Chna s Regonal Convergence n Panels wh Mulple Srucural Breaks (Runnng le: Chna s Regonal Convergence n Panels wh Mulple Breaks) akash Masuk * and Ryoch Usa Deparen of Econocs, Osaka Gakun Unversy, -36- Kshbena, Sua- Cy, Osaka, 564-85, Japan Absrac hs sudy nvesgaes he exsence of regonal convergence of per capa oupus n Chna fro 95 4, parcularly focusng on consderng he presence of ulple srucural breaks n he provncal-level panel daa. Frs, he panel-based un roo es ha allows for occurrence of ulple breaks a varous break daes across provnces s developed; hs es s based on he p-value cobnaon approach suggesed by Fsher (93). Nex, he es s appled o Chna s provncal real per capa oupus o exane he regonal convergence n Chna. o oban he p-values of un roo ess for each provnce, whch are cobned o consruc he panel un roo es, hs sudy assues hree daa generang processes: a drfless rando walk process, an ARMA process, and an AR process wh cross-seconally dependen errors n Mone Carlo sulaon. he resuls obaned fro hs sudy reveal ha he convergence of he provncal per capa oupus exss n each of he hree geographcally classfed regons he Easern, Cenral, and Wesern regons of Chna. * Correspondng auhor. E-al: asuk@ogu.ac.jp
. Inroducon One of he poran ssues n Chna, whch has acheved hgh econoc growh raes snce he end of 978, s he exsence of large dfferenals n oupu per capa beween provnces. Reducng hese gaps s one of he an objecves se n he Elevenh Fve-Year-Plan (6 ). herefore, fro he perspecve of polcy akng by he Chnese governen, s exreely poran o undersand he behavour of provncal per capa oupus, parcularly observng wheher hese per capa oupus can converge. Los of sudes, ncludng Mankw, Roer and Wel (99), Bernard and Durlauf (995, 996), and Quah (993a, b, 996), have been conduced on he convergence of per capa oupus snce Barro (99) and Barro and Sala--Marn (99). Aong hese, soe eprcal sudes have ulzed nonsaonary e seres echnques such as un roo ess and conegraon ess. On he oher hand, Evans and Karras (996), Lee, Pesaran and Sh (997), Evans (998), Flessg and Srauss (), and McCoskey () have used un roo ess exended for panel daa ses o nvesgae convergence across counres and he saes of US; soe of hese ess have been proposed by I, Pesaran and Shn (3) (hereafer, IPS) and Maddala and Wu (999) (hereafer, MW). Wh regard o he convergence hypohess of provncal per capa oupus n Chna, here are several conrbuons o he leraure, such as Zhang, Lu and Yao () and Pedron and Yao (6), ha use un roo esng ehods for a sngle e Bernard and Durlauf (995), Oxley and Greasley (995), Hobjn and Franses (), Pesaran (4), L and McAleer (4), ec.
seres and panel daa ses. Zhang e al. () aggregaed he real per capa GDP of 3 Chnese provnces fro 95 997 no hree regons (he Easern, Cenral, and Wesern regons) n accordance wh he offcal classfcaon and hen appled he Augened Dckey Fuller -es and he un roo es wh a break suggesed by Perron (994) o he relave regonal and naonal per capa GDPs of each of he hree regons. hen, hey concluded ha wo of he hree regons (he Easern and Wesern regons) are convergng o her own specfc seady saes. Pedron and Yao (6) ulzed he panel-based un roo ess, he IPS and MW ess, o nvesgae convergence of he annual real per capa GDP across all he provnces of Chna. hey spl each provncal seres no he pre-refor saple (95 977) and he pos-refor saple (978 997) o consder he pacs of he econoc refors snce 978. he resuls revealed convergence n he pre-refor saple, bu no n he pos-refor saple. Whle esng un roos or conegrang relaonshps usng a sngle e seres, he saple sze used n he analyss needs o be suffcenly large o oban hgher power of he es. Slarly, he e seres denson of panel ses for each cross-seconal un should be large n panel un roo ess, especally n he ess based on cobnaons of separae un roo ess such as he IPS and MW ess. However, panels wh longer e spans have a hgher possbly of ncludng srucural changes caused by wars, supply shocks, sgnfcan polcy changes, and so on. Perron (989), Leybourne, Mlls and he sudes on regonal growh n Chna whch have no adoped he nonsaonary e seres or panel echnques are Chen and Flesher (996), Jan, Sachs and Warner (996), Gundlach (997), Raser (998), and Weeks and Yao (3).
Newbold (998), and I, Lee and eslau (5) have shown ha gnorng he exsng srucural breaks n e seres or panel daa ses ay lead o a subsanal loss of power or serous sze dsorons n coonly used un roo ess such as he Dckey Fuller es and he IPS es. akng such evdence no accoun, s desrable o eploy ess ha allow for srucural changes n daa. 3 Syh and Inder (4) ascrbe he logc behnd he ncluson of ulple srucural breaks n he offcal oupu of Chna o he occurrence of sgnfcan polcal and econoc evens: he Grea Leap Forward fro 958 96; he sudden suspenson of he econoc asssance fro he Sove Unon n 96; he crop falures fro 959 96; he Culural Revoluon fro 966 976; econoc refors fro 978 979; and Deng Xaopng s souhern our n 99. Fgures o 3 show he flucuaons of log of real per capa oupus of he provnces, wheren each seres s subraced fro he ean value of each of hree regons, wh weny-nne provnces. 4 5 In all he fgures, we observe apparen shfs n he level of he seres correspondng o 3 Carlno and Mlls (993), Greasley and Oxley (997), and L and Papell (999) have exaned convergence usng un roo ess whch can deal wh a breakng e seres. 4 hs classfcaon of provnces s nearly dencal o ha of Zhang e al. (), bu he aggregaon of provncal seres s no conduced n hs paper. he deals wll be descrbed n Secon 4.. 5 Sudes on ul-counry convergence ofen use he devaon fro he cross-seconal ean and look no s nonsaonary (Evans and Karras, 996; Lee e al., 997; and Evans, 998), whch wll be descrbed n he laer secons. 3
each provnce for he followng e perods: 959 6, 967 7, and he early 99s (shown as grey areas n he fgures). hese shfs concde wh he occurrences of he evens enoned above. Based on hese fndngs, soe sudes have focused on he presence of srucural changes n he annual seres n Chna (L, ; Zhang e al., ; and Syh and Inder, 4); hese sudes have adoped un roo esng ehods perng one or wo breaks n a sngle e seres for he analyss of he nonsaonary of he acroeconoc or provncal e seres. However, wh regard o he convergence hypohess n regonal panel daa n Chna, publshed papers whch explcly deal wh he exsence of ulple srucural breaks occurrng a dfferen break daes n he panels have been few n nuber. 6 6 In general, exsng panel-based un roo ess whch allow for breaks ay be oo resrcve for eprcal applcaons based on he convergence hypohess. Specfcally, hese ess are based on wo ajor assupons: he presence of a lnear e rend n a seres and he absence of cross-seconal dependence beween error ers n he daa generang process (DGP). In he case of he forer assupon, he ess defned under he DGP wh a e rend are no drecly applcable o nvesgaons on convergence. In hese nvesgaons, he dfference beween wo seres or he devaon fro he ean value of all cross-seconal uns s usually used, and he dfference or he ean devaon s ofen assued o be zero ean saonary when absolue convergence exss, or level saonary when condonal convergence exss (Bernard and Durlauf, 995; Evans and Karras, 996). hus, hese analyses requre ess whch are defned under DGPs whou a e rend, nsead of DGPs wh a e rend. In he case of he laer 4
herefore, whle exanng convergence across provnces n Chna, hs sudy focuses on he presence of ulple srucural breaks n he panels. Based on he cobnng p-values ehod of Fsher (93), we frs develop a un roo es whch allows for ulple breaks n he panels. We hen nvesgae he exsence of regonal convergence n Chna by applyng he es o Chna s provncal per capa oupus. he p-values of he -ype un roo ess for each provnce, whch are cobned by he panel un roo ess based on Fsher s p-values cobnaon approach, are calculaed by Mone Carlo sulaon under hree daa generang odels he drfless rando walk odel, he ARMA odel, and he AR odel wh cross-seconally dependen errors. In parcular, n he case of cross-seconal dependence beween error ers, he boosrap ehod proposed by Maddala and Wu (999) and Wu and Wu () s eployed n order o correc he bases of he panel-based un roo ess. 7 he reanng secons of hs paper are organzed as follows: Secon defnes convergence; Secon 3 descrbes he econoerc ehodology; Secon 4 brefly enons he daa and dscusses he eprcal resuls; and Secon 5 presens he conclusons. assupon, cross-seconal correlaon beween error ers s a ajor ssue n dynac panel esaon because neglecng hs correlaon ay lead o a bas of an esaed paraeer and ncrease s varance (O connell, 998; Phllps and Sul, 3). 7 Banerjee and Carron--Slvesre (6) have deal wh several ssues on srucural breaks and cross-seconal dependence n a nonsaonary panel fraework. 5
