Supplemental Material for "Automated Estimation of Vector Error Correction Models"

Transcription

1 Supplmtal Matrial for "Automatd Estimatio of Vctor Error Corrctio Modls" ipg Liao Ptr C. B. Pillips y Tis Vrsio: Sptmbr 23 Abstract Tis supplmt icluds two sctios. Sctio cotais t proofs of som auxiliary lmmas usd i t mai txt. Sctio 2 provids som mor simulatio rsults wic ar ot rportd i t mai txt. Proofs of Auxiliary Lmmas I tis sctio, w provid t proofs of som lmmas wic ar usd i t mai txt to driv t asymptotic proprtis of t LS srikag stimator. For as of xpositio, w put t lmmas ad tir proofs of di rt sctios i t papr i di rt subsctios.. proofs of som lmmas i sctio 3 Assumptio. (WN) fu t g t is a m-dimsioal iid procss wit zro ma ad osigular covariac matrix u. Assumptio.2 (RR) (i) T dtrmiatal quatio ji (I + o )j as roots o or outsid t uit circl; (ii) t matrix o as rak r o, wit r o m; (iii) if r o >, t t matrix R I ro + o o as igvalus witi t uit circl. Dpartmt of Ecoomics, UC Los Agls, 8379 Buc Hall, Mail Stop: 4773, Los Agls, CA zipg.liao@co.ucla.du y Yal Uivrsity, Uivrsity of Aucklad, Uivrsity of Soutampto ad Sigapor Maagmt Uivrsity. Support from t NSF udr Grat No SES is gratfully ackowldgd. ptr.pillips@yal.du

2 T urstrictd LS stimator b st of o is b st arg mi 2R mm ky t Y t k 2 Y t Y t Y t Y t : (.) T asymptotic proprtis of b st ad its igvalus ar dscribd i t followig rsult. Lmma. Udr Assumptios. ad.2, w av: (a) rcall D diag( 2 I ro ; I m bst wr B m; ad B m;2 ar d d i Torm 3.5; ro ), t b st satis s o Q D d (B m; ; B m;2 ) (.2) (b) t igvalus of b st satisfy k ( b st ) p k ( o ) for k ; :::; m; (c) t last m r o igvalus of b st satisfy ( b st ); :::; m ro ( b st ) o; d ; :::; o;m ro ; (.3) wr t o;j (j ; :::; m I m r r o ) ar solutios of t followig dtrmiatal quatio db w2 Bw 2 B w2 Bw 2 : (.4) Proof of Lmma.. (a) From (.) bst o Q D u t Y t Q u t t D QY t Y t Q D D t t D : (.5) Rsult (a) follows dirctly from Lmma.. (b) Tis rsult follows dirctly by (a) ad t cotiuous mappig torm (CMT). (c) Lt k k( b st ) (k r o + ; :::; m), so tat k is by d itio a solutio of t quatio os () o o? S () i S () o o S () o o S () o? ; (.6) 2

3 wr S () I m b st. For ay compact subst K R, w ca ivok t rsults i (a) to sow os () o o o o bst uiformly ovr K. From Assumptio.2.(iii), w av o o o o o o o o o o o 6 : o o + o o o p o o o ; (.7) Tus, t ormalizd m r o smallst igvalus k (k r o + ; :::; m) of b st ar asymptotically t solutios of t followig dtrmiatal quatio, o? S () i S () o o S () o o S () o? ; (.8) wr os () o? bst o o o? ; (.9) o? S () o? I m r o bst o? o o? ; (.) o? S () o o? b st o p o? o o o : (.) Usig t rsults i (.7) ad (.9)-(.), w gt o? S () I m r o o? i S () o o S () o o S () o? I m o i o o o + o p () bst o o? : (.2) Not tat bst o o? bst i o Q D D Q o? u t t D t t D D u t 2;t 2 2;t 2;t d B w2 db u o;? o;? o;? o;? B w2 B w 2 o;? o;?: (.3) 3

