A Consistent Nonparametric Test for Causality in Quantile

Transcription

1 SFB 649 Discussion Paper 8-7 A Consisen Nonparaeric es for Causaliy in Quanile Kiho Jeong* Wolfgang Härdle** * Kyungpook Naional Universiy Daegu, Korea ** Hubold-Universiä zu Berlin, Gerany SFB E C O N O M I C R I S K B E R L I N his research was suppored by he Deusche Forschungsgeeinschaf hrough he SFB 649 "Econoic Risk". hp://sfb649.wiwi.hu-berlin.de ISSN SFB 649, Hubold-Universiä zu Berlin Spandauer Sraße, D-78 Berlin

2 A Consisen Nonparaeric es for Causaliy in Quanile Kiho Jeong School of Econoics and rade Kyungpook Naional Universiy Daegu 7-7, Korea Eail: Wolfgang Karl Härdle CASE - Cener for Applied Saisics and Econoics Hubold-Universiä zu Berlin Wirschafswissenschafliche Fakulä Spandauer Srasse, 78 Berlin, Gerany Eail: haerdle@wiwi.hu-berlin.de Absac his paper proposes a nonparaeric es of causaliy in quanile. Zheng (998) has proposed an idea o reduce he proble of esing a quanile resricion o a proble of esing a paricular ype of ean resricion in independen daa. We exend Zheng s approach o he case of dependen daa, paricularly o he es of Granger causaliy in quanile. he proposed es saisic is shown o have a second-order degenerae U-saisic as a leading er under he null hypohesis. Using he resul on he asypoic noral disribuion for a general second order degenerae U-saisics wih weakly dependen daa of Fan and Li (996), we esablish he asypoic disribuion of he es saisic for causaliy in quanile under β-ixing (absoluely regular) process. Key Words: Granger Causaliy, Quanile, Nonparaeric es JEL classificaion: C4, C5 We hank Jürgen Franke for his Malab code o copue a nonparaeric kernel esiaor of condiional quanile. he research was conduced while Jeong was visiing CASE-Cener for Applied Saisics and Econoics, Hubold-Universiä zu Berlin in suers of 5 and 7. Jeong is graeful for heir hospialiy during he visi. Jeong s work was suppored by he Korean Research Foundaion Gran funded by he Korean Governen (MOEHRD) (KRF-6-B) and Härdle s work was suppored by he Deusche Forschungsgeeinschaf hrough he SFB 649 "Econoic Risk".

3 . Inroducion Wheher oveens in one econoic variable cause reacions in anoher variable is an iporan issue in econoic policy and also for he financial invesen decisions. A fraework for invesigaing causaliy has been developed by Granger (969). esing for Granger causaliy beween econoic ie series has been sudied inensively in epirical acroeconoics and epirical finance. he ajoriy of research resuls have been obained in he conex of Granger causaliy in he condiional ean. he condiional ean, hough, is a quesionable eleen of analysis if he disribuions of he variables involved are non-ellipic or fa ailed as o be expeced wih financial reurns. he fixaion of causaliy analysis on he ean igh resul in any unclear resuls on Granger causaliy. Also, he condiional ean arges on an overall suary for he condiional disribuion. A ail area causal relaion ay be quie differen o ha of he cener of he disribuion. Lee and Yang (7) explore oney-incoe Granger causaliy in he condiional quanile by using paraeric quanile regression and find ha Granger causaliy is significan in ail quaniles, while i is no significan in he cener of he disribuion. his paper invesigaes Granger causaliy in he condiional quanile. I is well known ha he condiional quanile is insensiive o oulying observaions and a collecion of condiional quaniles can characerize he enire condiional disribuion. Based on he kernel ehod, we propose a nonparaeric es for Granger causaliy in quanile. esing condiional quanile resricions by nonparaeric esiaion echniques in dependen daa siuaions has no been considered in he lieraure before. his paper herefore inends o fill his lieraure gap. Recenly, he proble of esing he condiional ean resricions using nonparaeric esiaion echniques has been acively exended fro independen daa o dependen daa. Aong he relaed work, only he esing procedures of Fan and Li (999) and Li (999) are consisen and have he sandard asypoic disribuions of he es saisics. For he general hypohesis esing proble of he for E( ε z) = a.e., where ε and z are he regression error er and he vecor of regressors respecively, Fan and Li (999) and Li (999) all consider he disance easure of J = E[ ε E( ε z) f( z)] o consruc kernel-based consisen es procedures. For he advanages of using disance easure J in kernel-based

