Fractional Dickey-Fuller tests under heteroskedasticity

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1 Fracional Dickey-Fuller ess under heeroskedasiciy Hsein Kew and David Harris Deparmen of Economics Universiy of Melbourne May 9, 007 Absrac In a recen paper, Dolado, Gonzalo and Mayoral (00) inroduce a fracional Dickey-Fuller (FD-F) -saisic for esing a uni roo agains he alernaive of a mean revering fracional uni roo process. This -saisic is based on he assumpion ha he errors are uncondiionally homoskedasic. However, Busei and Taylor (003), Mc- Connell and Perez-Quiros (000), and van Dijk e al. (00) have found compelling evidence ha such an assumpion is unlikely o hold in many macroeconomic and - nancial ime series, especially hose obained a a longer ime span. In his paper, we invesigae he nie-sample properies of he FD-F saisic when he errors are uncondiionally heeroskedasic. We nd ha, depending on he form of heeroskedasiciy, he FD-F saisic su ers from subsanial size disorions. In order o correc for such disorions, we propose he use of Whie sandard errors (Whie (980)) when compuing he FD-F saisic. This yields a es ha is robus o heeroskedasiciy of unknown form. We demonsrae ha he FD-F saisic ha employs Whie sandard error has a sandard normal limiing disribuion under he uni roo null hypohesis as in he FD-F saisic wih homoskedasic errors. Mone Carlo resuls sugges ha: (i) Whie s correcion is e ecive in reducing he size disorions; and (ii) he power loss of using Whie sandard error in he case of homoskedasiciy is very small. These resuls sugges ha i is pruden o use he Whie robus sandard errors regardless of wheher he errors are heeroskedasic or no. Keywords: Fracional uni roo ess, heeroskedasiciy, srucural breaks J.E.L. Classi caion: C30, C3 Inroducion Several aricles have provided Mone Carlo evidence on he performance of he uni-roo ess, saionary ess and persisence change ess under uncondiional heeroskedasiciy. The general nding from hese aricles is ha hese ess su er from severe size disorions when here is an abrup change in he error variance. In sandard uni roo esing agains saionariy alernaives, Hamori and Tokihisa (997) sudy he Dickey-Fuller (D-F) es, Cavaliere and Taylor (007) sudy he M es of Sock (999) and Beare (007) invesigaes he Phillips-Perron es. In he saionariy esing framework, where he null hypohesis

2 of saionariy is esed agains a uni roo alernaive, Cavaliere (004) examines he properies of he KPSS es, and Busei and Taylor (003) examines he properies of he es saisics due o Busei and Harvey (00). In he persisence change esing framework, where saisical procedures are developed o es he null hypohesis of a consan uni roo process agains he alernaive of a change in persisence eiher from a saionary process o a uni roo process or vice versa, Cavaliere and Taylor (006) invesigae he properies of he raio-based ess inroduced by Kim (000), Kim e al. (00) and Busei and Taylor (004). However, o he bes of our knowledge, no sudy has examined he e ecs of uncondiional heeroskedasiciy on exising saisical procedures designed o es he null hypohesis of a uni roo process agains he alernaive of a mean revering fracional process. I is well known ha alhough he D-F es is consisen agains such an alernaive, he power of D-F es is generally quie low. To circumven his problem, Dolado, Gonzalo and Mayoral (00, hereinafer DGM) recenly inroduced he Fracional Dickey-Fuller (FD-F) es for a uni roo ha has high power agains fracional alernaives. Le = ( L), where L is he lag operaor. The FD-F saisical procedure is based on he usual -saisic of in he ordinary leas squares (OLS) regression y = d y +e : The null hypohesis ha y has a uni roo is hen H 0 : = 0 agains H : < 0: If = 0; hen y = e and so y has a uni roo. Under he alernaive ha < 0; i can shown ha y is fracionally inegraed of order d : The regression canno be made operaional wihou a value of d : DGM demonsrae ha if he value of d is obained by using a p T -consisen esimaor of he rue fracional di erencing parameer, he resuling -saisic of has a limiing sandard normal disribuion under he uni roo null hypohesis. In DGM, he error erm e is assumed o be independen and uncondiionally homoskedasic. In his paper, we invesigae he robusness of his -saisic in he presence of uncondiional heeroskedasic errors. Our Mone Carlo experimens show ha, depending on he form of heeroskedasiciy, he FD-F saisic su ers from subsanial size disorions. In order o correc for such disorions, we propose o replace he usual OLS homoskedasic sandard errors by Whie s heeroskedasiciy robus sandard errors when consrucing he -saisic (Whie (980)). We demonsrae ha he FD-F saisic ha employs Whie sandard errors sill reains is sandard normal limiing disribuion. We obain his resul by exploiing he fac ha under he uni roo null hypohesis he FD-F -saisic, unlike he sandard uni roo ess, possesses sandard limi heory. Whie s resuls are esablished under he assumpion ha he regressors are exogenous. Nicholls and Pagan (983) and more recenly Phillips and Xu (006) show ha Whie s resuls remain valid in a dynamic regression model, provided ha he ime series are covariance saionary. Our resuls herefore illusrae he general applicabiliy of Whie s resuls in uni roo esing lieraure. Robusi ed versions of he uni roo ess menioned above have been suggesed in he ime series lieraure. Kim, Leybourne and Newbold (00) sugges pre-esimaing he variance break poin ogeher wih he pre- and pos-break variances. These esimaes are hen employed in modi ed varians of he Perron-ype uni roo ess (Perron (989, 990)).

