N (0, W ) + TN −1/2 BPC , √ √ which holds as N, T → ∞ with NT −1 → 0 and TN −1 → 0. √ Next, consider the denominator of NT ( βˆ ( Fˆ PC ) − β), which we expand as + Op(N

1 NT

N

∑ xi′ MFˆ

PC

xi =

i=1

1 NT

N

1

N

∑ xi′ MFH xi − NT ∑ xi′ ( MFH − MFˆ

i=1

(74)

PC

) xi .

(75)

i=1

√

By Lemma PC3, NT −1 ||dPC ||2 = O p (1), implying

NT −1/2 ||dPC || = O p (1). Similarly, since

T −1 || F ||2 = T −1 ∑tT=1 || Ft ||2 = O p (1), we have T −1/2 || F || = O p (1). Hence,

|| M FH − M Fˆ PC || ≤ T −1 ||dPC ||2 ||( T −1 ( Fˆ PC )′ Fˆ PC )−1 || + 2T −1 || H ||||dPC || || F || ||( T −1 ( Fˆ PC )′ Fˆ PC )−1 || ′

+ T −1 || H ||2 || F ||2 T ||(( Fˆ PC )′ Fˆ PC )−1 − ( H F ′ FH )−1 || = O p ( T −2 ) + O p ( N −1/2 ). implying 1 N 1 ′ − M ) x ( M x PC ˆ i ≤ || M FH − M Fˆ PC || ∑ i FH F NT i=1 NT 31

N

∑ ||xi ||2 = O p (T −2 ) + O p ( N −1/2 ).

i=1

By using this and ′

′

xi′ M FH xi = ηi′ M FH ηi = ηi′ ηi − T −1/2 ηi′ FH ( T −1 H F ′ FH )−1 T −1/2 H F ′ ηi = ηi′ ηi + O p (1), we obtain 1 NT

N

∑ xi′ MFˆ PC xi =

i=1

1 NT

N

∑ xi′ MFH xi + O p (T −2 ) + O p ( N −1/2 )

i=1

= Ση +

1 N

N

∑ (T −1 ηi′ ηi − Ση,i ) + O p (T −1 ) + O p ( N −1/2 )

i=1 −1

= Ση + O p ( T

) + O p ( N −1/2 ).

By adding all the results, as N, T → ∞ with

√

NT ( βˆ ( Fˆ PC ) − β)

= =

1 NT −1 Ση

N

∑ i=1

√

xi′ M Fˆ PC xi

1

N

∑ NT

! −1

ηi′ ǫi

+

√

√

NT −1 → 0 and

N

1 NT

TN

√

∑ ( xi′ MFˆ

i=1

−1/2

BPC

i=1

!

PC

(76)

√

TN −1 → 0,

−

ǫi − xi′ M Fˆ PC dPC H λi )

√ √ + O p ( NT −1 ) + O p ( TN −1 )

) + O p ( T −1/2 ) −1 −1 −1 √ ∼ N (0, Σ η WΣη ) + Ση TN −1/2 BPC . + Op(N

−1/2

This completes the proof for the PC estimator. The proof for the CA estimator uses the same steps as in the above with Fˆ PC , dPC , and H replaced by Fˆ CA , dCA , and C, respectively. The main differences are with the derivations of K1 , K2 and L3 , from which the three bias terms (B1PC , B2PC and B3PC ) are obtained. Consider K1 . By Lemmas CA1 and CA2,

√ √ √ || NT −1/2 (dCA )′ Fˆ CA || ≤ TN −1/2 || NT −1 (dCA )′ dCA || + || NT −1/2 (dCA )′ F ||||C || √ = O p ( TN −1/2 ) + O p (1), (77) giving ′

T ||(( Fˆ CA )′ Fˆ CA )−1 − (C F ′ FC)−1 ||

′

′

≤ ||( T −1 ( Fˆ CA )′ Fˆ CA )−1 ||T −1 ||(dCA )′ Fˆ CA + C F ′ dCA ||||( T −1 C F ′ FC)−1 || = O p ( N −1 ) + O p (( NT )−1/2 ),

(78)

32

and therefore

||T −1 (dCA )′ ( M FC − M Fˆ CA )dCA || ≤ ||T −1 (dCA )′ dCA ||2 ||( T −1 ( Fˆ CA )′ Fˆ CA )−1 || + 2||C ||||T −1 (dCA )′ dCA || ||( T −1 ( Fˆ CA )′ Fˆ CA )−1 || ||T −1 F ′ dCA ||

′ + ||C ||2 ||T −1 (dCA )′ F ||2 T ||(( Fˆ CA )′ Fˆ CA )−1 − (C F ′ FC )−1 ||

= O p ( N −2 ) + O p ( N −1 )O p (( NT )−1/2 ) + [O p ( N −1 ) + O p (( NT )−1/2 )]O p (( NT )−1 ) = O p ( N −2 ) + O p ( N −3/2 T −1/2 ). Hence, 1 √ NT

− ′ CA ′ CA − − M ) d C ) ( d ) ( M C λ Λ ( i ∑ i Fˆ CA FC i=1 N

√

≤

(79)

−

NT ||C ||2 || T −1 (dCA )′ ( M FC − M Fˆ CA )dCA ||

√

= O p ( TN

−3/2

) + Op(N

−1

1 N

N

∑ ||Λi ||||λi ||

i=1

).

(80)

By using this and 1 N − ′ CA ′ ′ ′ ′ − Λi (C ) (d ) FC(C F FC)−1 C F ′ dCA C λi √ ∑ NT i=1

√

≤

√ 1 − ′ ||C ||2 ||C ||2 || NT −1/2 (dCA )′ F ||2 ||( T −1 C F ′ FC )−1 || N NT 1

N

∑ ||Λi || ||λi ||

i=1

= O p (( NT )−1/2 ), we obtain

√ T 1 √ NN

K1 =

+ Op(N

N

∑ Λi (C

√ − ) NT −1 (dCA )′ dCA C λi + O p ( TN −3/2 )

− ′

i=1 −1

) + O p (( NT )−1/2 ).