4 esing procedures, see Li and Wang (998) and Hsiao and Li (). A feasible es saisic based on he easure J has a second order degenerae U-saisics as he leading er under he null hypohesis. Generalizing Hall s (984) resul for independen daa, Fan and Li (999) esablish he asypoic noral disribuion for a general second order degenerae U-saisics wih dependen daa. All he resuls saed above on esing ean resricions are however irrelevan when esing quanile resricions. Zheng (998) proposed an idea o ransfor quanile resricions o ean resricions in independen daa. Following his idea, one can use he exising echnical resuls on esing ean resricions in esing quanile resricions. In his paper, by cobining he Zheng s idea and he resuls of Fan and Li (999) and Li (999), we derive a es saisic for Granger causaliy in quanile and esablish he asypoic noral disribuion of he proposed es saisic under he bea-ixing process. Our esing procedure can be exended o several hypoheses esing probles wih condiional quanile in dependen daa; for exaple, esing a paraeric regression funcional for, esing he insignificance of a subse of regressors, and esing seiparaeric versus nonparaeric regression odels. he paper is organized as follows. Secion presens he es saisic. Secion 3 esablishes he asypoic noral disribuion under he null hypohesis of no causaly in quanile. echnical proofs are given in Appendix.. Nonparaeric es for Granger-Causaliy in Quanile o siplify he exposiion, we assue a bivariae case, or only{ y, w } are observable. Denoe U = { y, L, yp, w, L, wq} and W = { w, L, wq}. Granger causaliy in ean (Granger, 988) is defined as (i) w does no cause y in ean wih respec o U if E( y U ) = E( y U W ) and (ii) w is a pria facie cause in ean of y wih respec o U if E( y U ) E( y U W ), Moivaed by he definiion of Granger-causaliy in ean, we define Granger causaliy in

5 quanile as () w does no cause y in quanile wih respec o U if Q ( y U ) = Q ( y U W ) and () () w is a pria facie cause in quanile of y wih respec o U if Q ( y U ) Q ( y U W ), () where Q ( y ) inf { y F( y ) } is he h( < < ) condiional quanile of y. Denoe x ( y, L, yp), z ( y, L, yp, w, L, wq), and he condiional disribuion funcion y given v by ( ) yv F y v, v ( x, z) =. Denoe Q ( v ) Q ( y v ). In his paper, F ( y v ) is assued o be absoluely coninuous in y for alos all ( x, z) yv v=. hen we have { } =, v= ( x, z) F Q ( v ) v yv and fro he definiions () and (), he hypoheses o be esed are H : { Fyz Q x z } H: { Fyz Q x z } Pr ( ( ) ) = = (3) Pr ( ( ) ) = <. (4) Zheng (998) proposed an idea o reduce he proble of esing a quanile resricion o a proble of esing a paricular ype of ean resricion. he null hypohesis (3) is rue if and only if E I{ y Q ( x) z} = or I{ y Q ( x) } = + ε where E( z) ε = and I() is he indicaor funcion. here is a rich lieraure on consrucing nonparaeric ess for condiional ean resricions. Refer o Li (998) and Zheng (998) for he lis of relaed works. While various disance easures can be used o consisenly es he hypohesis (3), we consider he following disance easure, { } J E Fyz( Q ( x) z) fz( z), (5) where fz ( z ) be he arginal densiy funcion of z. Noe ha J and he equaliy holds if and only if H is rue, wih sric inequaliy holding under H. hus J can be 3

6 used as a proper candidae for consisen esing H (Li, 999, p. 4). Since { } E( ε z ) = F Q ( x ) z, we have y z { ε ε } J = E E( z ) f ( z ). (6) z he es is based on a saple analog of E{ ε E( ε z) f ( z)}. Including he densiy funcion f ( z ) is o avoid he proble of riing on he boundary of he densiy funcion, z see Powell, Sock, and Soker (989) for an analogue approach. he densiy weighed condiional expecaion E( ε z) f ( z) can be esiaed by kernel ehods z ˆ Eˆ( ε z ) fz ( z ) = Ksε s, (7) ( ) h where p q s = + is he diension of z, K K{ ( z z )/ h} s s h is a bandwidh. hen we have a saple analog of J as J s s ( ) h = s = s K ε ε z = is he kernel funcion and = Ks I{ y Q ( x )} I{ ys Q ( xs ) } ( ) h (8) he -h condiional quanile of y given x, Q ( x ), can also be esiaed by he nonparaeric kernel ehod Qˆ x = Fˆ x, (9) ( ) y x ( ) where Fˆ ( y x ) = yx s LI( y y) s s s L s () is he Nadaraya-Wason kernel esiaor of Fyx ( y x ) wih he kernel funcion of L x L x a s s = esiaed as: and he bandwidh paraeer of a. he unknown error ε can be 4

