靴帶抽樣預測績效檢定量
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(2) 1. û˝œDñí. ã¿øòul¾%ÈçðÉ·íÿõ …Î7ªàVR¿%ÈCÀ¢‰b„V•‘Õ, °v6uð%ÈÜD‡il¾_iší™Ä5ø ø__çÇ,ã¿ô^íÄ †DÃãíR˙åZïé½b. Êo‚, ×Öbíõ„û˝îN¬š…Õã¿, Jã¿ÌjÏ (mean square prediction error, MSPE), Ì"úÏÏ (mean absolute error, MAE) Ć‘²_çíã¿. _ Wà: Meese and Rogoff (1983) Ê−0ã¿2, Sà MSPE ªœ_íã¿ô ^ Akgiray (1989) ‚à MAE Ç,$Ñ{‰æ (volatility) ã¿M Swanson and White (1995) Ñû SIC(Schwarz information criterion) Dš…Õ R2 , WJ«n±‚. ‚0u´ªJã¿„V¹‚‚0 Engel (1994) †ÑûÄüã¿−0‰“íŸb. ,Hj¶|×í½æÊk: Ìu MSPE Cu MAE îÑ Óœ‰b, Bb. ?òQâ…bíø ,lM (estimate) í×üV‡ìã¿_íiš Ü,Bý@ vû|…bíך…}º, yJ$lRíjªWcqì Ê¥*-, ~>| sÇ,_ã¿ô^íì˙å, wøuâ Diebold and Mariano (1995) FT| íã¿Äüì¾ (J DM [ý), àJìs__ã¿ÌjÏCwF%ȸ܃b (economic loss function) u´ó ùÇ,j¶uâã¿°Q (forecast encompassing) íhõVÇi_iš ?¹, ‡i¨ÖÊúG_2 (competing model) í. Ôìm7u´ŒkTôã¿ô^ (Chong and Hendry, 1986 Clements and Hendry, 1993). ÇøjÞ, Àr †\¼Drù. (2003) T|ø_híì˙å, ˚5Ñ AR−tc ì¾ k5, ¤˙ålJ AIC ‘²ø_Öiáí AR _, yJ|üj¶ (least squares, LS) ,lñÍ2í¡b, |(Jf$í t $l¾ðiáu´éOæ. kÉ ¥˙å.cªJ°vàVã¿Äü4Dã¿°Q4ì, °vx–1 ÀDÍTñqíiõ ÇøjÞ, 6.}êÞéN DM D°QìJÌ‚b,lÅ‚ 2 第五屆全國實證經濟學論文研討會 The 5th Annual Conference of Taiwan's Economic Empirics.
(3) ‰æbv, ú ƒb (kernel) . Uí½æ (Clark, 1999) ÖÍ,Hú˙åîJÃãí$lj¶‡i_ã¿ô^íiš, Ou Harvey et al. (1997, 1998) J_Ò}&êÛ: çš…Õã¿_b.Öv, ,H‡sì¾î}. ßÞýíøÏÏ… ÇøjÞ, Óã¿‚È (forecast horizon) ÓÅ, øÏÏ… íÛïý °vÀr †\¼Drù. (2003) í_Ò!‹6éý, Ê׶} í8”- AR − tc øÏÏ[Ûikã¿Äü4Dã¿°Q4ì, OÊ,H8$?æÊøÏÏ…í½æ ¥<âÊ,H8”-, J쾇"™Ìcq, BbÌ* ‡ì˝§uÄÑs__íã¿ÄüÏæ4, Cuâì¾Ã½øÏÏ…F¨A ¥øU,Hì¾íõà4×H~p Ĥ, J;b)ƒœ£üí$lR, Bý@S àx U (robust) øÏÏ[Ûí쾪Wcqì …dô/Àr †\¼Drù. (2003) íû˝, ÒT|ø ¬˙Ö¥º‚ š ˙å (sieve bootstrap), $ÇZ¾ AR − tc ì¾íüš…øÏÏRÏ âk AR − tc íì˙媰vàkã¿Äü4Dã¿°Q4ì, 6ªJfn‘². ƒbí½æ, /üš…øÏÏik(s6 Jòg7k, â AR − tc ì¾|ê, J ZZ¾ã¿Äü4Dã¿°Q4ìíøÏÏ…N˛yxªW4 ÇøjÞ, ¬˙ Ö‚š˙åuJ ¼ŸÓš…bÓ‹ í AR(p) _¡NøOí(4¬˙ (linear process) 1 , 1J LS ,l AR(p) 2Fí„ø¡b, y½ì¾íüš…}º, |(. J¤}ºªWcqì Ĥ, ú׶}íu‰æ0ì¬˙, Wà̼Ÿí ARMA, ¹ªà,H AR(p) ‹J¡N M)øTíu, Ê AR − tc íZ˙å2, 6uJ AR ¬˙¡NbíåóÉ, ¥£ßD¬˙Ö¥º‚šxó°íbç!Z !k,H Üâ, 'AÍË, …dZSà¬˙Ö¥º‚šZ¾ AR − tc ì¾øÏÏ… ;W Horowitz (2000) íõ, d.2àkTÜåóÉíÖ‚š¶þ: ¡b “Ö¥º‚š (parametric bootstrap, PB) D –)Ö‚š¶ (moving blocks 1. ;W B¨ uhlmann (1997), p.124 ízp, ¥½µ‚šjD–)‚š¶°ÑÌ‚b‚š¶. 3 第五屆全國實證經濟學論文研討會 The 5th Annual Conference of Taiwan's Economic Empirics.
(4) bootstrap, MBB) ‡6lø’eº_ø__çí_ (¦Ñ ARMA _), ,l_. 2F„ø¡b(yªW½; w!‹.cú_íqìÝÜ>, °v…−ƒü š…-Ý(4,l, âk¡b,l.Äüª7à‚ší!‹ ¤Õ, MBB øhôM }A _–), âÓœ‚¦–)‚šš… Öͳ‡6_Ïqí½æ, ºª ?âk‚š˙å2, I7–)ÈíóÉ4, û_½(íbÑÝ0ì IA³\í u, …ú–)Å í²ÏÝÜ>, /d.2?Ìø_íj¶²Ï–)Å ÄÑ DM Dã¿°Q4ìú ƒb.x U4, /âJ,ínªJø−, ¬˙Ö¥º‚š ¶uJ AR(∞) ¡N ARMA, Ĥª PB yøO“, 7/³Ý(4,lí½æ, [Û óú ì, °v…?³,H MBB íÿõ ¥<£u…dSà¬˙Ö¥º‚š¶7 .Uà PB D MBB íŸÄ. …dFT|í¬˙Ö‚š¶, .cÊÍT,qjZ, ÊÜ,yb¯¯ø_4 (bootstrap consistency) íÃãb° LSÖ‚š˙åFÞ@í½bܽæÊk:. ÓOš…bÓ‹, Ö‚šì¾×š…}ºu´Y¹ƒú@íÚ¡}º (asymptotic distribution) 2 ? ¥uÖ‚š˙åu´£üíÉœFÊ ÄÑøO7k, Bb1.ø. −ì¾üš…}º, û˝6ªàÚ¡}ºCÖ}º (bootstrap distribution) ¡N … ÄÑÖ}ºªeÑì¾üš…}ºí,l, AÍ.âb°Ö}ºxø_ 4 JÊך…-, ì¾íÖ}º.°kwÚ¡}º, ¤vUàÖ‚šì¾ª Wcqì, AÍ̶×)£üí!‹ ¥¿Ö¤Ö‚š˙å1./ç. Ê AR í=1-, Ö‚šì¾íÚ¡WÑ1.àøO;dÀÓ 9õ,, Basawa et. al. (1989) {„p, Êêà AR(1) _qì-, Ö¥º‚šÀc¦,l. íך…}º}Y¹ƒwÚ¡}º ÇÕ, Basawa et. al. (1991a) „p, ÊÀ;_q ì-, ¹Uß×áÑG, Ö‚šÀc¦,lך…}º.°kwÚ¡}º O uÊÇø¹dı2 (Basawa et al., 1991b), FbT|Çø½º‚š˙å, 1„p_à k,Híqì ¥[ýÎÝ?„pw£ü4, ´†Ö‚š¶øÏàí8”êÞ B 2. ;W Nankervis and Savin (1996) íì2, UàÚ¡@äMªWì˚ÑÚ¡Üì¾ (asymptotic test), J½(}ºl@äMªWì˚ÑÖ‚šì¾ (bootstrap test). 4 第五屆全國實證經濟學論文研討會 The 5th Annual Conference of Taiwan's Economic Empirics.
