A CUSUM of squares test for cointegration using OLS regression residuals

Transcription

1 A CUSUM of squares es for coiegraio usig OLS regressio residuals Julio A. Afoso-Rodríguez Dearme of Ecoomic Saisics, ad Ecoomerics Uiversiy of La Lagua, Teerife. Caary Islads. Sai Worksho o Dyamic Models drive by he Score of Predicive Likelihoods 9-h Jauary 24. Teerife

2 Sice he work by Egle ad Grager (EG) (987), ad wih ime series daa, here has bee a growig ieres ad use i modellig, imlicaios ad reame of he coce of coiegraio whe dealig wih highly ersise daa i regressio ad VAR-ye models. Some rece ad releva exesios: Fracioal coiegraio ad ubalaced coiegraig regressios Co-summabiliy Time-varyig coiegraio Fucioal-coefficie coiegraio Threshold coiegraio ad coiegraio wih hreshold effecs A grea amou of research has bee devoed o he crucial quesio of esig for he exisece of a cerai umber of saioary (liear) combiaios of he o-saioary sysem variables, ha is, o esig for he exisece of coiegraig relaioshis, also kow as sable log-ru relaioshis. There are hree mai aroaches o esig for coiegraio: () a sysem based FIML esimaio of a VECM (Johase (989)), (2) a wo-se rocedure based o a sigle-equaio regressio (EG), ad (3) a sigle-equaio codiioal ECM (Baerjee e.al. (986)). Zivo (2) discusses various examles for which ecoomic heory ca imly a sigle coiegraig vecor ad exlais why i is reasoable o es for coiegraio i sigle equaio regressio models i such cases. I he coex of a sigle-equaio regressio model, here are maily wo yes of esig rocedures for coiegraio: (a) T-ye es saisics (DF/ADF or PP) for he ull hyohesis of o coiegraio based o a auxilary regressio for he OLS residuals, ad (b) Flucuaio-ye es saisics for esig he ull of coiegraio based o regressio residuals (OLS/FM- OLS/CCR/ DOLS). This work coribue o his las secod family of esig rocedures for he ull of coiegraio wih a relaively simle o comue ad easy o use ew flucuaio-ye es saisic.

3 A CUSUM of squares es for coiegraio usig OLS regressio residuals Coes:. The model ad basic assumios 2. OLS esimaio ad residual-based ess for coiegraio 3. A ew CUSUM-ye esig rocedure for he ull of coiegraio 4. The effecs of highly auocorrelaed regressio errors ad es cosisecy 5. Fiie samle ower resuls

4 . The model ad basic assumios The se of k+ observable iegraed of order, I(), series admis he decomosiio Y d η,, X = + = k, d k, η k, d d α,..., where d Is he uderlyig deermiisic comoe give by a olyomial red fucio, such as,,, = =, = d k, Ak, τ, ad η is he sochasic red comoe defied as η, ε, η = = η + ε, ε = η k, ε k, τ, τ (,,..., ), Uder codiios esurig a mulivariae ivariace ricile for he zero mea ε sequece [ r ] [ r ] 2 ε, C( r) (/ ) (/ ) ( ) ( ), k r γ ε = C = = =,, ( ) kk > k k r BM Γ Γ γ Γ = = ε C γ k Γ kk if here exiss a k-vecor β k such ha u ( η ) η (, ), = β k = ck η k, is saioary, he he uderlyig sochasic red comoes (ad hece he observed series) are coiegraed i he sese of Egle ad Grager (987), ad he log-ru covariace marix Γ is sigular. These resuls jusify he secificaio of he (saic ad liear) coiegraig regressio equaio model give by Y = α τ, + β kxk, + u =,..., α = α, A, β k k