3 The criical values for hese modi ed ess are given in Perron (989), and hey depend on he locaion of he break. Kim, Leybourne and Newbold (00) assume he exisence of a single break in he error variance process. In a recen paper, Cavaliere and Taylor (007), based on he work of Cavaliere (004), relax his assumpion. Their procedure does no require a parameric speci caion of he error variance, bu only requires he error variance o be uniformly bounded and o display a counable number of jumps. They demonsrae ha heeroskedasiciy induces a ime-deformaion in he limiing disribuion of he uni roo saisics. Using a consisen non-parameric esimaor of his ime-deformaion, he resuling limiing disribuion can hen be simulaed o obain asympoically valid criical values. Thus he correc criical values o use depend on he precise naure of he heeroskedasiciy. More recenly, Beare (007) suggess an alernaive approach ha does no require case-by-case numerical abulaion of criical values. Beare s approach is o ransform he daa in such a way ha he ransformed daa are approximaely homoskedasic. The Phillips-Perron uni roo es is hen applied o he homoskedasic ransformed daa. This yields, under condiions more resricive han hose used by Cavaliere and Taylor (007), a uni roo es which has a pivoal asympoic null disribuion. Since he FD-F es involves sandard limi heory, our soluion o he size disorion problem is very simple o implemen in pracice, is robus agains unknown heeroskedasiciy, and does no require pre-esimaion of some non-parameric funcions of he heeroskedasiciy. Unlike Cavaliere and Taylor (007), our error variance is no required o display counable number of jumps. We only assume ha he error variance is uniformly bounded away from zero and in niy. We noe here ha he procedures suggesed by Kim, Leybourne and Newbold (00) and Cavaliere and Taylor (007) allow for he presence of deerminisic componens in he daa generaing process, while our procedure assumes ha he series conains no deerminisic componens. Neverheless, our procedure could be exended in a sraighforward manner o include deerminisic regressors. p Throughou his paper, he symbols! and! d denoe, respecively, convergence in probabiliy and in disribuion. We wrie X = O p () o denoe a sochasic sequence fx g ha is uniformly bounded in probabiliy for all : We wrie X = o p () o denoe ha X converges in probabiliy o zero. The indicaor funcion () is equal o one if or o zero oherwise. The noaion bxc denoes he larges ineger below x: The noaion (x) denoes he Euler gamma funcion de ned for any real value of x excep negaive inegers. We use a _ b as shorhand for max (a; b). The res of his paper is organised as follows. Secion presens he model and assumpions. Secion 3 describes he FD-F saisical procedure and analyses he properies of his es under uncondiional heeroskedasiciy. The Mone Carlo experimens presened in Secion 3. serve o illusrae ha he presence of uncondiional heeroskedasiciy in he error erm can cause invalid saisical inferences. Secion 4 suggess a modi ed version of he FD-F saisic ha is robus o he presence of uncondiional heeroskedasiciy. I also esablishes he asympoic heory of his modi ed es. Mone Carlo evidence on he small-sample properies of his modi ed es are presened in Secion 5. The nal secion concludes. The proofs are colleced in he Appendix. 3