7 { ˆ } ˆ ε I y Q ( x ). () Replacing ε by ˆε, we have a kernel-based feasible es saisic of J, Jˆ K ˆˆ ε ε s s ( ) h = s = s { ( )} { ( )} = K ˆ ˆ s I y Q x I ys Q xs ( ) h () 3. he Liiing Disribuions of he es Saisic wo exising works are useful in deriving he liiing disribuion of he es saisic; one is heore.3 of Franke and Mwia (3) on he unifor convergence rae of he nonparaeric kernel esiaor of condiional quanile; anoher is Lea. of Li (999) on he asypoic disribuion of a second-order degenerae U-saisic, which is derived fro heore. of Fan and Li (999). We resae hese resuls in leas below for ease of reference. Lea (Franke and Mwia) Suppose Condiions (A)(v)-(vii) and (A)(iii) of Appendix hold. he bandwidh sequence is such ha a= o() and S% a s for p = ( log ) soe s. Le S = a + S %. hen for he nonparaeric kernel esiaor of / condiional quanile of Qˆ ( x ) of equaion (9), we have ˆ sup Q( x) Q( x) = O( S ) + O p x G a a.s. (3) Lea (Li / Fan and Li) Le L ε z = (, ) be a sricly saionary process ha saisfies he condiion (A)(i)-(iv) of Appendix, ε R and z R, () K be he kernel funcion wih h being he soohing paraeer ha saisfies he condiion (A)(i)-(ii) of Appendix. Define σ ( z) = E[ ε z = z] and (4) ε 5

8 J K ε ε s s ( ) h = s (5) hen h J N(, σ ) in disribuion, (6) / 4 where σ E { σ ε ( z) fz( z) }{ K ( u) du} = and f () is he arginal densiy funcion of z z. echnical condiions required o derive he asypoic disribuion of J ˆ are given in Appendix, which are adoped fro Li (999) and Franke and Mwia (3). In he assupions we use he definiions of Robinson (988) for he class of kernel funcions ϒ v and he class of funcions A v, defined in Appendix. Condiions (A)(i)-(iv) and (A)(i)-(ii) are adoped fro condiion (A) and (A) of Li (999), which are used o derive he asypoic noral disribuion of a second-order degenerae U-saisic. Condiions (A)(v)-(vii) and (A)(iii) are condiions (A), (A), (B), (B), (C) and (C) of Franke and Mwia (3), which are required o ge he unifor convergence rae of nonparaeric kernel esiaor of condiional quanile wih ixing daa. Finally Condiions (A)(iv)-(v) are adoped fro condiions of Lea of Yoshihara (976), which are required o ge he asypoic equivalence of nondegenerae U-saisic and is projecion under he β -ixing process. We consider esing for local deparures fro he null ha converge o he null a he rae h / /4 H :. More precisely we consider he sequence of local alernaives: Fyz { Q ( x) dl( z) z} + =, (7) where d h / /4 = and he funcion () l and is firs-derivaives are bounded. heore. Assue he condiions (A) and (A). hen (i) Under he null hypohesis (3), h J ˆ N(, σ ) in disribuion, where / 6

9 { z } 4 σ = E σ ε ( z ) f ( z ) K ( u) du and σ ( ) ( ε z = E ε z) = ( ). (ii) under he null hypohesis (3), ˆ σ ( ) K s ( ) h 4 = E ( z ) f ( z ) K ( u) du. esiaor of σ { σ ε z } s is a consisen (iii) under he alernaive hypohesis (4), ˆ p {[ y z( ( ) ) ] z( )} J E F Q x z f z >. (iv) under he local alernaives (7), h J ˆ N( μ, σ ) in disribuion, where / { } μ = E fyz Q ( z) z l ( z) fz( z). heore generalizes he resuls of Zheng (998) of independen daa o he weakly dependen daa case. A deailed proof of heore is given in he Appendix. he ain difficuly in deriving he asypoic disribuion of he saisic defined in equaion () is ha a nonparaeric quanile esiaor is included in he indicaor funcion which is no differeniable wih respec o he quanile arguen and hus prevens aking a aylor expansion around he rue condiional quanile Q ( x ). o circuven he proble, Zheng (989) appealed o he work of Sheran (994) on unifor convergence of U-saisics indexed by paraeers. In his paper, we bound he es saisic by he saisics in which he nonparaeric quanile esiaor in he indicaor funcion is replaced wih sus of he rue condiional quanile and upper and lower bounds consisen wih unifor convergence rae of he nonparaeric quanile esiaor, ( y Q ( x ) C ) and ( y Q ( x ) + C ). An iporan furher sep is o show ha he differences of he ideal es saisic J given in equaion (8) and he saisics having he indicaor funcions obained fro he firs sep saed above is asypoically negligible. We ay direcly show ha he second oens of he differences are asypoically negligible by using he resul of Yoshihara (976) on he bound of oens of U-saisics for absoluely regular processes. However, i is edious o ge bounds on he second oens wih dependen daa. In he proof we insead use he fac ha differences are second-order degenerae U-saisics. hus by using he resul on he asypoic noral disribuion of he second-order degenerae U-saisic of Fan and Li 7