Application of Lemma CA3 now yields K1 =

√

√ TN −1/2 B1CA + O p ( TN −3/2 ) + O p ( N −1/2 ),

−

′

(81)

′

where, via C = C (CC )−1 , B1CA =

1 N

N

−

−

∑ Λi (C )′ Σu C λi = i=1

1 N

N

′

′

′

∑ Λi (CC )−1CΣu C (CC )−1 λi . i=1

33

(82)

Next, consider K2 , which we again write as K2 = −K21 − ... − K24 with obvious definitions of K21 , ..., K24 . By using the same steps as before it is possible to show that

√ |K21 | = O p ( TN −1 ), |K23 | = O p ( N −1/2 ), √ |K24 | = O p ( TN −1 ) + O p ( N −1/2 ). Also, making use of Lemma CA4, K22 can be written as √ √ T 1 N ′ CA − K22 = √ ηi d C λi + O p ( TN −1 ) + O p ( N −1/2 ) ∑ N T i=1 √ √ T 1 N − = √ Ση,i ( β, Im )C λi + O p ( TN −1 ) + O p ( N −1/2 ) ∑ N N i=1 √ √ = TN −1/2 B2CA + O p ( TN −1 ) + O p ( N −1/2 ), ′

(83)

′

where B2CA = N −1 ∑iN=1 Ση,i [ βλ + ( ββ′ + Im )Λ] (CC )−1 λi . The last result holds because −

′

′

′

′

′

′

′

( β, Im )C = ( β, Im )C (CC )−1 = ( β, Im )(Λ β + λ, Λ )′ (CC )−1 = [ βλ + ( ββ′ + Im )Λ](CC )−1. The results for K1 and K2 imply

−√

1

N

∑ xi′ MFˆ NT

CA

−

dCA C λi

i=1

= K1 + K2 √ −1/2 √ √ = TN ( B1CA − B2CA ) + O p ( NT −1 ) + O p ( N −1/2 ) + O p ( TN −1 ). The appropriate version of L3 to consider in case of CA estimation is given by L3 =

= +

√ √ √

1

N

xi′ FC(( Fˆ CA )′ Fˆ CA )−1 (dCA )′ ǫi ∑ NT 1

i=1 N

′

xi′ FC(C F ′ FC )−1 (dCA )′ ǫi ∑ NT 1

i=1 N

′

xi′ FC[(( Fˆ CA )′ Fˆ CA )−1 − (C F ′ FC)−1 ](dCA )′ ǫi , ∑ NT i=1

34

(84)

where 1 N ′ xi′ FC[(( Fˆ CA )′ Fˆ CA )−1 − (C F ′ FC)−1 ](dCA )′ ǫi √ ∑ NT i=1 1 N T T ′ ′ ′ CA ′ CA − 1 − 1 CA ˆ ˆ x F = √ C [(( F ) F ) − ( C F FC ) ] d ǫ i,s ∑ ∑ i,t t s NT i∑ =1 t =1 s =1 !1/2 √ 1 T 1 ′ 2 ||C ||T ||(( Fˆ CA )′ Fˆ CA )−1 − (C F ′ FC )−1 || || Ft || T ≤ T t∑ T =1  2 1/2 T T N 1 1 ×  2 ∑ ∑ √ ∑ xi,t ǫi,s  T t =1 s =1 N i=1 √ T [O p ( N −1 ) + O p (( NT )−1/2 )][O p ( T −1/2 ) + O p ( N −1/2 )] = √ = O p ( N −1 ) + O p (( NT )−1/2 ) + O p ( TN −3/2 ). −

′

′

T

∑ s =1

2 ||dCA s ||

!1/2

′

and, via Lemma CA5 and (C )′ (1, 0)′ = (CC )−1 C(1, 0)′ = (CC )−1 (Λ β + λ),

√

1

N

′

xi′ FC(C F ′ FC )−1 (dCA )′ ǫi ∑ NT

= = + =

i=1

√

1

N

′

∑ Λi F′ FC(C F′ FC)−1 (dCA )′ ǫi + √

NT i=1 √ T 1 N − √ Λi (C )′ NT −1 (dCA )′ ǫi ∑ N N i=1 1 1 √ NN √ T 1 √ NN

N

1

N

′

ηi′ FC(C F ′ FC )−1 (dCA )′ ǫi ∑ NT i=1

′

∑ T −1/2 ηi′ FC(T −1 C F′ FC)−1 NT −1 (dCA )′ ǫi i=1 N

′

′

∑ σǫ,i2 Λi (CC )−1(Λ β + λ) + O p ( N −1/2 ).

i=1

′

′

2 Λ ( CC ) −1 ( Λ β + λ ), we obtain Hence, letting B3CA = N −1 ∑iN=1 σǫ,i i

L3 =

√

√ TN −1/2 B3CA + O p ( N −1/2 ) + O p ( TN −3/2 ).

(85)

The rest of the results are unaffected by the change of factor estimator. Therefore,

√

−1

−1 √

−1

NT ( βˆ ( Fˆ CA ) − β) ∼ N (0, Σ η WΣη ) + Ση

as N, T → ∞ with

√

NT −1 → 0 and

√

TN −1/2 BCA

(86)

TN −1 → 0, where BCA = B1CA − B2CA − B3CA . This

establishes the proof.



