A NECESSARY MOMENT CONDITION FOR THE FRACTIONAL FUNCTIONAL CENTRAL LIMIT THEOREM

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Ecoometric Theory, 28, 2012, 671 679. doi:10.1017/s0266466611000697 A NECESSARY MOMENT CONDITION FOR THE FRACTIONAL FUNCTIONAL CENTRAL LIMIT THEOREM SØREN JOHANSEN Uiversity of Copehage ad CREATES MORTEN ØRREGAARD NIELSEN Quee s Uiversity ad CREATES We discuss the momet coditio for the fractioal fuctioal cetral limit theorem (FCLT for partial sums of x t = d u t, where d ( 1 2, 1 2 is the fractioal itegratio parameter ad u t is weakly depedet. The classical coditio is existece of q 2 ad q > ( d + 1 2 momets of the iovatio sequece. Whe d is close to 1 2 this momet coditio is very strog. Our mai result is to show that whe d ( 1 2,0 ad uder some relatively weak coditios o u t, the existece of q ( d + 1 2 momets is i fact ecessary for the FCLT for fractioally itegrated processes ad that q > ( d + 1 2 momets are ecessary for more geeral fractioal processes. Davidso ad de Jog (2000, Ecoometric Theory 16, 643 666 preseted a fractioal FCLT where oly q > 2 fiite momets are assumed. As a corollary to our mai theorem we show that their momet coditio is ot sufficiet ad hece that their result is icorrect. 1. INTRODUCTION The fractioal fuctioal cetral limit theorem (FCLT is give i Davydov (1970 for partial sums of the fractioally itegrated process d ε t, where d = (1 L d is the fractioal differece operator defied i (3 i our Sectio 2 ad ε t is idepedet ad idetically distributed (i.i.d. with mea zero. Davydov We are grateful to Beedikt Pötscher, three aoymous referees, ad James Davidso for commets ad to the Social Scieces ad Humaities Research Coucil of Caada (SSHRC grat 410-2009-0183 ad the Ceter for Research i Ecoometric Aalysis of Time Series, (CREATES, fuded by the Daish Natioal Research Foudatio for fiacial support. Address correspodece to Morte Ørregaard Nielse, Departmet of Ecoomics, Duig Hall Room 307, Quee s Uiversity, 94 Uiversity Aveue, Kigsto, Otario K7L 3N6, Caada; e-mail: mo@eco.queesu.ca. c Cambridge Uiversity Press 2011 671

672 SØREN JOHANSEN AND MORTEN ØRREGAARD NIELSEN (1970 proved the fractioal FCLT uder a momet coditio of the form E ε t q < for q 4 ad q > 4d/ ( d + 1 2, which was subsequetly improved by Taqqu (1975 to q 2 ad q > q 0 = ( d + 1 2. The stadard momet coditio from Dosker s theorem is q 2 (see Billigsley, 1968, Ch. 2, Sect. 10, ad because the coditio q q 0 is oly stroger tha q 2 whe d < 0 we cosider oly d ( 1 2,0. The fractioal FCLT has bee exteded ad geeralized i umerous directios. For example, Mariucci ad Robiso (2000 replace ε t by a class of liear processes, assumig the momet coditio q > q 0. The latter authors proved FCLTs for so-called type II fractioal processes, whereas Davydov (1970 ad Taqqu (1975 discussed type I fractioal processes, but the distictio betwee type I ad type II processes is ot relevat for our discussio of the momet coditio. Davidso ad de Jog (2000, heceforth DDJ claim i their Theorem 3.1 that for some ear-epoch depedet (NED processes with uiformly bouded qth momet the fractioal FCLT holds, but (icorrectly, as we shall see subsequetly they assume a much weaker momet coditio tha previous results, amely, q > 2. To the best of our kowledge, Theorem 3.1 of DDJ is the oly fractioal FCLT that claims a momet coditio that is weaker tha the earlier coditio. I the ext sectio we give some defiitios ad costruct a i.i.d. sequece ad a fractioal liear process that are cetral to our results. I Sectio 3 we preset our mai results, which state that if the fractioal FCLT holds for ay class of processes U(q with uiformly bouded qth momet cotaiig these two processes, the it follows that q q 0 if the fractioal FCLT is based o fractioal coefficiets ad q > q 0 if the coefficiets are more geeral. The proofs of both results are based o couterexamples that are costructed i a similar way as a couterexample i Wu ad Shao (2006, Rmk. 4.1. I Sectio 4 we discuss the results ad give two applicatios. I particular, it follows from our results that if the FCLT holds for NED processes with uiformly bouded q momets, the q q 0. Hece DDJ s Theorem 3.1 ad all their subsequet results do ot hold uder the assumptios stated i their theorem. Throughout, c deotes a geeric fiite costat, which may take differet values i differet places. 2. DEFINITIONS DEFINITION 1. We defie the class U 0 (q as the class of processes u t that satisfy the momet coditio sup E u t q < for some q 2 (1 <t< ad have log-ru variace

