Journal of Economercs 46 (8 59 73 Conens lss avalable a ScenceDrec Journal of Economercs ournal homepage: www.elsever.com/locae/econom he lm dsrbuon of he esmaes n conegraed regresson models wh mulple srucural changes Mohosh Kerwal a, Perre Perron b, a Kranner School of Managemen, Purdue Unversy, 43 Wes Sae Sree, Wes Lafayee, IN 4797, Uned Saes b Deparmen of Economcs, Boson Unversy, 7 Bay Sae Road, Boson, MA 5, Uned Saes a r c l e n f o a b s r a c Arcle hsory: Receved Augus 7 Receved n revsed form March 8 Acceped 4 July 8 Avalable onlne July 8 JEL classfcaon: C Keywords: Change-pon Break daes Un roos Conegraon Confdence nervals We sudy esmaon and nference n conegraed regresson models wh mulple srucural changes allowng boh saonary and negraed regressors. Boh pure and paral srucural change models are analyzed. We derve he conssency, rae of convergence and he lm dsrbuon of he esmaed break fracons. Our echncal condons are consderably less resrcve han hose n Ba e al. [Ba, J., Lumsdane, R.L., Sock, J.H., 998. esng for and dang breaks n mulvarae me seres. Revew of Economc Sudes 65, 395 43] who consdered he sngle break case n a mul-equaons sysem, and perm a wde class of praccally relevan models. Our analyss s, however, resrced o a sngle equaon framework. We show ha f he coeffcens of he negraed regressors are allowed o change, he esmaed break fracons are asympocally dependen so ha confdence nervals need o be consruced only. If, however, only he nercep and/or he coeffcens of he saonary regressors are allowed o change, he esmaes of he break daes are asympocally ndependen as n he saonary case analyzed by Ba and Perron [Ba, J., Perron, P., 998. Esmang and esng lnear models wh mulple srucural changes. Economerca 66, 47 78]. We also show ha our resuls reman vald, under very weak condons, when he poenal endogeney of he non-saonary regressors s accouned for va an ncreasng sequence of leads and lags of her frs-dfferences as addonal regressors. Smulaon evdence s presened o assess he adequacy of he asympoc approxmaons n fne samples. 8 Elsever B.V. All rghs reserved.. Inroducon Issues relaed o srucural change have receved consderable aenon n he sascs and economercs leraure (see Perron (6, for a survey. In he las ffeen years or so, subsanal advances have been made n he economercs leraure o cover models a a level of generaly ha allows a hos of neresng praccal applcaons n he conex of unknown change pons. hese nclude models wh general saonary regressors and errors ha can exhb emporal dependence and heeroskedascy. Andrews (993 and Andrews and Ploberger (994 provde a comprehensve reamen of he problem of esng for srucural change assumng ha he change pon s unknown. Ba (997 sudes he leas squares esmaon of a sngle change pon n regressons nvolvng saonary and/or rendng regressors. He derves he conssency, rae of Perron acknowledges fnancal suppor for hs work from he Naonal Scence Foundaon under Gran SES-64935. We are graeful o wo referees for useful commens. Correspondng auhor. el.: + 67 353 4389. E-mal addresses: mkerwa@purdue.edu (M. Kerwal, perron@bu.edu (P. Perron. convergence and he lmng dsrbuon of he change pon esmaor under general condons on he regressors and he errors. Ba and Perron (998 exend he esng and esmaon analyss o he case of mulple srucural changes, whle Ba and Perron (3 presen an effcen algorhm o oban he break daes correspondng o he global mnmzers of he sum of squared resduals. Perron and Qu (6 consder he case n whch resrcons whn or across regmes are mposed. Qu and Perron (7 cover he more general case of mulple srucural changes n a sysem of equaons allowng arbrary resrcons on he parameers. When dealng wh non-saonary varables, he leraure s less exensve. Wh respec o esng, Hansen (99b develops ess of he null hypohess of no change n models where all coeffcens are allowed o change. An exenson o paral changes has been analyzed by Kuo (998. he ess consdered are he Sup and Mean LM ess dreced agans an alernave of a one me change n parameers. Hao (996 also suggess he use of he exponenal LM es. Seo (998 consders he Sup, Mean and Exp versons of he LM es whn a conegraed VAR seup. he Sup and Mean LM ess n hs seup are shown o have a smlar asympoc dsrbuon as he Sup and Mean LM ess of Hansen (99b. Kerwal and Perron (8 show ha such ess can suffer 34-476/$ see fron maer 8 Elsever B.V. All rghs reserved. do:.6/.econom.8.7.
6 M. Kerwal, P. Perron / Journal of Economercs 46 (8 59 73 from mporan lack of power n fne samples and be subec o a non-monoonc power funcon such ha he power decreases as he magnude of he break ncreases. hey sugges modfed Sup Wald ype ess ha perform consderably beer. Wh respec o esmaon, Perron and Zhu (5 analyze he properes of parameer esmaes n models where he rend funcon exhbs a slope change a an unknown dae and he errors can be eher saonary or have a un roo. Wh negraed varables, he case of mos neres s ha of a framework n whch he varables are conegraed. Accounng for parameer shfs s crucal n conegraon analyss snce normally nvolves long spans of daa whch are more lkely o be affeced by srucural breaks. he goal s hen o es wheher he conegrang relaonshp has changed and o esmae he break daes and form confdence nervals for hem. In hs respec, an mporan paper s ha of Ba e al. (998 who consder a sngle break n a mulequaons sysem and show he esmaes obaned by maxmzng he lkelhood funcon o be conssen. hey also oban a lm dsrbuon of he esmae of he break dae under a shrnkng shf scenaro assumng ha he coeffcens assocaed wh he rend and he non-saonary regressors shrnk faser han hose peranng o he saonary regressors. he am of hs paper s o provde a comprehensve reamen of ssues relaed o esmaon and nference wh mulple srucural changes, occurrng a unknown daes, n conegraed regresson models. Our work bulds on ha of Ba and Perron (998 who underake a smlar reamen n a saonary framework. Our framework s general enough o allow boh saonary and non-saonary varables n he regresson. he assumpons regardng he dsrbuon of he error processes are mld enough o allow for general forms of seral correlaon and condonal heeroskedascy, as well as mld forms of uncondonal heeroskedascy. Moreover, we analyze boh pure and paral srucural change models. A paral change model s useful n allowng poenal savngs n he number of degrees of freedom, an ssue parcularly relevan for mulple changes. I s also mporan n emprcal work snce helps o solae he varables whch are responsble for he falure of he null hypohess. he parameer esmaes of he regresson coeffcens and he break daes are obaned by mnmzng he sum of squared resduals. We derve he conssency, rae of convergence and lmng dsrbuon of he esmaed break fracons under much weaker condons han hose n Ba e al. (998. We show ha f he coeffcens of he negraed regressors are allowed o change, he esmaed