Bootstrap Unit Root Test
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(5) Abstract In recent years the bootstrap unit root tests have become increasingly popular and they have been applied to a wide range of topics, including nonstationary time series. In this paper we review the basic knowledge of the unit root tests and. 库. the bootstrap. We mainly discuss general methods and theories of the bootstrap. 要. DF tests and the bootstrap ADF tests. We summarize the latest achievements of the bootstrap unit root tests, including detrending bootstrap unit root tests.. 摘. On the basis of theory, we use the method of Monte-Carlo Simulation to do lots. 文. of empirical analysis. We compare the differences between the DF tests and the. 论. bootstrap DF tests and differences between the ADF tests and the bootstrap ADF tests on the rejection probabilities and the power of test. We use the unit. 士. root tests and the bootstrap unit root tests to analyze the national debt rates. 硕. and the Chinese GDP dates during the period from 1978 to 2009. We prove the. 博. superiority of the bootstrap from the angle of probability.. 学. Key words:Bootstrap Unit Root Tests; Bootstrap DF Test; Bootstrap ADF. 厦. 门. 大. Test.. ii.
(6) 8¹ ¥©Á . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i =©Á . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ii 1Ù Úó . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. 库. §1.1 ïĵ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. 要. §1.2 ©Ìó . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. 摘. §1.3 ©( . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. 1Ù ïÄnØ9{ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. 文. §2.1 ý£ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. 论. §2.2 gÏü u{ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19. 士. §2.3 gÏDFu u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 §2.4 gÏADFu u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22. 硕. §2.5 Gت³ü u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26. 博. 1nÙ ¢y©Û . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31. 学. §3.1 [' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31. 大. §3.2 ¥I IÅ|Ç©Û . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 §3.3 ¥IGDP© ©Û . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35. 门. 1oÙ o(Ð" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38. 厦. ë©z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .39 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42. iii.
(7) Contents Abstract (in Chinese) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i Abstract (in English) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Chapter 1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. 库. §1.1 Research Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. 要. §1.2 Main Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. 摘. §1.3 Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Chapter 2 Research Theory and Method . . . . . . . . . . . . . . . . . . . . . 4. 文. §2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. 论. §2.2 Bootstrap Unit Root Tests Method . . . . . . . . . . . . . . . . . . . . . . . . . 19. 士. §2.3 Bootstrap DF Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 §2.4 Bootstrap ADF Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22. 硕. §2.5 Detrending Bootstrap Unit Root Tests . . . . . . . . . . . . . . . . . . . . . 26. 博. Chapter 3 Demonstrational Analyses . . . . . . . . . . . . . . . . . . . . . . . . . 31. 学. §3.1 Simulation Compare . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31. 大. §3.2 Chinese National Debt Rates Analyses . . . . . . . . . . . . . . . . . . . . . 34 §3.3 Chinese GDP Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35. 门. Chapter 4 Conclusion and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38. 厦. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42. iv.
(8) gÏü u. 1. 1Ù Úó §1.1 ïĵ ü umSü ïÄ´mS©Û9:¯K. mS ÝA5C1¢SþN mS²5. é²mS. 库. ?n{´òÙ=C²S, ùÒ±A^k'²mS. 要. {5?1AïÄ. émSü uÒ´émS²5u. 摘. , ²mSXJ3ü , K±ÏL ©{5Øü , ²S. éu3ü mS, Ñw«Ñ²wPÁ5Ú. 文. ÅıY5, Ïdü u´?Øk''X35uÚSÅı. 论. Y5?ØÄ:. 3¢S²LÚ7K+k2A^. ¿ÂþgÏ{ Ò´UÄ{õgÄ , ?1u{. Ï~3ÕáÓ©Ù . 士. ¥#Ä. ü u´émSu, ÏdXÛémS ?1. 硕. #Ä´gÏü u¥7L)û¯K. kü«{, «´©¬ Ä{, ,«´seiv{. 3. ©¥ò[0 .. 博. @3þVlÊc, <Òm© égÏü ïÄ, ¿ . 学. U ¤ Ò. R.L.Taylor(1991)[1] ? Ø g £ 8 . g Ï {. Juan. 大. Romo(1993)[2] &? g£8 .gÏü u. égÏü uÑÈ4. 5kþêÆ[. z, Joon Y.Park(2000)[3] , (2003)[4] éugÏü. 门. ìC5Ñ nØþy². A.M.Robert Taylor(2008)[5] ?Ø k. 厦. ²ÅÄÇmSü u. c'ugÏü u®²k´a¤J, )Giuseppe Cavaliere(2009)[6] g. ÏM u, Dilem Yildirim(2009)[7] 5 .gÏü u, Stephan. Smeekes(2009)[8] Gت³gÏü u, ±9Nikolay Gospodinov(2009)[9] Ø.