Remark 1. It is interesting to compare the proofs of the PC and CA estimators. In particular, not only are proofs of the CA estimator substantially shorter but there are also not as many 35

remainder terms. For example, consider Lemmas PC5 and CA5. While the remainder in the √ former lemma is O p ( N −1/2 ) + O p ( NT −1 ) + O p ( T −1/2 ) + O p ( NT −3/2 ), in the latter lemma the remainder is O p ( T −1/2 ), which is obviously less demanding in the sense it is o p (1) regardless of the relative expansion rate of N and T. In fact, N does not even have to go to infinity. Remark 2. Consider

√

NT [ βˆ ( Fˆ PC ) − βˆ ( Fˆ CA )]. By using the same steps as in the proof of

Theorem 1, we can show that

√

NT [ βˆ ( Fˆ PC ) − βˆ ( Fˆ CA )] √ √ = NT ( βˆ ( Fˆ PC ) − β) − NT ( βˆ ( Fˆ CA ) − β) √ √ −1 √ = Ση TN −1/2 ( BPC − BCA ) + O p ( NT −1 ) + O p ( TN −1 ) + O p ( N −1/2 ) + O p ( T −1/2 ) −1 √ = Ση TN −1/2 ( BPC − BCA ) + o p (1)

as N, T → ∞ with

√

NT −1 → 0 and

√

TN −1 → 0. Hence, if in addition

√

TN −1/2 → 0,

then the two estimators do not only have the same asymptotic distribution, but are in fact asymptotically equivalent.

3 On the asymptotic distributions of the estimated factors The asymptotic distributions of the PC and CA factor estimators can be easily obtained given the results reported in Section 2. In case of the PC estimator, we have ′ FˆtPC − H Ft = ( NT )−1 V −1 ( Fˆ PC )′ (uu′t + FC ′ ut + uCFt ),

where, by Lemma A.2 of Bai (2003),

( NT )−1 ( Fˆ PC )′ uu′t = O p ( N −1 ) + O p ( T −1 ), ( NT )−1 ( Fˆ PC )′ uCFt = O p ( N −1 ) + O p (( NT )−1/2 ). Hence, since ||V || = O p (1), as N, T → ∞ with

√

√

NT −1 → 0,

′ N ( FˆtPC − H Ft ) √ √ = N ( NT )−1 V −1 ( Fˆ PC )′ FC ′ ut + O p ( N −1/2 ) + O p ( T −1/2 ) + O p ( NT −1 ) √ ′ −1 = H Q N −1/2 C ′ ut + O p ( N −1/2 ) + O p ( T −1/2 ) + O p ( NT −1 ) ′

∼ N (0, H Q

−1

SQ

−1

H ).

(87) 36

In case of the CA estimator, we have

√

′ N ( FˆtCA − C Ft ) =

√

Nut ∼ N (0, Σ u )

(88)

√

NT −1 = o(1). √ ′ The fact that the asymptotic distribution of N ( FˆtPC − H Ft ) depends on the rotation

as N, T → ∞, which does not require

matrix H means that the relative variance of the estimators cannot be compared. However, while Ci and Ft are not identifiable, their product is. Proposition 1 therefore provides the limiting distributions of the PC and CA estimators of the common component, Ci′ Ft . The PC and CA estimators of Ci , henceforth denoted Cˆ iPC and Cˆ iCA , respectively, are obtained from a LS regression of zi,t onto the corresponding factor estimate. For instance, Cˆ iPC is obtained as the LS slope estimator in a regression of zi,t onto FˆtPC . Proposition 1. Under Assumptions JOINT, PC1, PC2 and CA, as N, T → ∞, 1 −1/2 ˆ p ′ ˆ p ( N −1 Ci′ Ω p Ci + T −1 Ft′ Σ− [(Ci ) Ft − Ci′ Ft ] →d N (0, 1), F Ft Σ u,i )

where p ∈ { PC, CA} and →d signifies convergence in distribution. Proof of Proposition 1. We begin by considering the PC estimator. The PC estimator of Ci is Cˆ iPC = (( Fˆ PC )′ Fˆ PC )−1 ( Fˆ PC )′ zi , where T −1 ( Fˆ PC )′ zi = T −1 ( Fˆ PC )′ FCi + T −1 ( Fˆ PC )′ ui ′ = T −1 H F ′ FCi + T −1 (dPC )′ FCi + T −1 ( Fˆ PC )′ ui ′

′

= T −1 H F ′ FCi + T −1 H F ′ ui + T −1 (dPC )′ FCi + T −1 (dPC )′ ui .

(89)

By Lemma PC2,

||T −1 (dPC )′ F || = O p ( T −1 ) + O p (( NT )−1/2 ), with || T −1 (dPC )′ ui || being of the same order (see, for example, Bai, 2003, Lemma B.1). Moreover, from the proof of Theorem 1, ′

T ||(( Fˆ PC )′ Fˆ PC )−1 − ( H F ′ FH )−1 || = O p ( T −1 ) + O p ( N −1 ) + O p (( NT )−1/2 ). 37

It follows that Cˆ iPC = (( Fˆ PC )′ Fˆ PC )−1 ( Fˆ PC )′ zi −

′

′

= H Ci + ( T −1 H F ′ FH )−1 T −1 H F ′ ui + O p ( T −1 ) + O p ( N −1 ) + O p (( NT )−1/2 ), − which can be used to show that, letting aiPC = Cˆ iPC − H Ci ,

(Cˆ iPC )′ FˆtPC − Ci′ Ft ′

= Ci′ ( H )− dtPC + ( aiPC )′ FˆtPC ′

′

= Ci′ ( H )− dtPC + ( aiPC )′ H Ft + ( aiPC )′ dtPC ′

′

′

′

= Ci′ ( H )− H (C ′ C )−1 C ′ ut + u′i FH ( H F ′ FH )−1 H Ft + O p ( T −1 ) + O p ( N −1 ) + O p (( NT )−1/2 ) = Ci′ (C ′ C )−1 C ′ ut + u′i F ( F ′ F )−1 Ft + O p ( T −1 ) + O p ( N −1 ) + O p (( NT )−1/2 ),