break fracons are asympocally dependen so ha confdence nervals need o be consruced only. Mehods o consruc such confdence nervals are dscussed. If, however, only he nercep and/or he coeffcens of he saonary regressors are allowed o change, he esmaes of he break daes are asympocally ndependen as n he saonary framework analyzed by Ba and Perron (998. hough our heorecal resuls hold under much weaker condons han hose of Ba e al. (998 and allow for mulple breaks, our analyss s resrced o a sngle conegrang vecor unlke hers whch s vald n a mulequaons sysem whch, hereby allows mulple conegrang vecors. In he mulple break case, he fac ha he esmaed break fracons are asympocally dependen complcaes he analyss consderably and he exenson o a mul-equaons sysem s ousde he scope of hs paper. hs arcle s organzed as follows. Secon presens he model and assumpons. In Secon 3, we derve he conssency, rae of convergence and lmng dsrbuon of he esmaes of he break daes. Secon 4 presens he resuls of smulaon expermens o assess he adequacy of he asympoc approxmaons n fne samples. Secon 5 offers concludng remarks and all echncal dervaons are ncluded n a mahemacal Appendx.. he model and assumpons Consder he followng lnear regresson model wh m breaks (m + regmes: y c + z f δ f + z b δ b + x f β f + x b β b + u ( +,..., ( for,..., m +, where, m+ and s he sample sze. In hs model, x f (p f and x b (p b are vecors of I( varables whle z f (q f and z b (q b are vecors of I( varables defned by z f z f, + u f z z b z b, + u b z x f µ f + u f x x b µ b + u b x where z f and z b are assumed, for smplcy, o be eher O p ( random varables or fxed fne consans. For ease of reference, he subscrp b on he error erm sands for break and he subscrp f sands for fxed (across regmes. By labelng he regressors x f and x b as I(, we mean ha he paral sums of he assocaed nose componens sasfy a funconal cenral lm heorem. he condons mposed are dscussed below. We hen label a varable as I( f s he accumulaon of an I( process. he break pons (,..., m are reaed as unknown. hs s a paral srucural change model n whch he coeffcens of only a subse of he regressors are subec o change whle he remanng coeffcens are effecvely esmaed usng he enre sample. When p f q f, a pure srucural change model s obaned where all coeffcens are allowed o change across regmes. We can express ( n marx form as: Y Gα + Wγ + U where Y (y,..., y, G (Z f, X f, Z f (z f,..., z f, X f (x f,..., x f, U (u,..., u, W (w,..., w, w (z b, x b, γ (δ b, β b,..., δ b,m+, β b,m+, α (δ f, β f and W s he marx whch dagonally parons W a he m- paron (,..., m, ha s, W dag(w,..., W m+ wh W (w +,..., w for,..., m+. he daa generang process s assumed o be Y Gα + W γ + U ( where α, γ and (,..., m are he rue values of he parameers and he marx W s he one ha parons W a (,..., m. As a maer of noaon, p denoes convergence n probably, d convergence n dsrbuon and weak convergence n he space D[, ] under he Skorohod merc. Also, x (x, f x b, u x (u f x, u b x, z (z, f z b, u z (u f z, u b z, ξ (u, u f z, u b, z uf x, u b x, µ (µ, f µ b and λ {λ,..., λ m } s he vecor of break fracons defned by λ / for,..., m. We make he followng assumpons on he regressors and he elemens of he nose componen ξ. Noe ha ( assumes a parcular normalzaon of he conegrang vecor. Ng and Perron (997 sudy he normalzaon problem n a wo varable models. hey show ha he leas squares esmaor can have very poor fne sample properes when normalzed n one drecon bu can be well behaved when normalzed n he oher. hs occurs when one of he varables s a weak random walk or s nearly saonary. hey sugges o use as regressand he varable for whch he specral densy a frequency zero of he frs dfferences s smalles.
M. Kerwal, P. Perron / Journal of Economercs 46 (8 59 73 6 Assumpon A: Le f (z, f x, f z, b x, b F (f,..., f and F be he dagonal paron of F a (,..., m such ha F dag(f,..., F m+. Defne he marx D dag(( I qf, ( / I pf, ( I qb, ( / I pb. We assume ha for each,..., m +, D F F D converges o a random marx no necessarly he same for all. Assumpon A: Defne he marx D dag( / I qf, I pf, / I qb, I pb. here exss an l > such ha for all l > l, he mnmum egenvalues of A l (/l D +l f f D + and of A l (/l D f f D l are bounded away from zero (,..., m +. Assumpon A3: he marx B kl l w k w s nverble for l k q b + p b. Assumpon A4: Le ξ (u f z, u b, z uf x, u b x and p p f + p b + q f + q b. he vecor {ξ u } sasfes Assumpon A4 n Qu and Perron (7. Defne he L r -norm of a random marx X as X r ( E X r { /r for r and F σ - feld..., ξ, ξ,..., } u, u. If ξ { u s weakly saonary } whn each segmen, hen (a ξ u, F forms a srongly mxng (α-mxng sequence wh sze 4r/ (r for some r >, (b E(u and ξ < M < for some δ >, u r+δ (c Le S k, (l +l+k ξ +l+ u,,..., m +, for each e ( R n of lengh, var e, S k, ( v (k for some funcon v (k as k (wh, he usual nner produc. If ξ u s no weakly saonary whn each segmen, we assume ha (a (c holds, and n addon, ha here exss a posve defne marx Ω [ ] w,s such ha for any, s,..., p, we have, unformly n l, (( k E S k, (l ( Sk, (l s w,s C k ψ, for some C, ψ >. I s also assumed ha {ξ } sasfes he condons saed n hs assumpon. Assumpon A5: E(u x u. Assumpon A6: < λ < < λ < m wh [ ] λ. Assumpon A7: Le γ, γ γ,+,, hen γ, dag( / I qb, I pb γ v, for some γ ndependen of, where v > s a scalar sasfyng v and / v as. Assumpon A s needed for mulple lnear regressons nvolvng boh saonary and negraed regressors and smply ndcaes ha sample momens of he regressors exss when scaled as saed. Assumpon A ensures ha here s no local collneary problem so ha he break pons can be denfed. he use of he weghng marces D and D s due o he presence of boh I( and I( regressors. Assumpon A3 s a sandard nverbly requremen o have well defned esmaes. Assumpon A4 deermnes he dependence srucure of he processes ξ u and ξ. In parcular, hey mply ha ξ u and ξ are shor memory processes havng bounded fourh momens. he assumpons are mposed o oban a funconal cenral lm heorem, a generalzed Háek and Rény (955 ype nequaly and a srong law of large numbers ha allow us o show he esmaes o be conssen and o derve her rae of convergence. he condons are mld n he sense ha hey allow for subsanal condonal heeroskedascy and { auocorrelaon. Also, f no auocorrelaon s presen,.e., ξ } u and {ξ } are marngale dfference sequences wh respec o he flraon F, hen even he weak saonary assumpon can be dropped and ξ allowed o be uncondonally heeroskedasc wh bounded fourh momens. he condons for hs o hold are very general (see, e.g., Davdson (994. I can be shown o apply o a large class of lnear processes ncludng hose generaed by all saonary and nverble ARMA models. Noe ha Assumpon A4 could be replaced by oher suffcen condons ha can yeld he man ngredens saed above. Assumpon A5 specfes ha he saonary regressors are conemporaneously uncorrelaed wh he