NECESSARY MOMENT CONDITION FOR FRACTIONAL FCLT 673 σ 2 u = lim T T E ( T 2 u t, 0 <σu 2 <. (2 For such processes we defie two subclasses of U 0 (q as follows. (i U LIN (q U 0 (q is the class of liear processes u t U 0 (q satisfyig u t = =0 τ ε t, where =0 j= τ j 2 < ad ε t is i.i.d. with mea zero ad variace σε 2 > 0. (ii U NED (q U 0 (q is the class of zero mea covariace statioary processes u t U 0 (q that are L 2 -N E D of size 1 2 o v t with d t = 1, where v t is either a α-mixig sequece of size q/(1 q or a φ-mixig sequece of size q/(2(1 q; see Assumptio 1 of DDJ. Note that the coditio =0 j= τ j 2 < i the defiitio of the class U LIN (q either implies or is implied by (2. Also ote that if u t U LIN (q the u t is zero mea ad statioary. Next we defie the fractioal ad geeral fractioal processes. DEFINITION 2. For ay u t U 0 (q we we defie the fractioal process x t = d u t = j=0 b j (du t j for 1 < d < 0, (3 2 where b j (d = ( j( d j = d(d + 1...(d + j 1/j! cj d are the fractioal coefficiets, i.e., the coefficiets i the biomial expasio of (1 z d, ad meas that the ratio of the left- ad right-had sides coverges to oe. The geeral fractioal process is defied by x t = j=0 a j (du t j for 1 < d < 0, (4 2 where a j (d cl( j j d ad l( j is a (ormalized slowly varyig fuctio; see Bigham, Goldie, ad Teugels (1989, p. 15. Note that the b j (d coefficiets from the fractioal differece filter are a special case of a j (d. The processes (3 ad (4 are well defied because, with x 2 deotig the L 2 -orm, we have from (1 that x t 2 u t 2 + c j=1 l( j j d u t j 2 c for d < 0 usig Karamata s theorem; see Bigham, Goldie, ad Teugels (1989, p. 26. We base our results o the costructio of the followig two specific processes. DEFINITION 3. Let ε t be i.i.d. with mea zero, variace σε 2 > 0, ad fiite qth momet for some q 2 to be chose later. For such ε t we defie the followig two processes.