. Defnon of Convergence A frs, we consder convergence as proposed by Evans (998). Suppose ha y s a log per capa oupu for provnce (cross-seconal un) a e ( =, K, N, =, K, ). Nex, consder he dfference beween y and he ean value of y over =, K, N, whch s denoed as y% y y, where y N N y =. As proved by Evans (998), snce N y ( ) y = N y j y = j, f y y j s saonary for all pars of and j, y y s saonary for all. A converse proof s also avalable: snce y yj = ( y y ) ( yj y ), f y y s saonary for all, y yj s saonary for all pars (, j ). hese resuls equae bvarae convergence whn all pars of provnces, refleced by he saonary of y yj for all pars of and j, o he saonary of y y for all. hs equvalence enables us o focus on nvesgang he sochasc properes of y% = y y for all nsead of y y j for all pars of and j. In he nex secon, we wll specfy srucural changes a soe e perods n a seres as ulple shfs n he level of he seres. Accordngly, convergence s defned as follows: For all, f y% s saonary wh shfs n s level a soe, hen convergence exss across all he provnces. 8 8 Evans and Karras (996) have posulaed ha convergence s absolue f y% has a zero ean for all, or condonal f y% has a non-zero ean for soe. Accordng o Evans and Karras, when all he seres of y% are saonary and have soe srucural breaks, convergence can be consdered as beng absolue f y% has a zero ean for all 6
hs sudy does no allow he rend saonary of y% for each because he presence of a lnear e rend ples ha soe of he dfferences beween y and y j for fxed and all j wll dverge as e approaches nfny unless he e rends are he sae for all he pars (Bernard and Durlauf, 995). Furher, wh he excepons of Laonng n Fgure and Helongjang and Hube n Fgure, none of he fgures show a dsnc upward or downward endency for any seres durng he enre saple perod. herefore, we consder y% as a seres whou a e rend n he laer secons. 3. Econoerc Mehodology 3.. Models and es Sascs for a Sngle e Seres We assue y% for each provnce o be nonsaonary whou breaks under he null hypohess, and saonary wh breaks under he alernave hypohess. As dscussed n Secon, alhough each seres exhbs no lnear e rend, conans soe shfs n he level. herefore, hs sudy assues ha y% s generaed by he followng daa generang process (DGP). Under Null Under Alernave ~ y ~ + ε () = y = y + δ DU + δ DU ~ y ρ ~ + ε, ρ < () =, K, N, =, K, afer he las break dae, or condonal f y% has a non-zero ean for soe afer he las break dae. 7
where ε s ndependenly and dencally dsrbued across and wh a zero ean and a fne varance; δ h denoes he sze of he h h break ( h =, ); DU h denoes he h h break n he level of a seres ( h =, ), where DU = for >, and zero oherwse; and h s he fracon of he h h break ( h =, ) n < < <, whch s defned as B h / for all, where B h s he dae of he h h break ( h =, ). In hs DGP, he seres s a drfless rando walk process under he null hypohess, whereas s a saonary process and has up o wo-e level shfs under he alernave hypohess. Nex, he regresson odel ness Models () and (). ~ y = ˆ α d + ˆ φ ~ y + ˆ δ DU + ˆ δ DU error =, (3) + where ~ y = ~ ~ y y, ˆ φ = ˆ ρ, and d denoes he deernsc er, where d = { } for = and { } for =. Le be he -sasc esng he null hypohess ˆ φ = and ˆ δ ˆ δ agans he alernave hypohess ˆ φ and = = ˆ δ, ˆ δ n each regresson odel ( =, ) for each. As carred ou n Zvo and Andrews (99) and Lusdane and Papell (997), he break daes { B, B } are endogenously deerned o be where he one-sded -sasc s nzed n sequenal esaons over all possble break daes whn he range of < < <. For fxed, when Model () has a consan and Model (3) has boh a h h consan and lnear e rend, he -sasc s he counerpar of he one proposed by Zvo and Andrews (99) for a sngle break ( δ and ˆ δ = ) and of ha proposed by Lusdane and Papell (997) for double breaks, where he asypoc = behavour of he sasc as can be found. On he oher hand, no leraure provdes he exac asypoc behavour of he es consdered here. herefore, as for fxed, we derve he lng dsrbuon of he sasc for each case of 8
breaks n he followng heores n whch he subscrp s oed for splcy. heore. For Models () and (3), wh δ = and ˆ δ =, as, he lng dsrbuon of he nu -sasc s gven as follows: ( ) nf / {( + b ) W ( r, ) dr} W ( r, dw ) =, (4) where denoes weak convergence n dsrbuon; W r, ) denoes he resduals ( fro he projecon of a sandard Wener process W (r) ono he subspace generaed by he funcons { r, )} du for = ( and {, ( r, )} du for =, where du ( r, ) = for r >, and zero oherwse. b s gven by b b = ( ) = ( { W ( r) dr} ) { W ( r) dr} where W (r) s a deeaned sandard Wener process defned as W ( r) W ( r) oed. 9 W ( r) dr. he proof of heore s analogous o he followng heore and s, herefore, heore. For Models () and (3), as, he lng dsrbuon of he nu -sasc s gven as follows: (, ) nf, / {( + b + c ) W ( r,, ) dr} W ( r,, dw ) =, (5) 9 hs proof s avalable on reques. 9
where W r,, ) denoes he resduals fro he projecon of W (r) ono he ( subspace generaed by he funcons { r, ), du ( r, )} {, ( r, ), du ( r, )} du for = and ( du for =, where duh ( r, h ) = for r > h, and zero oherwse ( h =, ). b and c are gven by b c = ( ) = ( ) { W ( r) dr W ( r) dr} { W ( r) dr + ( )( ) W ( r) dr} { W ( r) dr + W ( r dr} { W ( r) dr ( )( ) W ( r dr} b = ( ) ) c = ( ) ). he proof of heore s gven n Appendx. 3.. Consrucon of Panel Un Roo es wh Breaks In hs subsecon, we consruc a panel un roo es wh breaks by cobnng he ndvdual nu -es; hs es s based on Fsher s (93) su of log p-value approach, whch has been nroduced and used by Maddala and Wu (999). Suppose ha p s he p-value fro he h es sasc aong N connuous es sascs. herefore, snce each p s an ndependen unfor (, ) varable, log has he ch-square dsrbuon wh wo degrees of freedo. Furher, he p suaon of log p fro = o N also has he ch-square dsrbuon wh N degrees of freedo. Fsher (93) ulzed hs fac o develop he es (hereafer, Fsher es). By applyng Fsher s p-value cobnaon ehod o N augened Dckey Fuller -ess, Maddala and Wu (999) has bul a panel-based un roo es whch does no allow breaks. In hs sudy, we use Fsher s approach o consruc panel un roo ess ha allow ulple breaks. Le p denoe he p-value fro he
ndvdual nu -es. herefore, he su of log p s defned as follows: N Fsher _ B = log =, (6) = p he Fsher_B es (he Fsher es wh breaks) also has he ch-square dsrbuon wh N degrees of freedo. In he presen case, however, he degree of freedo of he ch-square dsrbuon s ( N ) due o he resrcon of ~ y =. he null and N = alernave hypoheses of he es are specfed as H : φ = and δ = δ = for all and H : φ < and δ, δ for soe respecvely. here are wo noeworhy feaures of he ess based on Fsher s p-value cobnaon approach: () Snce he ess have an exac (ch-square) dsrbuon, hey do no requre a large cross-seconal denson of panel daa. Hence, hey are expeced o perfor well n he analyses usng panels wh a relavely large e denson and a sall cross-seconal denson such as counry-level, sae-level, or provncal-level panels. () Even f soe of he N un roo ess gve larger p-values han convenonal sgnfcance levels, e.g. 5 or per cen, whch ples he non-rejecon of he un roo null n each es, f hese p-values ndcae a slgh endency o rejec he un roo null (e.g..5 or.), he ess based on Fsher s p-value cobnaon approach can capure. o calculae he Fsher_B es sasc, we need o copue he p-value of he Alhough Becker (997) copared he perforance of 6 p-value cobnaon ess, ncludng he Fsher es, he concluded ha here was no es ha was he os accurae or effecve.