errors. hs s a sandard requremen o oban conssen esmaes. I s mporan o noe ha no such assumpon s mposed on he correlaon beween he nnovaons o he I( regressors and he errors. Hence, we allow endogenous I( regressors. Assumpon A6 mples asympocally dsnc breaks,.e. each regme conans a posve fracon of he sample even n he lm. Assumpon A7 mples a shrnkng shfs asympoc framework where he magnudes of he shfs converge o zero as he sample sze ncreases. Specfcally, we assume ha he coeffcens assocaed wh he I( varables shrnk a a faser rae han hose assocaed wh he I( varables. Noe ha Ba and Perron (998 assume all coeffcens o shrnk a he same rae snce all regressors n her framework are assumed o be saonary. Moreover, snce breaks of larger magnude are easer o denfy, conssency of he break fracons assumng a small magnude of shf mply conssency for breaks of larger magnude. Our se of assumpons s consderably weaker han hose of Ba e al. (998 who mpose he followng condons: (a he errors u are ndependen of he regressors a all leads and lags, whch precludes, among oher hngs, endogenous I( regressors, (b he nose componens are lnear processes wh..d. errors, (c some bound on he expecaon of some funcons of he squared regressors (see her Assumpons 3.3 and 3.4, (d zero mean saonary regressors. Hence, our framework allows a much wder varey of models ha are of neres n appled work. For he rae a whch he magnude of he breaks shrnk o zero, Ba e al. (998 also mpose he requremen ha / v / log( as. Our condon s slghly weaker. Esmaes of he parameers are obaned by mnmzng he global sum of squared resduals. For each m-paron (,..., m, denoed { }, he assocaed leas squares esmaes of α and γ are obaned by mnmzng SSR (,..., m m+ + [ y c z f δ f x f β f z b δ b x b β b]. (3 Le ˆα({ } and ˆγ ({ } be he resulng esmaes. Subsung hese no he obecve funcon and denong he resulng sum of squared resduals as S (,..., m, he esmaes of he break daes (ˆ,..., ˆm are such ha (ˆ,..., ˆm arg mn,..., m S (,..., m. (4 hroughou, we also mpose he followng assumpon on he se of permssble parons where ε acs as a rmmng parameer and mposes a mnmal lengh for each regme. Assumpon A8: he mnmzaon (4 s aken over all parons (,..., m (λ,..., λ m n he se Λ ε { (λ,..., λ m ; λ + λ ε, λ ε, λ m ε }. hs assumpon s no very resrcve gven ha ε can be small. he esmaes of he regresson coeffcens are hen ˆα ˆα({ˆ } Examples of such condons are dscussed by Dehlng and Phlpp (98, Alssmo and Corrad (3 and Lavelle and Moulnes (, among ohers. (5
6 M. Kerwal, P. Perron / Journal of Economercs 46 (8 59 73 and ˆγ ˆγ ({ˆ }. he esmaes of he parameers can be obaned usng he algorhm of Ba and Perron (3 wh no modfcaon snce he algorhm self s vald rrespecve of he naure of he regressors and errors gven ha s desgned o oban esmaes of he break daes ha mnmze he global sum of squared resduals n a regresson wh some or all coeffcens allowed o change across a pre-specfed number of regmes. Fnally, noe ha rends of he form (/ for,..., d, say, are allowed for he I( regressors. Exendng he analyss o I( and I( varables wh unscaled rends of he form s sraghforward for I( varables wh mnor modfcaons of he scalng marces n Assumpons A, A and A7. he resuls saed below abou he conssency and rae of convergence go hrough, hough no he resul abou he lm dsrbuon, whch would requre some modfcaons. We can allow I( regressors wh rend of he form z f ρ f + z f z b ρ b + z b by ncludng a lnear me rend as a regressor n (. Moreover, snce z b behaves asympocally lke a me rend, he rae of decrease of he shfs needs o be specfed as δ b,+ δ b, δ b, v and δ b, ρ b. Followng he argumens n Ba e al. (998, all resuls abou conssency, rae of convergence and even lm dsrbuon carry hrough. 3. Conssency and raes of convergence 3.. Conssency Le ˆλ (ˆλ,..., ˆλ m wh correspondng rue values λ (λ,..., λ m. he followng heorem saes he conssency of ˆλ for λ. heorem. Under A A8: ˆλ k λ k, k,..., m. p o prove he heorem, we need o esablsh wo lemmas. Le û y g ˆα w ˆγ k, for [ˆk +, ˆk ] and d g ( ˆα α + w ( ˆγ k ˆγ, for [ˆk +, ˆk ] [ +, ] (k,,..., m +. By he properes of proecons, û u. (6 Snce û u d, û u + d u d. (7 Now we have he followng frs lemma: Lemma. Assume A A8 hold, hen u d O p ( / v. o prove he heorem, we wll prove ha d > u d n he lm as. o do hs, we show ha d dverges a a faser rae han u d f any esmae of he break fracons s no conssen. hs gves us he desred conradcon from (6. he followng lemma saes he rae of dvergence of d n such a case. p Lemma. Assume A A8 hold and ha ˆλ λ for some ; hen ( lm sup P d > C δ b δ b+ > ɛ for some C > and ɛ >. Snce d > C δ b δ b+ > C v, d > u d f v / / v, whch follows from Assumpon A7. hs proves heorem. Remark. If he breaks occur only n he coeffcens assocaed wh he I( regressors, Lemma sll holds. Moreover, we can show ha d > C β b β b+ > C v whch also proves heorem n hs case. 3.. Raes of convergence We now show ha ˆλ k converges o λ k a rae v, whch s saed n he followng heorem. heorem. Under A A8, we have v (ˆλ k λ k O p( for k,..., m. Remark. Noe ha Ba e al. (998 assume / v / log. hus our resuls on conssency and raes of convergence are vald under weaker condons. Gven he above rae of convergence, he lmng dsrbuon of he esmaes of he regresson coeffcens s he same as ha obaned when he break daes are known. Proposon. Le θ (α, γ, F (G, W. Defne he ((m + (q b + p b (m + (q b + p b marces D m+ dag( I qb, / I pb,..., I qb, / I pb D dag( I qf, / I pf, Dm+. Assume A A8 hold, hen we have D (ˆθ θ d H κ, where H p lm ( D F F D and κ s he lmng dsrbuon of D F U. 3.3. he lmng dsrbuon of he esmaes of he break daes We now consder he lm dsrbuon of he esmaes of he break daes. We frs mpose he followng condon: Assumpon A9: Le ; for,..., m, as, unformly n s [, ], ( Q +[s ] + [ Q ff Q bf Q fb Q ] x x p sq where s a nonrandom posve defne marx. Assumpon A9 rules ou rendng regressors wh a saonary nose componen. Noe also ha we have mposed he same dsrbuon for he I( regressors across segmens, conrary o wha s cusomary n he recen leraure. Relaxng hs assumpon would be relavely sraghforward followng he same argumens as n Ba and Perron (998, bu allowng for heerogeney n he dsrbuon of he errors underlyng he I( regressors would be consderably more dffcul. Insead of havng a lm dsrbuon n erms of sandard Wener processes, we would have me-deformed Wener processes accordng o he varance profle of he errors hrough me; see, e.g., Cavalere and aylor (7. hs would lead o mporan complcaons gven ha, as shown below, he lm dsrbuon of he esmaes of he break daes depends on he whole me profle of he lm Wener processes. For hese reasons, we resrc he analyss o he case of homogeneous dsrbuons across segmens, and do so for boh he I( and I( regressors as well as he error of he regresson. Gven hs, Assumpon A4 mples he followng dsrbuonal resuls whch also ses up he noaon o be used.