4 Usig t rsults i (.2), (.3) ad t quality o ( o o ) o + o;? ( o;? o;?) o;? I m; (.4) w dduc tat o? I m o i o o o + o p () bst d ( o;? o;?) B w2 dbw 2 o o? B w2 B w 2 o;? o;? (.5) T, from (.8)-(.5), w obtai o? S () d I m ro o S () o o S () o o S () o? B w2 dbw 2 B w2 Bw 2 ; (.6) uiformly ovr K. T rsult i (c) follows from (.6) ad by cotiuous mappig..2 proofs of som lmmas i sctio 4 Assumptio.3 (LP) Lt D(L) P j D jl j, wr D I m ad D() as full rak. Lt u t av t Wold rprstatio X u t D(L)" t D j " t j j, wit X j 2 jjdj jj < ; (.7) j wr " t is iid (; "" ) wit "" positiv d it ad it fourt momts. T followig lmma is usful i stablisig t asymptotic proprtis of t srikag stimator wit wakly dpdt iovatios. Lmma.2 Udr Assumptio.2 ad.3, (a), (b) ad (c) of Lmma. ar ucagd, wil Lmma..(d) bcoms 2 ut ;t uz () d N(; V uz ); (.8) 4

5 wr uz () P j uu(j) o R j < ad V uz is t log ru variac matrix of u t ;t ; ad Lmma..() bcoms u t 2;t d B w2 db u + ( uu uu ) o? : (.9) Proof of Lmma.2. Gragr rprstatio, Rcall tat Assumptio.2 lads to t followig partial sum Y t C tx u s + o ( o o ) R(L) ou t + CY ; (.2) s wr C o;? ( o;? o;?) o;?. From t partial sum xprssio i (.2), w gt ;t oy t R(L) ou t, wic implis tat f oy t g t is a statioary procss. Not tat E u t ;t X E u t u t j j o R j X uu (j) o R j < : Usig a CLT for liar procss tim sris (.g. t multivariat vrsio of torm 8 ad Rmark 3.9 of Pillips ad Solo, 992), w dduc tat j 2 ut ;t uz () d N(; V uz ); wic stabliss (.8). T rsults of (a)-(c) ad () ca b provd usig similar argumts to tos of Lmma.. Lt P (P ; P 2 ) b t ortoormalizd rigt igvctor matrix of ad b a r r diagoal matrix of ozro igvalus of, wr P is a m r matrix (of igvctors of ozro igvalus) ad P 2 is a m (m r ) matrix (of igvctors of zro igvalus). By t igvalu dcompositio, (P ; P 2 ) m r Q Q 2 P Q (.2) 5

6 wr Q (Q ; Q 2 ) ad Q P. By d itio Q Q 2 (P ; P 2 ) Q P Q P 2 Q 2 P Q 2 P 2 I m (.22) wic implis tat Q P I r. From (.2), witout loss of grality, w ca d P ad Q. By (.22), w dduc tat Q P ad P Q wic imply tat ad ar osigular r r matrix. Witout loss of grality, w lt? P 2 ad? Q 2, t?? I m r ad udr (.22),? Q 2 P wic implis tat? as? (? ; o? ). Lt [ ( b st ); :::; m ( b st )] ad [ ( ); :::; m ( )] b t ordrd igvalus of b st ad rspctivly. For t as of otatio, w d N N(; V uz ) + uz () z z N(; V z z ) z z o wr N(; V uz ) is a radom matrix d d i (.8) ad N(; V z z ) dots t matrix limit distributio of p bs z z. W also d N 2 db u Bu + ( uu uu ) o? B w2 B w 2 o? : T xt lmma provids asymptotic proprtis of t OLS stimat ad its igvalus w t data is wakly dpdt. Lmma.3 Udr Assumptio.2 ad.3, w av t followig rsults: (a) t OLS stimator b st satis s bst Q D O p () (.23) wr Q H o Q + o ; (b) t igvalus of b st satisfy k ( b st ) p k ( ) for k ; :::; m; 6

7 (c) t last m r o ordrd igvalus of b st satisfy [ ro+( b st ); :::; m ( b st )] d [ r o+; :::; m] (.24) wr j (j r o + ; :::; m) ar t ordrd solutios of ui m r o o? N 2 + N??N??N 2 o? ; (.25) (d) b st as r o r igvalus satisfyig p [r +( b st ); :::; ro ( b st )] d [ r +; :::; r o ] (.26) wr j (j r + ; :::; r o ) ar t ordrd solutios of ui ro r?n? : (.27) Proof of Lmma.3. (a). By d itio, bst Q u t t u t t t t Q H o t t uz () z z ; m(m ro) " u t t # uz (); m(m ro) + uz (); m(m ro) 2 4 t t t t z z 3 5 : (.28) 7