10 (999), we can derive he asypoic variance which is based on he i.i.d. sequence having he sae arginal disribuions as weakly dependen variables in he es saisic. Wih his lile rick we only need o show ha he asypoic variance is o () in an i.i.d. siuaion. For deails refer o he Appendix. 4. Conclusion his paper has provided a consisen es for Granger-causaliy in quanile. he es can be exended o esing condiional quanile resricions wih dependen daa; for exaple, esing isspecificaion es, esing he insignificance of a subse of regressors, esing soe seiparaeric versus nonparaeric odels, all in quanile regression odels. 8

11 Appendix Here we collec all required assupions o esablish he resuls of heore. (A) (i) { y, w } is sricly saionary and absoluely regular wih ixing coefficiens β ( τ) = O( ρ τ ) for soe < ρ <. (ii) For soe ineger v, f y, f z, and f x all are bounded and belong o A v (see D). (iii) wih probabiliy one, E z z =. [ ε μ ( ), μ ( )] E 4 ε +η < + ξ i i il E εε ε l < L for soe arbirarily sall η > and ξ >, where l 4 is and an ineger, i j 4 and l i j 8. σε ( z) = E( ε z), j= μ z = E z = z all ( ) 4 ε 4 ε saisfy soe Lipschiz condiions: au ( + v) au ( ) Du ( ) v wih for soe sall ' f τ,, τ l (iv) Le ( ) hen ( ) η >, where () σ ( ), μ ( ) a ε ε4 =. E D( z) K be he join probabiliy densiy funcion of ( zτ zτ ) f τ,, τ l,, l +η ' < K ( l 3). K is bounded and saisfies a Lipschiz condiion: ( +, +, K + ) (,, K ) ( ) f z u z u z u f z z z, K, l l l, K, l l τ τ τ τ D,,,, τ K τ z K z l l u, where K τ ( ) is inegrable and saisfies he condiion ha (,,,, l ) D τ,, l (,, ) (,, ), K, l l, K, l Dτ τ z K z fτ τ z K z dz < M < for soe ξ >. l D z z z ξ τ M K τ K < <, l (v) For any yx, saisfying < Fyx ( y x) < and fx ( x ) > ; for fixed y, he condiional disribuion funcion F yx and he condiional densiy funcion f yx belong o A 3 ; fyx ( Q ( x) x) > for all x ; f yx is uniforly bounded in x and y by c f, say. (vi) For soe copac se G, here are ε >, γ > such ha f x γ for all x in he ε -neighborhood { x x u ε, u G } < of G ; For he copac se G and soe 9

12 copac neighborhood Θ of, he se { v Q ( x) x G, } Θ = = + μ μ Θ is copac and for soe consan c >, f ( v x) c for all x G, v Θ. (vii) here is an yx increasing sequence s of posiive inegers such ha for soe finie A, s /(3 ) ( s ) A s β, s for all. (A) (i) we use produc kernels for boh L( ) and K ( ), le l and k be heir corresponding univariae kernel which is bounded and syeric, hen l() is non-negaive, l() ϒ, v k() is non-negaive and k() ϒ. (ii) h ' = O( α ) for soe α ' (7/8) < <. (iii) a = o() and % p = ( log ) for soe S a s s (iv) here exiss a posiive nuber δ such ha for r = + δ and a generic nuber M r z z K dfz( z) dfz( z) M < h h and r z z E K M h h < (v) for soe δ ' ( < δ ' < δ), β ( + δ ')/ δ ' ( ) O( ) =. he following definiions are due o Robinson (988). Definiion (D) ϒ, λ is he class of even funcions k: R R saisfying where λ i uk( u) du = δ R i ( i =,,, λ ) ( ++ ) K, ku ( ) = O( + u λ ε ), for soe ε >, δ ij is he Kronecker s dela. α Definiion (D) A μ, α >, μ > is he class of funcions g: R R saisfying ha