4 Model and Assumpions Suppose he ime series fy g is generaed by he fracionally inegraed model: d 0 y = e () e = T " () () where d 0 [0; ] : Since he fracional parameer d 0 can ake on any real value raher han only an ineger value, he fracional ler d 0 in () can be expanded o obain a runcaed auoregressive represenaion for fy g : X d 0 y = j (d 0 ) y j=0 j where j (d 0 ) = (j d 0 ) = ( (j + ) ( d 0 )). Saring a 0 (d 0 ) = ; j (d 0 ) can be wrien recursively as j (d 0 ) = ( j (d 0 ) (j d 0 )) =j for j : In () ; T and " are assumed o saisfy he following condiions. Assumpion V T is non-sochasic and sais es for all 0 < < T < where and are sricly posiive consans. Assumpion E " is an i.i.d. process wih E (" ) = 0; E " = and sup E j" j B < for some > : Under Assumpion E, he error erm e in () has zero uncondiional mean. Assumpion E normalises he variance of " o uniy so ha () implies ha E e = T ; signifying explicily ha he uncondiional variance of he error erm is no consan over ime. The funcional form of he error variance T is reaed as unknown, and hus, our framework is non-parameric. By Liapunov s inequaliy and Assumpion E, i follows ha for r 4; he error erm e sais es sup E je j r = sup E j T " j r = sup r T E j" j r < r B r < : (3) Under Assumpions V and E, he series fy g in ()-() is generaed by a fracionally inegraed model wih ime-varying innovaion variances. Since T depends on T; formally we should wrie e = e T and hus y = y T ; so ha he ime series in () () form a riangular array. However, his exra subscrip T does no play any role in he developmen of asympoic heory, and is hus suppressed for noaional simpliciy. 4

5 Assumpion V is slighly weaker han ha of Cavaliere (004) and Cavaliere and Taylor (007). We do no require he error variance o be imbedded in a variance funcion, o display a counable number of jumps, and o saisfy a Lipschiz condiion. The independence assumpion imposed in Assumpion E is sronger han necessary and is adoped for simpliciy. This assumpion rules ou he presence of condiional heeroskedasiciy of he ype inroduced by Engle (98) and Bollerslev (986). 3 The Fracional Dickey-Fuller Tes The FD-F es is a regression based saisical procedure. I involves OLS esimaion of he following regression equaion: y = d y + e : (4) For a given value of d [0; ), under he Type de niion of he fracionally inegraed model (see Robinson and Marinucci (00)), he regressor can be compued as d y = P j=0 j (d ) y j : Noice ha when d = 0; equaion (4) is he sandard D-F regression. Like he D-F es, he FD-F es is he convenional -saisic for esing he hypohesis H 0 : = 0 agains H : < 0 in (4) : Under he null, he process y has a uni roo because he regression in (4) becomes y = e : Under he alernaive ha < 0, he process y is fracionally inegraed of order d ; because DGM show ha he regression in (4) can be rewrien as d y = C (L) e where C (L) = d L and hey show ha C (L) does no conain uni roos. If we have some a priori knowledge as o he value of d ; hen i can be seen immediaely ha esing for he signi cance of in (4) is equivalen o esing he null hypohesis H 0 : d 0 = agains he simple alernaive H : d 0 = d : (5) Noe ha he sandard D-F uni roo esing procedure imposes a priori resricion on d by seing d = 0: This is heoreically jusi ed only when he process y is aken o follow eiher an I () process or an I (0) process. The FD-F es eliminaes such a resricion by generalising he D-F es o explicily allow for he possibiliy ha y is a mean revering fracional inegraed process I (d). If he rue daa generaing process for fy g is indeed an I (d 0 ) process wih 0 < d 0 <, i is expeced ha he FD-F will yield beer nie sample power properies han he D-F es. In a realisic case in which d is ypically unspeci ed in pracice, he simple alernaive hypohesis in (5) can be replaced by a composie alernaive H : 0 d 0 < : 5