8 By Lmma.2, w av " 2 O p (): u t t # uz (); m(m ro) t t D u t ;t uz () ; u t 2;t D t t D (.29) Similarly, w av 2 uz (); m(m ro) 4 uz (); m(m ro) uz () O p (): 2 t t 2 S b 2 S2 b 4D 2 S2 b S22 b ( S b S2 b S b b i 22 S 2 ) z z z z 3 5 D D z z bs S b bs 22 2 bs 2 b S b S D (.3) Form t rsults i i (.28), (.29) ad (.3), wic iss t proof. bst (b). Tis rsult follows dirctly from (a) ad t CMT. Q D O p () (.3) (c). If w dot u ;k k( b st ), t by d itio, u ;k (k 2 fr o + ; :::; mg) is t solutio of t followig dtrmiatal quatio S (u)? S (u) S (u) S (u) i S (u)? ; (.32) wr S (u) u I m b st. From t rsults i (a), w av S (u) u b st p ; (.33) 8

9 wr is a r r osigular matrix. Hc u ;k is t solutio of t followig dtrmiatal quatio asymptotically? S (u) S (u) S (u) i S (u)? : (.34) Dot T (u) S (u) S (u) S (u) i S (u), t (.34) ca b quivaltly writt as?t (u)? o? T (u) T (u)??t (u) i??t (u) o? : (.35) By?,? ad t rsult i (a), w av 2?S (u)? 2? bst? + o p (); (.36) 2?S (u) 2? bst O p (); (.37) 2 S (u)? 2 bst? O p (): (.38) From (.33), (.36), (.37) ad (.38), w gt 2?T (u)? 2?S (u)??s (u) S (u) i 2 S (u) i? i bst? Q D 2 D Q? + o p (): (.39) Usig t xprssios i (.28), w gt bst bst " " 2 2 i Q D 2 D Q i Q D o (m ro)m + o p () u t ;t b S 2 uz () z z # o + o p () ut ;t uz () # S b + uz () 2 bs z z o + o p (); (.4) 9

10 wr ad 2 ut ;t uz () b S o d N(; V uz ) z z o N ; (.4) uz () 2 uz () b S bs bs 2 z z o z z i z z o d uz () z z N(; V z z ) z z o N ;2 : (.42) From (.39)-(.42), w ca dduc tat wr N N ; + N ;2. ad ad Udr (.33) ad t rsult i (a), p?t (u)? d?n? 6 ; a:: (.43) o? T (u) o? o? S (u) o? o? S (u) S (u) i S (u) o? ui m ro b o? st o? b o? st S (u) i b st o? i ui m ro o? bst o? + o p () (.44)?T (u) o??s (u) o??s (u) S (u) i S (u) o? b? st o? b? st S (u) i b st o? i? bst o? + o p () (.45) 2 o? T (u)? 2 o? S (u)? 2 o? S (u) S (u) i S (u)? i o? 2 bst? + o p (): (.46)

11 By (.43), (.44), (.45) ad (.46), w av o? T (u) T (u)??t (u) i??t (u) o? o? T p (u) o? o? T (u) p??t (u) i??t (u) o? i ui m ro o? bst o? + o p () bst o???t (u) i i?? bst o? : (.47) Usig t xprssios i (.28), w ca dduc tat bst o? bst u t t Q i o? o? t t o? o? u t 2;t S22 b o? o? + o p () d N 2 o? ; (.48) wr N 2 R db u B u + ( uu uu ) o? R Bw2 B w 2 o?. From (.47) ad (.48), w dduc tat o? d ui m ro o? T (u) T (u)? Now, t rsults i (c) follow from (.49) ad t CMT.?T (u) i??t (u) o? N 2 + N??N??N 2 o? : (.49) (d) If w dot u ;k p k ( b st ), t by d itio, u ;k (k 2 fr + ; :::; r o g) is t solutio of t followig dtrmiatal quatio S (u)? S (u) S (u) S (u) i S (u)? ; (.5) wr S (u) u p I m b st.

