The periodogram of fractional processes

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1 The periodogram of fracioal processes Carlos Velasco y Deparameo de Ecoomía Uiversidad Carlos III de Madrid December 4, 2005 Absrac We aalyze asympoic properies of he discree Fourier rasform ad he periodogram of ime series obaied hrough (rucaed) liear lerig of saioary processes. The class of lers coais he fracioal di erecig operaor ad is coe cies decay a a algebraic rae, implyig log-rage depedece properies for he lered processes whe he degree of iegraio is posiive. These iclude fracioal ime series which are osaioary for ay value of he memory parameer ( 6= 0) ad possibly osaioary redig ( 0:5). We cosider boh fracioal di erecig or iegraio of wealy depede ad log memory saioary ime series. The resuls obaied for he momes of he Fourier rasform ad he periodogram a Fourier frequecies i a degeeraig bad aroud he origi are weaer compared o he saioary orucaed case for > 0; bu su cie for he aalysis of parameric ad semiparameric memory esimaes. They are applied o he sudy of he properies of he log-periodogram regressio esimae of he memory parameer for Gaussia processes, for which asympoic ormaliy could o be showed usig previous resuls. However oly cosisecy ca be showed for he redig cases, 0:5 < : Several deredig ad iiializaio mechaisms are sudied ad oly local codiios o specral desiies of saioary ipu series ad rasfer fucios of lers are assumed. MSC: Primary: 62M5; Secodary: 62M0, 60G8 Keywords: Discree Fourier rasform; Log-rage depedece; Log memory; Nosaioary series; Log-periodogram regressio; Asympoic ormaliy. This paper is a revised versio of he worig paper "Log-periodogram iferece for osaioary iegraed ad di ereced processes". y Correspodece address: Deparameo de Ecoomía, Uiversidad Carlos III de Madrid, Calle Madrid 26, Geafe (Madrid), Spai. Fax: Carlos.Velasco@uc3m.es. Research fuded by he Spaish Miiserio de Educació y Ciecia, Ref. o. SEJ /ECON.

2 Iroducio Mos models for osaioary redig ad log memory ime series are based o some uderlyig wealy depede saioary processes. Thus for saioary log memory or log-rage depede processes i is ofe assumed ha x = ( L) d " ; 0 < d < 0:5; = ; 2; : : : ; where he fracioal di erece ler ( formal expasio, 6= ; 2; : : : L) d is de ed i erms of he lag operaor L hrough he ( L) d := =0 ' (d) L ; ' (d) = ( + d) (d) ( + ) ; where (x) = R 0 z x e z dz is he Gamma fucio such ha (d) = for d = 0; ; 2; : : :. Here ", = 0; ; : : : ; is assumed o be covariace saioary ad wealy depede accordig o some sadard de iio, which usually eails ha he process " has a posiive bouded specral desiy a zero frequecy. For isace, Hosig (98) ad Grager ad Joyeux (980) origially proposed ha " is a ARMA process. The x is ermed iegraed of order d, is covariace saioary wih ie variace uder d < 0:5, ad is specral desiy has a pole (zero) a zero frequecy if d > 0 (d < 0). Tredig observaios ca be modelled by parial sums (Sowell (990); Hurvich ad Ray (995); Velasco (999)), y = y 0 + x ; = ; 2; : : : ; (2) = leadig o di erece-saioary models where he icreme series ( L)y = x is saioary iegraed of order d, 0:5 d < 0:5, saisfyig. The we say ha y is iegraed of order d, 0:5 d < :5. Usig successive parial sums i is sraighforward o de e higher order iegraed series for ay d 0:5. Aleraively, i is possible o cosider (e.g. Robiso ad Mariucci (200), Phillips (999)) ha a process is geeraed by a rucaed lerig as = ' ; > 0; = ; 2; : : : ; (3) =0 so all he pas wealy depede saioary iovaios ; 0, are igored. This coveio maes esseial he dae of sar of observaio. However his framewor ca easily be geeralized seig a warmig up period where he i ow of iformaio ca begi before we acually observe he process. The lered process ; hough wih ie variace for xed, is o-saioary for ay value of 6= 0: However if < 0:5, i coverges i mea square as! o he covariace saioary x obaied i for he same sequece of iovaios = ", = ; : : : ;. Usig Sirlig s formula, we d ha ' as!, where meas ha he raio of lefad righ-had sides eds o oe. Therefore, whe 0:5 he variace of ca grow wihou limi wih ad is osaioary log-rage depede i he sese of Heyde ad Yag (997). Of course, boh ideas ca be combied o de e fracioal iegraed processes, achievig complee geeraliy i he geeraio of ime series wih log rage depede behaviour. 2

3 The log-rage properies of he processes -(2) ad (3) are described by he memory parameers d ad, ad uder regulariy codiios ad appropriaely ormalized, such processes coverge o di ere versios of fracioal Browia moio of parameers d > 0:5 ad > 0:5 respecively (see Mariucci ad Robiso (999) for a discussio). The memory parameers ca be esimaed by a variey of mehods uder geeral assumpios. Mai focus i he lieraure has bee paid o saioary log memory series (e.g. Fox ad Taqqu (986), Robiso (995a, b)), hough di erecesaioary models have bee also cosidered (e.g. Velasco (999), Velasco ad Robiso (2000)). Usig he oher aleraive, Phillips (999) cosidered asympoic properies of he discree Fourier rasform (DFT) of give by (3), where is a liear process wih coe cies i he Wold decomposiio associaed o a uiformly smooh specral desiy, exedig ideas of Phillips ad Solo (992) i erms of a Barle s approximaio o he DFT of he liear iid iovaios. Robiso (2005) cosidered bouds bewee he DFT of ad is saioary versio x give by whe = d 2 [ 0:5; 0:5) ad " =, ad sudied he e ec of aperig whe hese series are possibly furher fracioally iegraed o achieve osaioary memory levels. These resuls are used o derive he properies of di ere parameer esimaes which are liear fucios of he periodogram, ad whose aalysis had bee coduced previously for (di erece) saioary processes. I his paper we geeralize ad improve he previous resuls o he momes of he DFT ad he periodogram for frequecy domai iferece o fracioal series give by (3) followig a di ere roue. I a similar spiri o Robiso (995a), we sudy direcly he DFT of he observed series ad se our local codiios i erms of he specral desiy of he wealy depede iovaios ad of a geeral osaioary log rage depede ler as i Robiso ad Mariucci (200). We also cosider i his paper several exesios of model (3). We cosider simulaeously series wih egaive memory ( < 0), which are releva for saisical iferece o fracioally di ereced daa; processes wih lers iiialized a a poi di ere from zero; ad fracioal di erecig ad iegraio of saioary log memory ime series wih = x give, e.g. by wih 0 < d < 0:5. Our resuls are weaer ha he oes ha ca be obaied i he orucaed case for boh < 0:5 ad 0:5 (cf. Robiso (995a) ad Velasco (999), respecively). However, hey are su cie o usify valid large sample log-periodogram iferece o ; < 0:5 (Gewee ad Porer- Huda (983), Robiso (995a)) for Gaussia processes, hough we ca oly obai cosisecy resuls whe 0:5 <. The resuls allow for saioary log memory iovaios, log-periodogram regressio esimaig he overall memory of he resulig ime series i his case. Previous resuls i Phillips (999) ad Robiso (2005) could o usify such asympoic properies. Our local codiios o he specral desiy of are also weaer ad more geeral ha hose of Kim ad Phillips (999) who sudied his semiparameric esimae whe i a relaed se-up followig Phillips (999) aalysis. The resuls obaied i his paper are also useful for oher problems ad, for isace, he arrow-bad While esimae ivesigaed i Robiso (995b) for saioary log memory liear processes has bee aalyzed i Marmol ad Velasco (2004) uder (3) for liear ad < 0:5. The res of he paper is orgaized as follows. Nex secio aalyzes he covariace properies of he DFT of fracioally iegraed series, while Secio 3 cosiders he log-periodogram esimae of : Secio 4 discusses he iiializaio of he process ad di ere deredig mechaisms. The case whe is a log-memory (saioary) process is sudied i Secio 5. Some echical lemmas ad proofs are coaied i a appedix. 3