674 SØREN JOHANSEN AND MORTEN ØRREGAARD NIELSEN (i u 1t = ε t. (ii u 2t = ε t + 1+d ε t. For these two processes we ote the followig coectio with the classes U LIN (q ad U NED (q. LEMMA 1. For d ( 1 2,0 ad for i = 1,2,u it U LIN (q U NED (q, ad the log-ru variace of u it is σ 2 ε. Proof. Clearly, u 1t = ε t is cotaied i both U LIN (q ad U NED (q ad has log-ru variace σε 2. The process u 2t = ε t + 1+d ε t = ε t + j=0 b j( d 1ε t j is a liear process ad =0 j= b j ( d 1 2 c =1 j= j 2d 4 c =1 2d 3 c so that u 2t is i U LIN (q. To see that u 2t is i U NED (q, we calculate u 2t E(u 2t ε t m,...,ε t+m 2 = =m+1 c ( =m+1 b ( d 1ε t 2 2d 4 1/2 cm d 3/2. for d ( 12,0, Because 3 2 +d > 1 2 for d ( 1 2,0, this shows that u 2t is L 2 -NED of size 1 2 o ε t, ad hece u 2t is also i U NED (q. The geeratig fuctio for u 2t is f (z = 1 + (1 z 1+d, ad for z = 1 we fid because 1 + d > 0 that f (1 = 1. Therefore the log-ru variace of u 2t is lim T T E T ( ( εt + 1+d 2 ε t = f (1 2 Var(ε t = σε 2. We ext give a geeral formulatio of the FCLT for fractioal processes. For this purpose we defie the scaled partial sum process X T (ξ = σt [T ξ] x t, 0 ξ 1, (5 where σt 2 = E(T x t 2 ad [z] is the iteger part of the real umber z. FRACTIONAL FCLT FOR U(q. Let X T (ξ be give by (5 ad suppose x t is liear i u t. We say that the FCLT for fractioal Browia motio holds for a set U(q U 0 (q of processes if, for all u t U(q, it holds that X T (ξ D X (ξ i D[0,1], (6 where X (ξ is fractioal Browia motio.

NECESSARY MOMENT CONDITION FOR FRACTIONAL FCLT 675 D Here, deotes covergece i distributio (weak covergece i D[0, 1] edowed with the metric d 0 i Billigsley (1968, Ch. 3, Sect. 14, which iduces the Skorokhod topology ad uder which D[0,1] is complete. 3. THE NECESSITY RESULTS Our first mai result is the followig theorem. THEOREM 1. Let X T (ξ be defied by (5 for x t give by the fractioal coefficiets b j (d i (3 with 1 2 < d < 0, ad let U(q U 0(q be such that u it U(q for i = 1,2. If the fractioal FCLT holds for all u t i U(q for some q 2, the q q 0. Proof. We prove the theorem by assumig that there is a q 1 [2,q 0 for which the FCLT holds for U(q 1 ad show that this leads to a cotradictio by a careful costructio of ε t ad therefore u 1t ad u 2t. For u it,i = 1,2, we defie x it ad X it by (3 ad (5. Because u it is i U(q 1 the fractioal FCLT holds by the maitaied assumptio for u it, ad hece X it (ξ coverges i distributio to fractioal Browia motio. (a The ormalizig variace for X 1T. The variace of T x 1t = T d u 1t = T d ε t ca be foud i Davydov (1970; see also Lemma 3.2 of DDJ: σ 2 1T = E ( T x 1t 2 σ 2 ε V d T 2d+1, (7 where ( 1 1 V d = Ɣ(d + 1 2 2d + 1 + ((1 + τ d τ d 2 dτ. 0 (b The ormalizig variace for X 2T. We write x 2t ad X 2T i terms of x 1t ad X 1T, usig (3 ad (5: x 2t = d u 2t = x 1t + ε t ε t, (8 X 2T (ξ = σ2t [T ξ] x 2t = σ 1T σ 2T X 1T (ξ + σ 2T (ε [T ξ] ε 0. (9 We ext fid that the variace of T x 2t = T d u 2t = T (x 1t +ε t ε t is ( T 2 T 2 ( 2 σ2t 2 T = E x 2t = E( (x 1t + ε t ε t = E ε T ε 0 + x 1t T 2 T = E(ε T ε 0 2 + E( x 1t + 2E( d ε t (ε T ε 0.