6 Wihou he value of d ; he OLS regression in (4) is no feasible and in order o make he FD-F es operaional, DGM sugges replacing he unknown parameer d in (4) by a p T -consisen esimae of d0 in () : The OLS esimaor of for equaion (4) is ^ = T P T d y y T P T (d y ) : (6) In DGM, he error erm fe g is assumed o be i:i:d: wih zero mean and unknown variance : Under hese assumpions, he sandard error of ^ (denoed as SE^ ) is given by SE ^ = ^ PT (d y ) = where ^ is he usual OLS esimaor of he unknown error variance : ^ = T T X The -raio for esing = 0 is herefore given by y ^ d y : (d ) = ^ : (7) SE ^ 3. Normaliy under he null hypohesis Under he uni roo null hypohesis, Theorem of DGM esablishes ha he OLS esimae ^ is consisen, converges a he sandard asympoic rae of p T and is asympoically normal only if d lies in he inerval = < d < : An ineresing feaure of his resul is ha he (d )-raio given in (7) has a sandard normal limiing null disribuion. This is in conras o he D-F uni roo es which has a non sandard asympoic disribuion. The inuiive reasoning behind his resul is quie simple. When he H 0 : d 0 = is rue, he daa generaing process for fy g is y = e and hence he sandardised and cenred OLS esimaor in (6) becomes p T ^ = T = P T z e T P T z where z = d y : The sequence fz g is fracionally inegraed of order d. To see his, we pre-muliply boh sides of () by d o obain d y = d e d (z ) = e : (8) 6

7 Thus, fz g I ( d ) as claimed. If d is resriced o lie in he inerval (=; ), he process fz g is asympoically saionary since 0 < d < =: This indicaes ha he process fz g may possess sandard asympoic properies. Alhough, under he Type model of fracional inegraion, he series fz g is non-saionary for any values of d, Lemma of DGM shows ha T T X = z (z s ) p! 0 (9) where z s is he non-runcaed fracionally inegraed process of order d corresponding o z and is expressed as an in nie order moving average of he innovaions: z s = z + X j (d ) e j : j= The resul in equaion (9) saes ha he di erence beween z and (z s ) ; when suiably cenred, disappears asympoically. This is he key ingredien in deriving he formula for he asympoic variance of ^: Using (9) ogeher wih he fac ha z s is a saionary ergodic process (see Sowell (990)), he probabiliy limi of he denominaor in (8) can be obained simply by appealing o he WLLN for saionary and ergodic process: X T T z p! E z s : (0) Noe ha E (z s ) is he variance funcion of z s ; which can be obained from Hosking (98) by replacing he parameer d in equaion (3:) of Hosking wih d : E (z s ) = ( ( d )) ( ( d )) = (d ) : () (d ) As for he numeraor in (8), Lemma of DGM shows ha i has he normal asympoic disribuion; ha is X T T = d z e! N 0; E z s e : () Puing ogeher Cramer s Theorem wih (0) and () yields he resul given in DGM Theorem ; ha is under he null hypohesis, p d T ^! N 0; AV ar ^ where AV ar ^ denoes he asympoic variance of ^ and is given by AV ar ^ = E z s E z s 7 e E z s : (3)