4 2 Discree Fourier rasform of fracioally iegraed ime series We follow he same oaio as i Robiso ad Mariucci (200) bu i a uivariae framewor o simplify he preseaio. We cosider liear lers more geeral ha he fracioal ( L), wih similar rs order asympoic properies. We cosider simulaeously he cases of posiive ad egaive memory parameer. The ler wih posiive memory parameer iduces posiive auocorrelaio for log lags ad is he releva case for mos daa i pracice, hough egaives values are liely o occur if some deredig mechaism, such as di erecig, has bee applied previously. We suppose ha he observed sequece is give by = = where he ler coe cies ; is rasfer fucio = ; = ; : : : ; ; =0 ad he process saisfy he followig Assumpio. exp (i) Assumpio.. f (0) g 2, where ; < ; is he class of sequeces f ; = 0; ; : : :g, such ha = ( = 0); ad + ; as! 0; (4) = O( ); as! 0:.2. (0) = P =0 = 0; ad 2 as! 0, < < 0:.3. is a zero mea covariace saioary process wih specral desiy f ; posiive ad coiuous i a ierval aroud = 0. I is easy o show ha he sequece ' give by he fracioal ler ( L) belogs o he class : I fac, he class is more geeral ha ha de ed i erms of he fracioal iegraio ler sice we allow for coribuios of a smaller order of iegraio ; > 0; which ca be of ieres i some applicaios regardig coiegraed ime series ad volailiy sequeces (see e.g. he survey i Velasco (2005)). This possibiliy will be e ecively accoued for i Assumpio 3.2. For < 0; he class could be de ed i erms of di ereces of lers belogig o ( + ): Thus = (+) (+) ; = ; 2; : : : ; 0 = ; as whe we ae fracioal di ereces of a ie legh vecor observed i = ; : : : ;. For > 0; he asympoic behavior of whe! 0, 4

5 is deduced i Lemma 2 i he Appedix, usig Assumpio., ad described by Assumpio.2 for < < 0: The codiios o imposed by Assumpio.3 are o resricive as we leave all he specral desiy f uparamerized excep a zero frequecy, where i is smooh, ad oly require iegrabiliy for covariace saioariy. We furher relax his codiio i Secio 5 ad cosider saioary log-rage depede whose specral desiy has sigulariies a = 0. We are maily cocered wih he asympoic properies of he DFT of ; = ; : : : ; ; w ( ) = (2) =2 = exp(i ); a he se of Fourier frequecies = 2=; i a degeeraig bad aroud he origi. These are he releva frequecies o describe log-rage properies of. To reproduce resuls parallel o Theorem 2 of Robiso (995a) or Theorem of Velasco (999), for saioary ad di erece saioary processes, respecively, we cosider a (possibly o-iegrable) bu saioary specrum for ; f = f 2 ; (see e.g. Velasco (999) for di erece-saioary processes). The specrum f is o-iegrable for 0:5 because we show i Lemma 2 (for > 0) ha f f (0) 2 ; as! 0; (5) wih 0 < f (0) < : This specrum describes he secod order dyamics properies of ad would be he referece arge for log-periodogram regressios because i equals he limi of he expecaio of he periodogram, as happes for (di erece) saioary process, see also Solo (992). The specrum f is he limi, as!, of he ime-varyig specral desiy f () = f 2 ; = exp(i); = which is direcly relaed o he expecaio of he periodogram I = w 2 : To see his we ca use ha E [w ( )w ( )] = 2 ; f d; (6) =0 see (8.) i Robiso ad Mariucci (200), where = [ ; ] ad ; = ( ; ) ( ; ); (; ) = exp (i( + )) ( = ): The we ca chec ha 2 ; d = ( ) 2 = ad, if f is smooh eough a = ; we ca approximae he periodogram expecaio E [I ( )] = E [w ( )w ( )] by f () ( ): Here, he fucio ; plays he oi role of 5

6 Feer s erel, K = (2) P = exp(i)2, ad of he fracioal rasfer fucio ' 2 = P =0 '(d) exp(i) 2 = exp(i) 2d i he (di erece) saioary case -(2) for d < :5, sice i his case E [I yy ( )] = K ( )f "" ' 2 d: The erel K eds o a Dirac s dela fucio as! ; so he above quaiy eds o f yy ( ) = f "" ( )'( ) 2 : The sudy of he properies of (6) based o he erel (; ) geeralizes usual saioary Fourier aalysis echiques for ime depede liear lers, possibly wih algebraic decay o zero. Now we impose some exra smoohess codiios o f a = 0 hrough ad f : Assumpio f is boudedly di ereiable for 2 ( ; ) ; some > 0: 2.2. is di ereiable i ( ; ) f0g, wih d d = O( ) as! 0: This assumpio is similar o Assumpio 2 of Theorem 2 of Robiso (995a), where he covariace marix of he DFT of saioary log memory series is aalyzed. I does o impose exra smoohess compared o his case ad i implies ha (d=d) log f = O( ) as! 0, i view of Lemma 2 ( > 0) ad Assumpio.2 ( < 0). We give ow a equivale resul for he DFT of a Fourier frequecies, whose proof ca be foud i he Appedix. Theorem Uder Assumpios ad 2, <, for < < m; m=! 0 as! ;.. E [w ( )w ( )] = f ( ) + O f ( ) + (log ) =0:5o ; > 0; = f ( ) + O f ( ) + log (log ) = 0:5o ; < 0:.2. E [w ( )w ( )] ; E [w ( )w ( )] ; E [w ( )w ( )] q = O f ( )f ( ) + (=) log + (log ) =0:5o ; > 0; q = O f ( )f ( ) + (=) log (log ) = 0:5o ; < 0: pf For = 0 all he bouds are replaced by O ( )f ( ) log ; : The bouds obaied i his heorem for he ormalized periodogram expecaio f ( )E [I ( )] he saioary oes, O( log ); of osaioary daa a Fourier frequecies are di ere for ay from 0:5 < < 0:5; give by Robiso (995a) ad repored here 6

7 for he = 0 case. They are also di ere from he di erece saioary resul, O( 2( ) log ); 0:5 < (see Velasco, 999), bu depede o he degree of iegraio as well. For isace, he ew coribuio i whe > 0 arises from he emporal ihomogeeiy of ; = ; : : : ; ; whe he ime ivaria specral desiy f is compared wih he ime-varyig specral desiy f () : However, whe 0:5 < < 0 he resuls i Theorem are he same as i he saioary case, up o a logarihmic facor, ad, as expeced, he e ecs of rucaio seem o be alleviaed i case of egaive memory. Robiso (2005) obaied for fracioal series as i (3) ha h f ( )E w ( ) w x ( ) 2i = O (log ) = 0:5 (7) for 2 [ 0:5; 0:5) whe x is give by for d ad he same sequece of iovaios. Usig his resul ad a riagle iequaliy i his equaio (C.4), ogeher wih Robiso (995a, Theorem 2(i)), we ca obai ha for 2 (0; 0:5) ; f ( )E [I ( )] = + O =2 : This resul is improved by he correspodig boud O( ) i our Theorem for ay < 0:5. This improveme is impora o allow for he asympoic ormaliy of he log-periodogram regressio esimae of ; for < 0:5; bu oly cosisecy ca be obaied whe 0:5; see ex secio. Usig he same argumes i he proof of Theorem ad cosiderig oly he coribuio from 2 i (2), we ca obai uder Assumpios ad 2 ha he lef had side of (7) is O + log for 2 (0; 0:5) ; see (24), which geeralizes Robiso s (2005) resul o log memory lers i he class by meas of local assumpios o he smoohess of f ad a = 0: Phillips (999) provides some probabiliy bouds o he di erece w ( ) e i w ( ); > 0; for fracioally iegraed a each frequecy ; eiher xed or i a degeeraig bad. Apar from some correcio erms, similar o he oes discussed i Secio 4 below, which are O p =2 whe < 0:5; he remaider is O p, which would correspod o a similar boud o he oe i our Theorem., > 0. Taig hose bouds as uiform, some properies of di ere esimaes are deduced i Phillips (999), ad some ad-hoc modi caios of he usual semiparameric procedures of memory esimaio ca be proposed, see also Phillips ad Kim (999). However, oe ha for he aalysis of he properies of saisics which are oliear fucios of he periodogram of a observed fracioally iegraed series, such as he log periodogram memory esimae sudied i ex secio, he aalysis of di ere approximaios of he DFT of daa is o su cie. Uiform bouds o he periodogram expecaios of uder Gaussiaiy are required i his case because i is o eough o corol he (liear) disace wih he DFT of he saioary versio of he origial process, cf. (7). For he aalysis of he periodogram properies a all frequecies i [0; ], icludig hose xed for which is of he same order as, we eed o mae Assumpio 2 uiform for ay 2 [0; ]: The he resuls of Theorem will hold ad full parameric esimaes for osaioary redig daa could be sudied as i Velasco ad Robiso (2000). I ex secio we cocerae o a paricular semiparameric esimae uder our local codiios. 7

8 3 The log-periodogram regressio esimae of The semiparameric esimae of he memory parameer based o he log-periodogram regressio esimae was proposed by Gewee ad Porer-Huda (983). Robiso (995a) showed ha he periodogram of saioary log memory series is asympoically ubiased ad ucorrelaed, as wih shor memory, whe evaluaed a he harmoic frequecies for growig wih sample size. This is he basis o wrie he logarihm of (5) as a liear regressio model wih approximaely homoscedasic ad ucorrelaed errors, log I ( ) log f (0) 2 log + log I ( ) ; = ; : : : ; m; (8) f ( ) where m is small compared wih : The log-periodogram regressio esimae b is he leas squares esimae of ; 0 m b (r r) 2 A = m = (r r) log I ( ); where he log-periodogram is he depede variable ad he regressor is r = 2 log, r = m P m = r. The asympoic properies of b were aalyzed rigourously for muliple saioary Gaussia series ( 0:5 < < 0:5) by Robiso (995a). He cosidered a poolig of coribuios from adace frequecies o achieve e ciecy gais ad showed ha he periodograms a he very rs frequecies do o have ice asympoic saisical properies, so followig Küsch (987) he proposed o exclude he rs ` frequecies from he regressio. Hurvich, Deo ad Brodsy (998), HDB heceforh, have show uder some addiioal codiios ha rimmig of very low frequecies may o be ecessary for he aalysis of he asympoic properies of b. HDB oly cosider fracioal processes wih wealy depede iovaios which possess a specral desiy wih hree bouded derivaives a = 0, ad he regressor i (8) which arises aurally for fracioal processes, z = log(4 si 2 ( =2)); as origially proposed by Gewee ad Porer-Huda. However, we geeralize heir resuls uder he followig assumpio. We say ha a fucio g (x) is Hölder, 0 < ; i a ierval of he origi if g (x) g (0) Cx uiformly for x 2 ( ; ) ; for some > 0 ad C < : Assumpio 3 Furher o Assumpio 2, we se 3.. For some 2 (; 3], for i a ierval of he origi, f is di ereiable wih f 0 Hölder( ), < 2; f is wice di ereiable wih f 00 Hölder( 2), 2 < For some 2 (0; 3], 2 = 2 ( + O( )) as! 0, 0 < 2; 2 = 2 ( + A 2 + O( )) as! 0, 2 < 3: Assumpio 3 imposes a rae o he approximaio (5), beig Assumpio 3.2 similar o Robiso s (995a) Assumpio. The reaso for cosiderig values of ad larger ha 2 is o esimae explicily he bias of ^, hough cosisecy oly requires mi f; g > 0 ad he bes possible rae for he asympoic bias is already achieved by mif; g = 2. Thus for fracioal ime series, we 8