nu -es for all (, ) by Mone Carlo sulaon because he nu -es has non-sandard lng dsrbuons for each shown n heores and. Under he un roo null hypohess, hs sudy consders he followng hree DGPs: Model (I) Model (II) ~ y ~ = y + ε ˆ θ ( L) ~ y = ψˆ ( L) ˆ σ ε Model (III) ~ y k * ~* ˆ = k y k k = γ + ε * where ε s an..d. N (, ) error across and ; ˆ θ ( L) = ˆ θ L ˆ θ L p L θˆp L and ψˆ ( L) = ψ L ψ L ˆ ˆ L ψˆ L q q, where ˆ θ, L, ˆ θ and p ψ ψˆ ˆ, L, are esaed paraeers; and L s he lag operaor such as q Ly = y. For Model (I), y~ s generaed for each by a drfless rando walk odel. For Model (II), for each, y~ s generaed by he opal auoregressve ovng average (ARMA) ( p, q ) odel wh esaed paraeers and N(, ˆ σ ) nnovaons, where ˆ σ s he esaed nnovaon varance of he ARMA odel. he selecon of he opal ARMA odel follows he Zvo and Andrews (99) procedure, whch fs ARMA ( p, q ) odel o y~ over he possble cobnaons of p and q wh p, q 5, hen fnds he bes fed odel accordng o he Akake nforaon creron and he Schwarz nforaon creron. When he wo crera choose dfferen odels, he os parsonous odel s seleced. For Model (III), ~* y s he boosrap saple for y~, whch s obaned by he boosrap ehod eployed by Maddala and Wu (999) and Wu and Wu (). he procedure followed heren s elaboraed below. Frsly, we esae he equaon ~ k y = ˆ γ ~ = y + ε for each by usng he OLS ehod, and hen we oban he k k k
resduals ε = [ ε ε, L ε ] ( =, K, ). Nex, we resaple,, N ε fro he obaned resduals by preservng her cross-seconal correlaon srucure based on he boosrap ehod of Maddala and Wu (999), wheren he vecor ε = [ ε L ε ] s resapled,, N nsead of ndvdual ε. In addon, we generae a rando nuber g whch akes neger values on [, ] wh probably /, by usng a unfor rando nuber. We hen draw a row of resduals ε = [ ε L ε ] accordng o he realzaons of g. g g,, Ng he boosrap saple ε ( =, K, ) s obaned by -e whdrawals fro he * resduals. he boosrap saple ~ y * s generaed by Model (III) wh esaed paraeers γˆ ( k =, K,k ) n he prevous OLS esaon. However, k ~ y, L, ~ y * * k + are replaced by he saple obaned by he block resaplng ehod of Berkowz and Klan (996). her ehod dvdes he acual saple y~ no k overlappng subsaplng blocks wh sze k + and randoly draws a block fro k blocks. hen, ~ y, L, ~ y are replaced wh hs block. * * k + In fac, n he case where he cross-seconally dependen errors are presen n he daa generang odel, he Fsher_B es does no belong o he ch-square dsrbuon under he null hypohess because he nu -ess are correlaed across. Accordngly, he es ay be based owards over- or under-rejecons of he null. In order o correc hese bases of he es, we frs capure he cross-seconal correlaon srucure n he panels accordng o he above resaplng schee. hen, wh he generaed boosrap saple ~ y * ( =, K, ), we oban he eprcal o reove cross-seconal dependence n he panels wh srucural breaks, he coon facor odel s also applcable. See Banerjee and Carron--Slvesre (6). 3
dsrbuon funcon of he Fsher_B es hrough sulaon, whch provdes he approprae sall-saple crcal values for he es. hese values are lsed n able. Based on hese sulaed crcal values, we can conduc un roo esng n an approprae anner. A Mone Carlo sulaon s perfored usng 5, replcaons under each DGP. he suary of he sulaon s as follows: () For each, he eprcal dsrbuon funcon of he nu -sasc s obaned hrough replcaons. In parcular, n Model (III), 5, boosrap saples are generaed and used n he sulaon. () For each, he p-value ( p ) of he acual nu -es, obaned fro he orgnal daa se, s evaluaed based on he eprcal dsrbuon funcon obaned n (). hen, he Fsher_B sasc s calculaed. (3) In each replcaon n Model (III), p of he sulaed nu -es, whch s copued fro each boosrap saple, s evaluaed for each based on he eprcal dsrbuon funcon obaned n (). hen, usng p,, K p, he value N N of log = p s calculaed. he eprcal dsrbuon funcon of he Fsher_B es can be obaned fro he calculaed values of log. p 4. Eprcal Analyss 4.. Daa Provncal daa have been sourced fro Chna Copendu of Sascs 949 4. We have used he annual real per capa oupus of 9 provnces fro 95 o 4; 4
hese oupus have been generaed by he chan ndex of he per capa gross regonal produc (GRP) wh 95 as he reference year. Hanan and Schuan provnces have been excluded due o he lack of daa. All he seres used n hs paper have been aken n naural logarhs. 3 As n Zhang e al. (), we dvde he 9 provnces accordng o her geographcal locaons no he followng hree regons: he Easern, Cenral and Wesern regons. 4 However, we have ncluded he Guangx Zhuang auonoous he chan ndex of he per capa GRP s copued as Y = ( Y5 /) ( Y53 /) L( /), where Y * s he chan ndex of he per * Y capa GRP. Furher, Y s he ndex of he per capa GRP (precedng year = ), and Y 5 s se o. 3 he qualy of offcal Chnese sascs has been argued by any researchers (e.g. Chow, 986; Rawsk, ; and Holz, 6). Currenly, s wdely recognzed ha offcal Chnese daa a he naonal and provncal levels have ceran nconssences and scalculaons due o facors such as he lack of echncal personnel for he collecon of sascs and polcal pressure o exaggerae sascs a he lower levels. However, our resuls, whch wll be presened n Secon 4.3, rean vald as long as he sochasc properes of he seres used n hs paper do no change even f here are ceran naccuraces n he. 4 he Easern regon has he followng en provnces: Bejng, anjn, Hebe, Laonng, Shangha, Jangsu, Zhejang, Shandong, Fujan, and Guangdong. he Cenral regon ncludes he followng nne provnces: Shanx, Inner Mongola, Jln, 5
regon n he Wesern regon, nsead of he Easern regon, because snce 978, s log of real per capa oupu has shown consderable devaon fro hose of he oher Easern provnces. In fac, he dfferences beween he recen daa of Guangx and oher Wesern provnces are consderably less copared o he dfferences beween Guangx and oher Easern provnces. herefore, s reasonable o nclude Guangx n he Wesern regon. 5 he panel for each regon used n hs sudy s coposed of he devaons of a log of real per capa oupu fro he ean value across all he provnces n he correspondng regon, whch s denoed by he nuber of provnces n he regon. ~ N ' y = y y = = y y, where ' N s 4.. es Procedure Model (3) shown n Secon 3. s regressed for each, ncludng lagged augenaon ers of he frs dfference of y~, n order o elnae he Helongjang, Anhu, Jangx, Henan, Hube, and Hunan. he Wesern regon consss of he followng en provnces: Guangx, Chongqng, Guzhou, Yunnan, be, Shaanx, Gansu, Qngha, Nngxa, and Xnjang. 5 For exaple, for he seres (n logarhs) n 4, he dfference beween he seres of Guangx and Hebe (he closes seres aong oher Easern provnces) s.87. In conras, he dfference beween he seres of Guangx and Yunnan (he closes aong he Wesern provnces) s.9. In addon, he seres of soe oher provnces n he Wesern regon (Guzhou, Qngha, and Xnjang) are also close o ha of Guangx. 6