(5) bJ B¨uhlmann (1997) D Park (2002) Ñ!, û|J AR ˙åFí¬˙Ö ì¾Ú¡}º v}ºDú@5Ú¡Üì¾íך…}ºó°, îÑ™ÄG ¥ —J„pv˙åí£ü4. Bbí_Ò}&êr¯¯BbíZñí â_Ò!‹êÛ, Êã¿ÏÏVA5” 4, Êך…-, …dí¬˙Öì¾Dã¿Äüì, ã¿°Qì, AR − tc ú6 íøÏÏ·Q¡9l#ìí±ñéO®Ä ¥[ý¬˙Ö‚šì¾D,Hú ì¾xó°íÚ¡}º, ªJeÑ…dÖ‚šìø_4íøábM„W Oy xõ<2íu, Êõ„cíš…Õã¿_b- (Wà: n = 16, 32, 64 ), ÔuÊ% ‡ã¿‚bÓ‹v, …díÖ‚š˙åªJ^Z¾ AR − tc ì¾íøÏÏR Ï, U5øÏÏóçQ¡k±ñí®Ä Ê|c(‰jÞ, ¬˙Ö‚šì¾D wFú_ì¾xéNí[ÛG ÄÑ AR − tc 쾪àV DM Dã¿° Qì, º¯¬˙Ö‚š˙å, xy UíøÏÏ[Û, Ĥúã¿íõ„û˝ xóç×í6Œ dıíé§à-, ù Ñã¿ô^Ç,ìíè; ú Ѭ˙Ö‚š˙å £wÚ¡}º; û Ñ_Ò}&, |(ø Ñ! Ê
(6) -í}&2, ∗ [ýÖ‚š š…, (Ω∗ , ∗ , P ∗) [ý‘KÊø š…-íÖœ0˛È(bootstrap probability space), p∗. d∗. E ∗ , −→, −→ }[ýú@ P ∗ í‚ M, œ0Y¹D}ºY¹ bç„p-kË“. 2. ã¿ìíè. Ñ°…dí©ø4, Ê… Bb–I4íè,HúÇ,ã¿ô^ì¾, ¨: ã ¿Äü4ì, ã¿°Q4ìD AR − tc ì¾ ÇÕ, 6†bzpd.2ú¥< ì¾íuÇ. 5 第五屆全國實證經濟學論文研討會 The 5th Annual Conference of Taiwan's Economic Empirics.
(7) 2. 1. ã¿Äü4ì. ˛ø‰b yt ís ã¿M, û˝6¦É-s ã¿MÄü4íªœ Diebold and Mariano (1995)(J-˚ DM) ílT|$lRí-Z, JÇ,s ã¿MíÏæ. u´VAk$lí‚šÏÏ cq (e1t , e2t ), t = 1, 2, ..., n, }[ý_ 1 D_ 2 % ‡ h ‚ (h-step) íã¿ÏÏ DMíÇ,juªœ¥s ã¿ÏÏí¸Üƒb g(eit ) (loss function) íÌbu´ó Ĥ, ã¿Äü4ìí™ÌcqÑ: H0 : E [g(e1t ) − g(e2t )] = 0 C E [dt ] = 0. (1). 7úcqÑ E[dt ] = 0, ¤T, dt = g(e1t ) − g(e2t ) H[¸ÜÏæ (loss differential) Ö͸܃b g(eit ) íqìjÝÖ, eû˝6í‡æ7ì ʤ, …dYrÖõ „d.íd¶, Jã¿ÌjÏ (MSPE) V©¾¸Ü ?¹, I g(eit ) = e2it , ¤v¸ÜÏ æÑ dt = e21t − e22t =MSPE1 - MSPE2 â (1) ªø, Bbbì dt í‚ñÌbu´ÑÉ DøOí$lRéN, J dt íš…Ìb (d) TÑì (1) íì¾ Ä¤, DMíì$l¾à-: d DM = V ar(d). (2). ¤T,Var(d) ÑÅ‚‰æb (long-run variance) íø_4,l (consistent estimator) ÄÑ%‡ h ‚ã¿ÏÏx (h − 1) ¼åóÉ 3 , Ĥ, ç h > 1 v, b dt øæÊABóÉ (autocorrelation) íÛï ÑTÜ¥_½æ, DM SàÌ‚bí,. lj¶J×)Å‚‰æb (long-run variance) íø_,l à,FH, Ê%‡ h ‚ íã¿ÏÏ2, ×kCk h ¼íABóɉæbîÑÉ, Ĥ, FbSàÌG ƒb (uniform kernel) 1qì (bandwidth) Å Ñ (h − 1) VlÅ‚‰æbí,l h−1 M ?¹, Var(d) = n1 k=−(h−1) γk , w2, γk [ý dt í k ¼š…ABu‰æb DM„p, Ê (1) í™Ìcq-, ¤$l¾ÑÚ¡™ÄG}º 3. w2øª?Ñ (h − 1) ¼ ̬˙ (MA(h − 1)). 6 第五屆全國實證經濟學論文研討會 The 5th Annual Conference of Taiwan's Economic Empirics.
(8) ¹U DM ì¾ÑÚ¡™ÄG}º, O Harvey et al. (1997, 1998) í_Ò!‹ éý: Êš…Õã¿_b (n) .Öí8”-, âkÅ‚‰æb,l.Äü, û_Ú¡} ºÌ¶^¡N DM ì¾üš…}º, û_vì¾ßÞýíøÏÏ… Ô Ë, Ó h Ó‹, øÏÏ…y$ý ¥[ýÊ,H8”-, J DM 쾇"™ Ìcq, BbÌ*)ø¤!‹uVAkøÏÏ…, CuVAk㿸ÜÏæ4 ¥ øUvì¾íõà4×H~p …døÊš(J¬˙Ö‚š¶Z¾¥_½æ, 1„ p¤˙åí£ü4. 2. 2. ã¿°Q4ì. ÇÕøÇ,ã¿ô^íj¶Ñã¿°Q4ì h1,, ã¿°Qìk²ìíu, ñ‡_íã¿Mu´¨Ö¬_2íFóÉm7 JBbEí_¨ÖF óÉím7, †³.bD¬_$A ¯ã¿ Ĥ, û˝6ªJ*ã¿m7í i VÇ,ã¿_íß; ã¿°Q4ìÎ7ªJ‡i¬_u´Öúã¿ àíçÕm7JÕ, …6ªJçTã¿Äü4ìí.Œì ã¿°Qìíh1Ê ¯ã¿5, I yt Ѫh¿íb, (f1t , f2t ) }u _ 1 D 2 FßÞís ã¿M, /_ 1 Ñû˝6¿=í_, 7_ 2 Ñw¬_ Døs ã¿M‹Ì, $A ¯ã¿M fct à-, fct = (1 − λ)f1t + λf2t. 0≤λ≤1. (3). ¤T, λ Ñb ¦, xœüÏωæbíã¿M@v#8œòíb, Ĥ, J λ → 0, [ýã¿M f1t œÑÄü, œüíã¿Ïωæb; ¥5, †[ý f2t œÄü. Jbk 0 D 1 5È, †ªN¬ ¯ã¿, )ƒikŸlã¿Mí!‹ Ĥ, N¬ b λ í,lDì, ¹ª‡iu´$A ¯ã¿í.b ÑzpàS,l λ, I eit = yt − fit , i = 1, 2 [ý yt ís ã¿ÏÏ, εt Ñ ¯ã 7 第五屆全國實證經濟學論文研討會 The 5th Annual Conference of Taiwan's Economic Empirics.
(9) ¿MíÏÏ, †(3) ªZŸA, εt = yt − fct = e1t + λ(f1t − f2t ). (4). ZŸ (4) à-: e1t = λ (f2t − f1t ) + εt. (5). ʤ, b λ ªj„A, _ 2 D_ 1 ã¿MÏú_ 1 íã¿ÏÏíj„?‰ J λ = 0, [ý_ 2 íçÕm7úkj„_ 1 íã¿ÏÏ1Ì6Œ ?¹, _ 1 ¨Ö. _ 2 ím7 J λ > 0, †[ý_ 2 íçÕm7ªJj„_ 1 íã¿ÏÏ, ?¹, _ 1 „êr¨Ö_ 2 íFm7 Ĥ, ã¿°Q$l¾¹Êì λ kÉí™Ìc qu´A, ú@5úcqÑ λ > 0. ÑjZ,l λ, ø (5) ZŸA, e1t = λ(e1t − e2t ) + εt. (6). J λ = 0 í™ÌcqA, Chong and Hendry (1986) D Clements and Hendry (1993) ˚¤8”Ñ f1t °Q f2t . ÄÑ (6) 2í εt ª?}åóÉ, ĤHarvey, Leybourne and Newbold (1998)(J-˚ HLN ) ‡, J DM íj¶V$Aã¿°Qì¾ I dt = e1t (e1t − e2t ), à HLN FH, J f1t °Q f2t , † E[dt ] = 0, Ñìí™Ìcq, 7úcqÑ E[dt ] > 0, [ý f2t ¨Öàím7 HLN íã¿°Qì¾à-, d ENC = V ar(d). (7). 2, øàÊ DM ì2íì2, d¯ Ñ dt 5š…Ìb, Var(d) ÑÅ‚‰æb5ø_ 4,l, wljD DM $l¾ó° HLN „pÊ™Ìcq-, ì¾ÑÚ¡™ ÄG}º D DM ì¾éN, Êš…Õã¿_b n .Ö, CÓã¿‚È h Ó‹, ENC øßÞýíøÏÏ…. 8 第五屆全國實證經濟學論文研討會 The 5th Annual Conference of Taiwan's Economic Empirics.
(10) 2. 3. ABc¦íì˙å. Ê… , Bb†bzp, Àr †\¼Drù. (2003) àS‚àABc¦Vlã ¿ô^Ç,$l¾ ((2) D (7) ) Ñ7yÀU7jABc¦íì˙å, Bbl Jø_ÀíWä«Hvì˙åí!… ÿ cq dt = c + ηt , ηt Ñ IID(0,ση2 ), t = 1, 2, · · · , n âJ, 2-1 D 2-2 ínªJêÛ, .
