β-spline Estimation in a Semiparametric Regression Model with Nonlinear Time Series Errors

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1 Amrca Joural of Appld Sccs, (9): , 005 ISSN Scc Publcatos β-spl Estmato a Smparamtrc Rgrsso Modl wth Nolar Tm Srs Errors Jhog You, ma Ch ad 3 Xa Zhou Dpartmt of ostatstcs, Uvrsty of North Carola at Chapl Hll Chapl Hll, NC , USA Dpartmt of Mathmatcs ad Statstcs, Uvrsty of Calgary Calgary, Albrta, Caada TN N4 3 Dpartmt of Appld Mathmatcs, Th Hog Kog Polytchc Uvrsty Hug Hom, Kowloo, Hog Kog, PR Cha Abstract: W study th stmato problms for a partly lar rgrsso modl wth a olar tm srs rror structur Th modl cossts of a paramtrc lar compot for th rgrsso coffcts ad a oparamtrc olar compot Th radom rrors ar uobsrvabl ad modld by a frstordr Markov blar procss asd o a -spl srs approxmato of th olar compot fucto, w propos a smparamtrc ordary last squars stmator ad a smparamtrc gralzd last squars stmator of th rgrsso coffcts, a last squars stmator of th autorgrsso paramtr for th rrors, ad a -spl srs stmator of th oparamtrc compot fucto Th asymptotc proprts of ths stmators ar vstgatd ad thr asymptotc dstrbutos ar drvd W also provd a cosstt stmator for th asymptotc covarac matrx of th smparamtrc gralzd last squars stmator of th rgrsso coffcts Our rsults ca b usd to mak asymptotcally ffct statstcal frcs I addto, a small smulato s coductd to valuat th prformac of th proposd stmators, whch shows that th smparamtrc gralzd last squars stmator of th rgrsso coffcts s mor ffct tha th smparamtrc ordary last squars stmator Ky words: Smparamtrc rgrsso modl, frst-ordr Markova blar procss, β-spls srs stmato, smparamtrc last squars stmators, asymptotc ormalty INTRODUCTION Partly lar rgrsso modls hav attractd a lot of rsarch trsts du to thr flxblty to allow both lar ad olar compots, as wll as srally corrlatd rrors, whch abls thm to bttr dscrb crasgly complx data from th ral world tha pur paramtrc or oparamtrc modls Cosdr a partly lar rgrsso modl of th followg form: y = x ' β + g( t ) + ε, =,,, () whr, y 's ar rsposs, x = (x,, x p )' ad t ar calld th dsg pots, β = ( β,, β p ) ' s a ukow paramtr vctor rprstg th lar compot, g( ) s a ukow fucto dfd o [0, ε 's ar ] for th olar compot, ad uobsrvabl radom rrors Th modl () has b xtsvly studd by may rsarchrs A brf rvw of rlvat ltratur s as follows Wh ε ar d Radom varabls [-5] usd varous stmato mthods, such as th krl mthod, spl mthod, srs stmato, local lar stmato, M-stmato ad two-stag stmato, to obta stmats of th ukow quatts () Thy also dscussd th asymptotc proprts of ths stmators Howvr, th dpdc assumpto for th rrors s ot always approprat applcatos, spcally for squtally collctd coomc data, whch oft xhbt vdt sral dpdc th rrors For xampl, th procss of fttg th rlatoshp btw tmpratur ad lctrcty usag [6] foud that th data ar srally corrlatd Wh ε s a autorgrssv (AR) srs [7] studd a stmator of th autocorrlato coffct [8] cosdrd th stmato problm for modl () wth lar tm srs rrors It s wll kow that ot all corrlatd rrors ca b fttd wll by lar tm srs rrors Thrfor, much attto has b shftd to olar tm srs modls th rct ltratur [9] ad th rfrcs thr Thr hav b may paprs cocrd wth th ordary lar modls wth olar tm srs rrors For xampl, udr th assumpto of radom coffct autorgrssv rrors [0] vstgatd th Corrspodg Author: Xa Zhou, Dpartmt of Appld Mathmatcs, Th Hog Kog Polytchc Uvrsty, Hug Hom, Kowloo, Hog Kog, PR Cha 343