13 g is ( d ) -ies parially differeniable for d μ d ; for soe ρ >, μ for all z, where φzρ = { y y z < ρ} sup g( y) g( z) G ( y, z) / y z D ( z) y φzρ g g ; G g = when d = ; G g is a ( d ) h degree hoogeneous polynoial in y z wih coefficiens he parial derivaives of g a z of orders hrough d when d > ; and gz ( ), is parial derivaives of order d and less, and D ( z ), has finie α h oens. g Proof of heore (i) In he proof, we use several approxiaions o J ˆ. We define he now and recall a few already defined saisics for convenience of reference. Jˆ J J J K ˆˆ ε ε s s ( ) h = s (A.) s s ( ) h = s K ε ε (A.) U s U su ( ) h = s K ε ε (A.3) L s L sl ( ) h = s where ˆ ε { ˆ I y Q ( x) } K ε ε (A.4) =, { ( )} ε = I y Q x, ε { ( )} = I y + C Q x, U { ( )} ε = I y C Q x and L C is an upper bound consisen wih he unifor convergence rae of he nonparaeric esiaor of condiional quanile given in equaion (3). he proof of heore (i) consiss of hree seps. Sep. Asypoic noraliy: h J N(, σ ), (A.5) /

14 where σ E { ( ) f ( z )}{ K ( u) du} = under he null. Sep. Condiional asypoic equivalence: Suppose ha boh h / ( J J ) = o () and U p h / ( J J ) = o (). U p hen h ( J J ) = o (). (A.6) / ˆ p Sep 3. Asypoic equivalence: h / ( J J ) = o () and U p h / ( J J ) = o (). (A.7) L p he cobinaion of Seps -3 yields heore (i). Sep : Asypoic noraliy. Since J is a degenerae U-saisic of order, he resul follows fro Lea. Sep : Condiional asypoic equivalence. he proof of Sep is oivaed by he echnique of Härdle and Soker (989) which was used in reaing riing indicaor funcion asypoically. Suppose ha he following wo saeens hold. h / ( J J ) = o () and (A.8) U p h / ( J J ) = o () (A.9) L p Denoe C as an upper bound consisen wih he unifor convergence rae of he nonparaeric esiaor of condiional quanile given in equaion (3). Suppose ha sup Qˆ ( x) Q( x) C. (A.) If inequaliy (A.3) holds, hen he following saeens also hold: { Q ( x) > y + C } { Qˆ ( x) > y } { Q ( x) > y C }, (A.-) ( Q ( x) > y + C ) ( Qˆ ( x) > y ) ( Q ( x) > y C ), (A.-) J Jˆ J, and (A.-3) U L ˆ (A.-4) / / / h ( J J) ax { h ( J JU), h ( J JL) }

15 Using (A.) and (A.-4), we have he following inequaliy; / { h J ˆ ˆ J δ Q x Q x C} / / Pr { ax { h ( J ), ( ) } > sup ˆ JU h J JL δ Q( x) Q( x) C} Pr ( ) > sup ( ) ( ), for all δ >. (A.) Invoking Lea and condiion A(iii), we have { Q ˆ x Q x C } Pr sup ( ) ( ) as. (A.3) By (A.8) and (A.9), as, we have / / { h J JU h J JL δ } Pr ax { ( ), ( ) } >, for all δ >. herefore, as, / he L.H.S. of he inequaliy (A.) { h J ˆ J δ } he L.H.S. of he inequaliy (A.). (A.4) Pr ( ) > and In suary, we have ha if boh h / ( J J ) = o () and U p h / ( J J ) = o (), U p hen h ( J J ) = o (). / ˆ p Sep 3: Asypoic equivalence. In he reaining proof, we focus on showing ha h / ( J J ) = o (), wih he proof U p of h / ( J J ) = o () being reaed siilarly. he proof of Sep 3 is close in lines L p wih he proof in Zheng (998). Denoe H (,, s g) K {( y g( x )) }{( y g( x )) } and (A.5) s s s Jg [ ] H ( sg,, ) ( ) h =. (A.6) s hen we have J J[ Q ] and JU J[ Q C ]. We decopose H (,, s g ) ino hree pars; H ( s,, g) = K {( y g( x )) F( g( x ) z )}{( y g( x )) F( g( x ) z )} s s s s s 3

16 + K {( y g( x )) F( g( x ) z )}{ F( g( x ) z ) } s s s + K{ Fgx ( ( ) z) }{ Fgx ( ( ) z) } s s s = H (,, sg) + H (,, sg) + H (,, sg) (A.7) 3 hen le J j[ g] = H j( s,, g) ( ) h =,,, 3 s i =. We will rea J [ Q ] J [ Q C ] for j =,,3 separaely. j j h J ( Q ) J ( Q C ) = o (): / [] [ ] By siple anipulaion, we have J Q J Q C ( ) ( ) p = ( ) h = s = s [ H (,, s Q ) H (,, s Q C )] = Ks Q F Q s Q Fs Q ( ) h { [ ( ) ( ) ][ ( ) ( ) ] [ ( Q C) F( Q C) ][ s( Q C) Fs( Q C) ] } (A.8) o avoid edious works o ge bounds on he second oen of J ( Q) J ( Q C ) wih dependen daa, we noe ha he R.H.S. of (A.8) is a degenerae U-saisic of order. hus we can apply Lea and have [ ] h J ( Q ) J ( Q C ) N(, ) in disribuion, (A.9) / σ where he definiion of he asypoic variance σ is based on he i.i.d. sequence having he sae arginal disribuions as weakly dependen variables in (A.8). ha is, [ (,, ) (,, )] σ = E% H s Q H s Q C, where he noaion E % is expecaion evaluaed a an i.i.d. sequence having he sae arginal disribuion as he ixing sequences in (A.8) (Fan and Li (999), p. 48). Now, o h J ( Q ) J ( Q C ) = o (), we only need o show ha he asypoic / show ha [ ] p 4

17 variance σ z is () ( ) o wih i.i.d daa. We have [ (,, ) (,, )] E% H sq H sq C { [ Q F Q ][ s Q Fs Q ] ΛE % ( ) ( ) ( ) ( ) [ ( Q C ) F( Q C )][ ( Q C ) F ( Q C )] } s s { ( ) [ ( ) ] s( ) [ s( ) ]} ΛE% F Q F Q F Q F Q { ( ) [ ( ) ] s( ) [ s( ) ]} + E% F Q C F Q C F Q C F Q C { [ ] E F(in( Q, Q C ) F( Q ) F( Q C ) [ Fs(in( Q, Q C) Fs( Q) Fs( Q C) ] } {[ ( ) ( ) ( ) ][ s( ) s( ) s( ) ]} =ΛE% F Q F Q F Q F Q F Q F Q Λ E% { F(in( Q, Q C) F( Q) F( Q C) [ ] [ Fs(in( Q, Q C) Fs( Q) Fs( Q C) ] } +Λ E% { F( Q C) F( Q C) F( Q C) [ ] [ Fs( Q C) Fs( Q C) Fs( Q C) ] } Λ E% { F(in( Q, Q C) F( Q) F( Q C) [ ] [ Fs(in( Q, Q C) Fs( Q) Fs( Q C) ] } Λ C = o(). (A.) where he las equaliy holds by he soohness of condiional disribuion funcion and is bounded firs derivaive due o Assupion (A.8). hus we have [ ] / h J Q J Q C op ( ) ( ) = () (A.) h J ( Q ) J ( Q C ) = o (): / [] [ ] Noing ha H (,, ) s Q = because of F ( Q ( x ) z ) =, we have p yz s s J Q J Q C ( ) ( ) 5

18 =J ( ) Q C z z = ( ) h h s K = s {( y Q ( x ) C ) F ( Q ( x ) C z )}{ F ( Q ( x ) C z ) } (A.) y z y z s s Denoe Sg ( ) Fg [ ]/ g. By aking a aylor expansion of F ( Q ( x ) C z ) around Q ( x s ), we have yz s s J Q J Q C ( ) ( ) z z = K {( y Q( x) C) Fy z( Q( x) C z)} ( ) h h s = s ( C ) S( Q ( x )) s = C {( y Q ( x ) C ) F ( Q ( x ) C )} S( Q ( x )) fˆ ( z ) y z s z = C ( ( )) ˆ u S Q x f ( z ), (A.3) s z = where Q is beween Q and Q C. hus we have E J Q J Q C ( ) ( ) ΛC ˆ E u f ( z ) z = { } ΛC ˆ E u f ( z ) z = ( ( ) ) = O C h, (A.4) where he firs inequaliy holds due o Assupion ()(v) and he las equaliy is derived by using Lea C.3(iii) of Li (999) ha is proved in he proof of Lea A.4(i) of Fan and Li (996c). hus, we have 6

19 [ ( ) ( )] / h J Q J Q C = O p / ( Ch ) = o p (). (A.5) h J ( Q ) J ( Q C ) = o (): / [3] [ ] 3 3 p Noing ha H (,, ) 3 s Q = because of FQ ( ( x) z) = for j= s,, we have j j J3 Q J3 Q C ( ) ( ) z z = ( ) h h s K = s { FQ ( ( x) C z) }{ FQ ( ( x) C z) } s s z z s = K ( ( )) ( ( )) CS Q x S Q xs ( ) = s h h ˆ (A.6) = C S( Q( x)) S( Q( xs)) fz( z) = hus, we have E J3 Q J3 Q C ( ) ( ) ΛC E fˆ ( z ) z = Λ C E f ( z ) +ΛC E fˆ ( z ) f ( z ) z z z = = { } Λ C Ef ( z ) +ΛC E fˆ ( z ) f ( z ) z z z = = ( ) = O C (A.7) Finally, we have [ ( ) ( )] / h J3 Q J3 Q C p / ( ) = O h C 7

20 = o p (). (A.8) By cobining (A.), (A.5) and (A.8), we have he resul of Sep 3 Proof of heore (ii) Since { z } σ = ( ) E f ( z ) K ( u) du and ˆ σ ( ) K, s ( ) h s i is enough o show ha σ Noe ha K s ( ) h s { } = E fz ( z) K ( u) du+ op() (A.9) σ is a nondegenerae U-saisic of order wih kernel z zs H( z, zs) = K h h. (A.3) Since Assupion (A)(iv)-(v) saisfy he condiions of Lea of Yoshihara (976) on he asypoic equivalence of U-saisic and is projecion under β -ixing, we have for γ = ( δ δ ') / δ '( + δ) > σ (, ) H z zs ( ) s = H ( z, z ) df ( z ) df ( z ) z z z z z p = + H ( z, z ) df ( z ) H ( z, z ) df ( z ) df ( z ) + O ( γ ) = H ( z, z ) df ( z ) df ( z ) + o () z z p z z = K dfz( z) dfz( z) op() + h h 8

21 ( ) z p (A.3) = K u du f ( z ) dz + o () he resul of heore (ii) follows fro (A.3). Proof of heore (iii) he proof of heore (iii) consiss of he wo seps. Sep. Show ha Jˆ = J + o () under he alernaive hypohesis (4). p Sep. Show ha J = J + o () under he alernaive hypohesis (4), p where J = E F Q x z f z. he cobinaion of Seps and yields {[ yz ( ( ) ) ] z( )} heore (iii). Sep : Show ha Jˆ = J + o () under he alernaive hypohesis. p We need o show ha he resuls of Sep and Sep 3 in he proof of heore (i) hold under he alernaive hypohesis. Firs, we show ha he resul of Sep in he proof of heore (i) sill holds under he alernaive hypohesis. We can show ha J ( ) () Q C = o by he sae procedures as in (A.4). hus we focus on showing ha J ( ) () Q = o p. As in he proof of heore (i), denoe Sg ( ) Fg [ ]/ g. By aking a aylor expansion of Fyz ( Q ( xs) zs) around Q ( z s ), we have J ( Q ) z zs = K {( ( )) y z( ( ) )} y Q x F Q x z ( ) = s h h SQ ( ( x, z)) s s p = {( ( )) ˆ y Q x Fy z( Q( x))} S( Q( xs, zs)) fz( z) = ( (, )) ˆ us Q xs zs fz( z), (A.3) = where Q ( x, z ) is beween Q ( x s ) and Q ( z s ). By using he sae procedures as in s s 9

22 (A.4), we have ( ) ( ) J Q = O h. (A.33) Nex, we show ha he resul of Sep 3 in he proof of heore (i) holds under he alernaive hypohesis. Since FQ ( ( x) z) for j= s, under he alernaive hypohesis, we have J3 Q J3 Q C ( ) ( ) j j z z = { ( ( ) ) }{ ( ( s) s) } ( ) h h FQ x z FQ x z s K = s z z ( ) h h s K = s { FQ ( ( x) C z) }{ FQ ( ( x) C z) } s s = { ( ( ) ) }{ ( ( ) ) } ˆ FQ x z FQ xs zs fz( z) = { ( ( ) ) }{ ( ( ) ) } ˆ FQ x C z FQ xs C zs fz( z). (A.34) = By aking a aylor expansion of F ( Q ( x ) C z ) around Q ( z ) for j =, s, we have J3 Q J3 Q C ( ) ( ) yz j j j = { ( ( ) ) } ( ( )) ˆ FQ x z CSQ xs fz( z) = + ( ( )){ ( ( ) ) } ˆ CSQ x FQ xs zs fz( z) = CSQ ˆ x SQ xs fz z = ( ( )) ( ( )) ( ). (A.35) We furher ake aylor expansion of F ( Q ( x ) z ) around Q ( z ) for j =, s and have J3 Q J3 Q C ( ) ( ) yz j j j

23 = SQ ( (, )) ( ( )) ˆ x z CSQ xs fz( z) = + CSQ ( ( )) ( (, )) ˆ x SQ xs zs fz( z) = CSQ ˆ x SQ xs fz z = ( ( )) ( ( )) ( ), (A.36) where Q ( x, z ) is beween Q ( x s ) and Q ( z s ). hen by using he sae procedures as in (A.7), we have s s 3 3 ( ) J ( Q ) J ( Q C ) = O C. (A.37) Now we have he resul of Sep for he proof of heore (iii). Sep : Show ha J = J + o () under he alernaive hypohesis. p Using (7) and unifor convergence rae of kernel regression esiaor under β -ixing process, we have J = s s ( ) h = s K ε ε = E ˆ( ε ) ˆ ( ) z fz z ε = = = E( ε z ) f ( z ) ε z = { E ˆ( z ) f ( z ) E ( z ) f ( z ) } + ˆ ε z ε z ε = E( ε z) fz( z) ε + op() = [ ε ε ] = E E( z ) f ( z ) + o () z p = J + o p () (A.38)