auocorrelaon of he error er. ~ y = ˆ α d + ˆ φ ~ y + ˆ δ DU + ˆ δ DU + aˆ ~ y + u (7) l l= l l where l s a lag order paraeer and u s a serally uncorrelaed error. We deerne he nuber of lagged augenaon ers by followng he general-o-specfc procedure descrbed n Perron (989) and suggesed n Ng and Perron (995). he axu lag order s se a 8. Nex, he procedure frs esaes he regresson odel wh l = 8. If he las lag s sgnfcan a per cen, where he crcal value s an asypoc noral value of.645 on he -sasc, he procedure selecs 8 as he opal lag order; oherwse, s elnaed fro he regresson odel. he seps enoned above are repeaed unl he las lag becoes sgnfcan. In he even of a sngle nsgnfcan lag, he opal lag order s se a. For each, he nu -es sasc s obaned by sequenally regressng Model ( =, ) over he possble break daes B, B } whn + < { l < B < B 53 for wo-e breaks and B } whn l < B < 53 for { + a one-e break. hen, for each of he hree regons, he Fsher_B es s consruced for each ( =, ) by cobnng he p-value of he ndvdual es ( p ), whch s obaned va sulaon. 4.3. es Resuls and Dscusson We frs eploy he coonly used panel un roo ess whou a break he Levn, Ln and Chu () es and he I e al. (3) es. he resuls are shown n able. For each regon, boh he ess can rejec he un roo null hypohess n a leas one 7
regresson odel a he per cen or beer sgnfcance level. Fro hs es resul, he convergence hypohess of he provncal oupus appears o be suppored for each regon. However, he IPS and LLC ess ay possbly suffer fro bases owards underor over-rejecons of he un roo null because hey do no rea he presence of boh srucural breaks and cross-seconal dependence aong error ers n he panels. 6 Nex, we apply he ess based on Fsher s p-value cobnaon approach he MW es and he Fsher_B es on seres wh breaks (he esaon resuls for each provnce n he presence of breaks are presened n ables A 4A n Appendx.). 7 able provdes he sall-saple crcal values a he, 5, and per cen levels of he MW and Fsher_B ess under Model (III), whch are obaned by usng he procedure descrbed n Secon 3.. able 3 repors he es resuls obaned under he hree DGPs. In he case of ess on seres whou a break (he MW es), here are en sgnfcan ess of regonal 6 Wh regard o hese ssues, Perron (989), Leybourne e al. (998), and I e al. (5) have revealed ha gnorng breaks n a sngle e seres or panel daa can lead o an erroneous nference n a es, whle O connell (998) and Phllps and Sul (3) have argued ha esaed paraeers end o be based by he presence of cross-seconally correlaed errors. 7 We have also obaned es resuls for cases n whch he ean devaons of log per capa oupus dsplay a lnear e rend for Laonng n he Easern regon and Helongjang and Hube n he Cenral regon. Snce hese resuls are que slar o hose abulaed n able 3, hey have no been repored bu are avalable on reques. 8
convergence of real per capa oupus. In hese ess, however, due o he osson of breaks, he es resuls gh be naccurae and, herefore, sleadng. We hen consder he possbly of srucural breaks occurrng a varous break daes across provnces. he fourh colun of able 3 shows he resuls of he Fsher_B es n he case of a one-e break. When Models (I) and (II) are used as DGPs, for he Wesern regon, he Fsher_B es rejecs he un roo null hypohess for boh he regresson odels ( =, ) a he per cen sgnfcance level. In addon, under boh he DGPs, sgnfcan rejecons of he null are observed a he per cen level for he Easern regon ( = ) and a he or 5 per cen level for he Cenral regon ( = ). In he case of Model (III), wheren here s he cross-seconal correlaon beween error ers, he es sascs for boh he regresson odels for he Wesern regon are sll hgher han he correspondng crcal values a he per cen sgnfcance level. Furher, he sasc of he regresson odel for he Easern regon where = s also sgnfcan a he per cen level. In he case of Cenral provnces, he Fsher_B es canno suppor he saonary alernave. In Model (III), he fndng ha convergence occurs whn all provnces n he Easern and Wesern regons appears o be conssen wh ha of Zhang e al. (). he las colun of able 3 presens he resuls for cases wh wo-e breaks. In Models (I) and (II), wh one excepon n he Cenral regon, all he es resuls for all he regons exhb sgnfcan rejecons of he un roo null hypohess a he 5 per cen or beer levels. Moreover, when he correlaon of error ers aong provnces n each regon s consdered n Model (III), he Fsher_B es also srongly suppors he saonary alernave wh wo-e shfs for all of he regons (for eher or boh of 9
he regresson odels). As copared o he case of a seres ha ncludes a sngle srucural break, under any DGP, hs case ndcaes he exsence of regonal convergence whn all he hree regons. herefore, should be concluded ha dealng wh ulple srucural breaks occurrng a dfferen break daes for each provnce provdes sronger evdence of he exsence of convergence whn regons n Chna. hs fac ay also accoun for he dscrepances n he resuls copared wh hose of Zhang e al. (), where one endogenous break pon s assued n her esaon. he coparson of he hree ess resuls shown n able 3 reveals ha hey grealy depend on he nuber of breaks allowed n he ess. As dscussed n Secon, due o he pac of ceran sgnfcan polcal and econoc evens, he provncal real per capa oupus n Chna are suspeced o have soe srucural breaks; herefore, n he analyss on regonal convergence n Chna, we consder approprae o exane he possbly of ulple srucural changes n he suded e perods. Consequenly, when he provncal log per capa oupus are allowed o have wo-e level shfs a varous break daes across he provnces, we observe convergence of he seres n all he hree regons. 4.4. es Resuls Based on Oher Regonal Classfcaons 8 As llusraed n Fgure, he ean devaon of he real per capa oupu for Shangha s uch larger han hose for oher Easern provnces. Snce hs ay be ndcave of he heerogeney of Shangha, he seres for nne Easern provnces, 8 All he es resuls dscussed n hs subsecon have been oed bu are avalable on reques.
excludng Shangha, have been esed. Consequenly, convergence s also observed n he Easern regon. Furher nvesgaons have been conduced based on oher daa classfcaons where he Easern regon (wh or whou Shangha) ncludes he neghborng provnces, whch are Guangx, Jln, and Helongjang. he cases where one, wo, or all of he provnces are classfed as belongng o he Easern regon are analyzed. As a resul, n he case of wo srucural breaks, he evdence of convergence has been found n all he classfcaons. hs fac sees o ply ha he neghborng provnces are on he sae pah of convergence as ha of oher Easern provnces; however, hs s no conclusve. 9 o ake he dscusson ore concree, n classfyng provnces no ceran regons, he use of classfcaon ehods such as cluser analyss would be desrable. he work of Hobjn and Franses () s one such applcaon. However, hs s beyond he scope of hs paper. Meanwhle, as dscussed n Secon 4., here appear o be subsanal grounds for our classfcaon of Chnese provnces. herefore, our fndngs obaned fro able 3 are eanngful. 9 In addon, he saple conssng of whole provnces has been esed; oreover, a sgnfcan rejecon of he un roo null hypohess has been obaned. However, we beleve ha furher nforaon (e.g. he hoogeney of provnces classfed no dfferen regons) s needed o arrve a a concluson.
5. Concluson In hs sudy, we nvesgaed he regonal convergence hypohess of he provncal per capa oupus n Chna whle consderng up o wo-e srucural breaks n he panels. Accordng o he p-value cobnaon approach of Fsher (93), he panel-based un roo es has been developed by cobnng he p-value of he ndvdual un roo es whch allows for breaks n a sngle e seres. hs approach allowed us o consder ulple breaks a varous break daes across he provnces. We used hree daa generang odels n he Mone Carlo sulaon he drfless rando walk odel, he ARMA odel, and he AR odel wh cross-seconally dependen errors o calculae he p-value of he ndvdual nu -ype un roo es fro s eprcal dsrbuon. In parcular, n he case of he AR odel wh cross-seconally dependen errors, he eprcal dsrbuon of he es for each provnce was generaed on he boosrap saples, whch were obaned by he resaplng procedure proposed by Maddala and Wu (999) and Wu and Wu (). On he bass of her geographcal locaons, he provnces were grouped no he followng hree regons: he Easern, Cenral, and Wesern regons. Subsequenly, he exsence of convergence whn each regon was esed by he panel un roo es wh breaks, whch was developed n hs paper. As a resul, when he presence of wo-e breaks was consdered n he es, we found sgnfcan evdence o sugges ha he convergence of he provncal per capa oupus exss whn each regon. Acknowledgeen We would lke o hank Chen Kuang-hu, wo anonyous referees and he senar
parcpans a Yokohaa Syposa n 7 for her helpful coens. References Banerjee, A. and Carron--Slvesre, J. L. (6) Conegraon n panel daa wh breaks and cross-secon dependence, EUI Workng Papers, ECO No. 6/5, -47. Barro, R. J. (99) Econoc growh n a cross secon of counres, he Quarerly Journal of Econocs, 6, 47-443. Barro, R. J. and Sala--Marn, X. (99) Convergence, Journal of Polcal Econoy,, 3-5. Becker, B. J. (997) P-values, cobnaon of, n Encyclopeda of Sascal Scences: Updae vol. (Ed.) S. Koz and N. L. Johnson, Wley, New York, pp.448-453. Berkowz, J. and Klan, L. (996) Recen developens n boosrappng e seres, Dscusson paper 96/54, Board of Governors of he Federal Reserve Syse. Bernard, A. B. and Durlauf, S. N. (995) Convergence n nernaonal oupu, Journal of Appled Econoercs,, 97-8. Bernard, A. B. and Durlauf, S. N. (996) Inerpreng ess of he convergence hypohess, Journal of Econoercs, 7, 6-73. Carlno, G. A. and Mlls, L. O. (993) Are U.S. regonal ncoes convergng? A e seres analyss, Journal of Moneary Econocs, 3, 335-346. Chen, J. and Flesher, B. M. (996) Regonal ncoe nequaly and econoc growh n Chna, Journal of Coparave Econocs,, 4-64. Chow, G. C. (986) Chnese sascs, he Aercan Sascan, 4, 9-96. Evans, P. (998) Usng panel daa o evaluae growh heores, Inernaonal Econoc 3
Revew, 39, 95-36. Evans, P. and Karras, G. (996) Convergence revsed, Journal of Moneary Econocs, 37, 49-65. Fsher, R. A. (93) Sascal Mehods for Research Workers, Olver & Boyd, Ednburgh, 4h Edon. Flessg, A. and Srauss, J. () Panel un-roo ess of OECD sochasc convergence, Revew of Inernaonal Econocs, 9, 53-6. Greasley, D. and Oxley, L. (997) e-seres based ess of he convergence hypohess: soe posve resuls, Econocs Leers, 56, 43-47. Gundlach, E. (997) Regonal convergence of oupu per worker n Chna: a neoclasscal nerpreaon, Asan Econoc Journal,, 43-44. Hobjn, B. and Franses, P. H. () Asypocally perfec and relave convergence of producvy, Journal of Appled Econoercs, 5, 59-8. Holz, C. A. (6) Why Chna s new GDP daa aers, Far Easern Econoc Revew, 69, 54-57. I, K.-S., Lee, J. and eslau, M. (5) Panel LM un-roo ess wh level shfs, Oxford Bullen of Econocs and Sascs, 67, 393-49. I, K.-S., Pesaran, H. and Shn, Y. (3) esng for un roos n heerogeneous panels, Journal of Econoercs, 5, 53-74. Jan,., Sachs, J. D. and Warner, A. M. (996) rends n regonal nequaly n Chna, Chna Econoc Revew, 7, -. Lee, K., Pesaran, M. H. and Sh, R. (997) Growh and convergence n a ul-counry eprcal sochasc Solow odel, Journal of Appled Econoercs, 4