676 SØREN JOHANSEN AND MORTEN ØRREGAARD NIELSEN The first term is costat, the ext is σ1t 2, ad lettig 1 {A} deote the idicator fuctio of the evet A, the last term cosists of T E( d T ε t ε T = b k (de(ε t k ε T k=0 = σ 2 ε ad T E( d u 1t ε 0 = Therefore, T k=0 T k=0 = σ 2 ε T k=0 b k (d1 {k=t T } = σ 2 ε b 0(d = σ 2 ε b k (de(ε t k ε 0 b k (d1 {k=t} c T t d c for d < 0. σ 2 2T σ 2 1T + c. (10 (c The cotradictio. We ow costruct the i.i.d. process ε t so that it has o momet higher tha q 1, i.e., E ε t q =for q > q 1, by choosig the tail to satisfy P( ε t q 1 1 c c(logc 2 as c. (11 I this case we still have E ε t q 1 <. We the fid P(σ 1T max ε t < c = P(σ1T ε 1 < c T = P( ε 1 q 1 < c q 1 σ q 1 1 t T 1T T = (1 P( ε 1 q 1 c q 1 T q 1/q 0 T ( 1 1 c q 1T q 1/q 0 (q 1 (logc + q0 log T 2 ( T 1 q 1/q 0 exp c q 1(q 1 (logc + q0 0 log T 2 as T because q 1 < q 0. Thus, σ1t max 1 t T ε t because P the ormalizig costat σ 1T = σ ε V 1/2 d T 1/q 0 = σ ε V 1/2 d T 1/2+d <σ ε V 1/2 d T 1/q 1 is too small to ormalize max 1 t T ε t correctly. The defiitio (9 implies the evaluatio max 0 ξ 1 ε [T ξ] max 0 ξ 1 ε [T ξ] ε 0 + ε 0 max 0 ξ 1 σ 2T X 2T (ξ + max 0 ξ 1 σ 1T X 1T (ξ + ε 0 T

such that σ 1T NECESSARY MOMENT CONDITION FOR FRACTIONAL FCLT 677 max ε [T ξ] max 0 ξ 1 0 ξ 1 σ 1T σ 2T X 2T (ξ + max 0 ξ 1 X 1T (ξ +σ 1T ε 0. (12 We have see i (7 ad (10 that σ2t 2 σ 1T 2 +c ad σ 1T 2 σ ε 2V d T 1+2d for d ( 1 2,0, so that σ 1T σ2t 1. Therefore, both σ1t σ 2T X 2T (ξ ad X 1T (ξ coverge i distributio by the previous results, ad it follows from (12 that σ1t max 0 ξ 1 ε [T ξ] is O P (1. This cotradicts that σ1t max P 1 t T ε t ad hece completes the proof of Theorem 1. The proof of Theorem 1 implies that the issue is that the rate of covergece, T (d+1/2 [T ξ],of d u 1t ca be very slow for d close to 1 2. Thus, more cotrol o the tail behavior of the u t sequece is eeded whe d ( 1 2,0, ad this is achieved through the momet coditio (1. We ed this sectio by givig a result that shows whe the momet coditio q > q 0 is ecessary istead of q q 0. The former is the momet coditio applied by, e.g., Taqqu (1975 ad Mariucci ad Robiso (2000. THEOREM 2. Let X T (ξ be defied by (5 for x t give by the geeral fractioal coefficiets i (4 with 1 2 < d < 0, ad let U(q U 0(q be such that u it U(q for i = 1,2. If the fractioal FCLT holds for all slowly varyig fuctios l( ad all u t i U(q for some q 2, the q > q 0. Proof. We assume that there is a q 1 [2,q 0 ] for which the FCLT holds for U(q 1 ad show that this leads to a cotradictio. For u it,i = 1,2, we defie x it ad X it by (4 ad (5 ad use the proof of Theorem 1 with the followig modificatios. (a From Karamata s theorem (see Bigham, Goldie, ad Teugels, 1989, p. 26, we fid that the ormalizig variace is σ1t 2 cl(t 2 T 2d+1 = cl(t 2 T 1/q 0. (c We choose the tail of ε t as i (11 i the proof of Theorem 1 ad take l(t = (log T ad fid P(σ 1T max ε t < c 1 t T = P(σ 1T ε 1 < c T = P( ε 1 q 1 < c q 1 σ q 1 1T T = (1 P( ε 1 q 1 c q 1 T q 1/q 0 l(t q 1 T ( 1 1 c q 1T q 1/q 0 l(t q 1(q 1 (logc + q0 log T + logl(t 2 ( T 1 q 1/q 0 l(t q 1 exp c q 1(q 1 (logc + q0 0 log T + logl(t 2 T