5 . The model ad basic assumios Give he secificaio of he coiegraig regressio equaio model Y = α τ, + β kxk, + u =,..., wih a se of + ( ) deermiisic redig regressors ad k sochasic redig iegraed regressors, he releva asymoic resuls are based o he followig exlici assumio cocerig he behavior of he error erms. Assumio L. (Liear rocess) (a) The regressio error is give by u = α u + υ, wih α, ad (b) ξ = ( υ, ε k, ), wih ξ = D( L) e, where e = ( e,, e k, ) is a k+-variae iid rocess wih zero mea, covariace marix 2 /2 j= j j j j [ r] [ r] / 2 B ( r) (/ ) ()(/ ) O( ) ( r) υ ξ = D e + B = = ( ), = () Σ () ( ) e k r BM Ω Ω D D = = B For α < : u B r ( ) B r, B ( r) W ( r) ( r) [ r ] / 2 u ( ) ( α) ( ) υ., ( ) u u u k u uk kk k k k r = = = ω + k ( r) = ε BM Ω ω Ω B B B Ω u e m Σ > ad mh-order fiie mome, E [ e ] < for some m 2. Also, j for he ifiie order olyomial marix i he lag oeraor L, D( L) = ( d( L), D k( L)) = j= D jl, i is assumed ha D() = De( D ()) (osigular), wih coefficies saisfyig he summabiliy codiio j D <, wih a marix orm defied as D = [ Tr ( D D )]. ω ( αl) ( L) = ω = = d 2 u uk () (), ( ) C Σ ec C L ω ku Ω kk Dk ( L)

6 . The model ad basic assumios A oe o a more geeral family of coiegraig regressio models where he roosed esig rocedure is also of alicaio: Fucioal-coefficie regressio Y = α ( Z ) τ + β ( Z ) X + u =,..., Z :, k k, fucioal variable (uivariae) Cai, Li ad Park (29). Local-liear kerel esimaor s () ( s) θ( z) θ ( Z ) = θ ( z) + θ ( z)( Z z), θ ( z) = s z Xiao (29). Kerel esimaor Su, Hsiao, ad Li (2). OLS/Local cosa kerel esimaor Piarakis (22). Piecewise local liear esimaor Su, Cai ad Li (23). Local-liear kerel esimaor X s Z u (a) I()/I() I() I() (b) I() I() I() I() I() I() I() I() I() I() I() I() I() I() I() Paricular cases: () Deermiisic srucural breaks/deermiisic ime-varyig coeffs. (2) Coiegraio wih hreshold effecs (Gozalo/Piarakis (26): θ ( Z ) = θ + λ I( q d > γ) Cosise esimaio: OLS, IM-OLS (Afoso-Rodríguez (23))

7 2. OLS esimaio ad residual-based ess for coiegraio Asymoics (Ref.: Hase (992)) τ, = Γ, τ, τ, Γ, +, k τ, m = = =, k, k,, W m X A Γ Ik, k η Γ, = diag(,,..., ) k, / 2 η k, = η k, τ ( r) m m( r) = : (/ ) m m m( r) m ( r) dr >, v ( v) Y α = m m m = + ( W ) (/ ) m, m, m, u k, = = βk = = αˆ ˆ β Θˆ [ r ],,, k ( r ) B = OLS esimaor: Scaled ad weighed OLS esimaio errors: α ˆ α Γ [( α ˆ α ) + A ( β ˆ β )] = = v v,,, k, k, k k, W ˆ / 2+ v β ˆ k, β k ( β k, β k ) ( ) ( v) + m, m, m, m m ( /2) m u v= = = ku Coiegraio = (/ ) u ( s) ( s) ds ( s) db ( s) + Also : β ˆ β = Ω ω k, k kk ku ( v=/2) No coiegraio Mea ad covariace marix mixure of o-gaussia (suercosise) disribuios ( ) m( s) m ( s) ds m( s) B ( s) ds a.s. (full - raked rocess) Ω kk ω k υ Scale mixure of ormals ceered a (Phillis (989)) u

8 2. OLS esimaio ad residual-based ess for coiegraio Give he sequece of OLS residuals ˆ α ˆ α uˆ ( k) = Y τ α ˆ X β = u m ˆ ˆ = u v m Θ,,,, k, k,, k, β k, β k he basic comoe is he arial sum rocess give by Flucuaio-ye es saisics: ˆ CI ( k) U ( k) ˆ 2, = 2 2, ωˆ u. k, ( m ) = Uˆ k uˆ k O / 2, ( ) = j= j, ( ) = ( ) 3/ 2 = O ( ) Rˆ, ( k) = max Uˆ ˆ, ( k) ( / ) U, ( k) ωˆ,...,., ( ) u k m = CSˆ ˆ, ( k) = max U, ( k) ωˆ,...,., ( ) u k m = ωˆ ( m ) uder he coiegraio assumio uder o coiegraio () Quadraic-oal variaio (QvM meric) (Shi(994), Harris & Ider(994), Leyboure & McCabe(994)) (2) Maximum variaio (KS meric) (Xiao (999), Wu ad Xiao (28)) Exce uder serially ucorrelaed regressio errors ad weakly exogeeous regressors, heir limiig ull disribuios, ha deed o he deermiisic comoes ad umber of iegraed regressors, do o allow for valid iferece 2 2 u. k, u. k wih a cosise esimaor uder coiegraio of he codiioal log - ru variace : (a) lug - i esimaor based o OLS residuals, (b) based o FM- OLS/CCR/DOLS residuals Oher similar roosals: Hase (992), Jasso (25) (PO es), McCabe, Leyboure ad Shi (997), Sock (999) ω