2 lmt dstrbuto of th last squars stmators of th rgrsso ad auto rgrsso paramtrs Morovr [] usd th framwork of th formato matrx (IM) tst to dvlop a tst for th lar rgrsso modl wh th rrors ar from a autorgrssv codtoal htroscdastc (ARCH) procss [] Drvd th Wald ad Rao's Scor tst statstcs for tstg th ffcts of addtoal rgrsso paramtrs Thr has, howvr, b lttl work o th partal lar rgrsso modl wth olar tm srs rrors th ltratur xcpt [3] [3] vstgatd th stmatg problms of partal lar rgrsso modls wth radom coffct autorgrssv rrors I ths study, by approxmatg th oparamtrc compot wth -spl srs w study th problms of stmatg th paramtrc ad oparamtrc compots of th partal lar rgrsso modl () wth a olar tm srs rror structur Mor spcfcally, w cosdr a frst-ordr Markova blar rror procss ε, whch s a statoary soluto of : ε = ( φ + θ ) ε +, =,,, () whr, { } s a zro ma procss cosstg of d radom varabls wth ft scod momts σ Obvously, th modl () cluds th usual AR () structur [4] dscussd th stmato ad tst problms of th ordary lar rgrsso modl wth rror structur () Modl () has b xtsvly dscussd th cotrol thory ltratur [5,6] [7] appld modl () to study th wll-kow Wolfr suspot umbrs for th yars from 700 to 955 ad a ssmc rcord obtad from a udrgroud uclar xploso that was carrd out th USA o Octobr 9th, 966 Rctly, modl () has b xtdd to th cas of spac tm [8,9] Mor rfrcs about th thortcal rsults, applcatos ad th xtsos of th modl () ca b foud th moograph of [9] asd o th approxmato of g( ) by a -spl srs ad last squars stmato, w costruct th followg stmators for modl () wth rror structur (): Smparamtrc ordary last squars stmator (SOLSE) of β, Last squars stmator of th auto rgrsso paramtr φ, Smparamtrc gralzd last squars stmators (SLSE) of β, v Estmator of th asymptotc covarac matrx of th SLSE of β, ad v -spl srs stmator of th oparamtrc compot fucto g( ) Am J Appl Sc, (9): , W wll furthr vstgat th asymptotc proprts of ths stmators ad drv thr lmtg dstrbutos I addto, a small smulato s coductd to valuat th prformac of th stmators Estmators: Throughout ths study w wll assum that th dsg pots x ad t ar fxd, ad thy ar rlatd va: x = h ( t ) + u, =,, ; s =,, p () s s s Th rasoablss of ths rlato ca b foud [] I addto, suppos that th vctor (,,)' s ot th spac spad by th colum vctors of X = (x,, x )', whch surs th dtfablty of modl () accordg to [] It s also assumd that th squc of dsgs t forms a asymptotcally rgular squc [0] th ss that: t / max p( t) dt o( ), = 0 whr, p( ) dots a postv dsty fucto o th trval [0,] x, t, y ; =,, b a th modlf obsrvd Lt { } data from modl (), X = ( x j ) ad p y = ( y,, y )' Dot by S, j th class of spl fuctos of dgr ν ad k() kots It s wll kow [] that S, j has a bass cosstg of k() + ν ormalzd -spls {, j ( ) : j =,, k( ) + ν } whr ( ), ( ) has support k j ν + + ν +, ad ( ) α k j [( j ), ( j ) ] approxmatd by a lar combato ( ), ( ) R k + g ca b ' ( ) of