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(11) gÏü u. 4. 1Ù ïÄnØ9{ §2.1 ý£ 2.1.1 bootstrap {[11]. 库. 2.1.1.1 ëêbootstrap { oN©ÙF , ®²kNþn5g©ÙF êâ , gù. 要. U£Ä{ÄNþn , ù« ¡bootstrap ½. 摘. ¡gÏ . U/, Õá/g© ¥éõbootstrap , |^ù. 文. éoNF ?1ÚOíä. ù«{¡ëêbootstrap{, q¡gÏ {. ù{±^u<éoN$¹, §´CÚO¥«^. 论. uêâ?n¦^{. ù«{¢yI3OÅþþO, . 士. XOÅ%åO, §®¤«61{.. Ïïá.. 硕. bootstrap{´Efron320V70c. ( ) OþIOØ bootstrap O. 博. b IÑùOθ b 3OoNëêθ, <ØθOθ, q. b. q. b D(θ)¡Oþ θb IOØ .. 大. θb =. 学. b b °Ý, Ï~·^Oþθb IO D(θ)5ÝþO°Ý. OþθIO. X1 , X2 , · · ·, Xn ´5g±F (x)©Ù¼êoN , θ´·a,. 门. b b ëê, ^θb =θ(X 1 , X2 , · · ·, Xn )θOþ. 3A^¥θÄ©Ù~´é. 厦. J?n. ù, q. q. b D(θ)~vk{üLª, ØL·±^OÅ [. b d, gF )éõNþB , éuz {5¦ D(θ)O..
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(13) gÏü u. 6. 2◦ , U / ! Õ á / ¦ ÑB(B≥1000)N þ nbootstrap , x∗i = (x∗1i , x∗2i , · · ·, x∗ni ),i = 1, 2, · · ·, B. é u 1ibootstrap , · O Ñθbi∗ = b ∗i , x∗i , · · ·, x∗i ),i = 1, 2, · · ·, B. (θb∗ ¡θ1ibootstrapO) θ(x 1 2 i n. 3◦ , O q. 1 B−1. PB. b∗ i=1 (θi. ∗. ∗. − θ )2 ,Ù¥θ =. 1 B. PB. b∗ i=1 θi .. 库. σb =. 要. ( ) OþþØ 9 bootstrap O. 摘. X = (X1 , X2 , · · ·, Xn )´5goNF , F , R = R(X)´a,. 文. ÅCþ, §6u X. b·F"OR©Ù, A. ~XR êÆÏ"EF (R), Ò±Uìþ¡¤`nÚ½1◦ , 2◦ , 3◦ ?1, ´32◦ ¥. 论. éu1ibootstrap x∗i = x∗1i , x∗2i , · · ·, x∗ni , ORi∗ = R(x∗i )OOθi∗ , . 士. 33◦ ¥Oa,RA, ~XXJF"OEF (R), ÒO B 1 X Ri∗ . B i=1. 博. (n n) bootstrap & « m. 硕. E∗ (R∗ ) =. 学. e¡0 «¦ëêθbootstrap&«m{.. 大. X = (X1 , X2 , · · ·, Xn )´5goNF Nþn , x = (x1 , x2 , · · ·, xn )´ b ® , F ¥¹këêθ, θb = θ(X 1 , X2 , · · ·, Xn )´θOþ, y. 门. 35¦θ&Y²1 − α&«m.. 厦. U/!Õá/l x = (x1 , x2 , · · ·, xn )¥ÄBNþnbootstrap. ∗ . ò§g , éuzbootstrap ¦ÑθbootstrapO: θb1∗ , θb2∗ , · · ·, θbB. üS, ∗ ∗ ∗ . ≤· · ·≤θb(B) ≤θb(2) θb(1).