M. Kerwal, P. Perron / Journal of Economercs 46 (8 59 73 63 A(v,..., v m m v m B(v,..., v m γ f v η ( (v η ( (v f v < η ( (v f v > f v W ( (v xb W ( Π γ (v xb, f v < W ( xb, f v > γ / W b z (λσ η( (v ( (v (Q xu / W ( (v xb / W b z (λ W b z (λ / / W b z (λ W b z (λ / µ b µ b W b z (λ / µ b Q / W b z (λ W b z (λ / / W b z (λ W b z (λ / µ b µ b W b z (λ / µ b Q / W b z (λ µ b γ / W b z (λ µ b γ / and Box I. he vecor ξ (u, u f z, u b, z uf x, u b x, of dmenson n q f + p f + q b + p b +, sasfes he followng mulvarae funconal cenral lm heorem: / [r] ξ B(r where B(r (B (r, B f z(r, B b z (r, B f x(r, B b x (r s a n vecor Brownan moon wh symmerc covarance marx σ Ω f z Ω b z Ω f x Ω b x Ω f z Ω ff Ω fb Ω fb zx Ω ff zx q Ω Ω b z Ω bf Ω Ω bf zx Ω f zx q b Ω f x Ω ff xz Ω fb xz Ω ff Ω fb p f p b Ω b x Ω bf xz Ω xz Ω bf Ω lm E(S S Σ + Λ + Λ where S ξ, Σ lm E(ξ ξ and Λ lm E(ξ ξ +. wh σ > and p lm u lm E[u] σ u. When he errors are saonary processes, we also have Ω πf ξ ( where f ξ ( s he specral densy funcon of {ξ } a zero frequency. / [r] (uf x, u b x u Q / xu W x (r, where W (r x (W xf (r, W xb (r s a (p f + p b vecor of ndependen Wener processes and ( Q xu E(u x u u x h h [ Q ff xu Q fb xu Q bf xu Q xu ]. We shall also mpose he followng condon: Assumpon A: For all and s: (a E(u x u z s ; (b E(u x u u s ; (c E(u x u u xs. Assumpon A resrcs somewha he class of models applcable bu s que mld. Suffcen, hough no necessary, condons for o hold are: for (a ha he I( regressors are uncorrelaed wh he errors conemporaneously even condonal on he I( varables; for (b ha he auocovarance srucure of he I( regressors be ndependen of he errors and, smlarly, for (c ha he auocovarance srucure of he errors be ndependen of he I( regressors. hs assumpon s needed o guaranee ha W x ( and B( are uncorrelaed and, beng Gaussan, are herefore ndependen. Whou hs condon, he analyss would be much more complex. Fnally, we need he followng assumpon whch rules ou conegraon among he I( regressors: Assumpon A: ( Ω Ω bf Ω fb Ω ff >. Conssen esmaes of he marces Σ and Λ (and hence Ω are ˆΣ ˆξ ˆξ and ˆΛ w(/l ˆξ ˆξ +, where ˆξ (û, z, f z, b (x f x f, (x b x b wh û he OLS resduals from regresson (, x x ( f, b and w(/l s a kernel funcon ha s connuous and even wh w( and k (xdx <. Also, l as and l o( /. Conssency of hese covarance marx esmaes has been shown n Hansen (99a. he followng proposon saes he lmng dsrbuon of he esmaes of he break daes under src exogeney of he I( regressors. heorem 3. For,..., m, le η ( (v and W ( (v xb are wo sded Wener processes ndependen of each oher. Also, η ( (v and η ( (v are ndependen, and η ( (v (, are ndependen across. Smlarly, W ( (v xb, and W ( (v xb, are ndependen, and W ( (v xb, (, are ndependen across. hen, under Assumpons A A and Ω f Ω b z z (v, Π (ˆ,..., v Π m(ˆm m arg max H(v,..., v m (v,...,v m wh H(v,..., v m B(v,..., v m A(v,..., v m where A(v,..., v m and B(v,..., v m are gven n Box I.
64 M. Kerwal, P. Perron / Journal of Economercs 46 (8 59 73 hs heorem shows how dfferen nference abou he break daes s when he coeffcen of an I( regressor s allowed o change. In hs case, he lm dsrbuon of he esmaes of he break daes are no asympocally ndependen even f he break daes are separaed by a posve fracon of he sample sze. hs can be seen by nong ha he funcon H(v,..., v m nvolves he same Wener processes (hose correspondng o he I( regressors evaluaed a v,..., v m. hs gves rse o he correlaon beween he esmaes of he break daes even n he lm. hs resul conrass wh he case of regresson models wh I( regressors, n whch case he lm dsrbuons of he esmaes of he break daes are asympocally ndependen (see Ba and Perron (998. As shown n Corollary 3 below, hs asympoc ndependence connues o hold when I( regressors are ncluded bu her coeffcens are no allowed o change. Noe ha from a compuaonal aspec, evaluang he lm dsrbuon nvolves consderng m cases correspondng o he possble combnaons of he sgns of v,..., v m. In he sngle break case, he lm dsrbuon s dfferen from ha n Ba e al. (998 because we do no assume zero mean saonary regressors. Hence, he lm dsrbuon s a funcon of µ b, whch can neverheless be conssenly esmaed. he cumulave dsrbuon funcon of he random varable argmax (v,...,v m H(v,..., v m does no have a racable analycal formula and hence needs o be obaned by smulaons. Accordngly, we frs generae a realzaon of H(v,..., v m by replacng he rue value of he parameers wh conssen esmaes and smulang he Brownan moon processes over an approprae range, say, [ M, M]. hen, we oban he global maxmum of he funcon H(v,..., v m over (v,..., v m [ M, M] [ M, M]. hs s repeaed unl we have an esmae of he on dsrbuon over an approprae range. A sandard mehod o consruc on confdence nervals s o use he so called Bonferron procedure. In hs case, we smulae he margnal dsrbuons of ˆ,..., ˆm and form ( α/m% confdence nervals for each. he on confdence nerval a sgnfcance level α s hen he nersecon of he nervals for each of he m break daes (see, e.g., Goureroux and Monfor (995, p. 8. Oher mehods of consrucng on confdence nervals are dscussed n Lehmann and Romano (5. Ofen, specal cases of he general regresson model ( are used. We classfy hem n wo caegores: (a models nvolvng only I( regressors; (b models nvolvng boh I( and I( regressors. Caegory (a, Models wh I( varables only (p f p b, for all cases:. c for all,..., m +, q f : y z b δ b + u ;. q f : y c + z b δ b + u ; 3. q b : y c + z f δ f + u ; 4. c c for all,..., m +, q f : y c + z b δ b + u ; 5. no resrcon: y c + z f δ f + z b δ b + u ; 6. c c for all,..., m + : y c + z f δ f + z b δ b + u. Caegory (b, Models wh boh I( and I( varables:. c c for all,..., m +, p f q b : y c + z f δ f + x b β b + u ;. c c for all,..., m +, p b q f : y c + z b δ b + x f β f + u ; 3. c c for all,..., m +, p f q f : y c + z b δ b + x b β b + u ; 4. p f q f : y c + z b δ b + x b β b + u ; 5. p b q b : y c + z f δ f + x f β f + u ; 6. p b q f : y c + z b δ b + x f β f + u ; 7. p f q b : y c + z f δ f + x b β b + u ; 8. q f : y c + z b δ b + x f β f + x b β b + u ; 9. q b : y c + z f δ f + x f β f + x b β b + u ; We now gve a bref descrpon of each of he models n he wo caegores. Consder frs Caegory (a. Case s a pure srucural change model whou an nercep n whch all I( coeffcens are allowed o change across regmes. Case s a pure srucural change model whch allows for a change n he nercep as well. Case 3 s a paral change model n whch only he nercep s allowed o change. Case 4 s agan a paral change model where he nercep s no allowed o change. Cases 5 and 6 are block paral models n whch a subse of he I( coeffcens s allowed o change. In Caegory (b, Cases o 3 are paral change models where he nercep s no allowed o change across regmes. Case 4 s a pure change model where all I( and I( coeffcens as well as he nercep s allowed o change. Case 5 s a paral change model, whch nvolves only an nercep shf. Case 6 s a paral change model where he I( coeffcens are no allowed o change. Smlarly, Case 7 s a paral change model where he I( coeffcens are no allowed o change. Cases 8 9 are block paral models n whch a subse of coeffcens of a leas one ype of regressor s no allowed o change. he lm dsrbuon saed n heorem 3 smplfes accordng o parcular subgroups of hese specal cases, as saed n he followng Corollares. Corollary. For Cases (, (4 and (6 n Caegory (a and Case ( n Caegory (b we have Π γ / W b z (λ W b z (λ / γ B(v,..., v m [ m γ / W b z (λσ η( (v { } γ / Wz b(λ W z b(λ / / γ he cases covered by hs Corollary are hose for whch he consan s held fxed (or consraned o be zero and only coeffcens on I( regressors are allowed o change. Corollary. For Cases ( and (5 n Caegory (a and Case (6 n Caegory (b, we have ( Π γ / W b z (λ W b z (λ / / W b z (λ γ W b z (λ B(v,..., v m m γ γ / / W b z (λσ η( (v ( (v / W b z (λ W b z (λ W b z (λ / / ]. / W b z (λ γ he cases covered by hs corollary are hose for whch he consan s allowed o change and none of he coeffcens on he I( regressors s allowed o change. he combnaon of Corollares and show ha he lm dsrbuon of he break daes s dfferen wheher he consan s allowed o change or no, when coeffcens on I( regressors change. hs s n conras o he saonary case where havng a pure or paral srucural change model mples he same lm dsrbuon. Corollary 3. For Case (3 n Caegory (a and Cases, 5 and 9 n Caegory (b, we have Π γ γ B(v,..., v m [ ] m γ σ η( (v { } γ /. γ /.