8 Since he error erm e is i.i.d., he formula for AV ar () can be re ned furher by applying he LIE o E (z s e ) and noing ha E e j= = E e =. I herefore follows ha he asympoic variance for ^; under he null hypohesis, becomes AV ar ^ = = E z s h E E E z s h E z s e j= i E z s z s i E e j= E z s = E z s : (4) Subsiuing for E z s using equaion () ; we obain he required formula for AV ar ^ as given in Lemma of DGM: AV ar ^ = (d ) (d ) : In he presence of heeroskedasiciy, he formula for he asympoic variance given in (4) is incorrec because he hird equaliy in (4) is obained by assuming homoskedasiciy. This will yield incorrec esimae of OLS sandard errors. As a resul, he FD-F -saisic compued using his homoskedasic OLS sandard errors will give misleading saisical inferences. 3. The e ecs of uncondiional heeroskedasiciy In order o invesigae how heeroskedasiciy can a ec he size properies of he FD-F es, we conduc a simple Mone Carlo sudy. Under he null hypohesis ha = 0 in (4) ; he daa generaing process is () () wih d 0 = : The f" g are sandardised normal random variables i:i:d:n (0; ) and were generaed using GAUSS normal random number generaor rndn. We concenrae on he case of a single abrup srucural change in he error variance: = ( bt c) + ( > bt c) (5) wih (0; ) gives he locaion of he break poin. In his case, he error variance shifs from o a ime bt c : Le = = be he parameer ha measures he magniude of he shif. Wihou loss of generaliy, we se =, hen, > ( < ) corresponds o a posiive (negaive) shif. The furher he value of di ers from uniy, he larger he magniude of he shif. For a posiive shif, we consider = ; 5; 0 and for a negaive shif, we consider = 0:; 0:; 0:5: The values of are in seps of 0.. The number of Mone Carlo replicaions is 0,000. We calculae he empirical size of he (d )-saisic in (7) wih d = 0:9 for a sample size of T = 50: The criical value is obained from he sandard normal disribuion N (0; ) which is :645 for 5% signi cance level. Figure repors he real size agains he nominal size a he 5% level of signi cance. On he horizonal axis nine break poins are marked and he verical axis gives he corresponding 8

9 Probabiliy of Type d = d = 5 d = 0 Probabiliy of Type d = 0.5 d = 0. d = break poins break poins Figure : Empirical Sizes a he 5 percen signi cance level percenage of rejecions under he uni roo null hypohesis. The lef graph presens he case of an upward shif and he righ graph presens he case of a downward shif. DGM have shown ha he empirical size of he (d )-es, under homoskedasiciy, are reasonably close o he 5% nominal level. In he upward shif case (lef graph), as moves from 0. o 0.9, rejecion frequency increases from abou 5% o abou 30%. The opposie is rue for he case of a downward shif (righ graph). Noice also ha as he magniude of he shif increases, he rejecion frequency increases. The degree of size disorion varies wih he direcion, iming and magniude of he break. Wihou knowledge of he iming and magniude of he break, he -raio given in (7) canno be relied upon o give valid inferences. 4 Tesing for a uni roo under uncondiional heeroskedasiciy I is imporan o noe ha in he heeroskedasic case we canno claim ha he correc formula for AV ar ^ is given in (3) unless we re-esablish he resuls saed in (9) ; (0) and () under Assumpion V. In order o correc for he size disorion shown in he Mone Carlo sudy we follow Whie s suggesion. The Whie s heeroskedasiciy robus -raio (denoed as W (d )) for esing he signi cance of in (4) is given by ^ W (d ) = (6) W SE ^ where W SE ^ is he Whie sandard error given by v P T W SE ^ = u (d y ) be PT (d y ) : The advanage of using Whie sandard errors is ha applied researchers are no required o specify, a priori, a model for he error variance process. This is especially useful since in 9

10 many pracical siuaions such model is unknown a priori. In he case of a single srucural break in he error variance, he Mone Carlo evidence presened in he previous secion reveals ha he -es is a eced by he locaion of he break in he sample. The use of Whie sandard errors o correc for heeroskedasiciy is paricularly useful since i does no require any informaion regarding he iming of he break. 4. Asympoic Disribuion of he FD-F Tes wih a xed d The following heorem is concerned wih he asympoic properies of he W (d )-raio de ned in (6) for a xed d : Theorem Le he ime series process fy g T = be de ned by () () : Under Assumpions E and V, he asympoic properies of he W (d )-raio wih = < d < for esing = 0 in (4) are given by: under he null hypohesis ha d 0 = ; W (d ) d! N (0; ) W (d ) p! under he alernaive hypohesis ha 0 d 0 < : The rs par of Theorem saes ha he W (d )-raio de ned in (6) is pivoal and has a sandard limiing null disribuion. The second par saes ha he W (d )-raio is able o discriminae beween he null hypohesis of a random walk process and he alernaive hypohesis of a fracional inegraed process wih probabiliy one in large samples. Thus for a given value of d 0 [0; ); he W (d )-raio, compued using any value of d in he inerval = < d < ; ends o negaive in niy and so he power of he W (d )-raio ends o uniy as T! : This means ha he W (d )-raio is a consisen es saisic even if d is chosen di erenly from d 0. Esablishing Theorem is more di cul han in he homoskedasic case where he ergodic saionary WLLN is a key ingredien. This is because uncondiional heeroskedasiciy renders invalid he applicaion of ergodic saionary WLLN for he various sample momens appearing in ^ and W (d ). The proof of Theorem is given in he Appendix. 4. Asympoic Disribuion of he FD-F Tes wih an Esimaed d So far he value of d is speci ed a priori under he simple alernaive hypohesis. In he absence of such a speci caion, as is usually he case, Theorem shows ha any available consisen esimaor of d 0 in () can be used o make he FD-F operaional. Le ^d be a consisen esimaor such ha ^d d 0 = o p () : Since FD-F regression requires he value of d o be sricly less han one, we follow DGM s suggession and de ne a rimming rule for ^d : ^d if ^d < c ^d = c if ^d (7) c 0