(11) uã¿Äü4ìCuã. ¿°Q4ì, ·Êì dt í‚ñÌbu´ÑÉ4 ĤÊ,Hcq-, ì dt í‚ ñÌbu´ÑÉóçkì H0 : c = 0 ;WÀíl¾Ü, BbÉÛø dt ú bác¦(, Jf$ t $l¾ªWì, ¤v t $l¾ÑÚ¡™ÄG}º ÊyøO“í8”, J dt æÊABóÉ, øà ADF À;ì¾íZ˙å, 'A Í˪Sà AR ¬˙^£wåóÉ y‚à LS ,l¡b, 1J t ì¾ðiá u´éOækÉ ?¹lJ AIC ²Ï_çí¼Ÿ p, yJ AR(p) _º_ dt à-: dt = c +. p . αp,k dt−k + p,t ,. (8). k=1. QON¬ LS ,l¡b, 1Jf$í t $l¾ì (8) 2 c u´éOækɹª â Àr †\¼Drù. (2003) ÊJ-øOícq-„p, ¥l˙å2 t $l¾ YÍÑÚ¡™ÄG}º cq 1: dt = µd + ψ(L)t , w2 µd Ñb, / ψ(L) =. ∞. j j=0 πj L ,. π0 = 1 1xJ. -4”: 1. t Ñ IID (0, σ 2 ) íÓœ‰b/æÊ r ≥ 4 U) E|t |r < ∞ 2. úF |L| ≤ 1, ψ(L) = 0 / æÊ s > 1 U). ∞. j=0 j. s. |πj | < ∞. cq 2: p = p(n) Å— p → ∞ / p = o(nκ ), w2 κ < 1/2. 4. Êã¿Äü4ìDã¿°Q4ì- dt }Ñ e21t − e22t D e1t (e1t − e2t ). 9 第五屆全國實證經濟學論文研討會 The 5th Annual Conference of Taiwan's Economic Empirics.
(12) ìÜ 1: (Àr †\¼Drù., 2003) I cˆ D SE(ˆc) }[ý(8) 2¡b c í LS ,lD™ÄÏ Êcq 1,2 D (1) í ™ÌcqA-, ç n −→ ∞, AR − tc =. cˆ d −→ N(0, 1) SE(ˆ c). (9). cq 1 D 2 M)ªø¥zp cq 1 zp dt uø_ÝøO“í0ì¬˙, …°QF ̼Ÿí ARMA(p,q), J£"׶}x (h − 1) ¼åóɬ˙ 7cq 2 [ý J AR(p) ¡N dt ¬˙v, b°w¼Ÿ.âÓš…b (n) Ó‹7Ó‹, /Óŧ .â ük n1/2 OÊõÒÍT,, cq 2 1³TXLSj¶²ì AR í¼Ÿ ¹UÊd .2, AIC D BIC àk‘²_í_缟, OuÀr †\¼Drù. (2003) ‡J AIC ‘² AR í¼Ÿ ¥uÄÑSà AIC Ć‘² AR í¼Ÿ}Å—cq 2 í‘K, Jü\ìÜ 1 í£ü4 ã¯J,Fn, Ê AR − tc ì¾íõÒÍT,, ÉâJ AIC ‘²_çí¼Ÿ p, 1JÖiáí AR(p) º_ dt ; QOJ LS ,l¡b(, yJ t $l¾ìiáu ´éOækɹª ⤪ø, AR − tc .cª°vàk DM D ENC ì¾, w ŸÜÀq], 7/׶}í$l,ñîÛA˙àJÏW,H˙å, Ĥ AR − tc ××Ó‹õ„û˝íjZ4 M)·<íu, DM D ENC uJÌ‚blÅ‚‰æ b, BýÊÜ,úæ”4x U4 (robust), 7cq 1 †³5?ƒæ”4í8” .¬Ê-ø í_Ò2, Bb?5?æ”4íqì, àJðì¾í[Û âÀr †\¼Drù. (2003) í_ÒêÛ, ÖÍÊ׶}í8$, AR − tc øÏÏ[Ûik DM D ENC ì¾ OÊüš…-YÍæÊøÏÏ…, /¤Û ï}Ó h Ó‹y‹Ã½ ÔW7k, hô…d[ 1-1 í_Ò!‹, ç n = 64, h = 1 Ó ‹ƒ 8 v, AR − tc øÏÏâ 0.083 MÚÓ‹ƒ 0.120, ±¬ 0.05 íéO®Ä5 ¥ <_Ò!‹DÀr †\¼Drù. (2003) íÑûó", îN| AR − tc ª?í øÏÏ…Ûï.ñe ¨A¥ÛïíŸÄª?Êküš…-, âk¡b,l.D 5. àS×)¥<!‹í_Òql, økû Ì Ü. 10 第五屆全國實證經濟學論文研討會 The 5th Annual Conference of Taiwan's Economic Empirics.
(13) Äü, ª7U™ÄG̶^¡N AR − tc íüš…}º Ê- , BbÒJ¬˙Ö ‚š¶Z¾¥_½æ. 3. ¬˙Ö¥º‚šì. Efron (1979) ílT|Ö¥º‚š¶íh1, JªW$lR Ö¥º‚š¶uø. ½º‚š (resampling) ˙å, àJ,lÊ׶}8”-ì¾í„øüš…}º â kÖ‚šì¾ (bootstrap test) uòQJš…hôMÑ‚ñ, yY½µ‚š˙å Ö‚š}º, óœkך…}ºí¡Nj, øO7k, ¤j¶Êüš…v, øyQ ¡ì¾íüš…}º, wøÏÏ[ÛøikÚ¡Üì¾ Bbí¬˙Ö‚šwõD AR − tc Z ÿDd¶éN ÄÑʬ˙Ö‚š½ Öš…í¬˙2, à°l AR − tc ì¾øO (;W (8)), 6uJ p Óš… bÓ‹í AR(p) ¡N’eßÞ¬˙ ÄÑ AR − tc íD¬˙Ö‚š˙ås6x ó°íbç!Z, ¥£u…dSà¬˙Ö‚š¶Z¾ AR−tc øÏÏí3bŸÄ.
(14) -, BbílT|¬˙Ö‚š˙åíÏW¥£wzp, QO„pv˙åí} ºø_4. 1. cìø hôM dt , Êã¿Äü4Dã¿°Q4ì}Ñ: e21t −e22t D e1t (e1t − e2t ), Sà LS ,lJ- AR(p) _, w2 p â AIC F²ì: dt = c + αp,1 dt−1 + αp,2 dt−2 + · · · + αp,p dt−p + p,t. 1l AR − tc ì¾íM 2. J LS ,lJ- AR _: dt = α ˆ p,1 dt−1 + α ˆ p,2 dt−2 + · · · + α ˆ p,p dt−p + ˆp,t 11 第五屆全國實證經濟學論文研討會 The 5th Annual Conference of Taiwan's Economic Empirics. (10).
(15) 3. *J-íõ„}º (empirical distribution) Óœ‚¦{Ï, )ƒø. {∗t }nt=1 . 1 I(−∞,x] (ˆ p,t − ¯n ) n t=1 n. F (x) =. ,2, I ÑN™ƒb (indicator function), ¯n = 1/n. (11) n. p,t t=1 ˆ. . 4. ‚à {∗t }nt=1 /Y-ªW½, ×)ø Ö‚šš… {d∗t }nt=1 : d∗t = α ˆ p,1d∗t−1 + α ˆ p,2 d∗t−2 + · · · + α ˆ p,p d∗t−p + ∗t. (12). 5. J {d∗t }nt=1 º_J- AR(p) _, °vJ LS ªW,l: ∗ ∗ ∗ ˆ p,1 d∗t−1 + α ˆ p,2 d∗t−2 + · · · + α ˆ p,p d∗t−p + eˆ∗p,t d∗t = cˆ∗ + α. (13). ∗ w2 cˆ∗ , αˆp,i , i = 1, 2, · · · , p [ýw LS ,lM, eˆ∗p,t [ýw{Ï. 6. l AR − t∗c à-: AR − t∗c =. cˆ∗ SE(ˆ c∗ ). w2 SE(ˆc∗ ) Ñ cˆ∗ í™ÄÏ 7. ½º¥ (3) ƒ (6) NB Ÿ 8. }J,H NB _ AR − t∗c íMçTì¾í}º, 1v|@äM, ˚Ñ Ö. ‚š @äM(bootstrap critical value) ªœ¥ (1) 2 AR − tc íMD Ö ‚š @äM, Jì™Ìcq 9. ½º¥ (1) ƒ¥ (8)ND Ÿ, 1l‡"™ÌcqíŸb, J NR [ý 10. J dt VAk™Ìcq, †. NR ND. [ýÖ‚šøÏÏ (bootstrap size), ´†Ñ. Ö‚š‰ (bootstrap power). 12 第五屆全國實證經濟學論文研討會 The 5th Annual Conference of Taiwan's Economic Empirics.