( Π γ / W b z (λ W b z (λ / µ b W b z (λ / B(v,..., v m m M. Kerwal, P. Perron / Journal of Economercs 46 (8 59 73 65 / W b z (λ µ b γ Q / W b z (λσ η( (v (Q xu / W ( (v xb ( γ { ( γ / W b z (λ W b z (λ / µ b W b z (λ / Box II. / W b z (λ µ b Q } / γ he cases covered by hs Corollary are hose for whch no coeffcen assocaed wh I( regressors are allowed o change. Noe ha n hs case, he lms do no nvolve he Wener processes correspondng o he I( regressors. hs s because he coeffcens of he I( regressors are no allowed o change and, hence, hey do no maer asympocally as far as he lm dsrbuon of he break daes are concerned. Here, he esmaes of he break daes are herefore asympocally ndependen and we have, for,..., m, γ σ v (ˆ arg max v where η ( (v { η ( η ( (v f v < (v f v > { η ( (v v } whch reduces o he lm dsrbuon saed n Ba and Perron (998 for he saonary case. Corollary 4. For Case 3 n Caegory (b, we have n Box II. Here coeffcens on he I( and I( regressors are allowed o change bu he consan s no. he followng Corollary shows how allowng he consan o vary or no affecs he lm dsrbuon when boh I( and I( regressors are allowed o change. Corollary 5. he lm dsrbuon for Cases 4 and 8 n Caegory (b s he same as ha for he general case gven by regresson (. Here all coeffcens assocaed wh I( varables are allowed o change as well as he consan and he coeffcens on some I( regressors. When combned wh he oher resuls, hs pons o he fac ha, when a leas one coeffcen assocaed wh an I( regressor s allowed o change, wha nfluences he lm dsrbuon are: ( he number of I( regressors whose coeffcens are allowed o change; ( he number of I( regressors whose coeffcens are allowed o change, and (3 wheher he consan s allowed o change. Of parcular neres s he fac ha he lm dsrbuon obaned holdng a subse of he coeffcens fxed s he same ha prevals when no ncludng hese regressors. hs s dfferen from wha occurs when dong hypohess esng abou wheher breaks are presen or no (see, Kerwal and Perron (8. In hs case, ncludng regressors whose coeffcens are no allowed o change affecs he lm dsrbuon of he ess. 3.4. Exenson o he dynamc OLS regresson model o deal wh he possbly of endogenous I( regressors, a popular mehod s o use he so-called dynamc OLS regresson where leads and lags of he frs-dfferences of he I( varables are added as regressors, as suggesed by Sakkonen (99 and Sock and Wason (993. he regresson model s hen y ĉ + z f ˆδ f + x f ˆβ f + z b ˆδ b + x b ˆβ b + k k z ˆΠ + ˆv ( +,..., (8 for,..., m +, where v v + >k ζ z, Π v + e. Noe ha snce hese addonal regressors are nroduced o modfy he lm dsrbuon of he esmaes of he man parameers of he models, he coeffcens assocaed wh he leads and lags of he frs-dfferenced I( varables are held fxed across regmes. 3 I s well known ha o oban esmaes ha are asympocally opmal and es sascs ha have he usual ch-square lm dsrbuon, k mus ncrease a some rae as ncreases. hs makes he problem dfferen from ha dscussed so far, snce n he prevous secons he number of regressors s held fxed as he sample sze ncreases. he am of hs secon s o show ha, under some condons, he resuls dscussed so far reman vald n hs conex. o esablsh hs, we need he followng assumpon on he rae of ncrease of k. Assumpon A: As, k, k /, k Π >k, and k / v. Excep for he las condon, hs assumpon s he same as used n Kerwal and Perron (n press who showed ha he resuls saed n Sakkonen (99 under more resrcve condons reman vald. For nsance, f he Π are evenually exponenally decayng (as n he case of an ARMA process, he lower bound condon k Π >k perms a logarhmc rae of ncrease for k so ha a daa dependen rule based on an nformaon creron can be used o selec he number of leads and lags. Gven he fac ha he magnude of he breaks s decreasng as ncreases, here hngs are more complex and here s a need for an addonal lower bound condon gven by he requremen ha k / v. he relevan resul s saed n he followng heorem. heorem 4. Suppose ha he break daes are obaned by mnmzng he sum of squared resduals from regresson (8, hen under Assumpons A A, heorems 3 reman vald. Remark 3. Ba e al. (998 assume ha / v / log. Suppose ha k s seleced usng daa dependen rules based on nformaon crera such as he AIC or BIC so ha k O(log. hen we need (log / v. We can show ha he laer condon mples / v / log, bu no vce-versa. Hence, her assumpon s no suffcen o guaranee ha he same lm dsrbuon apples when addng an ncreasng sequence of leads and leads of he frs-dfferenced I( regressors. 4. Smulaon expermens In hs secon, we conduc smulaon expermens o assess he adequacy of he asympoc dsrbuon as an approxmaon o he fne sample dsrbuons. We nvesgae he fne sample coverage rae of asympoc confdence nervals based on he 3 Noe ha he number of leads and lags of z need no be he same. We specfy he same value for smplcy. Alernavely, one can nerpre k as he maxmum of he number of leads and lags.