11 where c is a xed consan such ha 0 < c < =. The leas squares regression in (4) can now be made operaional by replacing he unspeci ed d by ^d : y = b d y + e : (8) The heeroskedasiciy robus -saisic, compued in he same way as before excep using he regression (8), is now denoed as W ^d : The following Theorem esablishes ha he W ^d -raio has a sandard normal limiing null disribuion under heeroskedasic errors. The proof of he Theorem is given in he Appendix. Unlike DGM, we do no require he esimaor ^d o converge in probabiliy o d 0 a a p T -rae. Insead of relying only on parameric esimaion, which has a sandard p T -rae of convergence, Theorem opens up he possibiliy ha semi-parameric esimaion, which converges slower han he p T -rae, can be used o make he FD-F es operaional. Theorem Le ^d saisfy he rimming rule in (7) wih ^d d 0 = o p () : Suppose Assumpions E and V hold. Under he null hypohesis ha y is generaed by () () wih d 0 =, he asympoic disribuion of he W ^d -raio of he OLS esimaor of in he regression y = ^d y + e is given by W ^d d! N (0; ) : We follow he DGM approach and consider he parameric GMD esimaion mehod. Harris and Kew (007) examine he GMD esimaion in he presence of uncondiional heeroskedasiciy in a fracional inegraed model. The GMD esimaion procedure relies only on he absence of auocorrelaions and no of heeroskedasiciy in he residuals and hence his esimaor can be poenially robus o he presence of heeroskedasiciy. We nex describe he GMD esimaor. 4.3 GMD esimaion mehod This secion describes he GMD esimaion procedure. To describe his esimaor, we de ne he residuals e (d) for some d as X e (d) = d y = j (d) y j ; = ; :::; T: (9) The Minimum Disance Esimaion (MDE) of d 0 is de ned as j=0 ^d = arg min d[0;] m= kx ^ m (d)

12 where ^ m (d) is he m h sample auocorrelaion of he residuals e (d) ; for m = ; ; :::; k: The noaion arg min denoes he value of d such ha he argumen of P k m= ^ m (d) is minimised Given he residuals e (d) ; :::; e T (d), he ^ m (d) is a funcion of he parameer d and i can be calculaed as follows: ^ m (d) = T P T =m+ e (d) e m (d) T P T = e (d) : Noe ha he residual in (9) can be re-wrien as e (d) = d d 0+d 0 y = d d 0 d 0 y = d d 0 e and he populaion counerpar of ^ m (d) is de ned o be m (d) = T P T =m+ E (e (d) e m (d)) T P T = E : (0) e (d) Despie he fac ha e (d 0 ) are uncondiionally heeroskedasic, he populaion auocorrelaions, m (d) ; evaluaed a he rue parameer d 0 can sill be zero since m (d 0 ) = T P T =m++p E (e (d 0 ) e m (d 0 )) T P T =+p E e (d 0) = T =m++p E (e e m ) T P T =+p E = 0: e In order for he GMD mehod o be a reliable esimaor under uncondiional heeroskedasiciy, he populaion auocorrelaion m (d) should no equal zero if d 6= d 0 and herefore he residuals, e (d) ; are auocorrelaed. To verify his, we wrie he numeraor of m (d) in (0) as E (e (d) e m (d)) = E d d 0 e d d 0 e m 0 X = = = j=0 Xm j (d d 0 ) e j i=0 i (d d 0 ) e m i A X X j (d d 0 ) i m (d d 0 ) E (e j e i ) j=0 i=m X i m (d d 0 ) i i=m and so m (d) is no equal zero when d 6= d 0. I is clear ha he GMD esimaor can be poenially robus o an unknown form of uncondiional heeroskedasiciy. The assumpion of consan uncondiional variances over