12 Not tat?s (u)? 2 u??? b st? ; (.5)?S (u)? b st ad S (u)? b st? : (.52) Usig xprssios i (.5), (.52) ad t rsult i (a), w av 2? S (u) S (u) S (u) i S (u)? u?? 2? b st? 2? b st S (u) i b st? ui m r 2? bst? + o p (): (.53) From (a), w gt p bst o;? o p (): (.54) As a rsult, p bst? p bst? ; o? p bst? ; p bst o? i p bst? ; m(m ro)i + o p () (.55) ad 2? p? bst? p o;? bst? m ro A + o p (): (.56) Usig (.53) ad (.56), w av 2? S (u) S (u) S (u) i S (u)? u o;? o;? ui ro r 2 bst?? + op (): (.57) 2

13 Usig t xprssios i (.28), w ca dduc tat 2 bst 2 " u t t 2 t t Q 2 uz ()z z o u t ;t b S 2 uz () z z # o + o p () d N ; + N ;2 N ; (.58) wr N ; ad N ;2 ar d d i (.4) ad (.42) rspctivly. From (.57) ad (.58), w ca dduc tat p? S (u) S (u) S (u) i S (u)? d jui m r j ui ro r?n? : (.59) Not tat t dtrmiatal quatio jui m r j ui ro r?n? (.6) as m r zro igvalus, wic corrspod to t probability limit of p k ( b st ) (k 2 fr + ; :::; r o g), as illustratd i (c). Equatio (.6) also as r o r o-trivial igvalus as solutios of t stocastic dtrmiatal quatio wic iss t proof. ui ro r?n? ;.3 proofs of som lmmas i sctio 5 Assumptio.4 (GRR) (i) T dtrmiatal quatio jc()j as roots o or outsid t uit circl; (ii) t matrix o as rak r o, wit r o m; (iii) t (m r o ) (m r o ) matrix is osigular. m px j B o;j A o;? (.6) 3

14 W rst stablis t asymptotic proprtis of t OLS stimator ( b st ; b B st ) of ( o ; B o ) ad t asymptotic proprtis of t igvalus of b st. T stimat ( b st ; b B st ) as t followig closd-form solutio wr bst ; B b st bsy y b Sy x b Sy y b Sy x bs x y b Sx x ; (.62) bs y y Y t Yt ; Sy b x Y t Xt ; bs y y Y t Yt ; Sy b x Y t Xt ; bs x y S b y x ad S b x x X t Xt. (.63) Dot Y (Y ; :::; Y ) m, Y (Y ; :::; Y ) m ad wr X (X ; :::; X rprstatio b st cm I X b S x x X, ) mp, t b st as t xplicit partitiod rgrssio Y M c Y Y M c Y o + UM c Y Y M c Y ; (.64) wr U (u ; :::; u ) m. Rcall tat [ ( b st ); :::; m ( b st )] ad [ ( o ); :::; m ( o )] ar t ordrd igvalus of b st ad o rspctivly, wr j ( o ) (j r o + ; :::; m). Lt Q b t ormalizd lft igvctor matrix of b st. Lmma.4 Suppos Assumptio. ad Assumptio.4 old. (a) Rcall D ;B diag( 2 I ro+mp; I m ro ), t ( b st ; B b i st ) ( o ; B o ) Q B D ;B as t followig partitiod limit distributio N(; u z 3 z 3 ); R dbu B w 2 ( R B w2 B w 2 ) i ; (.65) (b) T igvalus of b st satisfy k ( b st ) p k ( o ) for 8k ; :::; m; 4

15 (c) For 8k r o + ; :::; m, t igvalus k ( b st ) of b st satisfy Lmma..(c). Lmma.4 is usful, bcaus t rst stp stimator ( b st ; b B st ) ad t igvalus of b st ar usd i t costructio of t palty fuctio. Proof of Lmma.4. (a). W start by d ig b S uy P u t Xt. From t xprssio i (.63), w gt ( b st ; B b i st ) ( o ; B o ) Q B D ;B " bsuy Sux b Q Sy b y BD ;B D ;B Q Sy b x B bs x b y Sx x P u t Y t ad b S ux Q BD ;B # : (.66) Not tat " bsuy Sux b Q BD ;B U Q B Y X # D ;B 2 U 3 U 2 (.67) ad D ;B Q B b Sy y b Sy x bs x y b Sx x Q BD ;B 3 2 P 3;t 3;t P 3 2 3;t 2;t P 2;t 3;t 2 P 2;t 2;t C A ; (.68) wr 3 ( 3; ; :::; 3; ) ad 2 ( 2; ; :::; 2; ). Now t rsult i (.65) follows by applyig Lmma.5. (b). Tis rsult follows dirctly by t cosistcy of b st ad CMT. (c). D S () I m st b, t js ()j o S () o o o? S () S () o o S () o o S () o? : (.69) Lt k k( b st ) (k r o + ; :::; m), usig similar argumts i t proof of Lmma..(c), w dduc tat k is a solutio of t quatio o o? S () S () o o S () o o S () o? ; (.7) 5