9 3. A ew CUSUM-ye esig rocedure for he ull of coiegraio Moivaio. (From Xiao ad Lima (27), Tesig covariace saioariy ) u u u [ r ] / 2 [ r ] / 2 = 2 2, (/ ) / 2 [ r ] 2 2 j v = σ = = = ( u ) σ j= Uder saioariy (i.e., uder coiegraio), he scaled arial sum of ceered squared regressio errors admis he decomosiio: [ r] v ( u ) ( u ) [ r] [ r ] / 2 / / = σu σu = = = Uder o saioariy (i.e., uder o coiegraio), we have ha: (/ ) [( / ) (/ ) ] σ [ r ] [ r ] (/ ) (/ ) ( u/ ) O( ) 5/ 2 u = [ r ] [ r ] 2 2 v = = 2 2 u σ = (/ ) [( u/ ) (/ ) ] = = so ha he behavior uder o saioariy is domiaed by he secod comoe, reflecig he violaio of he covariace saioariy assumio iduced by he ui roo. Some oher aleraive esig rocedures based o he squared OLS residuals from a coiegraig regressio: Lu, Maekawa, ad Lee (28) (CUSUM of squares es for srucural sabiliy i ARX() wih iegraed regressors) Xiao (29)

10 3. A ew CUSUM-ye esig rocedure for he ull of coiegraio The scaled arial sum of squared ad ceered OLS residuals from he esimaio of a (liear) coiegraig regressio model: [ r] [ r] ˆ, = ˆ, σˆ, σ ˆ, = ˆ j, = = j= (/ ) v ( k) (/ ) ( u ( k) ( k)) ( k) (/ ) u ( k) = (/ ) v + [ r] 2 [ r] u [ r ] [ r] / 22v ˆ ˆ Θ k, m, m, m, m, Θ k, = = = [ r] / 22 v ˆ ( v) ( v) Θ k, m, = = u B ( r) (/ ) v B ( r), B ( r) W ( r) ( r), [ r ] u v v = ω v. k v + γ kvbk γ kv = Ω kk ω kv = k, k ( r) ε B [ r] [ r] / 2 ˆ, = + v v = = (/ ) v ( k) (/ ) v O ( ) B ( r) rb () m Uder he liear rocess assumio o he error erms, m > 4, ad uder coiegraio: (modified) CUSUM of squared OLS residuals:, = ω ( W ( r) rw ()) + γ ( B ( r) rb ()) v. k v v kv k k ˆ K ( k) = max vˆ ( k) γˆ ( m )( X ( / ) X ) ωˆ ( ), j, kv, k, k,,..., v. k, m = j = Kˆ ( k) su W ( r) rw () (suremum of a sadard Browia Bridge), v v r [,] u

11 3. A ew CUSUM-ye esig rocedure for he ull of coiegraio A oe. Scaled arial sum of squared ad ceered FM-OLS residuals (Phillis ad Hase (99)) from he esimaio of a (liear) coiegraig regressio model Fully-modified (FM)-OLS esimaor: +, +, ˆ ˆ Y + ˆ Y + + = m m m + = Y ku, ( m ) k, k, ku, m γ Z = = αˆ ˆ ( ) β [ r ] [ r ] ˆ, = σ = γ ku ε k, = = [ r] / 2 [ r] / 2 + γ ˆ ˆ ku, γ ku ε k, ε k, Σ kk ε k, ε k, Σ kk γ ku, γ ku = = [ r ] / 2 2v ˆ + [ r] Θ k, m, m, m ˆ +, m, Θ k, = = [ r] / 2 [ r] / 2 2( γ ˆ ku, γ ) ku ( ε k, u σ ku ) ( ε k, u σ ku ) = = [ r] / 2 [ r] / 2 + 2( γ ˆ ku, γ ku ) ( ε k, ε k, Σ kk ) ( ε k, ε k, Σ kk ) γ ku = = [ r] / 2 2 v ˆ + (v ) [ r] ( v) 2 Θ k, m, z m, z = = [ r] v ˆ + / 2 [ r] / Θ ˆ k, m, ε k, m, ε k, ( γ ku, γ ku ) = = (/ ) v ( k) (/ ) ( z ) z u ( ) ( ) ( ) ( )