13 ime is unnecessarily resricive for he consisency of he GMD esimaor. Harris and Kew (007) show ha he GMD esimaor is consisen and converges a p T -rae under Assumpion V. This resul implies ha he GMD esimaor urns ou o be very useful in implemening a feasible FD-F saisic under heeroskedasiciy. 5 Mone Carlo Simulaion Resuls This secion uses Mone Carlo (MC) experimens o examine he nie sample performance of he FD-F ess wih Whie sandard errors when he errors are uncondiionally heeroskedasic. The simulaed daa se for fy g is generaed according o () - () : The pseudo random numbers for " are generaed using he rndn funcion in Gauss 7. For all of he MC experimens, he number of replicaions is 0000 and he seed used for he rndn funcion is 999. The sample sizes considered are T = 50; 500 and 000. The larger values of T are chosen since empirical sudies of srucural breaks in he error variance use daa colleced over an exended period of ime. Apar from he single srucural break model considered in Secion 3., we consider wo addiional models for he error variance process, : These heeroskedasiciy models are based on ha used by Cavaliere (004), Cavaliere and Taylor (007) and Phillips and Xu (006). Double Variance Shifs. Two abrup shifs in he error variance a rs from o occurring a ime b T c and hen follow by anoher abrup shif from o 3 a ime b T c : The dynamics of can be wrien as = ( b T c) + (b T c < b T c) + 3 (b T c < T ) where ; (0; ) : A special case arises when = and 3 = : In his case, he muliple variance shifs are symmeric and and hence he double variance shifs model reduces o = ( bt c) + (bt c < bt c) + ( bt c < T ) () where (0; ) : Trending Variances. The variance of he innovaions changes monoonically from a ime = 0 o a ime = T: Noe ha, he variance may no necessarily rends linearly. The dynamics of can be wrien as = + m T where m = ; ; :: < : The variance changes coninuously in a linear fashion when m = and i changes in a non-linear way oherwise. 3

14 For each of he variance models, a wide range of parameer seings are used o generae he di eren paerns of variance dynamics. Following Secion 3., we de ne = = and normalise =. As for he single srucural break model in (5), we allow he shif o occur owards he beginning, middle and end of he sample by seing = 0:; 0:5 and 0:9: Two values of were used: = 5 (posiive shif) and = 0: (negaive shif). We consider early posiive shif ( = 0: and = 5), early negaive shif ( = 0: and = 0:); posiive shif occurring mid-way hrough he sample ( = 0:5 and = 5), negaive shif occurring mid-way hrough he sample ( = 0:5 and = 0:), lae posiive shif ( = 0:9 and = 5); and lae negaive shif ( = 0:9 and = 0:): As for he double variance shifs model, we consider he special case where he double breaks occur symmerically. We le = 0:05; 0:45 and = 5; 0:: Here, four di eren ypes of muliple break poins are generaed: early posiive break hen followed by lae negaive break ( = 0:05 and = 5); early negaive break hen followed by lae posiive break ( = 0:05 and = 0:); posiive shif occurring near he middle of he sample hen immediaely followed by a negaive shif ( = 0:45 and = 5) and a negaive shif occurring near he middle of he sample hen immediaely followed by a posiive shif ( = 0:45 and = 0:): In he rending variances model, we allow he rending variance o increase coninuously in a linear and non-linear fashion. In he linear case where m = ; we consider boh upward ( = 5) and downward ( = 0:) rends. In he non-linear case (m = ), we consider boh upward and downward rending variances. 5. Size Properies wih known d Tables o 3 repor he percenage of rejecions under he null hypohesis (empirical size) when d 0 = in () for he (d )- and W (d )-ess wih d = 0:6; 0:7; 0:8; 0:9; 0:95 a he nominal 5% level. Since boh ess have sandard normal limiing disribuions, he criical value, a he 5% signi cance level, is All oher hings equal, he (d ) and W (d ) ess display roughly he same rejecion frequencies for all values of d, alhough i is worh nohing ha size disorions are slighly smaller he closer d is o : For comparison purposes, we include he acual sizes of he (d ) es under homoskedasic error and hese are repored in he rs row of Table. In erms of empirical size, he W (d ) es performs jus as well as he (d ) es in he absence of heeroskedasiciy. We will discuss rs he performance of he (d ) es in he presence of uncondiional heeroskedasiciy. Tables o 3 clearly show ha he (d ) ess are no robus o deparures from he homoskedasiciy assumpion. The degree of size disorions can vary, depending on he variance srucure. For example, in he case of a single abrup shif (see Table ), subsanial size disorion occurs when he abrup shif is eiher an early negaive shif or a lae posiive shif, while he opposie is rue when he abrup shif is eiher an early posiive shif or a lae negaive shif. When he posiive or negaive shif occurs owards he middle of he sample, he (d ) ess overrejec moderaely wih empirical sizes of around %. 4