9 have ha 2 = (2 si =2) 2 = O( 3 ) ; as! 0; so = 3 ad followig Robiso (995a), for hese models we have ha f = f (0) 2 f 00 (0) + f (0) O( ) ; as! 0; 2 for > 2, usig ha f(0) 0 = 0. For example, he presece of addiive oise i of smaller order of iegraio ha would lead o values of smaller ha 2. Therefore, we ca adap he bias esimaio resuls of HDB, hough hey de ed he log-periodogram regressio esimae usig he fracioal regressors z. Theorem 2 Le 0:5 < < 0:5. Uder Assumpios, 2 ad 3, = mif; g 2 (2; 3], ad m + (m ) 2 log m! 0 as! ; (9) for Gaussia E(^) = f 00 (0) m 2 f (0) + A 2 ( + o) + m Var(^) = 2 24m + o(m ); where m := O(m log m) >0 + O(m log 2 m) <0 + O(m log 3 m) =0 : Proof of Theorem 2. Follows from HDB, us usig our Theorem isead of Theorem 2 or Robiso (995a). The all Lemmas of HDB hold uder he codiios of he heorem, us seig i heir Lemma 3 ha = +(=) log ; 0 < < 0:5; log m < m; m +m! 0 ad he correspodig modi caios for 0 from Theorem. The HDB s Lemma 4 follows ow from he appropriae modi caio of Lemma 4. of Robiso ad Mariucci (200). I Lemmas 6 ad 7 of HDB we obai ha E[" ] = where = O ; > 0; = O ; < 0; ad Var[" ] = 2 =6 + O( ); log m m; whereas i Lemma 8 of HDB, he boud is ow O(m log m); > 0; O(m log 2 m); < 0; uder (9). The codiio (m ) 2 log m! 0 ad our Hölder assumpio are used isead of hree bouded derivaives i HDB, while he codiio m log m! 0 i HDB is implied by (9), 3: If we use z isead of r i he log-periodogram regressio ad 2 = e i 2 ; he he erm A does o show up i Theorem 2. Whe 0:5; i is o possible o show he p m-cosisecy of ^; bu us adapig he mehods of he previous heorem we ca prove he followig resul which gives codiios o m for he cosisecy of ^ for ay value > 0 ad 0:5 < <, 6= 0:5. Theorem 3 Le 0:5 < <, 6= 0:5. Uder Assumpios, 2 ad 3, = mif; g 2 (0; 2], ad m + (m ) log m! 0 as! ; (0) 9

10 for Gaussia E(^) = O (m ) log m + m ; Var(^) = O m + O m 2( ) log 2 m : The badwidh codiios (9) ad 0 hold if m K a for 0 < a < : However we are o able o show he cosisecy whe = 0:5 ad 0:5 ad all Fourier frequecies ; : : : ; m are used because of he erms growig wih log ad i Theorem. However, if we wish o rim ou he rs ` frequecies i he log-periodogram regressio, for some ` growig slowly wih m bu faser ha log i he asympoics, he i is easy o adap HDB s argumes ad a similar resul o Theorem 3 is valid also for = 0:5 ad 0:5: Fially if 0:5 < < 0:5 we ca adap Robiso s (995a) ceral limi heorem as i HBD o avoid ay rimmig of low frequecies, ad usig a badwidh m K a such ha 0 < a < 2=(2 + ): Theorem 4 Le 0:5 < < 0:5. Uder Assumpios, 2 ad 3, = mif; g 2 (0; 2], ad m log 2 + m 2+ 2! 0 as! ; for Gaussia p m(^ )! d N(0; 2 24 ): Proof of Theorem 4. Follows as Theorem 2 of HDB from Robiso (995a), adapig for a geeral smoohess codiio o f i erms of. The aalysis of o-gaussia ad apered series ca also be pursued usig he ideas pu forward here (see Velasco (2000), Hurvich, Moulies ad Soulier (2002)). 4 Periodogram modi caios ad iiial codiios I his secio we discuss some modi caios of he periodogram proposed i he lieraure, which are aleraives o he usual deredig procedure of osaioary series cosisig o aig di ereces. We also brie y explore he implicaios of he de iio of fracioally iegraed processes i erms of a rucaed ler i he previous asympoic heory. Phillips (999) proposed he followig correcio for he DFT a each usig a represeaio of he DFT ha ivolves he las observaio (we assume o iiial codiios, 0 0 i his oaio), w ( ) := w ( ) + p e i 2 e ; 6= 0mod: i Phillips (999) ad Kim ad Phillips (999) moivaed such correcio i erms of improvig he properies of he DFT for redig processes wih memory aroud = ; ad i is suiable o deal wih he cases where, sice i his case he usual log periodogram regressio esimae is o cosise. I fac, his correcio ca be see as a liear deredig i he ime domai seig = ; = ; : : : ; ; 0

11 so = 0 ad w ( ) is acually he DFT of. If we se addiioally = 0; so ( is easy o chec ha e i w ( ) = w ( ) e ; i L) = 0; i for he icremes = ( L). Therefore, he aalysis of Secios 2 ad 3 is valid for osaioary series wih < 2 replacig by = because, accordig o (5), has memory ad we ca se := e i ( e i ) where = ( e i ). I his sese he rasformaio is equivale o aig rs di ereces, ad higher order correcios ca be developed for furher di erecig. However whe 0 < < ad he modi ed DFT is used isead of w ( ), he implici process has egaive memory (because ypically has) ad he properies of is DFT ad log-periodogram esimaes are immediae cosequeces of our previous resuls wih < 0. We ow explore he iiial codiios problem. Le assume for some T for = (T ) ; = = T : This icludes he se-ups of Robiso ad Mariucci (200), T = he followig model ; Phillips (999), T = 0; ad oher possibiliies wih T xed wih or wih T icreasig wih, e.g. T = b ; b > 0; which implies ha he srech of iiial codiios icreases wih he widow of observed daa. As far as T is xed, i is easy o chec i he Appedix ha all he resuls of Secios 2 ad 3 hold as if T = us modifyig he boud for 2 +T + 2 used i he proof of Theorem. We oly cosider he case where 0; sice egaive values of usually origiae from some di erecig process give a sample of size : Corollary The coclusios of Theorems -4 are valid for ay T 0, 0: I fac, if T is xed or O(), he asympoic bouds give i Theorem ca o be easily improved furher, bu if T is icreasig fas eough wih ; some improvemes seem possible. I paricular, he boud O( ) i Theorem, > 0; ca be muliplied by a facor (T ) which is o as! if = o(t ); 0 < < : Noe ha if T! as! (iiial codiios growig faser ha observed daa) he we are closer o he (di erece) saioary framewor of Hurvich ad Ray (995) ad Velasco (999), ad he coribuio of he pas iovaios o each observaio, P (T ) := P 0 = T = P T + =, ca be showed o be O p ( 2 ); 0 < < 0:5; or O p ((T + ) 2 ) for 0:5; uder some regulariy codiios o : Thus, assumig ha Cov[ ; ] = O( ); > ; we have