(16) BbQO†bzp,H¥í<2 ¥ (2) íñíuÑ7½hßÞ dt íåó É!Z, N¬,l_¡b(, jZJ-½º‚šíT ¥ (3) øc¦{ÏpÎ Ìb(y‚š, ñíuÊ^£ABóÉ[b,lv}í Q, RÏ (downward bias) Ûï, °v?5?ƒ‚ñ}ºxÌbÑÉí4”, ¥D(10) 2‹piá. íTàó° ¥ (4) ¹F‚í½˙å, ñíÊßÞÖ‚šš… d∗t ¥ (3) B (7) uÑ7°| AR − t∗c Ö‚š}ºí½˙å M)øTíu, (12) í½¬˙. 2uøiáqÑÉ, ¥uÄÑBbubì H0 : c = 0 í™Ìcq, .â™Ì cqA-ì¾í}º, Ĥø c = 0 í8ÑÑp½í¬˙2 ªœ AR − tc í Z˙å (;W (8) ) D,H¬˙Ö‚ší½¬˙ ((12) ), 'péË, s6x ó°íbç!Z, îJ AR(p) ¡N’eíåóÉ ¥6u…dSà¬˙Ö‚š Z¾ AR−tc øÏÏí3bŸÄ BkwF¥uÖ‚š¶™Ä˙å, ʤ.;H Êd.2, AIC D BIC àk‘²_í_缟, OÊ,H¬˙Ö‚š˙å 2, Bb‡J AIC ‘² AR í¼Ÿ J˛ø dt uâ̼Ÿí AR(p) ßÞ, ;W An, Chen and Hannan (1982), Uà BIC ª)ƒ p íø_,l O9õ,, dt 1Ý. â̼Ÿí AR ßÞ, Ê…d2BbuJ AR ¡Nw’eö£íßÞ¬˙ ʤ8 ”-, Shibata (1980) ‡J AIC ‘²¼Ÿª BIC y?×)œ_çí¼Ÿ Î7¬˙Ö‚š¶, ¡b“Ö¥º‚šD –)Ö‚š°šuTÜvÈb í¥º‚š¶ ;W Horowitz (2000) íõ, ¡b“Ö¥º‚šJ ARMA ¡ N’eíåóÉ, .cú_íqìÝÜ>, 7/%%ÄÑüš…-, Ý(4,l ̶×)Äüí¡b,lM, û_½!‹.ÝÜ; 7(6I–)ÈíóÉ4, / ½(íbxÝ0ìí4” ÇøjÞ, …ú–)Å .x U4, d.2ÿþˇ ø_íĆ‘²_çí–)ŠĤ, ¬˙Ö‚š¶³Ý(4,lí½æ, °v …?³,H –)Ö‚šíÿõ. 13 第五屆全國實證經濟學論文研討會 The 5th Annual Conference of Taiwan's Economic Empirics.
(17) FíÖ‚š˙å, ¨…dí¬˙Ö‚š¶, ·.âÅ—ø_4íÜb°, ¥uÖ‚š˙åu´£üíÉœFÊ ÄÑøO7k, ì¾íüš…}º„ø, û ˝6ªàÚ¡}ºCÖ}º¡N… ?¹, Ö}ºuì¾üš…}ºí,l, AÍ.âb°Ö}ºxø_4 JÊך…-, ì¾íÖ}º.°kwÚ¡} º, ¤vUàÖ}ºªWcqì, AÍ̶×)£üí!‹ ¥[ývÖ‚š˙ å1.£ü Ê AR í=1-, Ö‚šì¾íÚ¡WÑ1.àøO;dÀÓ 9õ,, Basawa et. al. (1991a) „p, ÊÀ;_qì-, ¹Uß×áÑG, Ö‚šÀc. ¦,lך…}º.°kwÚ¡}º OuÊÇø¹dı2 (Basawa et al., 1991b), FbT|Çø½º‚š˙å, 1„p_àk,Híqì ¥[ýÎÝ?„pw£ü4, ´†Ö‚š¶øÏàí8”êÞ
(18) -ìÜ 2 û| AR − t∗c Ú¡}º ìÜ 2: Êcq 1,2 D (1) í™ÌcqA-, ç n −→ ∞, d∗. AR − t∗c −→ N(0, 1), prob.. (14). ìÜ 2 éý: Êך…-, Ö‚šì¾ (AR − t∗c ) Dú@íÚ¡ì¾ (AR − tc ) xó°íÚ¡}º, ?¹, ‡6u(6íø_,l ¥°v6„pvÖ‚š˙å£ üÌÏ. 4. _Ò}&. 4. 1. _Òql. âìÜ 2 í!‹ªø, ‚à,HÖ‚š˙åF)ƒí AR − t∗c ì¾u AR − tc í ø_,l JJ@àíi 7k, AR − t∗c íüš…4”nuõ„û˝6É-íÿõ Ĥʅ , Òâ_Ò}&xÍ, e…dí AR − t∗c ì¾íøÏÏ (size) D ‰ (power) [Û Bb°v5? AR − tc , DM D ENC ì¾, ñíʪœ¥<ì 14 第五屆全國實證經濟學論文研討會 The 5th Annual Conference of Taiwan's Economic Empirics.
(19) ¾D AR − t∗c íøÏÏ, jZBb7j AR − t∗c ì¾Z¾øÏÏí?‰ Ñ7 DŸVí DM D ENC ìTªœ, _Òql"Î HLN (1997, 1998) íd¶, 15? ã¿ÏÏx GARCH(1,1) í8” 5?¥qìjíÜâu, ׶}í‹b ‰æbÌsˇDò&¼íÔ, 7 GARCH(1,1) u‘Kæ”_2, |àV š,H‹’eíÔ4 Ĥ, 5?æ”4ã¿ÏϪJð, ç AR − t∗c àk‹ ’ev, u´YÍxGßí4” Àr †\¼Drù. (2003) ¹5?¥qì j _Òíš…×ü (n) qìA n = 16, 32, 64, 128, 256, %‡ã¿‚b (h) Ñ h=1,2,...,8, _ÒŸbÑ 2000 Ÿ, ?¹ ND = 2000 ÊÖ‚š2, ©øŸí_ÒÌ. ÏW 1000 ŸíÖ¥º‚š, ?¹ NB = 1000 ±ñíéO®ÄÑ 5% ך…@ä MVAk™ÄG}º ã¿Äü4ìÑ®ì, 7ã¿°Q4ìѬ®ì ¤Õ, ÊUà DM íj¶lã¿ô^Ç,$l¾, BbYÎ DM íd¶, UàÌG ƒb1ø qÑ h − 1 V,lÅ‚‰æb; Êl AR − tc D AR − t∗c v, ø AR |×r(¼ŸqìÑ 5 ¼, yâ˙YW|ü AIC MV²ì|_r(¼Ÿ p ã¿Äü4ìí’eßÞ¬˙ (DGP) YÎ HLN (1997) íd¶ ÑßÞs ã . ¿ÏÏ (e1t , e2t ), Iwu‰æä³Ñ R = . . 1 0. , ã¿ÏφY-ßÞ,. 0 k . . e1t e2t. v1t 0 = √ k v2t 0 . 1. (15). ¤T, Ñ7éã¿ÏÏx Ì (MA) D‘Kæ”íÔ4, Bbé vit â MA(h−1) ßÞ ?¹, Ê h = 2 v, ã¿ÏÏâ MA(1) ßÞ, Ê h = 3 v, ã¿ÏÏâ MA(2) ßÞ, J¤éR y üËz, ø (15) 2 vit qìA: vit =. h−1 . πl it−l , i = 1, 2. h = 1, 2, · · · , 8.. l=0. 15 第五屆全國實證經濟學論文研討會 The 5th Annual Conference of Taiwan's Economic Empirics. (16).
(20) w2 MA í¡bqìA, (π0 , π1 , ..., π7 ) = (1,0.1,-0.1,0.2,-0.2,0.3,-0.3,0.4) °v, Ó œ‰b (1t , 2t ) †Yà-jßÞ, . 1t. . . . . iid ∼ N 0, . 1 0. . . C . 0 1. 2t. 1t |Ψt−1 2t |Ψt−1. . . . ∼ N 0, . h1t. 0. 0. h2t. (17). ¤T, Ψt−1 [ý t−1 ‚5‡ím7Õ¯, ‘K‰æb hit Y GARCH(1,1) ¬˙, qì Ñ hit = 1.5+0.3hit−1 +0.22it−1 , i = 1, 2 6 Ĥç it â N(0, 1) ßÞ, ã¿ÏÏcx MA íÔ4, OEx5”4 ÇøjÞ, ç it â GARCH(1,1) ßÞ, ¤vã¿ÏÏ°. vx MA D‘Kæ”íÔ4, ¥qìy¯¯‹’eíÔ ÊløÏÏv, k = 1, J¯¯™Ìcq E(dt ) = E(e21t ) − E(e22t ) = 0 íb°; Êl‰v, ÑU.. °š…Å í‰óç, ;W HLN (1997), k M@Óš…×ü n = 16, ..., 256 |c, q Ñ k = 2, 1.5, 1.375, 1.25, 1.1875 J,úcq E(dt ) = E(e21t ) − E(e22t ) < 0 íb° ã¿°Qìí_Òql;W HLN (1998) íT¶ ÑßÞs ã¿ÏÏ (e1t , e2t ), . Iwu‰æä³Ñ R = . . . 1 δ. , w Choleski ä³Ñ P = . δ ω. ÏA-ßÞ, . . e1t e2t. =. δ . 1. 0. √ ω − δ2 δ. . 1. . 0 √. ω−. , ã¿Ï δ2. v1t. . (18). v2t. ¤T, [ v1t , v2t ] íßÞjà° (16) , cov(e1t , e2t ) = δ, var(e2t ) = ω > δ 2 Ê løÏÏv, δ = 1, 7 ω Mí×ü1.à_Ò!‹, ]qÑ 5 Ê¥šíqì-, E [e1t (e1t − e2t )] = E (e21t ) − E (e1t e2t ) = 0, ¯¯™Ìcqíb° Êl‰v, I δ = 0.5 < 1, OÑU.°š…Å í‰óç, ;W HLN (1998), ω D δ MíqìÛÅ √ √ ω−δ2 —ä k = 1−δ , ?¹, ω − δ 2 = k(1 − δ) k Mº¯š…×ü n = 16, ..., 256 |c,. qÑ k = 3, 4.25, 6.25, 9, 12.75, ]E [e1t (e1t − e2t )] = 1−δ > 0, ¯¯úcqíb° 6. ¤øqìÅ— GARCH(1,1) ¬˙ÑÝŠD0ìíb°. 16 第五屆全國實證經濟學論文研討會 The 5th Annual Conference of Taiwan's Economic Empirics.
(21) 4. 2. _Ò!‹. [ 1 Ñ DM, AR − tc D AR − t∗c í_Ò!‹ w2[ 1-1 Ñã¿ÏÏVAk MA(h − 1), [ 1-2 †uã¿ÏÏ°vVAk MA(h − 1) D GARCH(1,1) í!‹ hô[ 1-1, ç n = 256, DM, AR − tc D AR − t∗c úì¾øÏÏ[Ûóç, ·Q¡ 0.05 íé. O®Ä ÄÑ AR − tc D DM îÑÚ¡™ÄG}º, ĤªJR AR − t∗c í}º ?Í ÇøjÞ, #ì h, AR − t∗c øÏÏ}Ó n Ó‹7Y¹B 0.05 í‘ Wà, ç h = 2 v, n = 16, 32, 64, 128 D 256, wøÏÏ}Ñ: 0.068, 0.077, 0.07, 0.053 D 0.049 ¥u AR − t∗c ÑÚ¡™ÄG}ºíÇøábM„W, °v6uìÜ 2 A íÇøá1„ œ½bíu, Î7 h = 1 í8”JÕ, AR−t∗c øÏÏpéik AR−tc D DM ì¾ ÔuÊüš…D h œ×v, ‡6øÏÏik(s6y‹pé Wà n = 16, ç h = 2 v, ¥úì¾øÏÏ}Ñ 0.068, 0.337 D 0.123 ç h Ó‹ƒ 8 v, øÏÏ}Ñ 0.066, 0.398 D 0.363 ¥pé[ý…dFT|íÖ‚š˙åª J^Z¾ AR − tc íøÏÏ… cñ7k, AR − t∗c íøÏÏ[Ûóç U, D DM C AR − tc øÏω“óç×í8”bæ /׶}8”¢ËóçCœò í‰[Û QOhô[ 1-2, øO7k, D[ 1-1 í8”éN, AR − t∗c øÏÏ[Û ik DM D AR − tc , ÔÊüš…D h œ×í8”, ¥ÛïÔpé Wà, ç n = 16, h = 8 v, AR − t∗c , DM D AR − tc øÏÏ}Ñ: 0.076, 0.415 D 0.394. ¥éý AR−t∗c ªJ^Z¾ AR−tc øÏÏ… ʉjÞ, wGD[ 1-1 éN ÇøjÞ, [ 2-1 D[ 2-2 Ñã¿°Qìí_Ò!‹ Êüš…v, à n = 16, AR − t∗c øÏÏpéik AR − tc D ENC, ÓO h Ó‹, ¥8”pé Wà: ç h = 2 Ó‹ƒ h = 8, AR − t∗c , AR − tc D ENC íøÏÏ}â 0.078, 0.221 D 0.095 Z‰A 0.080, 0.246 D 0.219 ¥[ý AR − t∗c ªJ^Z¾ AR − tc íøÏ. Ï…, U5Q¡ 0.05 íéO®Ä Êך…-, ú6†xéNíøÏÏ[Û Ç ÕÊ GARCH(1,1) íqì-, hô[ 2-2, wøÏÏD‰í[ÛGD[ 1-1, 2-1 17 第五屆全國實證經濟學論文研討會 The 5th Annual Conference of Taiwan's Economic Empirics.
(22) éN, ʤ.;H ã¯J,FH, AR − t∗c øÏÏÊ׶}8”, ¨üš…æ”4, wøÏÏí [Ûóç U, îQ¡ 0.05 íéO®Ä ¥D AR − tc , DM D ENC í[Ûbæ ¥ —Jéý AR − t∗c üõªJ^Z¾ AR − tc íøÏÏ…, °v‰[Û¢D AR − tc óç. 5. !. š…Õã¿ô^íªœ, ª6Œû˝6ð„%ÈÜí£ü41²ì_çí_qì ªœíj¦ã¿Äü4ì, ã¿°Q4ìD AR − tc ì¾ ÖÍ AR − tc íì˙åªJ°vàkã¿Äü4ìDã¿°Q4ì, 7/³(s6ú ƒb. Uí½æ; Ouüš…-, D(s6éN, îæÊøÏÏ…í½æ Ô Ë, Ó%‡ã¿‚bÓ‹, ¬ øÏÏíÛïyý Ê…d2, BbT|ø_¬˙Ö¥º‚š˙å, àJZ¾ AR − tc ýøÏÏ …í½æ, ˚Ñ AR − t∗c ì¾ Bb1„p, AR − t∗c D AR − tc îÑÚ¡™Ä G}º ¥[ý, AR − t∗c íÖ‚š}ºu AR − tc ‚š}ºíø_,l, 6°v¿ Ö…díÖ‚š˙å£üÌÏ ÇøjÞ, â_Ò!‹éý, .
(23) ã¿ÏÏÊ MA C ‘Kæ”íqì-, AR − t∗c îªJZ¾ AR − tc øÏÏ…í½æ «wuÊüš …D%‡ã¿‚bœ×v, ¥Ûïy‹pé ĤBbwÑ, Êã¿ô^Ç,í@à ,, AR − t∗c uøá.˜í²Ï. 18 第五屆全國實證經濟學論文研討會 The 5th Annual Conference of Taiwan's Economic Empirics.
(24) ¡5d. Àr, †\¼, rù. (2003), “ã¿ô^ì: Àc¦5@à”, Working Paper Akgiray, V. (1989), “Conditional heteroscedasticity in time series of stock returns: evidence and forecasts”, Journal of Business, 62, 55-80. An, H.-Z., Z.-G. Chen and E.J. Hannan (1982), “Autocorrelation, autoregression and autoregressive approximation”, Annals of Statistics, 10, 926-936. Correction: 11, 1018. Basawa, I. V., A. K. Mallik, W. P. McCormick and R. L. Taylor (1989), “Bootstrapping explosive autoregressive process”, Annals of Statistics, 17, 14791486. Basawa, I. V., A. K. Mallik, W. P. McCormick and R. L. Taylor (1991a), “Bootstrapping unstable first order autoregressive process”, Annals of Statistics, 19, 1098-1101. Basawa, I. V., A. K. Mallik, W. P. McCormick and R. L. Taylor (1991b), “Bootstrapping test of significance and sequential bootstrap estimation for unstable first order autoregressive process”, Commun. Stat. Theory Methods, 20, 1015-1026. Berk, K. N. (1974), “Consistent Autoregressive Spectral Estimates”, Annals of Statistics, 2, 489-502. B¨ uhlmann, P. (1997), “Sieve Bootstrap for Time Series”, Bernoulli, 3, 123-148. Chang, Y. and J.Y. Park (2003) “A Sieve Bootstrap for the Test of A Unit Root”, Journal of Time Series Analysis, 24, 379-400. Chong, Y. Y., and Hendry, D. F. (1986), “Econometric Evaluation of Linear Macro-Economic Models,” Review of Economic Studies, 53, 671-690. Clements, M. P. and Hendry, D. F. (1993), “On the Limitations of Comparing Mean Square Forecast Errors”, Journal of Forecasting 12, 617-637.. 19 第五屆全國實證經濟學論文研討會 The 5th Annual Conference of Taiwan's Economic Empirics.