12 3. A ew CUSUM-ye esig rocedure for he ull of coiegraio A oe. Scaled arial sum of squared ad ceered FM-OLS residuals (Phillis ad Hase (99)) from he esimaio of a (liear) coiegraig regressio model: Uder coiegraio: [ r ] [ r ] ˆ, = σ + = = [ r] 2 2 [ r] 2 2 = (/ ) ( u σu ) (/ ) ( u σu ) = = [ r] / 2 [ r] / 2 + γ ku ( ε k, ε k, Σ kk ) ( ε k, k, kk ) ku = ε Σ γ = (/ ) v ( k) (/ ) ( z ) o () [ r] / 2 [ r] 2 γ ku ( ε k, u σ ku ) = ( ε k, u σ ku ) / 2 ε σ = The limiig ull disribuio will differ from ha he scaled arial sum of OLS residuals, wih he aearace of wo addiioal addiive erms ha deed o he secod order roeries of he error drivig he iegraed regressors ad o he k-dimesioal samle covariace bewee hese ad he regressio error.

13 3. A ew CUSUM-ye esig rocedure for he ull of coiegraio ˆ K ( k) = max vˆ ( k) γˆ ( m )( X ( / ) X ) ωˆ ( ), j, kv, k, k,,..., v. k, m = j = Uer quailes of he ull disribuio of he CUSUM of squares-ye saisic ˆK, ( k ) for esig he ull of Table 7. Fiie samle-adjused emirical size a 5% omial level. coiegraio Case of o deermiisic comoe, ˆK ( k ), wih Barle kerel ad samle size- Table. Case of o deermiisic comoe, ˆK ( ) Number of iegraed regressors, k Samle size, k = = = = = = deede badwidh give by ( ) [ ( /) / 4 ] d m d =, φ =.5, σ kυ =.75 Samle size, = Number of iegraed regressors, k d k = α = α = α = α = The simle ad hoc correcio for edogeeiy works well, ad for srogly correlaed regressio errors ad errors drivig he iegraed regressors he required badwidh is small o avoid uder-rejecio.

14 3. A ew CUSUM-ye esig rocedure for he ull of coiegraio Table 4. Fiie samle-adjused emirical size a 5% omial level. Case of o deermiisic comoe, ˆK ( k ), wih Barle kerel, samle sizedeede badwidh give by ( ) [ ( /) / 4 ] d m d =, ad (α,φ,σ kυ ) = (.5,.5,.75) Heavy-ailed disribuio for he coiegraig error erm: Sude-T, T(q) Number of iegraed regressors, k d k = q = 4 = = = q = 3 = = = Number of iegraed regressors, k d k = q = 2 = = =

15 3. A ew CUSUM-ye esig rocedure for he ull of coiegraio Table 5. Fiie samle-adjused emirical size a 5% omial level. Case of o deermiisic comoe, ˆK ( k ), wih Barle kerel, samle size-deede badwidh give by / 4 m ( ) [ ( /) ] d = d, ad (α,φ,σ kυ ) = (.,.,.). 5.. N(,)-GARCH(,), υ = zh, h = α + αυ + β h Number of iegraed regressors, k (α, α, β ) Samle size, d k = (.,.5,.65) = = = (.,.5,.75) = = = (.,.5,.9) = = =

16 3. A ew CUSUM-ye esig rocedure for he ull of coiegraio Table 5. Fiie samle-adjused emirical size a 5% omial level. Case of o deermiisic comoe, ˆK ( k ), wih Barle kerel, samle size-deede badwidh give by / 4 m ( ) [ ( /) ] d = d, ad (α,φ,σ kυ ) = (.,.,.) T(q)-GARCH(,), υ = zh, h = α + αυ + β h, q = 5 Number of iegraed regressors, k (α, α, β ) Samle size, d k = (.,.5,.65) = = = (.,.5,.75) = = = (.,.5,.9) = = =