12 ha Var [P (T )] = T + T + = = Cov[ ; ] T + C T + = =+ T + C = T + 2( ) = T + C 2( ) ; = ad he claim follows. For redig observaios wih > 0:5; he ime-depede iiial codiios ca have a domia coribuio compared o he iformaio accumulaed i he curre periods ; 2; : : : ; ; as log as =T! 0: 5 Fracioal iegraio ad di erecig of saioary lograge depede ime series We cosider i his secio he siuaio where a ie observed srech, = ; : : : ;, of a saioary log memory ime series x, wih memory 0:5 < d < 0:5 is fracioally iegraed (di ereced) of order > 0 ( < 0). These are he basic operaios performed whe usig fracioal values of o boh asympoically saioary ad redig processes, where we use a esimae close o d such ha he lered series is ear saioary ad shor-rage depede. We oly cosider 0:5 < 0:5 sice he fracioal lers are cumulaive i he sese ha if 2, ; > 0 he =0 2 ( + ); 2, ad (cf. Robiso ad Mariucci s (200) Lemma 3.), as moivaed i Secio 2 whe preseig he lers wih egaive. Oherwise we ca rs iegrae or ae di ereces, ad he properies of he DFT of hese processes ca be obaied as a direc cosequece of he resuls of his secio. Assumpio 4 For f g 2 as i Assumpio, <, = x ; (2) = wih d d = O( ) as! 0; ad where x is covariace saioary wih specral desiy saisfyig f xx = O( 2d ); d < 0:5; ad d d f xx = O(f xx ) as! 0: 0:5 2

13 The class of processes x de ed Assumpio 4 is more geeral ha, sice we do o require x o be a fracioal series, oly o have a specral desiy wih a similar asympoic behaviour as! 0. Therefore, he se of fracioally iegraed processes described by Assumpio 4 is more geeral ha ha of Phillips (999) where oly uiformly smooh specral desiies f xx are cosidered, so x is ecessarily shor memory. Usig our previous de iio, i holds ha f = 2 f xx G 2(d+) as! 0; ad i order o esimae he expoe h := d + we aalyze he asympoic properies of he momes of he DFT of log memory -fracioally iegraed or di ereced ime series. I order o dered a observed series, he ideally d; so h 0; ad upper bouds o h should o be very resricive i pracical applicaios. The followig heorems ca be cosidered ypically for series wih d < 0 ( > 0) ad d > 0 ( < 0) respecively, for which h is small. The proof is i he Appedix. Theorem 5 Uder Assumpio 4, 0:5 < 0:5, d 2 [ 0:5; 0:5); h < 0:5; h = d + ; for < < m; m=! 0 as! ; 5.. E [w ( )w ( )] = f ( ) + O f ( ) ; > 0; = f ( ) + O f ( ) (log ) = 0:5 ; < 0: 5.2. E [w ( )w ( )] ; E [w ( )w ( )] ; E [w ( )w ( )] q = O f ( )f ( ) + (=) +d log o ; > 0; q h = O f ( )f ( ) (log ) = 0:5 + (=) +d log i ; < 0: pf For = 0; all he bouds are O ( )f ( ) log ; : Noe ha whe d = 0 we basically recover he bouds of Theorem, us chagig by i he secod par of his heorem, because Assumpio 4 replaces he bouded di ereiabiliy of f a he origi, so ow (d=d) log f xx = O : Furhermore, we also recover he resuls from Robiso s (995a) Theorem 2 for saioary log memory ime series ( = 0), excep for he erm whe > 0; so i is sill possible o reproduce all he resuls of HBD o he asympoic properies of he log-periodogram esimae of he memory h = + d for saioary Gaussia ime series wih memory d possibly di ere from zero, iegraed or di ereced -imes (by he ( L) operaor i = ; : : : ; ). To esimae he MSE ad obai he asympoic disribuio of ^h we iroduce his ew assumpio, which geeralizes Assumpio 3. We omi he proof because is similar o ha of previous resuls. Assumpio 5 Furher o Assumpio 4, we se f xx = 2d f xx ad 5.. For some 2 (; 3], 2 (0; ); > 0; f xx is di ereiable wih f 0 xx Hölder( ), < 2; f xx is wice di ereiable wih f 00 xx Hölder( 2), 2 < 3: 3

14 5.2. For some 2 (0; 3], 2 = 2 ( + O( )) as! 0, 0 < 2; 2 = 2 ( + A 2 + O( )) as! 0, 2 < 3: Theorem 6 Le < 0:5; h = d + 2 ( 0:5; 0:5). The he coclusios of Theorems 2 ad 4 hold for he log periodogram esimae ^h of Gaussia series (2) uder Assumpio 5 (replacig f by fxx), for 2 (2; 3] ad 2 (0; 2] respecively; = mi f; g ; ad he same choices of m: The heory developed i his secio does o iclude he siuaios where a iiial esimae bd is compued ad he sequece ^d ; = ; : : : ;, is obaied by fracioal di erecig. This operaio should approach he wea depede iovaios d if ^d coverges fas eough o he rue d; bu i geeral he limi migh deped o he disribuio of ^d. This problem also relaes o memory esimaio based o residuals, obaied eiher from deermiisic ime deredig or by sochasic deredig i coiegraed sysems, as is aalyzed i Marmol ad Velasco (200) ad Hassler, Marmol ad Velasco (2006) respecively. Similar echiques o hose of his paper ca also be used for he aalysis of log-rage depede ime series geeraed by di ere simulaio algorihms. 6 Appedix: Proofs of resuls I he followig lemma we collec some echical resuls abou ad o be used i he proof of he resuls coaied i his appedix ad which ca be of idepede ieres for Fourier aalysis wih osaioary lers. Noe ha he boud for improves upo (8.3) of Robiso ad Mariucci (200), wha is ey for may of our resuls. Lemma 3 below will exed hese resuls for < 0. I he sequel C deoes a geeric cosa ha may chage each ime is used. Lemma 2 Uder Assumpio, for f g 2 ; 0 < < ; as! 0; 2 2 ; C mi f ; g ; C mi ; ; where = ; ad (; ) C mi ; + mi f ; [ + ]g : Proof of Lemma 2. The bouds for 2, ad are a direc cosequece of Lemma 3.2 i Robiso ad Mariucci (200). To obai he bouds for (; ); rs we have ha, wih C ; (; ) = e i(+) ( ) C ( = while, usig C ; = ) C C + ; (3) = (; ) C = C : (4) Now, de ig D (x) = P = eix ad usig summaio by pars, (; ) = ( )D ( + ) e i D ( + ) (5) = 4