(25) Diebold, F. X., and Mariano, R. S. (1995), “Comparing Predictive Accuracy,” Journal of Business and Economic Statistics, 13, 253-263. Efron, B. (1979), “Bootstrap Methods: Another Look at the Jackknife”, Annals of Statistics, 7, 1-26. Engel, C. (1994), “Can the Markov switching model forecast exchange rate?” Journal of International Economics, 36, 151-165. Harvey, D. I., Leybourne, S. J., and Newbold, P. (1997), “Testing the Equality of Prediction Mean Squared Errors,” International Journal of Forecasting, 13, 281-291. Harvey, D. I., Leybourne, S. J., and Newbold, P. (1998), “Tests for Forecast Encompassing,” Journal of Business and Economic Statistics, 16, 254-259. Horowitz, J. L. (2000), “The Bootstrap,” Handbook of Econometrics, 5, 31593228. Meese, R. A. and K. Rogoff (1983), “Empirical exchange rate models of the seventies: do they fit out of sample?” Journal of International Economics, 14, 3-24. Nankervis, J. and N. Savin (1996), “The level and power of the bootstrap t test in the AR(1) model with trend”, Journal of Business & Economic Statistics, 14, 161-168. Park, J. Y. (2002), “An Invariance Principle for Sieve Botstrap in Time Series”, Econometric Theory, 18, 469-490. Shibata, R. (1980), “Asymptotically efficient selection of the order of the model for estimating parameters of a linear process”, Annals of Statistics, 8, 147-164. Swanson, N. R. and H. White (1995), “A model-selection approach to assessing the information in the term structure using linear models and artificial neural networks”, Journal of Business & Economic Statistics, 13, 265-275. Wu, C. F. J. (1986), “Jackknife, Bootstrap and Other Resampling Methods in Regression Analysis”, The Annals of Statistics, 14, 1261-1295. 20 第五屆全國實證經濟學論文研討會 The 5th Annual Conference of Taiwan's Economic Empirics.
(26) Ë“: bç„p . J-„p, · [ýr!¼)× (Euclidean norm), #ì²¾ x = [x1 , x2 , · · · , xn ] , A Ñ p × p íä³, ì2 x =. x21 + · · · + x2n , A = maxx Ax / x . .Ü 1: Êcq 1,2 A-, ª)J-!‹: (a) E ∗ |∗t |2 = σ 2 + o(1), a.s. (b) E ∗ |∗t |4 = Op (1). „p: Ìc Chang and Park (2003) Lemma 5 D Lemma 7 . .Ü 2: I x∗p,t = [d∗t−1 , d∗t−2 , · · · , d∗t−p ], w2 d∗t â (12) ßÞ Ê.Ü 1 ícq‘K. -, 1 ∗ x = Op∗ (n−1/2 p1/2 ), a.s. n t=1 p,t n. (a). 1 ∗ ∗ −1 x x ) = Op∗ (1), (b) ( n t=1 p,t p,t n. 1 ∗ ∗ x = Op∗ (p1/2 ), (c) √ n t=1 p,t t. prob.. n. 1 ∗ ∗ x x = Op∗ (1), (d). n t=1 p,t p,t. a.s.. n. prob.. „p (a):. I d¯j = n1 nt=1 d∗t−j , j = 1, 2, · · · , p. ÄÑ d∗t = ∞ ˆp,k ∗t−k , ‚à Minkowski . k=0 π. 21 第五屆全國實證經濟學論文研討會 The 5th Annual Conference of Taiwan's Economic Empirics.
(27) , ª)J-í!‹: ∗. E |d¯j |2 = n−2 E ∗ | −2. ≤ n (. n ∞ ( π ˆp,k ∗t−k )|2. t=1 k=0 ∞ . ∗. |ˆ πp,k |(E |. n . t=1 n . k=0. = n−2 ( −2. ∞ . ∗t−j−k |2 )1/2 )2. |ˆ πp,k |)2 E ∗ |. ∗t−j−k |2 (E1). t=1. k=0. = n O(1)O(n) a.s. (E2) = O(n−1 ) a.s (E3). ÄÑ ∗t Öó°}º/â.Ü 1(a) ª): ∗. E |. n . ∗t−j−k |2. t=1. =. n . E ∗ |∗t−j−k |2. t=1. = nE ∗ ∗2 t = O(n) a.s. ∞. πp,k | < ∞, Ĥ (E1) D (E2) A k=0 |ˆ â (E3) ª) E ∗ (n−1 nt=1 d∗t−j )2 = O(n−1 ) a.s., Ĥ E ∗ n−1 nt=1 x∗p,t 2 = O(n−1 p) a.s., ?¹ n−1 nt=1 x∗p,t = Op∗ (n−1/2 p1/2 ), ] (a) )„. ÇøjÞ, â Park (2002) p.487 ø. (b) D (c) Ìc Chang and Park (2003) Lemma 3. I Γ∗p,p = (Γ∗i−j )pi,j=1, w2, Γ∗k = E ∗ (d∗t d∗t−k ) â Chang and Park (2003) p.398 ø, 1 ∗ ∗ E. xp,t xp,t − Γ∗p,p 2 = Op (n−1 p2 ) n t=1 n. ∗. ÄÑ |E ∗. 1 ∗ ∗ 1 ∗ ∗ xp,t xp,t − E ∗ Γ∗p,p | ≤ E ∗. x x − Γ∗p,p. n t=1 n t=1 p,t p,t n. n. = Op (n−1/2 p). ÄÑ E ∗ Γ∗p,p = Γ∗p,p , Ébª„p Γ∗p,p = O(1), a.s., (d) ¹)„ I f ∗ (λ) D f (λ) }[ý d∗t D dt í spectral density â Chang and Park (2003) Lemma 6 ø, sup |f ∗ (λ) − f (λ)| = o(1), a.s. λ. 22 第五屆全國實證經濟學論文研討會 The 5th Annual Conference of Taiwan's Economic Empirics.
(28) I λ∗n Ñ Γ∗p,p |×íÔ;, ª)J-!‹:. Γ∗p,p = λ∗n ≤ 2π sup f ∗ (λ) λ. = 2π sup f (λ), a.s. λ. = O(1), a.s.. ,H.D|(ø_Ìc Berk (1974) p.491 ] (d) )„ . . ∗ ∗ ∗ ∗ .Ü 3: I α ˆ p = [ˆ αp,1, α ˆ p,2 , · · · , α ˆ p,p], α ˆ p∗ = [ˆ αp,1 ,α ˆ p,2 ,··· ,α ˆ p,p ], α ˆ p,i D α ˆ p,i D cˆ∗ } ì2k (10) D (13) σn∗2 = n1 nt=1 eˆ∗2 ˆ∗p,t Ñ (13) í{Ï Ê.Ü 1 í‘KA p,t , e. -, (a) α ˆ p∗ − α ˆ p = Op∗ (n−1/2 p1/2 ), prob.. (b). √. 1 ∗ nˆ c∗ = √ + op∗ (1), prob. n t=1 t n. p∗. (c) σn∗2 −→ σ 2 ,. prob.. „p: I A∗n , Bn∗ D Cn∗ à-: A∗n. =. n . ∗t. t=1. Bn∗ = n − ( Cn∗ =. −(. n . n . x∗p,t )(. t=1 . x∗p,t )(. n . n . x∗p,t x∗p,t )−1 (. t=1 . x∗p,t x∗p,t )−1 (. t=1 t=1 n n n( x∗p,t x∗p,t )−1 ( x∗p,t ∗t ) t=1 t=1. n . n . x∗p,t ∗t ). t=1. x∗p,t ). t=1 n . −(. t=1. x∗p,t )(. n . . x∗p,t x∗p,t )−1 (. t=1. †ª): A∗n Bn∗ Cn∗ = Bn∗. cˆ∗ = α ˆ p∗ − α ˆp. 23 第五屆全國實證經濟學論文研討會 The 5th Annual Conference of Taiwan's Economic Empirics. n t=1. ∗t ).
(29) ÄÑ 1 1 ∗ 1 ∗ 1 ∗ ∗ −1 1 ∗ ∗ | √ A∗n − √ t | ≤ |( x )( x x ) (√ x )| n n t=1 n t=1 p,t n t=1 p,t p,t n t=1 p,t t n. n. n. n. 1 ∗ 1 ∗ ∗ −1 1 ∗ ∗ ≤. xp,t. ( xp,t xp,t ). √ x . n t=1 n t=1 n t=1 p,t t n. n. = Op∗ (n−1/2 p),. n. prob. (E4). ‚à.Ü 2, (E4) ¹A °Üª): 1 ∗ B = 1 + Op∗ (n−1 p), prob. n n 1. Cn∗ = Op∗ (n−1/2 p1/2 ), prob. (E5) n. ÄÑ. α ˆ p∗ − α ˆ p ≤ n−1 Cn∗ |(n−1 Bn∗ )−1 | = Op∗ (n−1/2 p1/2 ),. prob. â(E5). ] (a) )„ °Ü, √. n−1/2 A∗n n−1 Bn∗ n 1 ∗ = √ + op∗ (1), n t=1 t. nˆ c∗ =. prob.. ] (b) )„ â σn∗2 íì2, ª)J-!‹: σn∗2. 1 ∗2 = eˆ n t=1 p,t n. 1 ∗ = (t − cˆ∗ − x∗p,t (α ˆ p∗ − α ˆ p ))2 n t=1 n. 1 ∗2 1 = t + cˆ∗2 + (α ˆ p∗ − α ˆp) ( x∗p,t x∗p,t )(α ˆ p∗ − α ˆp ) n t=1 n t=1 n. −2ˆ c∗. n. 1 ∗ 1 ∗ ∗ ∗ 1 t − 2 xp,t t (α ˆp − α ˆ p ) + 2ˆ c ∗ (α ˆ p∗ − α ˆp) x∗p,t n t=1 n t=1 n t=1 n. n. n. 24 第五屆全國實證經濟學論文研討會 The 5th Annual Conference of Taiwan's Economic Empirics.