16 wr S () I m o? b st. Usig t rsults i (a), w ca sow tat S () I m r o o? o S () o o S () o o S () I m o i o o o + o p () bst o? o o? : (.7) Usig t d itios of H ad H 2 i t proof of Lmma..(c), w ca dduc tat H Q bst o Q H2 H QUM c Y Q D D QY M c Y Q D H 2 (.72) wr udr Lmma.5, D QY M c Y Q D D D D X S b x x X D z z z x xx xz d R Bw2 Bw 2 (.73) ad UM c Y Q D U D UX S b x x X D d B u;z B u;x xx R xz Bw2 dbu : (.74) Usig t rsults i (.73) ad (.74), w obtai H Q bst o Q H2 d ( o;? o;?) T, from (.7)-(.75), w obtai o? S () d I m ro B w2 db w 2 B w2 B w 2 ( o;? o;?): (.75) o S () o o S () o o S () o? B w2 dbw 2 B w2 Bw 2 ; (.76) uiformly ovr K. T rsult i (c) follows from (.76) ad by cotiuous mappig torm. 6

17 .4 proofs of som lmmas i sctio 6 Lmma.5 Suppos tat t coditios of Corollary 5.3 ar satis d, t F ; (k) Q (k)t ;o db u BuT 2;o + o p () (.77) for k ; :::; m, wr T ;o u u o ( o u o ) o u ad T 2;o o;? ( o;? o;?) o;? ; furtr, for j ; :::; p, F b; (j) d 2 u B mm () 2 yj jz 3S (.78) wr B m;m N(; I m I m ), yj jz 3S E (Y t j j 3S ) Y t j 3S ad Yt j j 3S Y t j yj z 3S z 3S z 3S 3S;t : Proof of Lmma.5. As t ordr of t tuig paramtr surs oracl proprtis of t LS srikag stimat, usig similar argumts i t proof of Torm 5.2, w rwrit F ; (k) as F ; (k) Q (k)u " u t Y t ( b S; S;o )Q S S;t Y t # + o p () (.79) wr udr Lmma.5 P u t t D ;S P S;t t mro ; db u B u o;? + o p (), ad (.8) (mpo+ro)r o R o;? Bu Bu + o p (): (.8) o;? Usig t xprssio i t proof of Torm 5.2, w obtai ( b S; S;o )Q S D ;S 2 b U 2; o ( o o ) + U;; U3;; b U 2; o;? ( o;? o;?) i (.82) 7

18 wr U; p (b o ), U 2; ro ; U2;, wr U 2; bo p bbsb B o;sb. From (.8) ad (.82), w dduc tat O o ad U 3; ( b S; S;o )Q S S;t t mro ; b U 2; o;? ( o;? o;?) B w2 B w 2 (.83) wr from t proof of Torm 5.2, w kow tat U2; ( ou o ) ou db u B w 2 Now, rsults i (.8), (.83) ad (.84) implis tat F ; (k) Q (k) b u; Q (k) u " B w2 B w 2 ( o;? o;?) 2;o;? + o p(). (.84) u t t ( b S; S;o )Q S mro ; I m o ( o # S;t t Q u o ) ou db u Bu o;? Q Q (k) u u o ( ou o ) ou db u Bu o;? ( o;? o;?) o;? wic sows t rsult i (.77). W xt sow t scod claim. W ca rwrit F b; (j) as F b; (j) b u; p " u t Y t j bs; S;o Q S S;t Y t j # : (.85) 8