th bass, whr α ν ad, j ( ) = (, ( ),,, + ν ( ))' For + ν p α R, β R, ( β, α ) = ( ' β α ' ( )) = S y x t ( β, α ) gv by β Y X β ( ' ) ' s mmzd at = X M X X M Y ad k ( ) α = ( ' ) '( ), whr ( ( t ),, ( t ))',,, +ν k( ) k( ) M = = P ad k ( ) k ( ) P = ( ' ) ' Ths β s calld th smparamtrc ordary last squars stmator (SOLSE) of β Wh th rrors ar corrlatd, th SOLSE β s ot asymptotcally ffct as t gors th corrlato Hc w propos a smparamtrc gralzd last squar stmator (SLSE) of β Sc for a gv φ :

3 Am J Appl Sc, (9): , 005 φ σ ( ) Cov( εε ') = Ω( φ) φ ε = ( ε,, ε )', w df a SLSE as whr, follows: ɶ β = ( X ' M Ω ( φ) M X ) k( ) Ω φ k( ) X ' M ( ) M Y Whr: φ φ + φ φ ( ) φ + φ φ Ω φ = 0 0 φ + φ φ φ Rmark : y Lmma 3 th Appdx, ad φ k( ) X ' M () k ( ) X ' M Ω ( ) M X ar postv dft wh s larg Thrfor, wthout loss of gralty, w ca assum that th vrss of ths two matrcs xst Wh φ s ukow, th X ɶ β () s ot drctly usabl, ad w d a sutabl stmator, say Ω φ by Ω ( ) φ, of φ Th w ca rplac ( ) () ad obta a stmat of β that ca b computd from th data Notg that ε s uobsrvabl, a rasoabl stmator of φ s th last squars stmator : φ = ε ε ε = = basd o th stmatd rsduals: ε = y x ' β α ' ( t ), =,, (3) φ Cosqutly, w df our SLSE of β, dotd by β, as: β = ( X ' M Ω ( φ) M X) X ' M ( ) M Y Ω φ k( ) (4) g ( t) = ( ' ) '( Y X β ) (5) k( ) k( ) Furthrmor, a stmator of th asymptotc covarac matrx of β s gv by Σ, whr Σ = σ ( X ' M Ω ( φ ) M X ) ad: ε ε = y x t = σ ( ' β α ' ( )) (6) Larg sampl proprts: W bg wth th followg assumptos rqurd to drv th ma rsults, whch ar qut mld ad ca b asly satsfd (Rmark blow) Assumpto : For =,, ad j =,, p, u j satsfy: lm = =, for = 0, ±, ±, ;, =,, h ukuk + h, j bhj h j p k = whr, th matrx = ( b oj ) s osgular, ad: p ( ) / max Au O [tr(a'a] foraymatrx A whr, = = (3) Euclda orm u = ( u,, u )' ad dots th Assumpto : Th fuctos g( ) ad h ( ),,h p( ) ar ν tms cotuously dffrtabl o th trval [0,], whr ν > Assumpto 3: Th coffcts θ ad φ, ad th varac φ + θ σ < σ of { } modl () satsfy φ < ad Rmark : Th abov u j bhav lk zro ma, ucorrlatd radom varabls ad h j (t ) ar th rgrsso of x j o t Spcfcally, suppos that th dsg pots (x, t ) ar d Radom varabls, ad lt h j (t ) = E[x j t ] ad u j = x j h j (t ) wth E[u u '] = Th by th law of larg umbrs, (3) holds wth probablty Morovr, accordg to [] (3) holds wh u j bhav lk zro ma, ucorrlatd radom varabls Assumpto s mld ad holds for most commoly usd fuctos, such as th polyomal ad trgoomtrc fuctos Th frst thorm blow shows th asymptotc ormalty of th SOLSE β asd o ths SLSE β, w ca costruct th followg stmator of th oparamtrc compot fucto g( ) : Thorm 3: Suppos that Assumptos to 3 hold ν/(ν+ ) ν/(ν+ ) ad c k() c whr c ad c ar postv costats Th: 345