, 357-39. Levn, A., Ln, C.-F. and Chu, C. () Un roo es n panel daa: Asypoc and fne saple resuls, Journal of Econoercs, 8, -4. Leybourne, S. J., Mlls,. C. and Newbold, P. (998) Spurous rejecon by Dckey Fuller ess n he presence of a break under he null, Journal of Econoercs, 87, 9-3. L, Q. and Papell, D. (999) Convergence of nernaonal oupu: e seres evdence for 6 OECD counres, Inernaonal Revew of Econocs and Fnance, 8, 67-8. L, X.-M. () he Grea Leap Forward, econoc refors, and he un roo hypohess: esng for breakng rend funcons n Chna s GDP daa, Journal of Coparave Econocs, 8, 84-87. L, L. K. and McAleer, M. (4) Convergence and cachng up n ASEAN: a coparave analyss, Appled Econocs, 36, 37-53. Lusdane, R. L. and Papell, D. H. (997) Mulple rend breaks and he un-roo hypohess, he Revew of Econocs and Sascs, 79, -8. Maddala, G. S. and Wu, S. (999) A coparave sudy of un roo ess wh panel daa and a new sple es, Oxford Bullen of Econocs and Sascs, 6, 63-65. Mankw, N. G., Roer, D. and Wel, D. N. (99) A conrbuon o he eprcs of econoc growh, he Quarerly Journal of Econocs, 7, 47-437. McCoskey, S. K. () Convergence n Sub-Saharan Afrca: a nonsaonary panel daa approach, Appled Econocs, 34, 89-89. Naonal Bureau of Sascs (5) Chna Copendu of Sascs 949 4, Chna Sascs Press, Bejng. 5
Ng, S. and Perron, P. (995) Un roo ess n ARMA odels wh daa-dependen ehods for he selecon of he runcaon lag, Journal of Aercan Sascal Assocaon, 9, 68-8. O connell, P. G. J. (998) he overvaluaon of purchasng power pary, Journal of Inernaonal Econocs, 44, -9. Oxley, L. and Greasley, D. (995) A e-seres perspecve on convergence: Ausrala, UK and USA snce 87, he Econoc Record, 7, 59-7. Pedron, P. and Yao, J. Y. (6) Regonal ncoe dvergence n Chna, Journal of Asan Econocs, 7, 94-35. Perron, P. (989) he grea crash, he ol prce shock, and he un roo hypohess, Econoerca, 57, 36-4. Perron, P. (994) rend, un roo and srucural change n acroeconoc e seres, n Conegraon for he appled econoss, (Ed.) B. B. Rao, Macllan, Basngsoke pp.3-46. Pesaran, M. H. (4) A par-wse approach o esng for oupu and growh convergence, Meo, Unversy of Cabrdge. Phllps, P. C. B. and Sul, D. (3) Dynac panel esaon and hoogeney esng under cross secon dependence, Econoercs Journal, 6, 7-59. Quah, D. (993a) Eprcal cross-secon dynacs n econoc growh, European Econoc Revew, 37, 46-434. Quah, D. (993b) Galon s fallacy and ess of he convergence hypohess, he Scandnavan Journal of Econocs, 95, 47-443. Quah, D. (996) Eprcs for econoc growh and convergence, European Econoc 6
Revew, 4, 353-375. Raser, M. (998) Subsdsng nequaly: Econoc refors, fscal ransfers and convergence across Chnese provnces, Journal of Developen Sudes, 34, -6. Rawsk,. G. () Wha s happenng o Chna s GDP sascs? Chna Econoc Revew,, 347-354. Syh, R. and Inder, B. (4) Is Chnese provncal real GDP per capa nonsaonary? Evdence fro ulple rend break un, Chna Econoc Revew, 5, -4. Weeks, M. and Yao, J. Y. (3) Provncal condonal ncoe convergence n Chna, 953-997: a panel daa approach, Econoerc Revews,, 59-77. Wu, J.-L. and Wu, S. () Is purchasng power pary overvalued? Journal of Money, Cred, and Bankng, 33, 84-8. Zhang, Z., Lu, A. and Yao, S. () Convergence of Chna s regonal ncoes 95-997, Chna Econoc Revew,, 43-58. Zvo, E. and Andrews, D. W. K. (99) Furher evdence on he grea crash, he ol-prce shock, and he un-roo hypohess Journal of Busness & Econoc Sascs,, 5-7. 7
4 Fgure. Mean devaons of log of provncal real per capa oupus for he Easern provnces.5.5 998 996 994 99 99 988 986 984 98 98 978 976 974 97 97 968 966 964 96 96 958 956 954 95 -.5 - Year Bejng anjn Hebe Laonng Shangha Jangsu Zhejang Fujan Shandong Guangdong 8
Fgure. Mean devaons of log of provncal real per capa oupus for he Cenral provnces.5 4 998 996 994 99 99 988 986 984 98 98 978 976 974 97 97 968 966 964 96 96 958 956 954 95 -.5 - Year Shanx Inner Mongola Jln Helongjang Anhu Jangx Henan Hube Hunan 9
4 Fgure 3. Mean devaons of log of provncal real per capa oupus for he Wesern provnces.5 998 996 994 99 99 988 986 984 98 98 978 976 974 97 97 968 966 964 96 96 958 956 954 95 -.5 - -.5 Year Guangx Chongqng Guzhou Yunnan be Shaanx Gansu Qngha Nngxa Xnjang 3
able. he resuls for he Levn e al. () (LLC) es and he I e al. (3) (IPS) es Regon Regresson Model LLC es IPS es a Eas no consan & no rend.33**.46** consan.564.86 Cenral no consan & no rend.669**.569 consan.84.366* Wes no consan & no rend.947**.648** consan.486*.893** ***, **, and * denoe sascal sgnfcance a he %, 5%, and % levels, respecvely. a For boh regresson odels, he eans and he varances of he ndvdual augened Dckey Fuller -es for = 53 p were copued wh 5, replcaons, where p s he nuber of lagged augenaon ers of he frs dfference of a seres added n he ndvdual ADF equaon. 3
able. he crcal values of he Maddala and Wu (999) es and he Fsher B es n he case of cross-seconally dependen errors es Regon Regresson Model a % 5% % b MW es Eas = 9.5 3.93 4.346 = 9.48 3.96 38.84 Cenral = 5.955 8.588 34.47 = 6.4 8.986 35. Wes = 8.5 3.49 37.9 = 8.48 3.33 36.758 Fsher B es one break Eas = 3. 33.585 4.364 = 8.948 3.4 38.4 Cenral = 5.97 8.67 35.3 = 5.99 9. 34.58 Wes = 8.69 3.5 37.736 = 8.575 3.364 36.9 wo breaks Eas = 9.95 33.45 4.936 = 9.44 3.473 38.4 Cenral = 6.8 9.65 35.346 = 6.7 9.73 34.593 Wes = 8.394 3.736 38.7 = 8.79 3.94 38.53 a he regresson odel s ỹ = ˆα d + ˆφ ỹ + ˆδ DU + ˆδ DU + l l= âl ỹ l +u, =,..., N, =,..., 53, where N = for he Easern and Wesern regons and N = 9 for he Cenral regon, and d = { } for = and d = {} for = ; n addon, ˆδ = ˆδ = for all for he MW es and ˆδ = for all for he Fsher B es for he one break case. b he values are, 5, and per cen pons on he rgh al of he eprcal dsrbuons of he MW and Fsher B ess. hese dsrbuons are obaned as follows. For each and, under Model (III), he eprcal dsrbuon of he nu -sasc s obaned by a Mone Carlo sulaon wh 5, replcaons. Nex, he percenage pon (p ) of he nu -sasc copued for each replcaon s evaluaed on he eprcal dsrbuon obaned n he frs sep. Afer hs, he value of N = log p s calculaed for each replcaon. he eprcal dsrbuons of he ess can hus be obaned fro he calculaed values of log p. 3