19 Usig t argumts i t proof of Torm 5.2, 2 " u t Y t u t Yt 2 2 d N " u t Y t j j j bs; u t 3S;t S;o Q S S;t Y t j # 3S;t 3S;t 3S;t Yt u t 3S;t z 3S z 3S z3s y j # + o p () u t Y t j 3S;t z 3S z 3S z3s y j + op () j o p () ; u yj jz 3S : (.86) From t rsults i (.85) ad (.86), w dduc tat F b; (j) d u N wic iss t proof. d ; u yj jz 3S 2 u B mm () 2 yj jz 3S 2 Mor Simulatio Rsults I tis sctio, w coduct simulatio xprimt to ivstigat t it sampl prformacs of iformatio critria o joit slctio of laggd di rcs ad coitgratio rak i VECM. W us t tird modl i Sctio 7 of t mai txt to grat t simulatd data: Y ;t Y 2;t o Y ;t Y 2;t wr u t iid N(; u ) wit u ;o 2;o 2;o 22;o + B ;o Y ;t Y 2;t, :5 :5 :75 :5 :5 + B 3;o Y ;t 3 Y 2;t 3, o is spci d as follows ad :5 : :2 :4 + u t ; (2.) ; (2.2) 9

20 B ;o ad B 3;o ar tak to b diag(:4; :4) suc tat Assumptio 5. is satis d. T iitial valus (Y t ; " t ) (t 3; :::; ) ar st to b zro. T mpirical modl taks t followig form Y t o Y t + 3X B o;j Y t j + u t (2.3) j wic is stimatd by LS srikag stimatio i Sctio 7 of t mai txt. I t abov tr cass (i.., tr di rt spci catios of o ), w iclud 5 additioal obsrvatios to t simulatd sampl wit sampl siz to limiat start-up cts from t iitializatio. Bfor d ig t iformatio critrio, w itroduc som otatios. Lt i, i 2 ad i 3 b dummy variabls (wic oly tak valus or ). W us [r; (i ; i 2 ; i 3 )] to dot a spci c mpirical submodl wit coitgratio rak r (r ; ; 2) ad trasit dyamic structur (i ; i 2 ; i 3 ), wr i k (k ; 2; 3) idicats tat t k-t laggd di rc is icludd, ad i k mas t k-t laggd di rc is xcludd. W us b u; [r; (i ; i 2 ; i 3 )] to dot t stimator of u basd o t rducd rak rgrssio stimator of ( o ; B o ) i t mpirical modl idicatd by [r; (i ; i 2 ; i 3 )]. T iformatio critrio w cosidr taks t followig form i IC [r; (i ; i 2 ; i 3 )] l dt bu; [r; (i ; i 2 ; i 3 )] + c m 2 (i + i 2 + i 3 ) + r(m r) + mr wr c is a squc wic gos to zro wit gos to i ity. W c 2, l() or l(l()), t iformatio critrio bcoms AIC, BIC or HQ rspctivly(s,.g., Pillips ad McFarlad, 997 ad Cao ad Pillips, 999). T modl slctd by t iformatio critrio is idicatd by i br ; (bi ; ;bi 2; ;bi 3; ) arg mi IC [r; (i ; i 2 ; i 3 )] : (2.4) [r;(i ;i 2 ;i 3 )] Tabl 2. prsts t it sampl modl slctio probabilitis of di rt iformatio critria i di rt modl spci catios. It is clar tat BIC outprforms AIC ad HQ i all scarios cosidrd i tis simulatio dsig. Bot AIC ad HQ ar vry cosrvativ, particularly w t coitgratio matrix o dos ot av full rak ad t sampl siz is small. W t sampl siz icrass to 4, t probabilitis of AIC ad HQ slctig t tru modls ar improvd. O t otr ad, BIC dos rmarkably wll i slctig 2