4 ( φ ) σ ( β β) D N0, as φ whr, = U Ω φ U provdd t s postv lm ' ( ) dft, U = (u,, u ), s dfd Assumpto ad dots covrgc dstrbuto D Th asymptotc ormalty of φ s stablshd th scod thorm blow Thorm 3: Udr th codtos of Thorm 3, f, 4 addto, E [ ε ] <, th: σ [ ] [ ] E ε + θ σ E ε ( φ φ) D N0, as ( E[ ε ]) Th xt thorm shows that th SLSE ɶ β dfd () ad ts fasbl vrso β (4) hav th sam lmtg ormal dstrbuto Thorm 33: Udr th codtos of Thorm 3 w / hav ( β β ) = ( ɶ β β ) + O ( ) ad th p commo lmtg dstrbuto of β ad by: ( φ ) σ ( β β ) D N0, as φ Am J Appl Sc, (9): , 005 ɶ β s gv whr, = lm U ' Ω ( φ) U provdd t s postv dft Sc, accordg to Thorms 3 ad 33, β th asymptotc covarac matrxt tha β trms of asymptotc covarac matrx Lt ν ( ) dot th ν th roull polyomal, whch s rcursvly dfd by: ( t) =, ( t) = ( z) dz + b =,,, 0 0 whr, = t 0 0 t b ( z) dzdt s th th roull umbr [3] Th followg thorm stablshs th asymptotc ormalty of th -spl srs stmator g ( t) of g (t) Thorm 34: Udr th codtos of Thorm 3, for ay t ( /( k( ) + ν ),( + ) /( k( ) + ν )] : s g ( t) ( g( t) + b( t)) N(0,) as, { } D 346 whr, s = Var ( ' ) ' ε, ( ν ) ν b( t) = g ( t) ν ( tk( ) ) /[ ν! k( ) ], ε = ( ε,, ε )' For frc about β basd o th asymptotc dstrbuto of β, a stmator of ts asymptotc covarac matrx s dd Lt Σ b gv by (6)- -(7) Th w hav th followg rsult Thorm 35: Udr th codtos of Thorm 3 w hav : ( φ ) σ φ Σ = o () p From th rsults gv Thorms 33 ad 35, t follows that: ( β β )' Σ ( β β ) χ as D p p,α Thrfor, th st { β : ( β β ) ' Σ ( β β ) χ } costtuts a largsampl 00( α )% cofdc llpsod for β For small sampl szs F p ca b usd to substtut χ p, α p, p, α / Rmark 3: y applyg th tsor-product -spl tchqu [4] th abov rsults ca b asly xtdd to th cas of multvarat rgrssor t A smulato study: Ths prsts a smulato study to valuat th ft sampl prformac of th stmators Th obsrvatos ar gratd from: y = 35x + cos( πt ) + ε, ε = ( φ + 0 ) ε +, =,,, whr, t = ( 05) /, x = 5t + 05η ad { } ar d N (0,) For a rag of valus of φ, w grat 0,000 sampls of sz 00 from th abov modl (th x j valus ar gratd oc for a fxd φ valu) ad stmat φ, β ad g( ) for ach sampl W hr us th uform kots Accordg to [4] uform kots ar usually suffct wh th fucto g( ) dos ot xhbt dramatc chags ts drvatvs Thus, w just d to dtrm th umbr of kots to us W us th mthod [4] to do so ass ad sampl varacs (Var) of th smulatd stmats ar gv ɶ ar Tabls ad, whr ( ) g t, g ( t ) ad g ( t) basd o β, β ad ɶ β rspctvly

5 Tabl : Smulatd bass ad varacs of th stmators for φ ad β Am J Appl Sc, (9): , 005 φ as( φ ) Var( φ ) as( β ) Var( β ) as( β ) Var( β ) as( Tabl : Smulatd bass ad varacs of th stmators for g( ) ɶ β ) Var( ɶ β ) T g(t) as( g ) Var( g ) as( g ) Var( g ) as( (30-05)/ (60-05)/ (90-05)/ (0-05)/ (50-05)/ (70-05)/ gɶ ) Var( gɶ ) From Tabl w ca s that all cass th smparamtrc gralzd last squars stmator β has smallr bassmparamtrcs tha th smparamtrc ordary last squars stmator β Th advatag of β ovr β s mor sgfcat wh φ s larg (hgh sral corrlato), as o would xpct sc β taks th sral corrlato to accout whras β dos ot Morovr, as φ crass, th bas ad varac of β dcras, but ths s ot th cas for β I addto, th prformac of β s clos to that of th φ-kow dmostratd that th smparamtrc gralzd last squars stmator s mor ffct tha th smparamtrc ordary last squars stmator Appdx: Proofs of Thorms: I ordr to prov th thorms prstd arlr w frst troduc svral lmmas Lmma : Suppos that a fucto f ( ) satsfs Assumpto Th w hav: t [0,] k( ) k( ) k( ) k( ) sup f ( t) ( ' ) ' f = ν O( k( )) + O( k( ) ) smparamtrc gralzd last squars stmator β across th valus of φ Tabl also shows that th stmator φ of φ s