66 M. Kerwal, P. Perron / Journal of Economercs 46 (8 59 73 able Coverage raes for he sngle break case Model Values S S S3 S4 4 4 4 4 8% 9% 8% 9% 8% 9% 8% 9% 8% 9% 8% 9% 8% 9% 8% 9% c, δ, λ.5.9.94.95.97.87.9.9.9.9.93.95.97.93.94.95.96 c, δ.5, λ.5.8.87.9.93.78.8.85.88.79.86.88.9.86.89.9.9 A c, δ, λ.5.87.9.9.93.83.86.88.9.83.86.9.93.9.9.9.9 c, δ.5, λ.5.78.84.85.9.77.83.8.85.75.83.85.9.85.88.88.9 c, δ, λ.75.9.94.96.97.89.9.93.94.9.95.96.97.95.96.96.97 c, δ.5, λ.75.85.89.9.95.8.87.87.89.83.87.9.94.87.9.94.95 c.5, λ.5.86.9.87.93.9.97.9.96.8.86.86.9.8.86.85.9 c, λ.5.76.84.77.86.88.93.89.96.69.76.75.83.67.77.76.83 B c.5, λ.5.83.88.86.9.9.96.89.96.79.85.84.9.79.85.86.9 c, λ.5.74.83.8.87.84.9.89.95.66.75.77.84.68.76.77.84 c.5, λ.75.8.87.85.9.88.95.89.95.7.79.83.89.73.79.83.88 c, λ.75.7.78.75.83.84.9.87.95.58.67.7.8.59.67.7.79 δ, λ.5.9.94.95.96.88.9.9.93.9.94.94.96.93.94.95.96 δ.5, λ.5.84.88.87.9.78.84.83.86.8.86.89.9.87.89.9.93 C δ, λ.5.88.9.93.95.8.85.88.9.86.9.9.95.9.9.9.93 δ.5, λ.5.79.85.86.9.75.83.79.84.77.83.84.9.8.84.88.9 δ, λ.75.93.95.96.97.9.9.9.94.9.94.96.97.94.95.96.97 δ.5, λ.75.86.9.9.95.8.86.87.9.85.89.9.94.89.9.9.94 c, δ, λ.5.9.94.94.95.88.9.9.94.9.93.94.96.93.94.96.97 c, δ.5, λ.5.8.86.9.93.79.83.84.88.8.87.87.9.87.9.9.93 D c, δ, λ.5.88.9.9.94.85.88.87.9.85.89.9.93.9.93.94.96 c, δ.5, λ.5.76.83.85.9.7.77.78.83.73.8.83.88.85.88.87.9 c, δ, λ.75.93.95.96.97.88.9.93.94.93.95.94.96.94.95.96.97 c, δ.5, λ.75.86.89.9.93.78.84.87.9.83.88.9.93.88.9.9.93 lmng dsrbuon for cases where he Daa Generang Process (DGP nvolves one and wo breaks. Our choce of daa generang processes enables comparsons wh he resuls obaned by Ba e al. (998 for he sngle break case. he number of replcaons used s. he level of rmmng s se a ɛ.5. he sample szes used are and 4. hroughou, we consder a sngle I( regressor z generaed by: z z + η and an I( regressor x..d. N(,. Here η..d. N(, and ndependen of x. Also, le e..d. N(, where Cov(η, e Cov(x, e. 4.. he case wh one break We consder four models: he frs nvolvng shfs n boh he nercep and he conegrang coeffcen, he second nvolvng a shf n he nercep only, he hrd nvolvng only a shf n he conegrang coeffcen and he fourh whch s he same as he frs excep ha x s also ncluded n he regresson bu s coeffcen s no allowed o change. he models consdered are he followng: Model-A (Change n nercep and slope: y c + δ z + u f [ λ ] y c + δ z + u f > [ λ ]. Model-B (Change n nercep only: y c + z + u f [ λ ] y c + z + u f > [ λ ]. Model-C (Change n slope only: y + δ z + u f [ λ ] y + δ z + u f > [ λ ]. Model-D: Same as Model-A excep ha he (rrelevan I( regressor x, whose coeffcen s no allowed o change, s also ncluded. For each model, we consder he followng four specfcaons for u : S (..d. errors, exogenous regressor: u e ; S (MA( errors, exogenous regressor: u e.5e ; S3 (..d. errors, endogenous regressor: u.5η + e ; S4 (MA( errors, endogenous regressor: u.5v + η, v e.5e. For S, we correc for seral correlaon by consrucng he longrun varance esmaor based on he Quadrac Specral kernel and an AR( approxmaon for he bandwdh. For S3, we use he dynamc OLS esmaor dscussed n Secon 3.4 wh k. Fnally, for S4, we use he correcon for seral correlaon as n S and he endogeney correcon as n S3. We se c and δ. able presens fne sample coverage raes of 8% and 9% asympoc confdence nervals. For model A, when he magnude of he break n he slope s large, he confdence nervals are conservave, rrespecve of he sample sze. For a smaller change n slope, however, he coverage raes are usually lower for bu become conservave when he sample sze s doubled. A smlar pcure apples for model B excep ha here he coverage raes are usually lower han he correspondng asympoc confdence levels when he break s small and occurs a he begnnng or he end of he sample. For model C, he coverage raes are adequae for small breaks and. Wh larger breaks or wh 4, hey are somewha conservave. he resuls for model D are qualavely smlar o hose for model A. he smulaons presened n Ba e al. (998 show ha he confdence nervals are oo gh for mos of he cases consdered. hs s no he case n our smulaons. Indeed, our resuls show he asympoc approxmaon o be more accurae han repored n Ba e al. (998. Overall, he coverage raes are reasonably accurae or somewha conservave. 4.. he case wh wo breaks Wh wo breaks he models consdered are he followng: Model-E (Change n Inercep and Slope y c + δ z + u f [λ ]
M. Kerwal, P. Perron / Journal of Economercs 46 (8 59 73 67 able Coverage raes for he wo breaks case (λ /3, λ /3 Model Values S S S3 S4 4 4 4 4 8% 9% 8% 9% 8% 9% 8% 9% 8% 9% 8% 9% 8% 9% 8% 9% c.5, c 3, δ.5, δ 3.7.75.8.83.78.79.85.87.67.69.79.8.7.7.8.8 E c, c 3 3, δ, δ 3 3.89.9.9.94.9.9.9.93.84.86.9.93.87.88.9.9 c, c 3 3, δ.5, δ 3.74.77.8.84.77.79.83.84.66.7.79.8.7.74.79.8 F δ.5, δ 3.79.8.83.85.8.84.86.88.79.8.86.88.89.9.93.94 δ, δ 3 3.9.9.93.94.9.9.93.94.9.9.9.94.95.96.97.98 c.5, c 3, δ.5, δ 3.73.76.83.85.8.8.85.86.69.73.8.83.63.67.8.8 G c, c 3 3, δ, δ 3 3.89.9.9.93.9.93.93.94.86.88.9.9.83.86.93.94 c, c 3 3, δ.5, δ 3.7.74.8.84.79.8.86.88.7.7.79.8.65.68.8.84 able 3 Coverage raes for he wo breaks case (λ /4, λ 3/4 Model Values S S S3 S4 4 4 4 4 8% 9% 8% 9% 8% 9% 8% 9% 8% 9% 8% 9% 8% 9% 8% 9% c.5, c 3, δ.5, δ 3.7.76.8.84.76.78.84.86.65.69.77.8.64.67.8.83 E c, c 3 3, δ, δ 3 3.88.9.9.93.87.88.93.94.86.87.9.9.85.87.9.9 c, c 3 3, δ.5, δ 3.76.78.8.85.78.8.86.87.69.7.78.8.67.7.8.83 F δ.5, δ 3.76.79.84.86.77.8.87.88.77.8.86.87.88.89.93.94 δ, δ 3 3.9.9.93.94.89.9.93.94.89.9.93.95.95.96.97.98 c.5, c 3, δ.5, δ 3.7.75.83.86.76.79.8.85.69.7.8.83.67.7.8.8 G c, c 3 3, δ, δ 3 3.9.9.93.94.89.9.93.94.86.88.9.93.85.87.9.9 c, c 3 3, δ.5, δ 3.74.77.83.85.76.78.84.86.68.7.8.84.68.7.79.8 y c + δ z + u f [λ ] < [λ ] y c 3 + δ 3 z + u f [λ ] <. Model-F (Change n Slope only y + δ z + u f [λ ] y + δ z + u f [λ ] < [λ ] y + δ 3 z + u f [λ ] <. Model-G: Same as Model-E excep ha he (rrelevan I( regressor x, whose coeffcen s no allowed o change, s also ncluded. Agan, we se c and δ. As n he one break case, we consder he error specfcaons S S4 wh he correspondng correcons for seral correlaon and/or endogeney. he confdence nervals are consruced only usng he Bonferron procedure dscussed n Secon 3. he coverage raes are presened n ables and 3. Consder frs Model E. When he change n slope s small, he coverage raes are nadequae; however, he confdence nervals become conservave as he magnude of he change ncreases. An neresng feaure s ha he coverage raes reman almos unaffeced when he magnude of he nercep change ncreases bu he magnude of he slope change remans he same. For Model F, he confdence nervals are agan conservave provded he magnude of he breaks s large. Agan, he resuls for model G are smlar o hose for model E. For all models, he coverage raes ncrease when he sample sze ncreases. Noe ha, gven he use of an asympoc framework wh shrnkng shfs, he accuracy of he approxmaons need no mprove as he sample sze ncreases when he magnudes of he breaks are fxed. Overall, he coverage raes are reasonably accurae and, as expeced, somewha conservave wh larger sample szes. Hence, from hese lmed expermens, we can conclude ha he lmng dsrbuons derved provde useful approxmaons n fne samples. 5. Concluson hs paper has presened a comprehensve reamen of ssues relaed o esmaon and nference n conegraed regresson models wh mulple srucural breaks. We analyzed models wh I( varables only as well as models whch ncorporae boh I( and I( regressors. he breaks are allowed o occur n he nercep, he conegrang coeffcens, he parameers of he I( regressors or any combnaon of hese. he resuls show ha confdence nervals for he break daes need o be consruced only whenever he coeffcens assocaed wh he I( regressors are allowed o change even f he break daes are separaed by a posve fracon of he sample sze. A comparson of varous mehods o consruc such confdence nervals n erms of her performance n fne samples s an mporan avenue for fuure research. Appendx We use. o denoe he Eucldean norm,.e. x ( p x / for xɛr p. For a marx A, we use he vecor-nduced norm,.e. A sup x Ax / x. We have A [ r(a A ] /. Also, for a proecon marx P, PA A. W, W f z, W b z, W f x, W b x are ndependen Wener processes wh dmensons correspondng o hose of B, B f z, B b z, Bf x, B b x. We also defne he marces D dag( I qf, / I pf, D dag( I qb, / I pb and D 3 dag( / I qb, I pb. Henceforh, we wll refer o Ba and Perron (998 as [BP]. We frs sae a seres of lemmas whch wll be used subsequenly. Lemma A. (Qu and Perron, 7. Le (η be a sequence of mean zero R d -valued random vecors. Defne F k as an ncreasng σ -feld generaed by (η k. Suppose (η sasfes Assumpon A4. We have (a (Generalzed Haek Reny nequaly here exss an L <
68 M. Kerwal, P. Perron / Journal of Economercs 46 (8 59 73 such ha, for every c > and k >, P(sup k k k k η > c (L/c k ; (b (FCL / [r] η ΩW (r where W (r s a d-vecor of ndependen Wener processes and denoes weak convergence under he Skorohod opology; (c (SLLN k k η a.s. as k. he followng Lemma s a drec consequence of Lemma A.(b appled o ξ. Lemma A.. Under A4, we have unformly over all < r < s < : (a [s] [r] ξ O p ( /, (b [s] [r] z O p ( 3/, (c [s] [r] z z O p(, (d [s] [r] z ξ O p(. Lemma A.3. Under A, sup,..., m (D G M W GD O p (, where he supremum s aken over all possble parons such ha q b + p b (,..., m +. Proof. We have he deny G M W G G M W G + + G m+ M W m+ G m+. Each paron leaves a leas one rue regme nac. ha s, here exss an such ha (G, W conans (G, W as a sub-marx. Snce G M W G G (usng Lemma A. of M W G [BP], we have (D G M W GD (D G mples (D G M W GD max (D G M W M W all parons. he resul follows from Assumpon A. Lemma A.4. Under A, sup,..., m D G M W W O p (. G D. hs G D for Proof. Snce M Z s a proecon marx, we have D G M W W D G W unformly over all parons. he resul hen follows from he facs ha D G O p ( and W O p (, from Lemma A.. Lemma A.5. Under A4 and A8, sup,..., m P W U O p (. Proof. We shall prove U P W U O p ( unformly n,..., m. Noe ha U P W U s he summaon of m + erms ( + w + u ( + w + w ( + w + u, for,..., m. From Lemma A., D ( + w + w D + O p (. Also, D w + u O p (. Accordngly, U P W U O p ( unformly n,..., m. Lemma A.6. Under A A4, (a sup,..., m D G P W U O p (; (b sup,..., m W P W U O p (. Proof. hs follows from Lemma A.4, D G O p ( and D G P W U D G P W U. Smlar argumens can be used o prove par (b. Proof of Lemma. We have u d U G( ˆα α +U W ˆγ U W γ where W s he dagonal paron of W a some arbrary paron (,..., m. We have D ( ˆα({ } α (D G M W GD D G M W W γ + (D G M W GD D G M W U. (A. Now D G M W U D G U D G P W U. Snce D G U O p ( and D G P W U D G P W U O p ( so ha he second erm of (A. s O p (. From γ γ,+, D 3 O(v under Assumpon A7, we have γ γ,, D 3 O(v for all and. As n [BP], we use he fac ha ( W Wγ depends on changes n he parameers,.e., γ γ. hs mples ha D G M W ( W Wγ s a mos O p ( / v. By Lemma A.3, (D G M W GD D G M W ( W Wγ O p ( / v whch mples D ( ˆα({ } α O p ( / v + O p (. hus, U G( ˆα({ } α O p ( / v + O p ( over all parons. Nex, we have U W ˆγ ({ } U W γ U W( W MG W W MG W γ (A. + U W( W MG W W MG U U W γ. (A.3 Combne he frs and hrd erms of (A.3 and rewre hem as U W( W MG W W MG ( W Wγ + U ( W W γ (A.4 whch can be shown o be O p ( / v. Usng Lemmas A.3, A.5 and A.6 he second erm of (A.3 can be shown o be O p (. hus, U W ˆγ ({ } U W γ O p ( / v + O p (. (A.5 hus, from (A. and (A.5, we ge u d O p ( / v. Proof of Lemma. Followng [BP], we have d d + d ( g ˆα α g g w ( ˆγ k γ w g w w ( g ˆα α g g w ( + ˆγ k γ + w g w w ˆα α ˆγ k γ ˆα α ˆγ k γ + where exends over he se (λ η λ and exends over he se λ + (λ + η and η >. hen, wh D 4 dag((η I qf, (η / I pf, (η I qb, (η / I pb, d ( ˆα α γ k γ D ( ˆα α + ˆγ k γ D + where g g A D 4 w g g g A D 4 w g 4 A D 4 4 A D 4 ( ˆα α g w w w D 4 g w w w D 4. γ k γ ( ˆα α ˆγ k γ + + o p ( (A.6 Le ρ and ρ be he smalles egenvalue of A and A. hen, d + [ d ρ ( η ˆδ f δ f + (η ˆβ f β f + ( η ˆδ bk δ b + (η ˆβ bk β b ] + ρ [( η ˆδ f δ f + (η ˆβ f β f + ( η ˆδ bk δ b+ + (η ˆβ bk β b+ ]
η { mn ρ, ρ } ( ˆδ bk δ b + ˆδ bk δ b+ (/ η { mn ρ, ρ } δ b δ b+. By Assumpon A, ρ and ρ are bounded away from zero. Hence, d > C δ b δ b+ for some C > wh probably no less han ɛ >. Proof of heorem. he basc dea of he proof s same as ha of [BP]. We analyze he case wh hree breaks (m 3. Whou loss of generaly, we wll consder he case <. For each C >, defne V ɛ (C { (,, 3 ; < ɛ, 3, < C/v }. We nvesgae he behavor of he sum of squared resduals S (,, 3 on he se V ɛ (C. o prove heorem, s enough o show ha for each η, here exss C > and ɛ > such ha for large, P(mn{[S (,, 3 S (,, 3]/( } < η where he mnmum s aken over he se V ɛ (C. hs would mply ha wh large probably, ˆ C/v. We use he followng addonal noaon: ˆγ esmae of γ assocaed wh he regressor (,...,, w +,..., w,,..., ; ˆγ esmae of γ assocaed wh he regressor W (,...,, w +,..., w M. Kerwal, P. Perron / Journal of Economercs 46 (8 59 73 69,,..., ; ˆγ 3 esmae of γ 3 assocaed wh he regressor (,...,, w +,..., w 3,,..., ; SSR S (,, 3, SSR S (,, 3, and F (G, W. hen, (SSR SSR /( ( ˆγ ˆγ 3 [ ] W W ( ˆγ ˆγ 3 ( ( ˆγ 3 ˆγ [ W F ] [ F F] [ F W ] ( ˆγ 3 ˆγ ( ( ˆγ ˆγ [ W W / ] ( ˆγ ˆγ (. We have, unformly on he se V ɛ (C, z b z b Op (, + z b x b Op ( /, + x b x b Op (, + D ( ˆγ γ ɛo p ( / v and D ( ˆγ γ (W W W U + ɛo p( / v. Usng hese resuls, we can show ha (SSR SSR /( Av (W U (W W (W U ( ɛo p (v Av (q q + p +p / W l u l + W l + ɛo p (v where A s a posve consan and W l s he l-h componen of W. For every η >, we can choose a small ɛ > such ha P(ɛO p (v > Av/ < η. hen P(mn {(SSR SSR /( } s bounded by (q + p η + P max W l + q +p l > Av /. + W l u / o prove ha he laer probably s less han η, s suffcen o show ha for each l, / P max W l u < Cv W l + + > Av / < η. If W l s an I(, we have + W l ( mn W [ +, ] l ( O p (. hen (A.7 s bounded by P max < Cv O p ( + / W l u (A.7 > Av /. We can choose a B < such ha P(O p ( > B < η so ha hs probably s bounded by η + P max < Cv / W l u + > (A/B/ v / W η + P max l < Cv u + + / (W l W l u + > (A/B/ v η + P max O p ( < Cv u + > (A/8B / v + P max / < Cv (W l W l u > (A/8B/ v. +
7 M. Kerwal, P. Perron / Journal of Economercs 46 (8 59 73 Now, we can choose a B < such ha P( O p ( > B < η so ha hs probably s bounded by η + P max < Cv u + > (A/8BB / v + P max / < Cv (W l W l u + > (A/8B / v. he frs probably s bounded by η usng Lemma A.(a. Consder now he second one. We have P max / < Cv (W l W l u + > (A/8B/ v P max / v < Cv (W l W l u > (A/8B/. (A.8 Now, max + < Cv + (W l W l u O p( snce + (W l W l u (W l+ W l u + W u +, say, where W s an I( process. hen usng he fac ha / v, max / v (W l W l u o p(. < Cv + Gven ha A and B are consans, he probably (A.8 s also bounded by η, whch shows ha (A.7 holds when W l s an I( varable. If W l s an I( varable, hen we have / P max W l u < Cv P W l + max < Cv ( + > Av / ( W l + P max O p ( < Cv ( + > Av / + W l u / W l u > (A/ / v. We can choose B < such ha P(O p ( > B < η so ha he las probably s bounded by B η + P max < Cv ( W l u > (A/ / v + < η. he las nequaly follows applyng Lemma A.(a. hs complees he proof. Proof of Proposon. Le W be he marx W evaluaed a he esmaed break pons (ˆ,..., ˆm. he rue model can be wren as Y Gα + W γ + U, where U U + ( W W γ. hus, we have ( D (ˆθ θ D G GD D G W Dm+ D m+ W GD Dm+ W W Dm+ ( D G U + D G ( W W γ D m+ W U + Dm+ W ( W W. γ (A.9 We need o show ha he lm of he rgh hand sde of (A.9 s he same as he lm when W s replaced by W. Suppose < for all,..., m. Consder he erm Dm+ W ( W W γ. hs nvolves erms lke 3/ v z b z, b / v x b x b ec. We have 3/ v / v z b z b x b x b / v / v v o p ( O p ( o p ( v +sv +sv o p ( O p ( o p (. z b z b x b x b he oher erms can be handled smlarly, and he resul follows. Proof of heorem 3. We provde a dealed proof for he case of wo breaks. he exenson o m breaks s sraghforward. We have (ˆ, ˆ arg mn (, SSR(, or { (ˆ, ˆ arg max SSR(, SSR(, }. (, Followng Perron and Zhu (5, we have SSR(, SSR(, θ ( F F (I P F ( F Fθ θ ( F F (I P F U U (P F P F U where F (G, W, F (G, W, θ (α, γ, γ (c, δ, b β b. Le + [ ] s v, + [ ] s v. We hen have he followng four possble cases: ( s <, s <, ( s <, s >, (3 s >, s >, (4 s >, s <. We gve a dealed proof for Case only, he proof for he oher cases beng smlar. Consder he erm θ ( F F ( F Fθ γ, w w γ, + γ, w w γ, + +
M. Kerwal, P. Perron / Journal of Economercs 46 (8 59 73 7 θ ( F F ( F Fθ s γ / W b z (λ W b z (λ + s γ Ã(s, s / / W b z (λ W b z (λ / µ b µ b W b z (λ / µ b Q / W b z (λ µ b γ / W b z (λ W b z (λ / / W b z (λ W b z (λ / µ b µ b W b z (λ / µ b Q Box III. / W b z (λ µ b γ γ v z b z b / v z b / v z b x b + + + / v z b ( v v x b γ. + + / v x b z b v x b v x b x b + + We consder each of he erms n he above marces. For,, we have, ( v v +s v (z b z b o p (, we have v / ; ( / v s / W b z (λ ; ( / v + + z bz b s ( / z b ( / z b z b z b. Snce he second erm s + z bz b s + z b s / z b + z bx b + x b s µ b ; (v v (λ µ b ; (v v hus, we ge he equaon n Box III. Nex, consder / W b z (λ W b z (λ + o p ( s / W b z + x bx b s Q. θ ( F F U / v z b u / v z b u + + γ v u γ v u. + + x b u v x b u v + + We have ( v + u (s ; ( / v / W b z (λ (s ; ( v Hence, follows ha θ ( F F U γ / W b z (λ (s (s (Q xu / W ( xb, (s γ / W b z (λ (s (s (Q xu / W ( xb, (s B (s, s. + x bu + z bu (Q xu / W ( xb, (s. Nex, we need o esablsh ha θ ( F F P F ( F Fθ o p ( and θ ( F F P F U o p (. We wll show he former, he laer can be shown usng smlar argumens. Defne D dag( I qb, / I pb,..., I qb, / I pb and D dag( I qf, / I pf, D. hen, θ ( F F P F ( F Fθ θ ( F F F D ( D F F D D F ( F Fθ. We wll show ha D F ( F Fθ o p (, whch ogeher wh ( D F F D O p ( gves he desred resul. Now D F ( F Fθ nvolves he followng erms: ( 3/ v + z fz b O p ( / v o p (; ( v + z fx b O p ( v o p (; ( v + z f O p ( / v o p (, v + z b O p ( / v o p (; (v v + x fz b O p ( v o p (, v + x bz b O p ( v o p (; (v v + x fx b O p ( v o p (; (v / v + x f O p ( / o p (, / v + x b O p ( / o p (; (v 3/ v + z bz b O p ( / v o p (. Hence, D F ( F Fθ o p (. Fnally, U (P F P F U U ( F F D ( D F F D D F U + U F D ( D F F D ( D F F D D F F D ( D F F D ( D F U + U F D ( D F F D D ( F F U. We have U F D O p (, U F D O p (, ( D F F D O p (, ( D F F D O p (. Also, for,, we have + z b u O p ( / v o p (, / + u O p ( / v o p (, / + x b u O p ( / v o p (. Hence, U ( F F D o p (. Smlarly, we can show ha ( D F F D D F F D o p (. Hence, U (P F P F U o p (. We hus oban SSR(, ([ ] [ ] SSR(, H, s v, s v H (s, s wh H (s, s (/Ã(s, s + B (s, s. For Case wh s <, s >, we have B (s, s γ / W b z (λ (s (s (Q xu / W ( xb, (s + γ / W b z (λ 3(s 3 (s (Q xu / W ( xb, (s and SSR(, SSR(, H, ([ s v H (s, s, ] [ ], s v