15 (compare o (8.2) i Robiso ad Mariucci (200), which we are o able o reproduce here, ad wih (8.3), which is a sraighforward cosequece of our resuls). The, usig ha D (x) mifx ; g; we have ha ( ca obai ha e i D ( + ) C + C + = )D ( + ) C mif ; g mif + ; g: Similarly we = = C + : (6) Fially, usig ha D (x) = e ix e ix = e ix we have ha he secod erm i (5) is e i(+) e i(+) f ( )g (7) ad is modulus is bouded by C + f + g ad he lemma follows from his boud ad (3), (4) ad (6). Lemma 3 Uder Assumpio, for f g 2, < < 0; C f + g; C mi ; ; ad (; ) C f + + g mif; + g. Proof of Lemma 3. Firs we have for he bouds of ; = = 0 e i = = < < 0; ha A = O( ); while also = = + 0 e i = a= = 2 A = O : The, because (0) = P =0 = 0 we have ha (0) = P = = O( ); ad also = (0) + O 0 ( ) for some 2 [0; ]: The, we ca chec ha 0 ( P ) = O =0 = + ; so = O + + = O ( + ) : Therefore, if ; he = O ( ) ; while if > ; + C + C ; so C f + g : For he aalysis of (; ) we ca use (5) ad (7), wih D (x) Cx ; obaiig ha (; ) C + + C + f + g C C C ; ad usig D (x) C ad (x) ; (; ) C + + C C + : 5

16 Proof of Theorem. We oly prove he bouds for he rs wo expecaios, sice he proof for he res is similar bu simpler. We rs deal wih he case > 0: Proceedig as i equaio (8.2) of Robiso ad Mariucci (200), ; d = = e i( ) p pe i( p) D mi(;p) ( ) = 2 = 2 = e i( ) a=0 a e ia b=0 b e ib e i(+ )( ) ( ) ( ): (8) = The E [w ( )w ( )] f ( ) = 2 ; ff f ( )g d (9) ( ) +f ( ) ( ) 2 ( ) 2 ; (20) = because R ;d = 2 P = ( ) 2 : Wriig 2 2 = ( ) + ( ) + 2 ; (2) we obai from Lemma 2 for > 0; 2 2 C + 2 mif; () 2( o ) g ; (22) (compare o discussio o p. 968 of Robiso ad Mariucci (200)) ad herefore ( ) 2 ( ) 2o (23) C o 2 ( ) + mif; ( ) 2( ) g 8 9 C < 2 : = 2( ) + + ( ) ; = = = O f ( ) + ( + log f = 0:5g) + 2( ) 2( )o (24) = O f ( ) + log f = 0:5g : (25) I view of (25), (20) is O f ( ) + log f = 0:5g ; > 0 Fix oe > 0 such ha f is boudedly di ereiable i he ierval ( wrie (9) as ; ). The we ca 2 ; ( + ) ff ( + ) f ( )g d; (26) 6

17 where ; ( + ) = ( ; ) ( ; + ); ad from Lemma 2, ( ; ) CL mi f ; [ + + ]g ; > 0; where L = ; < ; = ; ; so L mif ; g. Now we cosider di ere iervals iside [ ; ] for he iegral i (26). Firs we have ha < =2 C C < =2 < =2 usig ha R < =2 L d = O (log ), ad ( ; ) 2 d 2 L d = O f ( ) log ; =2 3 =2 =2 C ( ; ) 2 d 3 =2 =2 C + 2 d + C 3 =2 2 =2 3 =2 d =2 C =2 2 d + C 2 C 2 = O f ( ) ; whe < =2; while for =2 < < we obai ha =2 3 =2 =2 C ( ; ) 2 d 3 =2 =2 C mi 2 ; + 3 = L 2 d =2 C mi 2 ; =2 2 d + 2 C 2 (log ) =0:5 = O f ( ) 2 (log ) =0:5 = O f ( ) 2( ) (log ) =0:5 : Nex =2 C ( ; ) 2 d C L d = O 2 log = O f ( ) log ; while he coribuio from he ierval [ ; 3 =2] ca be deal wih similarly. 7

18 Fially, usig he iegrabiliy of f ; C C = ( ; ) 2 ff ( + ) + f ( )g d ! 2 O + f ( + )d 2 ff ( + ) + f ( )g d = O f ( ) : Now o boud E [w ( )w ( )] = (2) R ;f d we disiguish wo cases, values of ad close ad far away. If =2 <, usig ha P ei(+ )( ) = P ) = 0; ei( 6= (mod ), (2) R ;d is by (8) equal o e i(+ )( ) ( ) ( ) ( ) 2o + e i(+ )( ) ( ) 2 ( ) 2o = e i(+ )( ) ( ) f( ( ) ( )) + (( ) ( ))g + (( ) ( )) e i(+ )( ) ( ) + e i(+ )( ) ( ) 2 ( ) 2o : (27) Usig d d C (MVT), as! 0; as = O( ); =2 < ; ad he Mea Value Theorem ( ) ( ) + ( ) ( ) C ; ( ) ( ) C ; while usig summaio by pars e i(+ )( ) ( ) = ) ei(+)( a= a ( ) a+ ( ) a a= = C a = O( ): e i( ) (28) The, as = O( ); =2 < ; ad usig (22), we obai e i(+ )( ) ( ) ( ) C 2 ( ) h i + mif; () 2( ) g + 2 C + ( + log 0:5 ) = O qf ( )f ( ) + log 0:5 : (29) 8

19 Whe < =2 he 2 ; ad usig summaio by pars, 2 ; d = ( ) ( ) + ( ) + ( ) = a= e i(+ a)( ) (30) + ( ) ( ) e i(+ )( ) : (3) Now (3) is exacly zero whe 6= (mod ); ad wih P g; he modulus of he righ had side of (30) is bouded by a= ei(+ a)( ) C mif; C ( ) ( ) + ( ) + + ( ) ( ) + ( ) = f ( ) + ( )g C = = O ( ) =! = O + = O qf ( )f ( ) : (32) Therefore, from (29) ad (32) we obai ha, < < =2; 2 ; d = O qf ( )f ( ) + log 0:5 : Nex, wih ; = ( ; ) ( ; ), ( ; ) CL ( ) mif ; + g; followig Robiso (995a, p. 063), we obai ha E [w ( )w ( )] = 2 2 ; ff f ( )g d (33) ( + )=2 + 2 (+ )=2 ; ff f ( )g d (34) (+ )=2 2 ff ( ) f ( )g ; d (35) ( + ) + 2 ; ff f ( )g d: (36) 2 Now usig he di ereiabiliy of f i 2 ( ; ); (33) is bouded by sup f ( ; ) ( ; ) d ( + )=22 ( + )=2 C ( ) 2 ( + )=2 L ( ) d = O ( ) log : 9

20 Nex (34) is bouded by C 2 2 sup f 0 ( + )=2 (+ )=2 L ( ) d = O 2 log : Now (35) is bouded by, 2 ( ) sup f 0 = O 2 (+ )=2 L ( (+ )=2 ) d! ( ; ) ( ; ) d = O 2 log : Fially, cosider (36). If < 0:5; he 2 ( ; ) ( ; ) ff f ( )g d C sup f 0 L ( ) + + d C 2 + C 2 ; ad if 0:5 2 ( ; ) ( ; ) ff f ( )g d C sup f 0 L ( ) ^ C 2 (log ) =0:5 + C ( ) (log ) =0:5 C ( ) : + ^ o C 2 2 2( ) (log ) =0:5 + d The bouds for he remaiig iervals follow ow i a simpler way. The usig O 2 log = pf O ( )f ( )(=) log he heorem follows for > 0. We ow deal wih he case < 0: Followig he proof for > 0, usig (2) ad Lemma 2 ow, we have ha (23) is bouded by, 0:5 < < 0; 8 C < 2 + : = = 2 + = 2 + = 2 9 = ; = O 2 = O f ( ) : (37) ad if < 0:5; we obai ha (23) is O 2 + (log ) f=0:5g = O f ( ) (log ) f=0:5go : (38) 20

21 We have o boud also he followig iegrals, " # ; ( + ) ff ( + ) < <2 2 < f ( )g d: Now ; (+ ) = ( ; ) ( ; +); ( ; ) C f + + g L, so < C < ( ; ) 2 d C 2 d = O f ( ) ; < <2 C <2 ( ; ) 2 d C 2 L d = O f ( ) log ; <2 2 < C C 2 < 2 < ( ; ) 2 d L d = O = O f ( ) 2 2 ; ad ally, usig he iegrabiliy of f ; C C ( ; ) 2 ff ( + ) + f ( )g d 2 ff ( + ) + f ( )g d! = O + f ( + )d = O f ( ) 2 2 : For he proof of he secod saeme of he heorem, we have ha E [w ( )w ( )] = (2) R ;f d. Cosider rs =2 < : Usig d d C = O( ); =2 < ; ad he MVT, ( ) ( ) + ( ) ( ) C ( ); as! 0; as ( ) ( ) C ; 2

22 while by summaio by pars e i(+ )( ) ( ) = ) ei(+)( C a= a a ( ) a+ ( ) a= + C a= + a a = e i( ) = O( ): The, from (27), because = O( ); =2 < ; ad followig he mehods used o obai (37) ad (38), e i(+ )( ) ( ) ( ) C ( ) 2 ( ) + + ( ) 2 ( ) 2 o = O f ( ) log ( = 0:5) : Whe < =2 he 2 ; ad usig summaio by pars as i (30)-(3), we have ha (3) is zero whe 6= (mod ); ad usig Lemma 2 he modulus of (30) is bouded by C ( ) ( ) + ( ) + + ( ) ( ) + ( ) = ^ C = f ( ) + ( )g + = + f ( ) + ( )g C = + C = + f ( ) + ( )g C C log ( = 0:5) + C o = O f ( ) log ( = 0:5) : The, ; = ( ; ) ( ; ), ( ; ) C f + g mif + ; g; < 0; so followig Robiso (995a, p. 063), we are led o esimae he coribuio of (33) o (36) whe < < 0. Usig he di ereiabiliy of f i 2 ( ; ); (33) is bouded by sup f ( ; ) ( ; ) d ( + )=22 ( + )=2 2 C 2 L ( ( + )=2 ) d = O 2 log : 22

23 Nex (34) is bouded by C 2 ad (35) by sup f 0 ( + )=2 (+ )=2 2 ( ) sup f 0 = O (+ )=2 (! ( ; ) d = O 2 L d (+ )=2 ; ) d! = O 2 log ; ( ; ) ( ; ) d = O 2 log : Fially (36) is, usig he iegrabiliy of f ; ( 2 ( C ) ) ( ; ) ( ; ) ff f ( )g d f d = O : The usig O pf = O ( )f ( )() 2 O (=) 2 () 2 pf = O ( )f ( )() 2 ad O 2 = p f ( )f ( ) he heorem follows. Proof of Corollary. I is eough o show ha Lemma 2 holds. The proof for Lemma 3 is similar. To show ha he approximaios of +T + by have he same bouds as whe T = ; we have ow ha ; d = 2 e i(+ )( ) +T + ( ) +T + ( ): The +T + C mif ; (T + ) g ad T ++ C (T + ) C 0 ; T ++ C ; > 0; so we ca obai 2 +T + 2 = +T + T ++ ( ) + +T + ( ) T T + 2 ad ha, 0 < < ; 2 +T + 2 o C 2 [(T + )] mif; ((T + )) 2( ) g C 2 [] + [] 2( )o : The ;T ( ; ) = P ei(+) +T + ( ) = P +T +e i( +T +) D ( + ) sais es he same bouds as ( ; ) i Lemma ad he corollary follows. 23

24 Proof of Theorem 5. We oly prove agai he boud for he rs wo expecaios. We cosider agai rs he case wih > 0: Now E [w ( )w ( )] f ( ) is E [I ( )] f ( ) = 2 ; ff xx f xx ( )g d (39) +f xx ( ) 2 ; d ( ) 2 ; where he secod lie is O f ( ) ; > 0; as i Theorem. For he aalysis of (39) we cosider he same iervals of iegraio as i Theorem ad oly replace f by f xx ; oig ha f xx is di ereiable i (0; ); for some > 0; wih f 0 xx = O( 2d ) as! 0+. The all bouds i Theorem for (9), 0 < < ; should be muliplied by ; obaiig a boud for (39) of order O f ( ) log = O f ( ) log : The oly ierval ha requires special cosideraio is =2 3 =2 =2 C (f xx ( ) + f xx ( + )) ( ; ) 2 d C 2 3 =2 = d d + C 2 =2 3 =2 f xx ( ) 2 d C 2 2 2d + C 2 2d = O f ( ) : For he secod expecaio cosidered i he heorem we ca cosider agai he same decomposiio (33)-(36) replacig f by f xx. Now proceedig as i he proof of Theorem 2(c) i Robiso (995a), cosiderig large ad small ; he bouds should be adaped by muliplicaio pfxx by 2d + 2d ; so we obai he boud O ( )f xx ( ) 2 (=) d log = pf O ( )f ( )(=) +d log : The oly ierval ha requires furher sudy is 2 ( ; ) ( ; ) ff xx f xx ( )g d C =2 2d + + d +C =2 f xx ( ) C 2d 2 = O + + q f ( )f ( )(=) +d log d : We ow cosider he case < 0: For aalyzig he coribuio of (20) we muliply he boud for (23) i Theorem by f xx ( ); obaiig O f ( ) : For he erm (9), we have o cosider 24

25 di ere iervals of iegraio of (26), < =2 C fxx( 0 ) ( ; ) 2 d < =2 C 2 2d L d = O f ( ) ; < =2 =2 2 =2 (f xx ( ) + f xx ( + )) L 2 d C 2 2 C 2 2 2d+ + =2 2 2d d! C 2 2d = O f ( ) ; ad ally, he ypical erm for he remaiig iervals is give by 2 C f xx ( ) L 2 2 d 2 +C max 2 d 0:5 f xx ( + ) C f xx ( ) 2 + C d 0:5 = O f ( ) : 2 L 2 0:5 d 2 d 0:5 d 2 For he secod expecaio, rs we ae (38) muliplied by p f xx ( )f xx ( ) = O The i is possible o esimae (33) o (36) for d d : 0:5 < < 0 usig he same mehod of Robiso (995a, pp ), because wih he boud ( ; ) C f + g L ( +); < 0; we obai he desired resul muliplyig a improved boud ha ca be obaied collecig Robiso s resuls for = 0; O ( )f xx ( ) (=) d log = O ( )f xx ( ) log ; pfxx pfxx by he facor (=) ; < 0: We omi he deails. Refereces Gewee, J. ad S. Porer-Huda (983). The esimaio ad applicaio of log memory ime series models. Joural of Time Series Aalysis 4, Fox, R. ad M.S. Taqqu (986). Large-sample properies of parameer esimaes for srogly depede saioary Gaussia imes series. Aals of Saisics 4, Grager, C.W.J. ad R. Joyeux (980). A iroducio o log-memory ime series ad fracioal di erecig. Joural of Time Series Aalysis, Hassler, U., F. Marmol ad C. Velasco (2006). Residual periodogram iferece for log-ru relaioships. Joural of Ecoomerics 30,

26 Heyde, C.C. ad Y. Yag (997). O de ig log-rage depedece. Joural Applied Probabiliy 34, Hosig, J.R.M. (98). Fracioal di erecig. Biomeria 68, Hurvich, C.M. ad B.K. Ray (995). Esimaio of he memory parameer for osaioary or oiverible fracioally iegraed processes. Joural of Time Series Aalysis 6, Hurvich, C.M., R. Deo ad J. Brodsy (998). The mea square error of Gewee ad Porer-Huda s Esimaor of he memory parameer of a log memory ime series. Joural of Time Series Aalysis 9, Hurvich, C.M., E. Moulies ad P. Soulier (2002). The FEP esimaor for poeially osaioary liear ime series. Sochasic Processes ad heir Applicaios 97, Kim, C.S. ad P.C.B. Phillips (999). Log-periodogram regressio: The osaioary case. Prepri, Cowles Foudaio for Research i Ecoomics, Yale Uiversiy. Küsch, H.R. (987). Saisical aspecs of self-similar processes. Proceedigs s World Cogress of he Beroulli Sociey, VNU Sciece Press. Mariucci, D. ad P.M. Robiso (2000). Wea covergece of mulivariae fracioal processes. Sochasic Processes ad heir Applicaios 86, Marmol, F. ad C. Velasco (200). Tred saioariy versus log rage depedece i ime series aalysis. Joural of Ecoomerics 08, Marmol, F. ad C. Velasco (2004). Cosise esig of fracioal coiegraio. Ecoomerica 72, Phillips, P.C.B. (999). Discree Fourier rasforms of fracioal processes. Cowles Foudaio for Research i Ecoomics Discussio Paper 243, Yale Uiversiy. Phillips, P.C.B. ad V. Solo (992). Asympoics for liear processes. Aals of Saisics 20, Robiso, P.M. (995a). Log-periodogram regressio of ime series wih log rage depedece. Aals of Saisics 23, Robiso, P.M. (995b). Gaussia semiparameric esimaio of log rage depedece. Aals of Saisics 23, Robiso, P.M. ad D. Mariucci (200). Narrow-bad aalysis of osaioary processes. Aals of Saisics 29, Robiso, P.M. (2005). The disace bewee rival osaioary processes. Joural of Ecoomerics 28, Solo, V. (992). Irisic radom fucios ad he paradox of /f oise. Siam Joural of Applied Mahemaics 52, Sowell, F.B. (990). The fracioal ui roo disribuio. Ecoomerica 58, Velasco, C. (999). No-saioary log-periodogram regressio. Joural of Ecoomerics 9,

27 Velasco, C. (2000). No-Gaussia log-periodogram regressio. Ecoomeric Theory 6, Velasco, C. (2005). Semiparameric Esimaio of Log-Memory Models. Forhcomig i Palgrave Hadboo of Ecoomerics, Vol.. Ecoomeric Theory, K. Paerso ad T.C. Mills eds, Palgrave, MacMilla. Velasco, C. ad P.M. Robiso (2000). While Pseudo-Maximum Lielihood Esimaes of No- Saioary Time Series. Joural of he America Saisical Associaio 95,