able 3. he resuls for he Maddala and Wu (999) es and he Fsher B es n he cases of one-e and wo-e breaks Regon Regresson Model a MW es b Fsher B es b (no break) one break wo breaks DGP Model (I): ỹ = ỹ + ɛ Eas = 37.5*** 7.5# c * 39.56#*** =.5 6.436 33.96** Cenral = 9.76 3.97 8.53** = 5.34* 8.73** 4.96*** Wes = 6.7* 37.85*** 5.676#*** = 6.75* 5.76#*** 58.674#*** DGP Model (II): ˆθ (L) ỹ = ˆψ (L) ˆσ ɛ Eas = 35.94*** 6.638#* 37.975*** =.5 6.4 3.379** Cenral = 7.85 9.875 9.74 = 3.36 3.88* 36.547*** Wes = 6.4* 38.9#*** 49.93*** = 7.49* 49.96#*** 58.59#*** DGP Model (III): ỹ = k k= ˆγ k ỹ k + ɛ Eas = 9.6 3.744#* 53.44*** = 5.4 8.74.887 Cenral = 7.4 3. 36.35*** = 7.9* 4.87 9.93** Wes = 8.99* 4.45*** 45.3*** = 8.88* 37.688*** 34.546** ***, **, and * denoe sascal sgnfcance a he %, 5%, and % levels, respecvely. a he regresson odel s ỹ = ˆα d + ˆφ ỹ + ˆδ DU + ˆδ DU + l l= âl ỹ l + u, =,..., N, =,..., 53, where N = for he Easern and Wesern regons and N = 9 for he Cenral regon, and d = { } for = and d = {} for = ; n addon, ˆδ = ˆδ = for all for he MW es and ˆδ = for all for he Fsher B es for he one break case. b Under Models (I) and (II), because of he resrcon of N = ỹ =, he degree of freedo of he ch-square dsrbuon of he es s (N ). c he sgn # ndcaes ha he p-values for soe provnces were esaed o be zero due o he fac ha for each of hese provnces, he realzaon of he nu -sasc lay far lef fro s eprcal dsrbuon whch was generaed by a Mone Carlo sulaon wh 5, replcaons. herefore, n order o calculae he Fsher B sasc, he obaned p-values for hese provnces were assgned a value of. (/5). hs ples ha we assue ha he nu -sasc for each of hese provnces ook a value whn he esaed eprcal dsrbuon only once n he 5, replcaons. 33
Appendx Proof of heore For splcy, we o he subscrp of a varable and denoe a e seres as erely y nsead of y~ used n he an ex. herefore, y s assued o be subjec o Models () and () wh an.. d. nnovaon ε wh a zero ean and a fne varance σ. In hs proof, we show he dervaon only for he case of = he case of = s obaned along he sae lnes. because ha for Le e be he OLS resdual obaned by regressng y on an nercep and wo duy varables ( DU and e DU ) for =, K,. hen, he resdual s expressed as ( DU DU ) ˆ ( DU DU ) = S ˆ δ δ (A) where S s he deeaned rando walk process such as S S = S, where S = ε, and s = s DU h = = DU h = h δ ˆ = (X'X) X'S, where ˆ = ( ˆ, ˆ )' δ δ ( h =, ). Now, we wre δ ; X ( DU DU DU ) =,, where DU DU = ( DU DU, L,DU DU ) ' ; S = (, ) ' h DU h h h h h S L S,. hen, we have / δˆ = ( ) 3/ = S + ( ) S = S =. + S + = ( S = S ) ( ) = hus, / ˆ δ σ ˆ δ W W ( r) dr + ( r) dr ( ) W ( ) where denoes weak convergence n dsrbuon. herefore, ( r) dr A = σ W ( r) dr B ( DU DU ) ( ) / δ DU DU / / / e S ˆ ˆ = δ σw r,, ) (A) ( 34
[ W r) A{ du ( r, ) ( )} B{ du (, ) ( )}] = σ r ( where W r,, ) denoes he resdual fro he projecon of W (r) ono he ( subspace generaed by he funcon {, ( r, ), du ( r, )} du, where du ( r, ) = for r > h, and zero oherwse ( h =, ). Fro he regresson of e on e, we can oban he -es sasc as = e = e (3A) s ( e ) / = where s = = ( e ˆ ρ e ), where ρˆ s he esaed coeffcen of e n he h h regresson of e on e. Now, we show he probably ls of = e e, = e and s. e e s expressed as { S + y y + ( ) y + ( ) } ˆ e ˆ e = S S δ + δ = = { ˆ S } ˆ + y y + + ( ) y + ( ) δ + ( + ) δ ˆ δ ˆ. δ Hence, we have he followng lng dsrbuon. = e e σ W { W ( r) dr + W () + ( ) A} ( r) dw ( r) A { W ( r) dr + W () + ( ) B} + ( + B ) AB. In he dervaon above, we used he followng facs ha = S S σ W ( r) dw ( r), / S σ W ( ) / y σw (), / y + σw ( h ), / y σw (). h h h 35
36 Based on he lng behavour of e / shown n Equaon (A) and he connuous appng heore, s sraghforward o show ha = ),, ( dr r W e σ. Fnally, snce ) ( ) ˆ ( p o e e e s + = = = = ρ, we show he probably l of he frs er n he las equaon. ( ) { } ) ( ˆ ) ( ˆ ˆ ˆ y y y y y e δ δ δ δ + + + = + + = = ( ) B A + + σ. Noe ha σ = p y, ) ( / + p h h y y where p denoes convergence n probably.
Esaon Resuls for Each Provnce he esaon resuls, obaned fro ndvdual regresson conduced for each provnce, are shown n he followng ables. Now, we brefly dscuss he esaon accuracy of he daes of breaks and sgns of break sze paraeers n he resuls. Heren, he e perods of nfluenal evens n Chna, as descrbed n Secon, are consdered as 958 6, 966 7, and 989 95 (.e. he e perods shown as grey areas n Fgures 3 wh a one-year lead and lag for each perod). In he case of a one-e break (shown n ables A and A), approxaely one-hrd o half of he esaed break daes are conssen wh he expeced e perods, for each regon. For he sze paraeers of he break varables ha have he expeced break daes, soe have he rgh sgn whle ohers do no. In he case of wo-e breaks (shown n ables 3A and 4A), approxaely half of he deeced break daes (of he oal nuber of he frs and second breaks) ach he expeced perods, for each regon. For he sgn of break sze paraeers, here s soe proveen n ers of accuracy. 37
able A. he esaon resuls for each provnce n he case of a one-e break ( = ) Regon Provnce ˆφ ˆδ Mn l Break Dae P-value Model (I) Model (II) Model (III) Eas Bejng.94.87.893 6 963.356.37.9 anjn.5.3. 6 96.678.74.5 Hebe..6.5 954.994.994.93 Laonng.8.47.65 8 994.459.464.494 Shangha.49.7 7.58 8 96.# a.#.# Jangsu.4..838 8 3.379.386.45 Zhejang.46.7. 955.937.939.89 Fujan.5.34.76 5 96.83.84.53 Shandong.36.8 3.6 8 3.6.33.378 Guangdong.9.6.986 955.78.748.386 Cenral Shanx.35.55 3.839 8 968.78.8.48 Inner Mongola.7.33.56 8 98.56.58.8 Jln.53.33.59 6 96.98.986.74 Helongjang.36.4.377 3 998.577.64.57 Anhu..67.5 954.865.874.65 Jangx.87.95 3.49 8 969.49.554.5 Henan.6.3.99 4 996.95.95.945 Hube.49..48 4 99.95.9.876 Hunan.9.34.844 8 98.376.395.76 Wes Guangx.57.7.58 96.859.866.739 Chongqng.64..556 99.854.856.85 Guzhou.6.4.74 7 983.989.99.898 Yunnan.7.9.83 8 96.384.45.4 be.53.7 3.449 959.7.83.48 Shaanx.379. 4.55 968..5.47 Gansu.9.75 7.6 7 97.#.#. Qngha.45.9.863 8 999.773.78.67 Nngxa.96. 3.89 957.79.76.36 Xnjang.89..496 8 98.58.58.54 a he sgn # ndcaes ha he p-value for he provnce was esaed o be zero due o he fac ha he realzaon of he nu - sasc lay far lef fro s eprcal dsrbuon whch was generaed by a Mone Carlo sulaon wh 5, replcaons. herefore, n order o calculae he Fsher B sasc, he obaned p-value was assgned a value of. (/5). hs ples ha we assue ha he nu -sasc ook a value whn he esaed eprcal dsrbuon only once n he 5, replcaons. 38
able A. he esaon resuls for each provnce n he case of a one-e break ( = ) Regon Provnce ˆφ ˆδ Mn l Break Dae P-value Model (I) Model (II) Model (III) Eas Bejng.75.64 3.68 6 967.385.4.555 anjn.56.66 3.49 6 986.479.57.439 Hebe.48.45 3.84 5 983.95.97.433 Laonng.349..786 6 99.779.765.466 Shangha.95.84 4.7 6 983.6.58.6 Jangsu.94.79.3 8 985.98.896.567 Zhejang.53.6.7 979.8.79.734 Fujan.44.9.768 5 964.788.779.785 Shandong.45.7 4.7 976.8.64.7 Guangdong.58.6.744 986.795.799.83 Cenral Shanx.58.65 4.34 5 993.3.87.83 Inner Mongola.34.48 3.999 7.3.76.34 Jln.6.57 5.479 6 986...8 Helongjang.4.66.83 3 98.743.87.64 Anhu.553.5 4.36 3 976.6.7. Jangx.574.79 4.46 974.4.45.33 Henan.86.65 3.965 3 979.39.84.79 Hube.6.5.9 4 976.7.79.63 Hunan.73.8.959 965.69.79.744 Wes Guangx.34. 3.79 4 966.64.589.63 Chongqng.99.59 3.8 99.556.55.493 Guzhou.475.64 4.346 7 97.4.6.6 Yunnan.339.6 3.56 969.68.64.63 be.595.9 7.793 7 985.# a.#. Shaanx.38.97 4.58 968.89.89.4 Gansu.96.88 6.39 7 973...8 Qngha.7.7.85 8 987.766.773.548 Nngxa.3.95 3.68 957.379.38.45 Xnjang.35. 4.566 7 966.84.86.6 a he sgn # ndcaes ha he p-value for he provnce was esaed o be zero due o he fac ha he realzaon of he nu - sasc lay far lef fro s eprcal dsrbuon whch was generaed by a Mone Carlo sulaon wh 5, replcaons. herefore, n order o calculae he Fsher B sasc, he obaned p-value was assgned a value of. (/5). hs ples ha we assue ha he nu -sasc ook a value whn he esaed eprcal dsrbuon only once n he 5, replcaons. 39
able 3A. he esaon resuls for each provnce n he case of wo-e breaks ( = ) Regon Provnce ˆφ ˆδ ˆδ Mn l Break Dae P-value s break nd break Model (I) Model (II) Model (III) Eas Bejng.66.59.9 3.394 6 966 97.7.743.4 anjn.4.9.44 3.96 8 963 98.485.558.6 Hebe.76.93.8 3.58 955 979.757.775.7 Laonng.8.33.55 4.77 8 969 978.58.63.75 Shangha.55.43.4 7.869 8 96 99.# a.. Jangsu.6.. 5.45 8 968 977.48.46.8 Zhejang.5.43.45 4.45 8 965 979.73.7.5 Fujan.75.6.5.439 6 96 987.945.94.57 Shandong.57.6.94 4.54 8 973 974.38.38.34 Guangdong.39.33.44 4.894 8 967 97.9.5.4 Cenral Shanx.78.63. 5.45 8 968 97.4.9.5 Inner Mongola.89.74.85 5.937 7 96 964.3..8 Jln.78.4.6.7 6 96 985.999..87 Helongjang.67.9.99 5.766 8 96 983..76. Anhu.37.76..4 955 956.967.97.69 Jangx.47.5.88 4.849 8 96 969..57.6 Henan.7.7.6.667 956 963.94.94.7 Hube.7.37.5 3.45 6 96 974.85.83.7 Hunan.99..5.99 8 98 983.854.884.468 Wes Guangx.9.5.9.33 4 964 966.974.979.863 Chongqng.3.3.6 3.39 3 964 989.77.743.667 Guzhou.9.5.8.38 958 97.954.958.389 Yunnan.7.3.33 3. 956 96.8.85.86 be.58.86. 7.848 959 985.#.. Shaanx.78.47.94 6.575 968 987.3.3.34 Gansu.36.4.35 7.6 7 965 97.#.6.6 Qngha.76.87.64 3.9 8 964 987.473.5.35 Nngxa.7.7.38 4.66 957 99.8.9.58 Xnjang.9.7. 3.58 3 957 96.67.636.67 a he sgn # ndcaes ha he p-value for he provnce was esaed o be zero due o he fac ha he realzaon of he nu -sasc lay far lef fro s eprcal dsrbuon whch was generaed by a Mone Carlo sulaon wh 5, replcaons. herefore, n order o calculae he Fsher B sasc, he obaned p-value was assgned a value of. (/5). hs ples ha we assue ha he nu -sasc ook a value whn he esaed eprcal dsrbuon only once n he 5, replcaons. 4
able 4A. he esaon resuls for each provnce n he case of wo-e breaks ( = ) Regon Provnce ˆφ ˆδ ˆδ Mn l Break Dae P-value s break nd break Model (I) Model (II) Model (III) Eas Bejng.473.5.83 6.668 6 967 99..7.6 anjn.85.5.8 3.875 6 967 986.833.854.77 Hebe.3.79.39 6.7 8 969 985.6.38. Laonng.89.97.87 3.65 96 988.898.894.684 Shangha.4.9.85 5.7 8 967 983.87.95.5 Jangsu.88.5.93.749 8 968 984.986.988.86 Zhejang.38.36.95 4.57 8 967 979.54.55.53 Fujan.8.73.56 4.73 5 966 987.436.454.46 Shandong.64.84.7 5.37 965 99.8.3.6 Guangdong.57.54.99 6.45 8 967 99.47.58.89 Cenral Shanx.5.87.6 4.78 955 993.43.539.49 Inner Mongola.765.5.49 4.876 8 966 983.366.46.53 Jln.44.9.9 7.76 6 96 986..4.3 Helongjang.38.74.59 3.74 4 98 99.865.93.79 Anhu.86.8.99 6.4 3 976 993.3..5 Jangx.657.89.7 5.493 97 994.45.48.7 Henan.3.7.45 5.694 3 963 986.99.38.6 Hube.489.65.88 4.78 955 977.65.666.67 Hunan.3..9 6.698 8 97 986..5.6 Wes Guangx.563.54.9 3.966 7 968 98.777.78.84 Chongqng.3.8.54 3.69 977 99.866.877.87 Guzhou.655.7.88 6.36 8 973 976.53.48.57 Yunnan.63.66.58 4.957 967 986.36.37.59 be.665.53.7 8. 7 98 985.# a..6 Shaanx.784.44.95 6.53 968 987..7.99 Gansu.43.73.79 8.84 7 97 974.#.#. Qngha.357.66.67 3.976 8 978 987.773.85.57 Nngxa.457.84.6 5.47 5 965 989.43.5.6 Xnjang.349.3.83 5.35 7 96 966.78.87.9 a he sgn # ndcaes ha he p-value for he provnce was esaed o be zero due o he fac ha he realzaon of he nu -sasc lay far lef fro s eprcal dsrbuon whch was generaed by a Mone Carlo sulaon wh 5, replcaons. herefore, n order o calculae he Fsher B sasc, he obaned p-value was assgned a value of. (/5). hs ples ha we assue ha he nu -sasc ook a value whn he esaed eprcal dsrbuon only once n he 5, replcaons. 4