(30) â Park (2002) Lemma 3.2 ). n. √1 n. ∗ t=1 t. √ . ˆp ) ( |(α ˆ p∗ − α. 1 n. n . d∗. −→ N(0, σ 2 ), y‚à.Ü 2 ª)J-!‹:. nˆ c∗ = Op∗ (1),. prob.. . x∗p,t x∗p,t )(α ˆ p∗ − α ˆ p )| = op∗ (1), prob.. t=1. |ˆ c∗. 1 ∗ | = Op∗ (n−1/2 ), n t=1 t n. prob.. 1 ∗ E | | = O(1), a.s. (ÄÑâ.Ü 1(a) ª)E ∗ |∗t | = O(1), a.s.) n t=1 t n. ∗. 1 ∗ ∗ ∗ | xp,t t (α ˆp − α ˆ p )| = Op∗ (n−1 p), prob. n t=1 n. ˆ p∗ − α ˆp ) |ˆ c ∗ (α. . 1 ∗ x | = Op∗ (n−3/2 p), n t=1 p,t n. prob.. Ĥª): 1 ∗2 = + op∗ (1), prob. n t=1 t n. σn∗2. ¢ÄÑ 1 ∗2 1 ∗2 ∗ ∗2 2 | t − σ 2 | ≤ | − E ∗ ∗2 t | + |E t − σ | n t=1 n t=1 t n. n. ≡ A+B. 2 â.Ü 1(a) ø, B = |E ∗ ∗2 t − σ | = o(1), a.s. ÇøjÞ,. 1 ∗2 2 ∗ ∗2 2 t − E ∗ ∗2 = n−2 E ∗ | (∗2 E | t | t − E t )| n t=1 t=1 ∗. n. n. = n−2. n . ∗ ∗2 2 E ∗ (∗2 ÄÑ∗t óÖ t − E t ). t=1. 1 = O( ), n. prob. ‚à.Ü 1. Ĥ, 1 ∗2 − E ∗ ∗2 A = | t | n t=1 t n. = op∗ (1),. prob.. ?¹, σn∗2 = σ 2 + op∗ (1), prob. 25 第五屆全國實證經濟學論文研討會 The 5th Annual Conference of Taiwan's Economic Empirics.
(31) ] (c) )„. „pìÜ 1: â.Ü 3(a) í„p¬˙ø 1 1 ∗ √ A∗n = √ + op∗ (1), prob. n n t=1 t n. 1 ∗ B = 1 + op∗ (1), prob. n n. y‚à. √1 n. n. d∗. ∗ 2 t=1 t −→ N(0, σ ) (Park, 2002, Lemma 3.2), D.Ü 3(c) ª)J-!. ‹: AR − t∗c = = =. =. cˆ∗ SE(ˆ c∗ ) A∗n ∗ Bn σn∗ (Bn∗ )−1/2 √1 A∗ n n. . 1 ∗ B n n n ∗ √1 t=1 t n σ2. σn∗. = N(0, 1),. + op∗ (1), prob.. prob.. ]ìÜ 1 )„. 26 第五屆全國實證經濟學論文研討會 The 5th Annual Conference of Taiwan's Economic Empirics.
(32) 表 1-1：預測準確性檢定之小樣本績效比較—MA( h − 1 )預測誤差 n = 16 AR − t c DM. n = 32 AR − t c DM. n = 64 AR − t c DM. n = 128 AR − t c DM. n = 256 AR − t c DM. h = 1 大樣本 型一誤差 檢力 靴帶反覆抽樣 型一誤差 檢力 大樣本 h=2 型一誤差 檢力 靴帶反覆抽樣 型一誤差 檢力 h = 3 大樣本. 0.349 0.076. 0.067 0.135 0.226 0.122. 0.059 0.083 0.178 0.185. 0.050 0.068 0.248 0.209. 0.053 0.059 0.239 0.249. 0.068 0.119. 0.079 0.172. 0.083 0.234. 0.059 0.245. 0.058 0.286. 0.337 0.072. 0.123 0.129 0.161 0.117. 0.086 0.083 0.158 0.194. 0.070 0.069 0.229 0.209. 0.059 0.058 0.231 0.247. 0.068 0.111. 0.077 0.161. 0.070 0.219. 0.053 0.235. 0.049 0.267. 型一誤差 檢力 靴帶反覆抽樣 型一誤差 檢力 h = 4 大樣本 型一誤差 檢力 靴帶反覆抽樣 型一誤差 檢力 大樣本 = h 5. 0.351 0.073. 0.177 0.138 0.133 0.107. 0.125 0.083 0.131 0.189. 0.083 0.065 0.210 0.208. 0.063 0.059 0.217 0.252. 0.070 0.111. 0.070 0.154. 0.064. 0.063. 0.227. 0.226. 0.060 0.275. 0.360 0.071. 0.210 0.140 0.128 0.122. 0.148 0.088 0.118 0.170. 0.099 0.069 0.168 0.196. 0.074 0.064 0.206 0.227. 0.074. 0.076. 0.084. 0.054. 0.121. 0.158. 0.216. 0.222. 0.054 0.251. 型一誤差 檢力 靴帶反覆抽樣 型一誤差 檢力 h = 6 大樣本 型一誤差 檢力 靴帶反覆抽樣 型一誤差 檢力 h = 7 大樣本. 0.357 0.071. 0.245 0.139 0.115 0.120. 0.167 0.093 0.115 0.176. 0.113 0.075 0.152 0.175. 0.079 0.065 0.176 0.216. 0.069. 0.083 0.162. 0.071 0.197. 0.073. 0.120. 0.213. 0.063 0.242. 0.367 0.072. 0.270 0.145 0.101 0.107. 0.181 0.100 0.103 0.167. 0.120 0.086 0.146 0.148. 0.089 0.072 0.156 0.198. 0.072. 0.080. 0.070. 0.090. 0.143. 0.176. 0.062 0.207. 0.067 0.208. 0.206 0.112 0.093 0.141. 0.134 0.090 0.130 0.149. 0.090 0.079 0.148 0.174. 0.074 0.163. 0.083 0.181. 0.074 0.205. 0.221 0.120 0.084 0.123. 0.143 0.103 0.113 0.128. 0.091 0.089 0.139 0.148. 0.087. 0.094 0.200. 0.069 0.207. 型一誤差 0.393 0.321 0.172 檢力 0.065 0.089 0.097 靴帶反覆抽樣 型一誤差 0.075 0.089 檢力 0.105 0.144 h = 8 大樣本 型一誤差 0.398 0.363 0.177 檢力 0.065 0.094 0.090 靴帶反覆抽樣 型一誤差 0.066 0.101 檢力 0.104 0.152 說明：資料產生過程與符號意義，請參考文章 4.1. 0.180 節說明。. 第五屆全國實證經濟學論文研討會 The 5th Annual Conference of Taiwan's Economic Empirics. 27. 0.048 0.277. 0.056 0.253. 0.056 0.259. 0.060 0.236. 0.064 0.215. 0.065 0.199. 0.065 0.178. 0.070 0.170.
(33) 表 1-2：預測準確性檢定之小樣本績效比較—MA( h − 1 )-GARCH(1,1) 預測誤差 n = 16 AR − t c DM. n = 32 AR − t c DM. n = 64 AR − t c DM. n = 128 AR − t c DM. n = 256 AR − t c DM. h = 1 大樣本 型一誤差 檢力 靴帶反覆抽樣 型一誤差 檢力 大樣本 h=2 型一誤差 檢力 靴帶反覆抽樣 型一誤差 檢力 h = 3 大樣本. 0.368 0.072. 0.134 0.147 0.150 0.093. 0.139 0.092 0.126 0.132. 0.141 0.073 0.156 0.136. 0.146 0.062 0.139 0.151. 0.063 0.095. 0.079 0.128. 0.080 0.165. 0.068 0.161. 0.064 0.187. 0.361 0.069. 0.148 0.143 0.118 0.089. 0.109 0.089 0.118 0.134. 0.097 0.071 0.156 0.127. 0.089 0.060 0.145 0.148. 0.065 0.083. 0.073 0.115. 0.067 0.152. 0.061 0.148. 0.060 0.172. 型一誤差 檢力 靴帶反覆抽樣 型一誤差 檢力 h = 4 大樣本 型一誤差 檢力 靴帶反覆抽樣 型一誤差 檢力 大樣本 = h 5. 0.374 0.061. 0.184 0.149 0.111 0.095. 0.127 0.094 0.110 0.131. 0.087 0.071 0.148 0.130. 0.075 0.062 0.138 0.145. 0.061 0.098. 0.062 0.113. 0.067. 0.064. 0.155. 0.164. 0.061 0.174. 0.379 0.062. 0.219 0.154 0.097 0.094. 0.148 0.094 0.102 0.108. 0.096 0.072 0.127 0.131. 0.074 0.061 0.146 0.140. 0.072. 0.077. 0.086. 0.057. 0.092. 0.120. 0.158. 0.152. 0.055 0.164. 型一誤差 檢力 靴帶反覆抽樣 型一誤差 檢力 h = 6 大樣本 型一誤差 檢力 靴帶反覆抽樣 型一誤差 檢力 h = 7 大樣本. 0.382 0.065. 0.257 0.152 0.096 0.100. 0.166 0.101 0.100 0.112. 0.111 0.074 0.108 0.121. 0.077 0.064 0.122 0.136. 0.072. 0.076 0.127. 0.068 0.151. 0.071. 0.094. 0.148. 0.065 0.155. 0.389 0.067. 0.293 0.160 0.088 0.087. 0.188 0.105 0.083 0.119. 0.122 0.083 0.116 0.107. 0.084 0.066 0.112 0.130. 0.070. 0.081. 0.077. 0.089. 0.119. 0.135. 0.060 0.146. 0.067 0.151. 0.210 0.116 0.079 0.103. 0.131 0.087 0.109 0.104. 0.088 0.070 0.114 0.120. 0.073 0.134. 0.078 0.137. 0.073 0.145. 0.232 0.129 0.075 0.096. 0.142 0.099 0.095 0.101. 0.092 0.082 0.110 0.118. 0.080. 0.092 0.155. 0.074 0.160. 型一誤差 0.409 0.347 0.179 檢力 0.063 0.076 0.084 靴帶反覆抽樣 型一誤差 0.074 0.080 檢力 0.089 0.116 h = 8 大樣本 型一誤差 0.415 0.394 0.188 檢力 0.057 0.082 0.079 靴帶反覆抽樣 型一誤差 0.076 0.087 檢力 0.092 0.133 說明：資料產生過程與符號意義，請參考文章 4.1. 0.159 節說明。. 第五屆全國實證經濟學論文研討會 The 5th Annual Conference of Taiwan's Economic Empirics. 28. 0.146 0.157. 0.083 0.149. 0.068 0.159. 0.065 0.144. 0.067 0.133. 0.067 0.139. 0.069 0.123. 0.073 0.123.