(90)

where the last equality uses that ′

′

u′i FH ( H F ′ FH )−1 H Ft ′

′

′

′

′

′

′

′

= u′i F ( H F ′ F )− ( H F ′ F ) H ( H F ′ FH )−1 ( H F ′ FH )( H F ′ FH )−1 H ( F ′ FH )( F ′ FH )− Ft = u′i F ( H F ′ F )− ( H F ′ FH )( F ′ FH )− Ft = u′i F ( F ′ F )−1 Ft (see Magnus and Neudecker, 1999, Miscellaneous Exercises 6, page 38). As for the first term on the right-hand side, we have

√

NCi′ (C ′ C )−1 C ′ ut = Ci′ ( N −1 C ′ C )−1 N −1/2 C ′ ut ∼ N (0, Ci′ Q

as N → ∞. Moreover, since

T

TE[u′i F ( F ′ F )−1 ( F ′ F )−1 F ′ ui ] = T ∑

−1

SQ

−1

Ci )

(91)

T

∑ E[ui,t Ft′ ( F′ F)−1 ( F′ F)−1 Fs u′is ]

t =1 s =1 T

= T ∑ E[ Ft′ ( F ′ F )−1 ( F ′ F )−1 Ft E(ui,t u′i,t | F )] t =1 !# " T

∑ Ft Ft′ ( F′ F)−1 ( F′ F)−1

= TE tr

= E[tr(( T

Σu,i

s =1 −1 ′

1 F F )−1 )]Σu,i → tr(Σ− F ) Σ u,i

as T → ∞, we get

√

1 Tu′i F ( F ′ F )−1 Ft = T −1/2 u′i F ( T −1 F ′ F )−1 Ft →d N (0, Ft′ Σ− F Ft Σ u,i ).

38

(92)

The fact that N −1/2 C ′ ut and T −1/2 u′i F are based on different summations implies that their asymptotic distributions are independent. Hence,

( N −1 Ci′ Q

−1

SQ

−1

1 1/2 ˆ PC ′ ˆ PC Ci + T −1 Ft′ Σ− [(Ci ) Ft − Ci′ Ft ] →d N (0, 1) F Ft Σ u,i )

(93)

as N, T → ∞. In case of the CA estimator, we simply replace H with C, giving − ′ − ′ (Cˆ iCA )′ FˆtCA − Ci′ Ft = Ci′ (C )′ ( FˆtCA − C Ft ) + (Cˆ iCA − C Ci )′ C Ft + o p (1) −

= Ci′ (C )′ ut + u′i F ( F ′ F )−1 Ft + o p (1) ′

= Ci′ (CC )−1 Cut + u′i F ( F ′ F )−1 Ft + o p (1), suggesting ′

′

′

1 1/2 ˆ CA ′ ˆ CA ( N −1 Ci′ (CC )−1 CΣu C (CC )−1 Ci + T −1 Ft′ Σ− [(Ci ) Ft − Ci′ Ft ] →d N (0, 1) F Ft Σ u,i )

as N, T → ∞.

(94)



Proposition 1 holds regardless of the relative expansion rate of N and T. However, the results simplify if either NT −1 = o(1) or TN −1 = o(1). On the one hand, if NT −1 = o(1), then p

p

−1/2 ˆ ′ ˆ 1 [(Ci ) Ft − Ci′ Ft ] ( N −1 Ci′ Ω p Ci + T −1 Ft′ Σ− F Ft Σ u,i ) √ p p −1/2 1 F Σ ) = (Ci′ Ω p Ci + NT −1 Ft′ Σ− N [(Cˆ i )′ Fˆt − Ci′ Ft ] t u,i F √ p p = (Ci′ Ω p Ci )−1/2 N [(Cˆ i )′ Fˆt − Ci′ Ft ] + o p (1). (95) √ p p suggesting that (Cˆ i )′ Fˆt is N-consistent for Ci′ Ft and that the asymptotic variance is given

by Ci′ Ω p Ci . The fact that the variance is given by Ci′ Ω p Ci means that the relative efficiency depends on the choice of estimator of Ft . On the other hand, if TN −1 = o(1), then −1/2 ˆ p ′ ˆ FPC 1 [(Ci ) Ft − Ci′ Ft ] ( N −1 Ci′ Ω p Ci + T −1 Ft′ Σ− F Ft Σ u,i ) √ p p −1/2 1 = ( Ft′ Σ− T [(Cˆ i )′ Fˆt − Ci′ Ft ] + o p (1). F Ft Σ u,i )

(96)

This is true regardless of whether p = PC or p = CA. Hence, in contrast to the case when 1 NT −1 = o(1), since the variances are now the same (equal to Ft′ Σ− F Ft Σ u,i ), in this case the

two estimators of the common component are equally efficient. A Monte Carlo study was carried out to investigate the small-sample accuracy of the results contained in Proposition 1. The results, which are not reported but available upon request, suggest that the N (0, 1) approximation is very accurate, even for as small samples as N = T = 50. 39

4 A comparison with a residual-based PC estimator As mentioned in Remark 7 of the main paper (Westerlund and Urbain, 2013) the PC estimator of F need not be based on z, but can also be based on an estimator of ei . Bai (2009a) considers such an approach where PC is applied to eˆi = yi − xi βˆ (0), where βˆ (0) is the LS slope estimator in a regression of yi onto xi . This estimator can then be used in a second step to obtain the associated PC estimator of β. These steps can then be repeated until convergence, leading to the residual-based PC (RPC) estimators of F and β, henceforth denoted Fˆ RPC and √ βˆ ( Fˆ RPC ). The asymptotic distribution of NT ( βˆ ( Fˆ RPC ) − β) is reported in Proposition 2. This result is established under the conditions of the main paper making it comparable with the results presented in Theorem 1 for PC and CA. Proposition 2. Under Assumptions JOINT, PC1 and PC2, as N, T → ∞ with TN −1 = O(1) and √ −3/2 TN → 0,

√

−1 √

NT ( βˆ ( Fˆ RPC ) − β) ∼ Ση

TN −1/2 BRPC + N (0, Ω F ),

where BRPC =

1 N

Ω RPC = P U = P =

N

∑ Λi (ΩRPC − σǫ,i2 P

i=1 −1 −1

UP

1 N 1 N

−1

)λi ,

,

N

∑ σǫ,i2 λi λ′i , i=1 N

∑ λn λ′n .

n =1

Proof of Proposition 2. Proposition 2 is almost an immediate consequence of Theorem 3 in Bai (2009a). Here it is shown that if TN −1 = O(1), then

√

NT ( βˆ ( Fˆ RPC ) − β) ∼

√

TN −1/2 B +

√

NT −1/2 C + N (0, D ( F )−1 DZ [ D ( F )′ ]−1 ),

(97)

where B, C, D ( F ) and DZ are given in Bai (2009a, pages 1240–1247). We now show that, √ provided TN −3/2 = o(1), the required result is implied by this. √ We begin by considering the variance of NT ( βˆ ( Fˆ RPC ) − β). D ( F ) can be written as D(F) =

1 NT

N

∑ Zi′ Zi , i=1

40

′ where Zi = M F ( xi − N −1 ∑ N j=1 x j ai,j ), with ai,j = λ i P

−1

λ j and P = N −1 ∑nN=1 λn λ′n . Note that

ai,j = O p (1). Moreover, under the assumptions of this paper, M F xi = M F ηi , and ηi′ M F ηj =

ηi′ ηj − T −1/2 ηi′ F ( T −1 F ′ F )−1 T −1/2 F ′ ηj = ηi′ ηj + O p (1), giving 1 N2 T

= = =

N

N

∑ ∑ xi′ MF xj ai,j i=1 j=1 N

N

1 N2 T

i=1 j=1

1 N2 T

∑ ηi′ MF ηi ai,i + N2 T ∑ ∑ ηi′ MF ηj ai,j

∑ ∑ ηi′ MF ηj ai,j N

1

i=1

1 1 N NT

= Op(N

N

N

i=1 j6 = i

N

1

1

= =

N

N

N

i=1 j6 = i

i=1

−1

) + Op(N

−1 −1/2

T

) + O p (( NT )−1 ) = O p ( N −1 ),

and, since N −1 ∑iN=1 ai,j ai,k = N −1 ∑iN=1 λ′k P 1 N3 T

N

∑ ηi′ ηi ai,i + N √ T N √ T ∑ ∑ ηi′ ηj ai,j + O p (( NT )−1 ) −1

λi λ′i P

−1

λ j = N −1 λ′k Pλ j = ak,j

N

∑ ∑ ∑ xk′ MF xj ai,j ai,k i=1 k=1 j=1

1 N3 T

N

N

N

∑ ∑ ∑ ηk′ MF ηj ai,j ai,k = i=1 k=1 j=1

1 1 N 2 NT

= Op(N

N

∑ ηj′ ηj aj,j + j=1

−2

) + Op(N

1 √

1 √

1 N3 T

N2 T N T

−2 −1/2

T

N

N

∑ ∑ ηk′ MF ηj ak,j k=1 j=1

N

N

∑ ∑ ηk′ ηj ak,j + O p ( N −2 T −1 ) k=1 j6 = k

) + O p ( N −2 T −1 ) = O p ( N −2 ).

This implies D(F) =

1 NT

N

∑ Zi′ Zi = i=1

= Ση + O p ( T

−1

1 NT

N

∑ ηi′ MF ηi + O p ( N −1 ) = i=1

) + Op(N

−1/2

1 NT

N

∑ ηi′ ηi + O p (T −1 ) + O p ( N −1 ) i=1

).

(98)

DZ is the (limiting) variance of ( NT )−1/2 ∑iN=1 Zi′ ǫi . Under the assumptions of the paper, 2 and E ( ǫ ǫ ) = 0 for all i 6 = j and t 6 = s. By using this and the same arguments E(ǫ2i,t ) = σǫ,i i,t j,k

as in the above, T −1 E( Zi′ Zi ) = T −1 E( xi′ M F xi ) + O( N −1 ) = T −1 E(ηi′ ηi ) + O( T −1 ) + O( N −1 )

= Ση,i + O( T −1 ) + O( N −1 ).

41

It follows that DZ = E

=

"

1 N

√

N

1

Zi′ ǫi

∑ NT

i=1

!

√

1

N

Zj′ ǫj

∑ NT

j=1

!′ #

1 NT

=

N

N

∑ ∑ E[Zi′ E(ǫi ǫ′j |Zi , Zj )Zj ]

i=1 j=1

N

∑ σǫ,i2 T −1 E(Zi′ Zi ) = W + O(T −1 ) + O( N −1 ),

(99)

i=1

−1

−1

2 Σ . Therefore, letting Ω = Σ WΣ , we have where W = N −1 ∑iN=1 σǫ,i F η,i η η

D ( F )−1 DZ [ D ( F )′ ]−1 = Ω F + O p ( T −1 ) + O p ( N −1/2 ).

(100)

It remains to consider the two bias terms, B and C. B is, in our notation, !′ N N N 1 1 1 1 −1 ai,k xk F ( T −1 F ′ F )−1 P λ j xi − B = − D ( F ) −1 ∑ ∑ ∑ N i=1 j=1 T N k=1 T !′ 1 N 1 N 1 −1 2 ai,k xk F ( T −1 F ′ F )−1 P λi σǫ,i xi − = − D ( F ) −1 ∑ ∑ N i=1 T N k=1

= D ( F ) −1 − D ( F ) −1

1 N 1 N

N

N

1

∑ NT ∑ ai,k xk′ F(T −1 F′ F)−1 P

i=1 N

−1

T

∑ E(ǫi,t ǫj,t ) i=1

2 λi σǫ,i

k=1

∑ T −1 xi′ F(T −1 F′ F)−1 P

−1

2 λi σǫ,i .

i=1

From xi = FΛ′i + ηi , 1 NT

N

∑

ai,k xk′ F =

k=1

=

1 N 1 N

N

∑ k=1 N

ai,k Λk T −1 F ′ F + √

1 NT

√

N

1 NT

∑ ai,k ηk′ F k=1

∑ ai,k Λk T −1 F′ F + O p (( NT )−1/2 ),

k=1

giving 1 N

N

∑ i=1

1 NT

N

∑

ai,k xk′ F ( T −1 F ′ F )−1 P

−1

2 = λi σǫ,i

k=1

N

1 N2

N

∑ ∑ ai,k Λk P

−1

2 + O p (( NT )−1/2 ) λi σǫ,i

i=1 k=1

= B1RPC + O p (( NT )−1/2 ). where B1RPC = N −1 ∑kN=1 Λk Ω RPC λk , Ω RPC = P

−1

UP

−1

2 λ λ ′ . The last and U = N −1 ∑iN=1 σǫ,i i i

equality follows from the definition of ai,k , as seen by writing 1 N

N

∑ ai,k Λk P

−1

2 λi σǫ,i = Λk P

i=1

−1

1 N

N

∑ σǫ,i2 λi ai,k = Λk P i=1

−1

1 N

N

∑ σǫ,i2 λi λ′i P

−1

λk = Λk P

−1

UP

−1

λk .

i=1

But we also have T −1 xi′ F ( T −1 F ′ F )−1 = T −1 ( FΛ′i + ηi )′ F ( T −1 F ′ F )−1 = Λi + T −1/2 T −1/2 ηi′ F ( T −1 F ′ F )−1

= Λi + O p ( T −1/2 ), 42

2 Λ P Hence, letting B2RPC = N −1 ∑iN=1 σǫ,i i

1 N

N

∑ T −1 xi′ F(T −1 F′ F)−1 P

−1

2 = λi σǫ,i

i=1

−1

λi ,

N

1 N

∑ Λi P

−1

2 + O p ( T −1/2 ) = B2RPC + O p ( T −1/2 ). λi σǫ,i

i=1

which, together with D ( F ) = Ση + O p ( T −1 ) + O p ( N −1/2 ), implies −1

B = Ση BRPC + O p ( T −1/2 ) + O p ( N −1 ),

(101)

where BRPC = B1RPC − B2RPC . Since M F F = 0, under our assumptions C = 0, as is seen by writing C = − D ( F ) −1

1 NT

= − D ( F ) −1

1 NT

N

∑ xi′ MF i=1 N

1 N

N

∑ E(ǫi ǫi′ ) F(T −1 F′ F)−1 P

−1

λi

i=1

∑ σǫ,i2 xi′ MF F(T −1 F′ F)−1 P

−1

λi = 0

(102)

i=1

Therefore, by putting everything together,

√

−1 √ NT ( βˆ ( Fˆ RPC ) − β) ∼ Ση TN −1/2 BRPC + N (0, Ω F ),

(103)

√ which holds as N, T → ∞ with TN −1 = O(1) and TN −3/2 → 0, where the last condition √ √ −1  is needed to ensure that TN −1/2 B = − TN −1/2 Ση BRPC + o p (1). As Proposition 2 and Theorem 1 of the main paper make clear, the only difference between the asymptotic distributions of the PC and RPC estimators of β is the bias; the asymptotic variance is the same. We also see that the condition for the RPC estimator to be unbiased, that is, T/N = o(1), is the same as for PC. Let us now consider the relative bias (when T/N > 0). Note how U and P in BRPC serve the same purpose as S and Q in BPC . BRPC therefore corresponds to the following partial PC bias: ( B1PC − B3PC ). B2PC is due to the fact that the estimated PC factor is correlated with ηi , which is not the case in RPC. In order to get a feeling for the factors determining the relative bias, it is useful to consider a restricted model. In particular, let us assume that r = m = 1 and β = 0 (as in the example considered in the main paper), such that BPC = BRPC =

1 N 1 N

N

∑

2 + Σ )( λ2 + Λ2 )]λ Λi [U + R − (σǫ,i η,i i

( λ2 + Λ2 )2

i=1 N

∑ i=1

2 λ2 ) λ Λi (U − σǫ,i i

( λ2 )2

,

,

(104) (105)

43

2 λ2 (as before), R = N −1 N Σ Λ2 , and λ2 and Λ2 are the average where U = N −1 ∑iN=1 σǫ,i ∑i=1 η,i i i

λ2i and Λ2i , respectively. Hence, even in this highly simplified case it is not possible to infer which estimator that is relatively more biased. What we can say, however, is that the relative bias depends critically on the relative importance of the common component in xi . To see this, suppose that (in addition to the above) Λi = Λ, in which case the expressions for BPC and BRPC become BPC =

1 1 ΛN

BRPC = Λ

1 N

2 + Σ )( λ2 /Λ2 + 1)]λ [U/Λ2 + Ση − (σǫ,i η,i i

N

∑

(λ2 /Λ2 + 1)2

i=1 N

∑

2 λ2 ) λ (U − σǫ,i i

i=1

( λ2 )2

,

.

Thus, if Λ → ∞, then BPC → 0 and | BRPC | → ∞, suggesting that PC should be relatively less biased, which is partly expected being based of a relatively large information set that includes xi . On the other hand, if Λ → 0, then both biases tend to zero, and hence the difference disappears. We also see that if Ση,i = Ση → ∞, while BRPC is unaffected, BPC → ∞. Remark 3. While the size of the bias depend positively on Ω PC and Ω RPC , this dependence is by no means the only factor determining the relative bias. Suppose, for example, that 2 2 2 = σ2 and Σ 2 2 2 2 r = m = 1, β = 0, σǫ,i η,i = Σ η . In this case, Ω PC = ( σǫ λ + Σ η Λ ) /( λ + Λ ) . ǫ

As explained in the main paper, Ω PC is the asymptotic variance of FˆtPC . The minimal value of this variance (for λ2 and Λ2 positive) is obtained by setting Ση = σǫ2 , giving Ω PC = σǫ2 /(λ2 + Λ2 ). The variance of the corresponding RPC factor estimator, FˆtRPC , is given by Ω RPC = σǫ2 /λ2 (see Bai, 2003, Theorem 1). Hence, since σǫ2 /(λ2 + Λ2 ) ≤ σǫ2 /λ2 , we have that the PC estimator is at least as efficient as the RPC estimator, which is partly expected as it based on a larger information set. Thus, since the bias depend critically on the variance of the factor estimator, one may expect the PC estimator to be relatively less biased. However, this is not the case. In fact, while BRPC = 0, we have BPC = −σǫ2 N −1 ∑iN=1 Λi λi /(λ2 + Λ2 ) 6= 0 (unless of course Λi and/or λi is zero).

5 Bias-correction As explained in Remark 8 of the main paper (Westerlund and Urbain, 2013), an obvious solution to the problem with bias in case T/N → τ > 0 is to use bias correction. Let us

44

therefore define the following bias-adjusted version of βˆ ( Fˆ p ) for p ∈ { PC, CA}: 1ˆ ˆ− βˆ BA ( Fˆ p ) = βˆ ( Fˆ p ) − N −1 Σ η Bp,

ˆ η = ( NT )−1 ∑iN=1 x′ M ˆ p xi , and Bˆ PC is BPC with β, Ση,i and σ2 replaced by βˆ ( Fˆ p ), where Σ F i ǫ,i ˆ η,i = T −1 x′ M ˆ p xi and σˆ 2 = T −1 (yi − xi βˆ p − Fˆ p λˆ p )′ (yi − xi βˆ p − Fˆ p λˆ p ), respectively. The Σ F ǫ,i i i i p

p

ˆ of λi and Λi , respectively, are obtained by simply picking the appropriestimators λˆ i and Λ i

p p ate elements in Cˆ i , which can be obtained from a LS regression of zi,t onto Fˆt , as explained

in Section 4 of this supplement. Proposition 3. Under Assumptions JOINT, PC1, PC2 and CA, as N, T → ∞ with √ and N/T → 0,

√

√

T/N → 0

NT ( βˆ BA ( Fˆ p ) − β) →d N (0, Ω F ).

Proof of Proposition 3. Write

√

√

NT ( βˆ BA ( Fˆ p ) − β) =

√

=

Consider

√

√

−

NT ( βˆ ( Fˆ p ) − β) − NT ( βˆ ( Fˆ p ) − β) −

√ √

1ˆ ˆ− TN −1/2 Σ η Bp 1 TN −1/2 Σ− η Bp −

√

1 ˆ ˆ− TN −1/2 Σ η (Bp − Bp)

1 −1 TN −1/2 (Σˆ − η − Ση ) B p .

−1/2 ) + O ( N −1 ), 1 ˆ ˆ ˆp ˆ− TN −1/2 Σ p η ( B p − B p ). From Theorem 1, ( β ( F ) − β) = O p (( NT ) p

p

and from the proof of Proposition 2 (Section 4), we have (Cˆ i − H p Ci ) = O p ( T −1/2 ) +

O p ( N −1 ), where HPC = H

−

− ˆ η − Ση ), (Σˆ η,i − Ση,i ) and (σˆ 2 − σ2 ) are and HCA = C . (Σ ǫ,i ǫ,i

all O p ( T −1/2 ) (details are available upon request). This implies that

|| Bˆ p − B p || = O p ( T −1/2 ) + O p ( N −1 ), 1 and therefore, with ||Σˆ − η || = O p (1),

√ √ −1/2 −1 1 ˆ || TN −1/2 Σˆ − TN ||Σˆ η |||| Bˆ p − B p || η ( B p − B p )|| ≤ √ = O p ( N −1/2 ) + O p ( TN −3/2 ),

which is o p (1) under our assumption that

√

(106)

TN −1 = o(1). Similarly, since || B p || = O p (1)

1 −1 −1/2 ), and, by Taylor expansion, ||Σˆ − η − Σ η || = O p ( T

√ √ −1/2 −1 1 −1 1 −1/2 || TN −1/2 (Σˆ − TN ||Σˆ η − Σ− ). η − Σ η ) B p || ≤ η |||| B p || = O p ( N 45

(107)

Together with Theorem 1 these results imply

√

NT ( βˆ BA ( Fˆ p ) − β) =

as N, T → ∞ with

√

√

NT ( βˆ ( Fˆ p ) − β) −

TN −1 → 0 and

√

√

NT −1 → 0.

1 TN −1/2 Σ− η B p + o p (1) → d N (0, Ω F )



According to Proposition 3, asymptotically the bias correction is successful. Moreover, the correction does not contribute to the limiting variance, which is the same as in Theorem 1. Remark 4. As explained in Remark 2 of the main paper, the DGP considered is not the most general one. This is deliberate, as there is a trade-off between generality and clearness of the results. Therefore, given our purpose to shed light on the differences between CA and PC, we have opted for a relatively simple model. This has important implications for applied work based on the CA and PC estimators. For example, while the DGP assumes no detereministic terms, in practice a constant (or unit-specific fixed effects) should always be included. Similarly, while we assume here that the data are serially uncorrelated, this will not be the case in practice. In terms of bias-corrected estimation, this means that all contemporaneous variance estimators should be replaced by the corresponding heteroskedasticity and autocorrelation consistent (HAC) variance estimators. Also, if the number of common factors are unknown, then this number must be estimated prior to the implementation of the estimators. One possibility towards this end is to use in information criteria. This approach has been shown to work in the context of PC estimation (see Bai, 2009b, Section C.3) and it is expected to work well also for CA. In order to assess the accuracy of the bias-corrected estimators in small samples, we use Monte Carlo simulation. The DGP is the same as in the main paper, and sets m = 1, r = 2 and ( Ft′ , ǫi,t , ηi,t )′ ∼ N (0, diag( I2 , σǫ2 , Ση )). λi and Λ′i are generated as λi = λ∗ e1 + [0.5 −

1(i > ⌊0.5N ⌋)]ι2 and Λ′i = Λ∗ e2 + [0.5 − 1(i > ⌊0.5N ⌋)]ι2 , where e1 = (1, 0)′ , e2 = (0, 1)′ , ι2 = (1, 1)′ , 1( A) is the indicator function for the event A, and ⌊ x⌋ is the integer part of x.

The DGP cases are the same as in the main paper; DGP1. λ∗ = 1, Λ∗ = 0.4 and σǫ2 = Ση = 1. DGP2. λ∗ = 1, Λ∗ = 4 and σǫ2 = Ση = 1. DGP3. λ∗ = 0.4 and Λ∗ = σǫ2 = Ση = 1. 46

DGP4. λ∗ = Λ∗ = σǫ2 = 1 and Ση = 0.1. DGP5. λ∗ = Λ∗ = σǫ2 = 1 and Ση = 10. DGP6. λ∗ = 1, Λ∗ = 0.4, σǫ2 = 0.1 and Ση = 1. DGP7. λ∗ = 1, Λ∗ = 4, σǫ2 = 0.1 and Ση = 1. DGP8. λ∗ = Λ∗ = 1, σǫ2 = 10 and Ση = 1. The data are generated for 3,000 panels with N = T = 50, 100. The results are reported in Table 1. It is seen that bias correction generally leads to a substantial reduction in bias, and hence also in size distortion. This is true for both estimation approaches. We also see that the bias of the bias-adjusted estimators is decreasing in the sample size, thus corroborating the theoretical prediction in this regard (see Proposition 3). The extent of bias depends critically on β. PC is more sensitive than CA; therefore, while PC performs best when β = 0, when β = −2 CA is the winner. In fact, when β = −2, for the sample sizes considered here, bias-correction is not enough to fully remove the bias of PC, leading to serious distortions in some cases. Thus, as expected, unless N and T are large enough, the bias-correction will not be complete.

47

References Bai, J. (2003). Inferential theory for factor models of large dimensions. Econometrica 71, 135–173. Bai, J. (2009a). Panel data models with interactive fixed effects. Econometrica 77, 1229–1279. Bai, J. (2009b). Supplement to “Panel data models with interactive fixed effects”. Econometrica Supplemental Material 77. Magnus, J. R., and H. Neudecker (1999). Matrix differential calculus: with applications in statistics and econometrics. Revised edition, John Wiley & Sons. Westerlund, J., and J.-P. Urbain (2013). Cross-sectional averages versus principal components. Unpublished manuscript.

48

Table 1: Bias and 5% size results for the bias-adjusted PC and CA estimators.

DGP

PC

Bias CA PC–BA

−0.129 −0.130 −0.113 −0.093 −0.093 0.030 −0.114 −0.846

1.630

−0.124 −0.114 −0.104 −0.067 −0.033 0.017 −0.111 −0.342

1.715 0.242 1.728 2.635

1 2

3.137 3.759

1.630 0.225

3 4 5 6 7

2.778 10.624 3.134 2.162 2.634

1.638 2.578 0.232 1.408 0.037

8

14.139

1 2 3 4 5 6 7 8

1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8

CA–BA

PC

β = 0, N = T = 50 −0.001 0.166 6.2

5% size CA PC–BA

CA–BA

37.7

6.4

8.0

5.6 5.1 5.8 6.7 6.1

5.7 36.7 15.3 11.1 95.6

5.3 5.0 7.2 5.6 6.3

5.6 7.5 8.3 6.0 16.8

7.3 6.7

5.4 13.9

5.2 7.2

5.5 7.3

β = 0, N = T = 100 0.001 0.093 6.2 −0.026 −0.018 5.4 0.022 0.109 5.9 −0.053 0.048 6.8

40.8 5.5 42.0 14.0

5.8 5.3 5.5 7.3

6.7 5.2 6.5 7.7

5.0 5.1 6.5 6.0

12.0 98.5 5.8 14.1

4.6 4.8 5.7 6.5

5.0 10.6 5.7 6.8

β = −2, N = T = 50 0.802 0.166 81.9 1.246 −0.030 91.4

33.2 4.4

14.7 25.5

8.0 5.6

74.1 88.8 99.9 99.7 99.9

34.0 14.6 2.6 87.8 0.2

18.5 13.4 99.7 24.5 47.0

7.5 8.3 6.0 16.8 5.5

2.280

6.776

98.5

13.4

55.1

7.3

2.890 3.270 2.452 10.661 3.562

1.715 0.242 1.728 2.635 0.257

0.404 0.586 0.521 0.618 2.883

0.093 −0.018 0.109 0.048 0.024

79.0 87.3 67.4 91.6 100.0

37.8 4.6 40.3 13.7 4.5

8.9 10.6 9.1 9.7 99.7

6.7 5.2 6.5 7.7 5.0

1.987 2.300 12.590

1.469 0.039 2.488

0.241 0.445 3.260

0.060 −0.001 0.261

100.0 100.0 97.2

96.9 1.7 13.7

10.9 23.0 22.2

10.6 5.7 6.8

0.225 1.638 2.578 0.232 1.408 0.037 2.280

0.257 1.469 0.039 2.488

−0.044 0.019 −0.063 −0.040 0.010 −0.009 −0.419

−0.006 −0.009 −0.005 −0.086

1.012 1.214 2.801 0.543 0.921

−0.030 0.179 0.151 0.033 0.147 −0.004 0.438

0.024 0.060 −0.001 0.261

0.179 0.151 0.033 0.147 −0.004 0.438

β = −2, N = T = 100

Notes: PC–BA and CA–BA refer to the bias-adjusted versions of the PC and CA estimators. See the text for a description of DGP1–DGP8.

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