17 4. The effecs of highly auocorrelaed regressio errors ad es cosisecy 4.. Highly auocorrelaed regressio errors. From ar (a) of Assumio L: u = α u, + υ α = α = λ, λ / 2 / 2 u[ r] ω υjλ( r) (a) u = o( ) (Phillis(987)) λr s ω ( e ζ + J ( r)) = ω M ( r) (b) u = α υ (Müller (25)), ζ N(,/2 λ) Proosiio 4.: [ r ] (/ ) v = O ( ) 4.2. Cosisecy υ λ λ υ λ s= s λ = [ r ] { r } υ λ υ λ υ υ λ λ = { r } ( ωυ/ λ) M ( s) dw ( s) r M ( s) dw ( s) (/ )( r) ( M ( r) rm ()) λ υ λ υ + λ ωυζ λ ωυ λ λ (/ ) v ( ω / λ) J ( s) dw ( s) r J ( s) dw ( s) ω ( J ( r) rj ()) Proosiio 4.2. Whe λ =, wih [ r] (/ ) vˆ ( k) = O ( ), k,, k, k,, = 2 2 ˆ ˆ v. k, ωkv,, λ =, wih uˆ ( k) = κ ˆ η, κ ˆ = (, β ˆ ) : ω ( m ) = O ( m ), ( m ) = O ( m ) Kˆ ( k) = O ( / m ) (a) (b)

18 5. Fiie samle ower resuls Noarameric kerel esimaio of he desiy fucio of he CUSUM-of squares es saisics comued uder he aleraive of o coiegraio Case A. No deermiisic comoe, wih samle size = 2, (σ kυ, φ) = (.75,.5), ad samle size-deede deermiisic badwidh m (d) = [d/(/) /4 ].6.5 CS(k,T) CS(k3,T) CS(k5,T) CS(k2,T) CS(k4,T).5.25 CS(k,T) CS(k3,T) CS(k5,T) CS(k2,T) CS(k4,T) d = Case B. No deermiisic comoe, wih samle size =, (σ kυ, φ) = (.75,.5), ad samle size-deede deermiisic badwidh m (d) = [d/(/) /4 ] d = 8.25 CS(k,T) CS(k3,T) CS(k5,T) CS(k2,T) CS(k4,T)..9 CS(k,T) CS(k3,T) CS(k5,T) CS(k2,T) CS(k4,T) d = d = 8

19 5. Fiie samle ower resuls Samle size =, badwidh d = 2 Samle size = 2, badwidh d = umber of iegraed regressors, k umber of iegraed regressors, k hi = hi =.25 hi =.5 hi =.75 hi = hi =.25 hi =.5 hi =.75 Samle size = 5, badwidh d = 2 Samle size =, badwidh d = umber of iegraed regressors, k umber of iegraed regressors, k hi = hi =.25 hi =.5 hi =.75 hi = hi =.25 hi =.5 hi =.75

20 5. Fiie samle ower resuls.4 Samle size =, badwidh d = 4.7 Samle size = 2, badwidh d = umber of iegraed regressors, k umber of iegraed regressors, k hi = hi =.25 hi =.5 hi =.75 hi = hi =.25 hi =.5 hi =.75 Samle size = 5, badwidh d = 4 Samle size =, badwidh d = umber of iegraed regressors, k umber of iegraed regressors, k hi = hi =.25 hi =.5 hi =.75 hi = hi =.25 hi =.5 hi =.75

21 Some cocludig commes: The roosed flucuaio-ye es for he ull of coiegraio i sigle-equaio coiegraig regressio models have he followig ieresig advaages: I is relaively simle o comue ad oly makes use of he OLS residuals from he esimaio of he coiegraig regressio, ad exlois heir iformaio coe as cosise esimaors of he regressio errors. The limiig ull disribuio does o deed o ay aricular feaure of he secified model, wheever all he model arameers are cosisely esimaed. This model-free limiig ull disribuio has he followig aealig imlicaios: (a) Simle o use i racice wih a sigle se of criical values (b) Preves for he icoveies caused by: subcoiegraio, surious coiegraio evidece caused by deermiisically redig iegraed regressors (see Hassler (2)), As usual, he cosisecy rae uder o coiegraio maily deeds o he badwidh used o esimae he log-ru variaces ad covariaces, bu he umerical resuls idicae ha whe based o he squared residuals relaively small values of he badwidh arameer are required. The umerical resuls o size ad ower i fiie samles show ha, as comared wih relaed exisig esig rocedures, his ew esig rocedure dislays good roeries eve i small samles ad eve uder some siuaios violaig he echical assumios required o develo he asymoics.

22 Thak you for your aeio!!!! ad ay comme or suggesio is welcome