(34) 表 2-1：預測涵蓋性檢定之小樣本績效比較—MA( h − 1 )預測誤差 n = 16 AR − t c DM. n = 32 AR − t c DM. n = 64 AR − t c DM. n = 128 AR − t c DM. n = 256 AR − t c DM. h = 1 大樣本 型一誤差 檢力 靴帶反覆抽樣 型一誤差 檢力 大樣本 h=2 型一誤差 檢力 靴帶反覆抽樣 型一誤差 檢力 h = 3 大樣本. 0.217 0.136. 0.068 0.109 0.301 0.232. 0.061 0.073 0.331 0.280. 0.054 0.061 0.346 0.324. 0.052 0.054 0.345 0.337. 0.085 0.222. 0.064 0.293. 0.055 0.331. 0.059 0.328. 0.052 0.342. 0.221 0.122. 0.095 0.105 0.254 0.224. 0.078 0.072 0.304 0.281. 0.064 0.061 0.328 0.315. 0.057 0.057 0.332 0.323. 0.078 0.206. 0.076 0.300. 0.064 0.339. 0.049 0.348. 0.061 0.321. 型一誤差 檢力 靴帶反覆抽樣 型一誤差 檢力 h = 4 大樣本 型一誤差 檢力 靴帶反覆抽樣 型一誤差 檢力 大樣本 = h 5. 0.219 0.125. 0.117 0.109 0.221 0.229. 0.093 0.073 0.270 0.283. 0.072 0.061 0.304 0.309. 0.061 0.058 0.320 0.317. 0.077 0.211. 0.079 0.286. 0.054. 0.060. 0.324. 0.341. 0.056 0.327. 0.221 0.120. 0.139 0.110 0.207 0.224. 0.106 0.076 0.250 0.281. 0.080 0.063 0.282 0.297. 0.063 0.056 0.308 0.312. 0.079. 0.063. 0.064. 0.047. 0.202. 0.295. 0.320. 0.297. 0.054 0.318. 型一誤差 檢力 靴帶反覆抽樣 型一誤差 檢力 h = 6 大樣本 型一誤差 檢力 靴帶反覆抽樣 型一誤差 檢力 h = 7 大樣本. 0.230 0.115. 0.155 0.111 0.195 0.218. 0.121 0.078 0.235 0.258. 0.090 0.067 0.271 0.280. 0.070 0.062 0.286 0.289. 0.085. 0.080 0.268. 0.061 0.304. 0.060. 0.200. 0.302. 0.051 0.305. 0.231 0.108. 0.177 0.118 0.184 0.210. 0.120 0.082 0.219 0.240. 0.091 0.068 0.244 0.258. 0.068 0.067 0.269 0.251. 0.074. 0.068. 0.068. 0.196. 0.258. 0.292. 0.062 0.295. 0.057 0.286. 0.141 0.090 0.192 0.218. 0.097 0.074 0.221 0.243. 0.074 0.068 0.236 0.245. 0.080 0.272. 0.060 0.285. 0.060 0.273. 0.145 0.095 0.198 0.210. 0.101 0.082 0.210 0.212. 0.076 0.079 0.208 0.215. 0.078. 0.075 0.253. 0.071 0.275. 型一誤差 0.238 0.200 0.126 檢力 0.116 0.173 0.197 靴帶反覆抽樣 型一誤差 0.081 0.077 檢力 0.187 0.261 h = 8 大樣本 型一誤差 0.246 0.219 0.138 檢力 0.117 0.163 0.182 靴帶反覆抽樣 型一誤差 0.080 0.082 檢力 0.189 0.260 說明：資料產生過程與符號意義，請參考文章 4.1. 0.278 節說明。. 第五屆全國實證經濟學論文研討會 The 5th Annual Conference of Taiwan's Economic Empirics. 29. 0.051 0.345. 0.054 0.331. 0.057 0.321. 0.053 0.326. 0.059 0.293. 0.065 0.257. 0.062 0.244. 0.067 0.222.
(35) 表 2-2：預測涵蓋性檢定之小樣本績效比較—MA( h − 1 )-GARCH(1,1) 預測誤差 n = 16 AR − t c DM. n = 32 AR − t c DM. n = 64 AR − t c DM. n = 128 AR − t c DM. n = 256 AR − t c DM. h = 1 大樣本 型一誤差 檢力 靴帶反覆抽樣 型一誤差 檢力 大樣本 h=2 型一誤差 檢力 靴帶反覆抽樣 型一誤差 檢力 h = 3 大樣本. 0.216 0.129. 0.069 0.106 0.311 0.233. 0.064 0.071 0.336 0.289. 0.055 0.061 0.346 0.319. 0.052 0.053 0.347 0.332. 0.085 0.208. 0.065 0.282. 0.055 0.336. 0.059 0.329. 0.050 0.346. 0.218 0.125. 0.091 0.105 0.267 0.229. 0.080 0.071 0.312 0.290. 0.063 0.062 0.336 0.312. 0.057 0.059 0.338 0.321. 0.067 0.202. 0.069 0.300. 0.063 0.333. 0.053 0.354. 0.063 0.330. 型一誤差 檢力 靴帶反覆抽樣 型一誤差 檢力 h = 4 大樣本 型一誤差 檢力 靴帶反覆抽樣 型一誤差 檢力 大樣本 = h 5. 0.220 0.123. 0.115 0.109 0.222 0.228. 0.093 0.076 0.274 0.285. 0.072 0.060 0.313 0.615. 0.060 0.058 0.325 0.315. 0.080 0.194. 0.076 0.288. 0.058. 0.053. 0.330. 0.344. 0.055 0.322. 0.220 0.127. 0.136 0.110 0.214 0.216. 0.106 0.076 0.267 0.282. 0.081 0.062 0.288 0.296. 0.064 0.057 0.311 0.311. 0.073. 0.065. 0.064. 0.049. 0.198. 0.289. 0.312. 0.305. 0.050 0.321. 型一誤差 檢力 靴帶反覆抽樣 型一誤差 檢力 h = 6 大樣本 型一誤差 檢力 靴帶反覆抽樣 型一誤差 檢力 h = 7 大樣本. 0.227 0.119. 0.155 0.110 0.203 0.213. 0.118 0.078 0.242 0.268. 0.089 0.068 0.265 0.286. 0.070 0.061 0.287 0.289. 0.078. 0.077 0.274. 0.059 0.295. 0.059. 0.188. 0.302. 0.054 0.313. 0.228 0.119. 0.175 0.115 0.186 0.212. 0.127 0.080 0.218 0.241. 0.089 0.068 0.253 0.263. 0.067 0.065 0.273 0.260. 0.069. 0.063. 0.073. 0.181. 0.261. 0.303. 0.061 0.293. 0.052 0.281. 0.137 0.089 0.195 0.224. 0.096 0.072 0.227 0.242. 0.072 0.067 0.245 0.243. 0.076 0.273. 0.060 0.283. 0.056 0.282. 0.144 0.094 0.196 0.212. 0.100 0.083 0.218 0.211. 0.077 0.078 0.211 0.211. 0.079. 0.071 0.252. 0.075 0.279. 型一誤差 0.238 0.198 0.122 檢力 0.115 0.177 0.202 靴帶反覆抽樣 型一誤差 0.082 0.077 檢力 0.196 0.261 h = 8 大樣本 型一誤差 0.248 0.216 0.134 檢力 0.114 0.158 0.189 靴帶反覆抽樣 型一誤差 0.080 0.077 檢力 0.184 0.244 說明：資料產生過程與符號意義，請參考文章 4.1. 0.279 節說明。. 第五屆全國實證經濟學論文研討會 The 5th Annual Conference of Taiwan's Economic Empirics. 30. 0.051 0.342. 0.057 0.335. 0.058 0.323. 0.054 0.323. 0.060 0.294. 0.066 0.258. 0.063 0.246. 0.066 0.219.
(36)
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