21 t tru modls v w t sampl siz is small (i.. ). If w compar t simulatio rsults displayd i Tabl.2 of t mai txt wit ts i Tabl 2., w ca s tat LS srikag stimatio prforms as wll as BIC, ad works muc bttr ta AIC ad HQ, i trms of slctig t tru modls. Tabl 2. Modl Slctio wit Iformatio Critria AIC Subst Slctio r o, o ( ) r o, o ( -.5) r o 2, o ( ) bm 2 T bm 2 C bm 2 I BIC Subst Slctio r o, o ( ) r o, o ( -.5) r o 2, o ( ) bm 2 T bm 2 C bm 2 I HQ Subst Slctio r o, o ( ) r o, o ( -.5) r o 2, o ( ) bm 2 T bm 2 C bm 2 I Tabl 2.: Rplicatios5. r o dots t rak of t tru matrix o ad o rprsts t igvalus of o. "T" dots t slctio of t tru modl, "C" dots t slctio of a cosistt modl (i.., t modl wic sts t tru modl), ad "I" dots t slctio of a icosistt modl (i.. t modl wic producs icosistt stimatio). T modl slctio procdur displayd i (2.4) is somtims calld as optimal subst slctio, bcaus it ivstigats all possibl submodls std by t largst mpirical modl (2.3) ad slcts t bst o (wic miimizs t iformatio critrio). I mpirical applicatios, t squtial slctio procdur is popular bcaus it is computatioally muc asir ta t optimal subst slctio. Spci cally, t squtial slctio procdur oly cosidr tr possibl valus for (i ; i 2 ; i 3 ), i.., (; ; ), (; ; ) ad (; ; ). By d itio, t squtial slctio procdur will always miss t tru modl, bcaus O t otr ad, t optimal subst slctio cosidrs all igt possibl valus for (i ; i 2; i 3). 2

22 t tru modl as t trasit dyamic structur (; ; ) wic is ot cosidrd by t squtial slctio. 2 T simulatio rsults of t iformatio critria combid wit t squtial slctio procdur ar prstd i Tabl 2.2 ad 2.3. Tabl 2.2 Modl Slctio wit Iformatio Critria AIC Squtial Slctio r o, o ( ) r o, o ( -.5) r o 2, o ( ) bm 2 F bm 2 C bm 2 I BIC Squtial Slctio r o, o ( ) r o, o ( -.5) r o 2, o ( ) bm 2 F bm 2 C bm 2 I HQ Squtial Slctio r o, o ( ) r o, o ( -.5) r o 2, o ( ) bm 2 F bm 2 C bm 2 I Tabl 2.2: Rplicatios5. r o dots t rak of t tru matrix o ad o rprsts t igvalus of o. "F" dots t slctio of t full modl, i.. t modl wit tru coitgratio rak ad all tr laggd di rcs. "C" dots t slctio of a cosistt modl (i.., t modl wic sts t tru modl), ad "I" dots t slctio of a icosistt modl (i.. t modl wic producs icosistt stimatio). From Tabl 2.2, w s tat AIC ad HQ slct all tr laggd di rcs wit probabilitis clos to, wic is also co rmd i Tabl 2.3. Morovr, AIC ad HQ ar vry cosrvativ, bcaus ty td to slct igr coitgratio raks i it sampls, wic is similar to t simulatio rsults wit optimal subst slctio displayd i Tabl 2.. W compard wit AIC ad HQ, BIC is mor accurat i slctig t tru coitgratio rak, but it also slcts all laggd di rcs wit probabilitis clos to. 2 Not tat t LS srikag stimatio dos ot su r from suc a problm bcaus its stimator is ivariat to t prmutatio of t laggd di rcs. 22

23 Tabl 2.3 Laggd Ordr Slctio wit Iformatio Critria AIC Squtial Slctio r o, o ( ) r o, o ( -.5) r o 2, o ( ) bp bp bp BIC Squtial Slctio r o, o ( ) r o, o ( -.5) r o 2, o ( ) bp bp bp HQ Squtial Slctio r o, o ( ) r o, o ( -.5) r o 2, o ( ) bp bp bp Tabl 2.3: Rplicatios5. r o dots t rak of t tru matrix o ad o rprsts t igvalus of o. bp dots t slctd ordr of t laggd di rcs. Rfrcs [] Cao, J. ad P.C.B. Pillips, "Modl slctio i partially ostatioary vctor autorgrssiv procsss wit rducd rak structur", Joural of Ecoomtrics, vol. 9, o. 2, pp , 999. [2] Pillips, P.C.B. ad J.W. McFarlad, "Forward xcag markt ubiasdss: T cas of t Australia dollar sic 984", Joural of Itratioal Moy ad Fiac, vol. 6 pp , 997. [3] Pillips, P. C. B. ad V. Solo, "Asymptotics for liar procsss", Aals of Statistics, vol. 2, o. 2, pp. 97-,