(14) gÏü u. 7. b ^ é A R(X ∗ ) = θb∗ © Ù R(X) © Ù C q, ¦ R(X) = θ, ∗ α, ¦ ÑR(X ∗ )©ÙCq© êθb∗α Úθb1− 2. 2. ∗ α ) = 1 − α. P (θb∗α <θb∗ <θb1− 2. (2 − 3). 2. Pk1 = [B ∗ α2 ], k2 = [B ∗ (1 − α2 )], 3(2)3)ª¥±θbk∗1 Úθbk∗2 ©O© 2. 要. 2. 库. ∗ α O, Cq ª êθb∗α Úθb1−. 摘. P (θbk∗1 <θb∗ <θbk∗2 ) = 1 − α,. u´dþªÒθ&Y²1 − αCq&«m(θbk∗1 , θbk∗2 ), ù«m¡. 文. θ&Y²1 − αbootstrap&«m. ù«¦&«m{¡© ê. 论. {.. 士. (o o) ^bootstrap-t{ {¦þµ bootstrap & « m. 硕. X = (X1 , X2 , · · ·, Xn )´5goNF Nþn , x = (x1 , x2 , · · ·, xn )´ ® , þµÚ σ 2 þëê, ·|^ x5Oµ.. t=. 学. 博. ļê. x−µ √ , S/ n. (2 − 4). 大. boNF äk©Ù, dt©ÙëêµÃ', §´Í¶þ, kt∼t(n − 1), |^Ͷþt, ÒU¦µ&«m. y3, oNF Øäk. 门. © Ù, ytE ´ Í ¶ þ, , t © Ù Ò Ø ´t(n − 1)© Ù, e ¡ · . 厦. ^bootstrap{5¦µCq&«m. ± © x = (x1 , x2 , · · ·, xn ) þ x= n1. tAͶþ. i=1 xi µ O,. Ä. ∗. X −x W = ∗ √ , S / n ∗. Pn. (2 − 5).
(15) gÏü u. 8. ∗. d?X , S ∗ ©OX, SAbootstrap þ IO . ^W ∗ ©ÙC ∗ α, ¦ qt©Ù, ¦ÑW ∗ Cq© êw∗α , w1− 2. 2. ∗. P {w∗α <. X −x ∗ √ <w1− α } = 1 − α, ∗ 2 S / n. P {w∗α <. X −x ∗ √ <w1− α } = 1 − α, 2 S ∗/ n. 2. ∗. ½=. 摘. S ∗ ∗ α √ <µ<X − w α } = 1 − α. P {X − w1− 2 2 n. 库. u´Cqk. (2 − 6). 要. 2. 文. òW ∗ BbootstrapgüS. (2 − 7). 论. ∗ ∗ ∗ . ≤· · ·≤w(B) ≤w(2) w(1). 士. ∗ ∗ Pk1 = [B ∗ α2 ],k2 = [B ∗ (1 − α2 )], 3(2)7)ª¥±w(k Úw(k ©O© 1) 2) ∗ α O, d(2)7)ªCq ª êw∗α , w1− 2. 硕. 2. (2 − 8). 博. S S ∗ ∗ √ <µ<X − w(k √ }, P {X − w(k 2) 1) n n. 大. 学. d(2)8)ªµ&Y²1 − αbootstrap&«m S S ∗ ∗ √ , X − w(k √ ). (X − w(k 2) 1) n n. 门. ù{¡bootstrap-t{.. 厦. ^ ë êbootstrap {, v k b ¤ ï Ä o N © Ù ¼ êF / ª,. bootstrap ´5g®êâ(© ), ¤±¡ëêbootstrap{..
(16) gÏü u. 9. 2.1.1.2 ëêbootstrap { b¤ïÄoN©Ù©Ù¼êF (x; β)/ª®, Ù¥¹ë êβ(β±´þ). y3®k5gF (x; β) . 库. X1 , X2 , · · ·, Xn . b 3F (x; β)¥±βOβF b b |^ù ¦Ñβq,Oβ. (x; β),. 文. ∗. 摘. b X1 , X2∗ , · · ·, Xn∗ ∼F (x; β).. 要. b X3F (x; β)¥)Nþn . ù« ±)éõ. ~X)B(B≥1000). Ò±|^ù . 论. éoN?1ÚOíä, Ù{ëêbootstrap{, ù«{¡ë. 士. êbootstrap{.. 硕. 2.1.2 ü u. 博. ü L§=I(1)L§, I(1)L§ ©²L§, «OI(0)ÚI(1)L §uѱI(1)b, duvkïÄ< ÊHÉäkÐk. 学. 5±I(0)bu, ùpÌ?ØbI(1)u. ~^. 大. ukDFu, ADFu, Phillipsu, Mu .. 门. 2.1.2.1 DF u[12]. 厦. ( ) v k å Ú m ª ³ / êâ)¤L§(DGP): yt = yt−1 + ut ,. Ù¥ut ´i.i.d.þ0, σ 2 .. (2 − 9).
(17) 厦. 门. 大. 学. 博. 硕. 士. 论. 文. 摘. 要. 库. Degree papers are in the “Xiamen University Electronic Theses and Dissertations Database”. Full texts are available in the following ways: 1. If your library is a CALIS member libraries, please log on http://etd.calis.edu.cn/ and submit requests online, or consult the interlibrary loan department in your library. 2. For users of non-CALIS member libraries, please mail to [email protected] for delivery details..
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