adquat From Tabl w ca s that th oparamtrc stmator g basd o th smparamtrc gralzd last squars stmator β s bttr tha th oparamtrc stmator g basd o th smparamtrc ordary last squars stmator β trms of bas ad varac CONCLUSION I ths artcl w hav studd th stmato problm of a partly lar rgrsso modl wth blar tm srs rrors Usg -spls to approxmat th oparamtrc compot, w hav costructd th smparamtrc ordary ad gralzd last squars stmators of th paramtrc compot, th last squars stmator of th autorgrssv paramtr, ad th -spl srs stmator of th oparamtrc compot W hav also drvd th asymptotc ormalty for ths stmats I both thory ad smulato, w hav whr f = (f(t ),, f(t ))' Lmma : For th bass of -spls, ν w hav: +ν, for all t; ad = ' All gvalus of c ad c 0 < c < c < Proof: Applyg Lmmas ad, th proof of Lmma 3 s smpl W hr omt th dtal 347 S ar btw for som costats Th proofs of Lmmas ad ca b foud [5] Lmma 3: Suppos that Assumptos ad hold Th as, X ' M X ad k ( ) ' ( φ) k( ) X M Ω M X k ( ) k ( )

6 Am J Appl Sc, (9): , 005 Cosdr th ormalzd wghtd sum of radom varabls { ε } dfd (), amly, / a aε = = costats It has th followg asymptotc proprty, whr {a } s a squc of ral Lmma 4: If h a a α for h = ±, ±,, + h h k= boudd, ad Assumpto 3 holds, th: / ε D σ = = whr a a N(0, ) as, σ α γ ( h) h= h h ad γ ( h) σ φ ( φ θ σ ) autocovarac fucto of { } {a } s = s th ε Cosqutly: ( β β ) = ( X ' M X ) X ' M ε + ( X ' M X ) X ' M k ( ) k ( ) k ( ) k ( ) g = U U U ε + o ( ' ) ' p() Th cocluso of Thorm 3 th follows from Lmma 3 ad Assumpto Proof of Thorm 3: It s asy to s that th followg quato holds: ε ε ε ε = ( ε ε )( ε ε ) + = = = ( ε ε ) ε + ( ε ε ) ε = = (A) Accordg to th dfto of ε (3), w hav: Proof: Accordg to th proprty of { ε } ad Lmmdffcult [6], t s ot dffculty to complt th proof W ar ow rady to prov th ma rsults Proof of Thorm 3: y () t s asy to s that ' ' ' u M ε = u ε u P ε + h ' ( I P ) ε, =,,, whr h = ( h ( t ),, hp ( t ))' Lt c0 = σ ( φ θ σ ) Sc ' ' ' 0 0 max Var( u P ε) = c u P Ω( φ) P u c λ [ Ω ( φ)] u P u = O( k( ) + ν ) = o( ), whr, λ max ( ) matrx, w hav dots th maxmum gvalu of a ' / p u P ε = o ( ) Morovr, by Lmma, thr xsts a ral vctor η such that: k( ) max h ' η = O( k( )) + O( k( ) ν ) Ths mpls: ' ' ' ε = ( ) 0 Ω φ ( ) ( ) 0λmax Ω φ k k k k ( ) Var( h M ) c h M ( ) M h c [ ( )] h M h ν = O( k( ) ) + O( k( ) ) ' / p Hc h M ε = o ( ) It follows that / = ε + p X ' M ε U ' o ( ) Nxt, by Lmma, thr xsts a ral vctor π such that: max g( t ) '( t ) π = O( k( )) + O( k( ) ν ) Thrfor ' ν k ( ) x M g = O( k( ) ) + O( k( ) ) for g = (g(t ),, g(t ))', whr x = ( x,, x )' 348 ( ε ε ) ε = ( β β )' x ε = = = = ( β β )' X ' ( ' ) ( t ) ε + [ g( t ) '( t )( ' ) ( t ) g ] ε = ε ' ( ' ) ( t ) ε = I I + I I, say 3 4 (A) y th root- cosstcy of β ad th proof of Thorm 3, w hav I = O p () Nxt, lt ( ( t ),, ( t ))', = (,, )' = ε ε ε, ad Ω ( φ ) = Var ( ε ) Th th Cauchy-Schwarz qualty gvs I ( ' ) ' ( )' X ' P X ( ) I I, say ' ε k( ) k( ) ε + β β β β = + Sc E[ I] = O( k( )), w hav I = O p ( k( )) y Thorm 3, t s asy to s I = O p () So / I = op ( ) Th Lmma ylds E[ I ] O( k( ) ν = ) + O( k( ) ) so 3 / ν / that I3 = O( k( ) ) + O( k( )) Th proof of I4 = Op ( k( )) s smlar to that for I Thrfor, by (A), th mddl trm th rght had sd of (A) s o p ( / ), ad smlar argumts show that th othr two trms ar o p ( / ) as wll Cosqutly, / ε ε = εε + p( ) = = hav / ε = ε + p( ) = = o y th sam raso, w o It follows that: ( φ φ) = ε εε φ + op () = =