24

25 References Cai, Z. (), Regression quaniles for ie series, Econoeric heory 8, Chen, X. and Y. Fan (999), Consisen hypohesis esing in seiparaeric and nonparaeric odels for econoeric ie series, Journal of Econoerics 9, Fan, Y. and Q. Li (999), Cenral lii heore for degenerae U-saisics of absoluely regular processes wih applicaions o odel specificaion ess, Journal of Nonparaeric Saisics, Franke, J. and P. Mwia (3), Nonparaeric esiaes for condiional quaniles of ie series, Repor in Wirschafsaheaik 87, Universiy of Kaiserslauern. Granger, C.W.J. (969), Invesigaing causal relaions by econoeric odels and cross-specral ehods, Econoerica 37, Granger, C.W.J. (988), Soe recen developens in a concep of causaliy, Journal of Econoerics 39, 99-. Györfi, L., W. Härdle, P. Sarda, and P. Vieu (989), Nonparaeric Curve Esiaion fro ie Series, Springer-Verlag: Berlin. Hall, P. (984), Cenral lii heore for inegraed square error of ulivariae nonparaeric densiy esiaors, Journal of Mulivariae Analysis 4, -6. Hansen, B.E. (4), Unifor convergence raes for kernel esiaion wih dependen daa, working paper, Universiy of Wisconsin-Madison, Härdle, W. and. Soker (989), Invesigaing sooh uliple regression by he ehod of average derivaives, Journal of he Aerican Saisical Associaion 84, Hsiao, C. and Q. Li (), A consisen es for condiional heeroskedasiciy in ie-series regression odels, Econoeric heory 7, 88-. Lee,. and W. Yang (7), Money-incoe Granger-causaliy in quaniles, unpublished paper, Universiy of California, Riverside Li, Q. (99), Consisen odel specificaion ess for ie series econoeric odels, Journal of Econoerics 9, -47. Li, Q. and S. Wang (998), A siple consisen boosrap es for a paraeric regression funcional for, Journal of Econoerics 87, Mwia, P. (3), Seiparaeric esiaion of condiional quaniles for ie series wih applicaions in finance, Ph D hesis, Universiy of Kaiserlauern. Powell, J.L., J.H. Sock and.m. Soker (989), Seiparaeric esiaion of index coefficiens, Econoerica 57, Robinson, P. (989), Hypohesis esing in seiparaeric and nonparaeric odels for econoeric ie series, Review of Econoic Sudies 56, Sheran, R. (994), Maxial inequaliies for degenerae U-processes wih applicaions o opiizaion esiaors, Annals of Saisics, Zheng, J. (998), A consisen nonparaeric es of paraeric regression odels under condiional quanile resricions, Econoeric heory 4,

26 SFB 649 Discussion Paper Series 8 For a coplee lis of Discussion Papers published by he SFB 649, please visi hp://sfb649.wiwi.hu-berlin.de. "esing Monooniciy of Pricing Kernels" by Yuri Golubev, Wolfgang Härdle and Roan ionfeev, January 8. "Adapive poinwise esiaion in ie-inhoogeneous ie-series odels" by Pavel Cizek, Wolfgang Härdle and Vladiir Spokoiny, January 8. 3 "he Bayesian Addiive Classificaion ree Applied o Credi Risk Modelling" by Junni L. Zhang and Wolfgang Härdle, January 8. 4 "Independen Coponen Analysis Via Copula echniques" by Ray-Bing Chen, Meihui Guo, Wolfgang Härdle and Shih-Feng Huang, January 8. 5 "he Defaul Risk of Firs Exained wih Sooh Suppor Vecor Machines" by Wolfgang Härdle, Yuh-Jye Lee, Dorohea Schäfer and Yi-Ren Yeh, January 8. 6 "Value-a-Risk and Expeced Shorfall when here is long range dependence" by Wolfgang Härdle and Julius Mungo, Januray 8. 7 "A Consisen Nonparaeric es for Causaliy in Quanile" by Kiho Jeong and Wolfgang Härdle, January 8. SFB 649, Spandauer Sraße, D-78 Berlin hp://sfb649.wiwi.hu-berlin.de his research was suppored by he Deusche Forschungsgeeinschaf hrough he SFB 649 "Econoic Risk".