678 SØREN JOHANSEN AND MORTEN ØRREGAARD NIELSEN as T because q 1 q 0. Note that eve with q 1 = q 0 (ad q 0 > 2 because d < 0 we have the factor exp( c(log T q1 2 0, which esures the covergece to zero. The cotradictio follows exactly as i the proof of Theorem 1. 4. DISCUSSION I this sectio we preset two corollaries that demostrate how our results apply to the processes i Mariucci ad Robiso (2000 ad to those i DDJ, respectively. We first discuss the implicatios of Theorem 1 for the results of DDJ, who state (i our otatio the followig claim (give as Theorem 3.1 i DDJ. DDJ CLAIM. If X T (ξ is defied by (3 ad (5, where d < 1 2 ad u t U NED (q for q > 2, the X T (ξ D X (ξ i D[0,1], where X (ξ is fractioal Browia motio. It is oteworthy that U NED (q allows u t to have a very geeral depedece structure through the NED assumptio, but i particular that DDJ assume oly that sup t E u t q < for q > 2, which is weaker tha q q 0 if d < 0. The followig corollary to Theorem 1 shows how our result disproves Theorem 3.1 i DDJ. COROLLARY 1. Let X T (ξ be defied by (3 ad (5 with 1 2 < d < 0.Ifthe fractioal FCLT holds for all u t i U NED (q the q q 0. Proof. From Lemma 1 we kow that u 1t ad u 2t are i U NED (q, which by Theorem 1 proves the corollary. It follows from Corollary 1 that Theorem 3.1 of DDJ (ad their subsequet results relyig o Theorem 3.1 does ot hold uder their Assumptio 1. We fiish with a applicatio of our results to the processes i Mariucci ad Robiso (2000. COROLLARY 2. Let X T (ξ be defied by (4 ad (5 with 1 2 < d < 0.Ifthe fractioal FCLT holds for all slowly varyig fuctios l( ad all u t i U LIN (q the q > q 0. Thus, the momet coditio (1 with q > q 0 is both ecessary ad sufficiet for Theorem 1 of Mariucci ad Robiso (2000. Proof. The ecessity statemet follows from Theorem 2 because u 1t ad u 2t are i U LIN (q by Lemma 1. The sufficiecy statemet follows because the uivariate versio of Assumptio A of Mariucci ad Robiso (2000 (traslated to type I processes was i fact used to defie the class U LIN (q ad the coefficiets a j (d. It follows from Corollary 2 that q > q 0 is both ecessary ad sufficiet for the fractioal FCLT whe usig the coefficiets a j (d to defie a geeral fractioal process ad u t is a liear process of the type i U LIN (q. However, it does ot follow from our results that q q 0 is ecessary for the FCLT whe u t is a i.i.d. or

NECESSARY MOMENT CONDITION FOR FRACTIONAL FCLT 679 autoregressive movig average (ARMA process because the process u 2t eeded i the costructio is either i.i.d. or ARMA. REFERENCES Billigsley, P. (1968 Covergece of Probability Measures. Wiley. Bigham, N.H., C.M. Goldie, & J.L. Teugels (1989 Regular Variatio. Cambridge Uiversity Press. Davidso, J. & R.M. de Jog (2000 The fuctioal cetral limit theorem ad weak covergece to stochastic itegrals II: Fractioally itegrated processes. Ecoometric Theory 16, 643 666. Davydov, Y.A. (1970 The ivariace priciple for statioary processes. Theory of Probability ad Its Applicatios 15, 487 498. Mariucci, D. & P.M. Robiso (2000 Weak covergece of multivariate fractioal processes. Stochastic Processes ad Their Applicatios 86, 103 120. Taqqu, M.S. (1975 Weak covergece to fractioal Browia motio ad to the Roseblatt process. Zeitschrift für Wahrscheilichkeitstheorie ud Verwadte Gebiete 31, 287 302. Wu, W.B. & X. Shao (2006 Ivariace priciples for fractioally itegrated oliear processes. IMS Lecture Notes Moograph Series: Recet Developmets i Noparametric Iferece ad Probability 50, 20 30.

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