7 Combg ths wth th rsult of [7], w complt th proof Proof of Thorm 33: y th dfto of β, Lmmas to 3 ad Thorm 3 w hav: β β U φ U U φ ε o ( ) = ( ' Ω ( ) ) ' Ω ( ) + p() Morovr, smlar to th proof of Thorm 43 [0], w hav: D ( U ' Ω ( φ) U ) U ' Ω ( φ) ε ( φ ) σ N0, as φ Thus th proof s complt Proof of Thorm 34: Combg th root- cosstcy of β wth Thorm [8] w ca asly complt th proof Proof of Thorm 35: Ths thorm follows from Lmma ad th proofs of Thorms 3 33 REFERENCES Ch, H, 988 Covrgc rats for paramtrc compots a partly lar modl A Statst, 6: Spckma, P, 988 Krl smoothg partal lar modls J Roy Statst Soc Sr, 50: Ch, H ad J Shau, 994 Data-drv ffct stmato for a partal lar modl lar modl A Statst, : Doald, ad K Nwy, 994 Srs stmato of smlar modls J Multvarat Aal, 50: Hamlto, A ad KTruog, 997 Local lar stmato partly lar modls J Multvarat Aal, 60:-9 6 Egl, RF, W J ragr, J Rc ad A Wss, 986 Smparamtrc stmats of th rlato btw wathr ad lctrcty sals J Amr Statst Assoc, 80: Schck, A, 994 Estmato of th autocorrlato coffct th prsc of a rgrsso trd Statst Probab Ltt, : ao, JT, 995 Asymptotc thory for partly lar modls Comm Statst Thory Mthods, A4 8: Tog, H, 990 Nolar Tm Srs Oxford Uvrsty Prss, Oxford 0 Hwag, SY ad I V asawa, 993 Paramtr stmato a rgrsso modl wth radom coffct autorgrssv rrors J Statst Pla Ifr, 36: Am J Appl Sc, (9): , ra, AK ad X L Zuo, 996 Spcfcato tst for a lar rgrsso modl wth ARCH procss J Statst Pla Ifr, 50: Dutta, H, 999 Larg sampl tsts for a rgrsso modl wth autorgrssv codtoal htroscdastc rrors Comm Statst Thory Mthods, A8: You, JH ad Ch, 00 Paramtr stmato partally lar rgrsso modls wth radom coffct autorgrssv rrors Comm Statst Thory Mthods, 3: Hwag, SY, 995 A study o a rgrsso modl wth olar tm srs rrors Kora J Appld Stat, 9: Rubbrt, A, A Isdor ad P d'allsadro, 97 Thory of lar Dyamcal Systms rl: Sprgr-Vrlag 6 Mohlr, RR, 973 lar Cotrol Procsss Nw York: Acadmc Prss 7 Rao, TS, 98 O th thory of blar tm srs modls J Roy Statst Soc Sr, 43: Da, Y ad L llard, 998 A spac tm blar modl ad ts dtfcato J Tm Sr Aal, 9: Da, Y ad L llard, 003 Maxmum lklhood stmato spac tm blar modls J Tm Sr Aal, 4: Sacks, J ad D Ylvsackr, 970 Dsg for rgrsso problms wth corrlatd rrors III A Math Statst, 4: d oor, C, 978 A practcal ud to Spls Sprgr-Vrlag, Nw York Moyd, R A ad P J Dggl, 994 Rats of covrgc sm-paramtrc modlg of logtudal data Austra J Statst, 36: arrow, DL ad PW Smth, 979 Effct L approxmato by spls Numr Math, 33: H, X ad P D Sh, 996 varat tsorproduct -spl a partly lar modl J Multvarat Aal, 58: urma, P, 99 Rgrsso fucto stmato from dpdt obsrvatos J Multvarat Aal, 36: Fullr, WA, 996 Itroducto to statstcal tm srs Wly, Nw York 7 Fg, PD ad RLTwd, 985 Radom coffct autorgrssv procsss: A Markov cha aalyss of statoarty ad ftss of momts J Tm Sr Aal, 6: Zhou, S, X Sh ad D A Wolf, 998 Local asymptotc for rgrsso spls ad cofdc rgos Aal Statst, 6: