Testing Structural Changes in Regression Error Variance

Appendix Proof of Theorem 1: Part (a) follows from Qu and Perron (2007a, Theorem 5) under A1. For part (b), sup LR2;T (ma ; na ; "jn = 0; ma ) ^ T (Tec ; :::; Tec ; Tev ; :::; Tev ) log L eT (T^c ; :::; T^c )] = 2[log L 1 ma 1 na 1 ma Pna +1 ev ev 2 2 = T log e Ti 1 ) log ^ i i=1 (Ti Pna ev 2 v Teiv ) log ^ 2i+1 ] + Te1v (log e21;1 Teiv log e21;i (Tei+1 = i=1 [Ti+1 log e1;i+1 PTeiv zt0et;j )2 with et;j = ej for T^jc 1 < t PTev Tjc0 ) (j = 1; :::; ma + 1) and ^ 2i = (Teiv Teiv 1 ) 1 t=i Tev where e21;i = (Teiv ) for Tjc0 1 < t 1 t=1 (yt x0t e i 1 log ^ 21 ) T^jc (also let 0t;j = 0j (y x0t ^ zt0 ^t;j )2 . +1 t Applying a Taylor expansion to log e21;i+1 , log e21;i and log ^ 2i+1 around log 20 , we obtain P a i i (F1;T + F2;T ) + op (1) sup LR2;T (ma ; na ; "jn = 0; ma ) = ni=1 where i v F1;T = ( 20 ) 1 [Tei+1 e21;i+1 Teiv e21;i h PTev = ( 20 ) 1 t=i+1 (yt x0t e Tev +1 v (Tei+1 zt0et;j )2 i and i F2;T = v (1=2)[Tei+1 ( e21;i+1 2 0 2 2 0 Teiv ( ) = (1=2)(I + II + III): e21;i 6 6 6 2 16 = ( 0) 6 6 4 (^ +( ^ +2( +2( 0 t;j e)0 X 0 Xi+1 ( ^ i+1 e)0 X 0 Zi+1 (^t;j i+1 ^ )0 X 0 Zi+1 (^t;j i+1 et;j ) + 2( e) e) + Zi+1 ( zt0 ^t;j )2 0 t;j ^) ^ ) + Zi+1 ( i 2 ^ Teiv )( i+1 0 t;j et;j )0 Z 0 Zi+1 (^t;j i+1 et;j ) + 2( ^ et;j ) + 2( ^ x0t ^ v (Tei+1 ) 2 0 e) + (^t;j ^t;j )0 Z 0 Zi+1 (^t;j i+1 (yt 2 0 2 i i We …rst show that F1;T = op (1). We can express F1;T as 2 (Ui+1 + Xi+1 ( 0 6 6 6 +Zi+1 ( 0t;j et;j ))0 (Ui+1 + Xi+1 ( 0 2 16 ( 0) 6 6 (Ui+1 + Xi+1 ( 0 4 +Zi+1 ( 0t;j ^t;j ))0 (Ui+1 + Xi+1 ( 0 2 Teiv )^ 2i+1 ] e)0 X 0 Ui+1 + 2(^t;j i+1 2 0 )] (A.1) 3 7 et;j )) 7 7 7 7 7 5 ^t;j )) ^ )0 X 0 Xi+1 ( ^ i+1 e)0 X 0 Zi+1 ( i+1 2 0 2 et;j ) 0 t;j e) ^t;j ) et;j )0 Z 0 Ui+1 i+1 3 7 7 7 7: 7 7 5 0 0 0 Zi+1 = Zi+1 = Op (T ), Xi+1 Xi+1 = Op (T ), Zi+1 The result follows using the facts that Xi+1 1=2 1=2 0 0 Op (T ), Xi+1 Ui+1 = Op (T ) and Zi+1 Ui+1 = Op (T ). Also, since under H0 with A1, the estimates of the break fractions converge to the true break fractions at a fast enough rate so that the estimates of the parameters of the models are consistent and have the same limit distribution as when the break dates are known, we have: 0 ^ = Op (T 1=2 ), 0 ^t;j = Op (T 1=2 ), ^ e = op (T 1=2 ) and ^t;j et;j = op (T 1=2 ). The last two quantities t;j p p 0 0 are op (T 1=2 ) since T ( ^ ) and T (e ) have the same limit distribution under H0 , p p 0 0 i T (et;j and likewise for T (^t;j t;j ) and t;j ). For F2;T , p v v PTei+1 PTei+1 v v ) 1=2 t=1 (ut = 0 )2 1 + op (1) ) 1=2 t=1 [f(yt x0t e zt0et;j )= 0 g2 1] = (Tei+1 I = (Tei+1 q p v ) W ( vi+1 )= i+1 p p v p by A1. Similarly, II ) W ( vi )= i and p v PTi+1 v III = [(Tei+1 Teiv )=T ] 1=2 T 1=2 t=T [(ut = 0 )2 1] + op (1) v i +1 v PTi+1 PTiv v = [(Tei+1 Teiv )=T ] 1=2 fT 1=2 t=1 [(ut = 0 )2 1] T 1=2 t=1 [(ut = 0 )2 1]g + op (1) q p v v [W ( vi+1 ) W ( vi )]= ) i+1 i: Therefore, i F2;T ) ( =2) = ( =2) ( W 2( v i+1 ) W 2 ( vi ) (W ( v v i+1 i v v v v 2 i W ( i+1 ) i+1 W ( i )) : v v v v i+1 i ( i+1 i) v i+1 ) v i+1 W ( vi ))2 v i This yields sup LR2;T (ma ; na ; "jn = 0; ma ) ) sup ( v v c 1 ;:::; na )2 v;" sup ( because c v;" v;" . = = where e2i = (T^iv x0t ^ T^iv For part (c), v v 1 ;:::; na )2 v;" ( vi W ( na P 2 i=1 2 i=1 na P ( v i+1 ) v v i+1 i ( v v i W ( i+1 ) v v i+1 i ( v i+1 W ( v v i+1 i) v i+1 W ( v v i+1 i) sup LR3;T (ma ; na ; "jm = 0; na ) ^ T (Tec ; :::; Tec ; Tev ; :::; Tev ) log L eT (T^v ; :::; T^v )] 2[log L 1 ma 1 na 1 na Pna +1 ^v ^v Pna +1 ev ev 2 Ti 1 ) log ei Ti 1 ) log ^ 2i i=1 (Ti i=1 (Ti PT^iv PTev 1 (y x0t e zt0e)2 and ^ 2i = (Teiv Teiv 1 ) 1 t=i Tev 1) t=T^v +1 t v 2 i )) i 1 +1 i 1 zt0 ^t;j )2 . Applying a Taylor expansion on log e2i and log ^ 2i around log P a +1 i i sup LR3;T (ma ; na ; "jm = 0; na ) = ni=1 (F1;T + F2;T ) + op (1) v 2 i )) 2 i0 , (yt we obtain T^iv 1 )(e2i = i where F1;T = (T^iv (1=2)[(T^iv i F2;T = Teiv 1 )(^ 2i = (Teiv 2 i0 ) T^iv 1 )([e2i 2 i0 ) 2 2 2 i0 ]= i0 ) i We …rst show that F2;T = op (1) as follows. We have: 2 i F2;T = (1=2)[(T^iv = (1=2)[T 1 2 i0 2 e T^iv 1 )( i ) 2 i0 2 e T^iv 1 )[T 1=2 ( i (T^iv (Teiv 2 i0 2 i0 and (Teiv Teiv 1 )([^ 2i 2 ^ Teiv 1 )( i )]2 T 1 2 2 2 i0 ]= i0 ) ]: 2 i0 2 )] 2 i0 (Teiv 2 ^ Teiv 1 )[T 1=2 ( i 2 i0 2 i0 )]2 ] p p 2 2 2 v v 2 2 2 e e ] and [( T T )=T ][ )= where [(T^iv T^iv 1 )=T ][ T (e2i T (^ 2i i0 i i 1 i0 i0 )= i0 ] have the i same limit distribution under A3. For F1;T , let 0 = 10 without loss of generality, then i h Pna +1 i Pna +1 ^v ^v 2 2 1 ev Tev )^ 2 ( T F = ( ) ( T T )e i i i i 1 1;T 0 i i 1 i=1 i=1 h i P 2 2 2 2 v v v v a +1 ^ ^ e e +( 20 ) 1 ni=1 ([ 2i0 ]= ) ( T T )e ( T T )^ 0 i0 i i 1 i i i 1 i : The …rst term becomes, Pna +1 h ^v ^v (Ti Ti 1 )e2i (Teiv i=1 P = ( 20 ) 1 Tt=1 [(yt x0t e zt0e)2 (yt ( 20 ) = ( 20 ) 1 j=1 t=1 (yt PTe1c x0t e t=1 (yt P r 1 a f m j=1 [D (1; j + +( 20 ) = ( 20 ) c Pma PTej+1 1 where Dr (1; j) = PTejc t=1 (yt Teiv 1 )^ 2i 1 x0t e zt0e)2 1) x0t e zt0e)2 PTejc t=1 (yt ( 20 ) Dr (1; j) 1 PTe1c t=1 (yt u x0t e x0t ^ x0t ^ zt0e)2 PTejc t=Tejc zt0 ^t;j )2 ] zt0 ^1 )2 D (j + 1)] + Dr (1; 1) zt0e)2 and Du (j) = i 1 +1 (yt c PTej+1 t=Tejc +1 (A.2) (yt x0t ^ Du (1)g; x0t ^ zt0 ^j )2 . The second term is op (1) by A3. Using similar derivations as in Qu and Perron (2007b), we obtain Dr (1; j + 1) = ) Dr (1; j) Du (j + 1) 0 0 0 0 0 0 U1:j+1 Z1:j+1 (Z1:j+1 Z1:j+1 ) 1 Z1:j+1 U1:j+1 + U1:j Z1:j (Z1:j Z1:j ) 1 Z1:j U1:j 0 0 1 0 +Uj+1 Zj+1 (Zj+1 Zj+1 ) Zj+1 Uj+1 + op (1); c c c c 2 j Wq ( j+1 ) j+1 Wq ( j ) c c c c j+1 j ( j+1 j) by A2. This yields sup LR3;T (ma ; na ; "jm = 0; na ) ) sup ( c c v 1 ;:::; ma )2 c;" sup ( c c 1 ;:::; ma )2 c;" ma X j=1 ma X j=1 zt0 ^j+1 )2 c c c c 2 j Wq ( j+1 ) j+1 Wq ( j ) ; c c c c j+1 j ( j+1 j) c c 2 c c j Wq ( j+1 ) j+1 Wq ( j ) ; c c c c j+1 j ( j+1 j) because = = = = v c;" c;" . For part (d), we have: sup LR4;T (ma ; na ; "jm = n = 0) ^ T T c ; :::; T c ; T v ; :::; T v eT ] 2[ sup log L log L 1 ma 1 na c v v c ( 1 ;:::; ma ; 1 ;:::; na )2 " h i ^ T (Tec ; :::; Tec ; Tev ; :::; Tev ) log L eT 2 log L 1 ma 1 na P na +1 ev T log e2 Teiv 1 ) log ^ 2i i=1 (Ti i h Pna ev 2 2 2 v v v e e e Ti log e1;i (Ti+1 Ti ) log ^ i+1 + Te1v (log e21;1 i=1 Ti+1 log e1;i+1 log ^ 21 ); PTei (yt x0t e zt0e)2 : Applying a Taylor expansion to log e21;i+1 , log e21;i where e21;i = (Teiv ) 1 t=1 and log ^ 2i+1 around log 20 , we obtain P a i i (F1;T sup LR4;T (ma ; na ; "jm = n = 0) = ni=1 + F2;T ) + op (1) v where the …rst term is the same as in (A.2), so that Pna i Pna 2 1 h ev 2 v Ti+1 e1;i+1 Teiv e21;i Tei+1 i=1 F1;T = i=1 ( 0 ) = ( 20 ) 1 PT t=1 x0t e yt P a r = ( 20 ) 1 f m j=1 [D (1; j + 1) 2 zt0e yt Dr (1; j) i 2 v e Ti ^ i+1 + ( 20 ) 1 Te1v (e21;1 x0t ^ zt0 ^t;j 2 Du (j + 1)] + Dr (1; 1) Du (1)g as shown in part (c). The second term is the same as (A.1) but with no changes in construct e21;i , i.e., LRv de…ned by (11). Hence, i F2;T = v (1=2)[Tei+1 ( e21;i+1 2 0 2 Teiv ( ) 2 0 e21;i 2 0 2 2 0 as shown in part (b). From the proof of part (c), Pna i i=1 F1;T ) Pma jj cj Wq j=1 c j+1 c c j+1 j ) 2 ^ Teiv ( i+1 v Tei+1 c j+1 Wq c c j+1 j c j ^ 21 ) to 2 0 2 2 0 )] jj2 under A2 and from that of part (b), i F2;T ) v iW 2 v i+1 v v i+1 i v v 2 i+1 W ( i ) v v i+1 i under A1. Hence, we obtain sup LR4;T (ma ; na ; "jm = n = 0) ) ( c 1 ;:::; 3 2 P c c c c 2 ma jj j Wq ( j+1 ) j+1 Wq ( j )jj c c c c j=1 6 7 j+1 j ( j+1 j) sup 4 v v v v 2 5: P i+1 W ( i )) a ( i W ( i+1 ) c v v + 2 ni=1 ma ; 1 ;:::; na )2 " v v v v i+1 i ( i+1 i)