15 As for he double variance shifs model (see Table ), serious size disorions arise when an early negaive shif is followed by a lae posiive shif of he same magniude as he earlier shif ( = 0:05, = 0:); and when a posiive shif occurring owards he middle of he sample is followed immediaely by a negaive shif of he same magniude ( = 0:45, = 5). In hese cases, he proporion of rejecions is abou 5% when he nominal size is 5%. However, he (d ) ess appear o have approximaely correc size when an early posiive shif is followed by a lae negaive shif ( = 0:05; = 5), and when a negaive shif occurring owards he middle of he sample is followed immediaely by a posiive shif ( = 0:45; = 0:). When he error variance follows a polynomial rend (see Table 3), he (d ) display smaller size disorions han hose in Tables and. In his case, he (d ) ess moderaely overrejec he uni roo null hypohesis wih empirical sizes varying from 7 o 0 per cen. Tables 3 abou here The W (d ) es which uses Whie sandard errors is very e ecive in reducing he observed size disorions. Take for example he case where here is an early negaive break (i.e. = 0: and = 0:). In his case, when d = 0:90 and T = 50; Table shows ha he Whie correcion can reduce he empirical size from 3.4% o 6.98%. This 6.98% empirical size of he W (d ) es is seen, as expeced, o fall owards he 5% nominal size as T grows. When he sample size is relaively large (T = 000); empirical sizes of W (d ) ess are always reasonably close o he 5% nominal level in all of he heeroskedasic models considered. Thus, Mone Carlo evidence reveals ha Whie s correcion works well in pracice for a wide range of models of uncondiional heeroskedasiciy. Tables 4 o 7 repor he raw power of he W (d ) es agains he alernaive of fracionally inegraed processes given in () wih values of d 0 chosen from [0:55; 0:95] in seps of 0:05. The W (d ) es is compued under he assumpion ha d is known a priori by seing d = d 0 : Under homoskedasiciy (i.e. = ), Table 4 shows ha he rejecion frequencies of he heeroskedasic sandard errors are comparable o hose of he homoskedasic sandard errors. This suggess here is no loss in power from using Whie s correcion when he errors are homoskedasic. Table 4 herefore can be used as a benchmark o compare he nie sample power resuls of he W (d ) under heeroskedasiciy. From Tables 5 o 7, here are cases where he power of he W (d ) ess under heeroskedasiciy are considerably lower han hose under homoskedasiciy. In hose cases, i urns ou ha he FD-F ess wihou Whie sandard errors su er from severe size disorions. To illusrae, ake an example of an early negaive break ( = 0:; = 0:). As noed before, he (d ) ess ends o overrejec subsanially. For his same variance model, Table 5 shows ha he power of he W (d ) es is considerably less han ha observed in Table 4, where he errors are homoskedasic. In some cases where he (d ) ess su er from severe size disorions, he Whie correcion loses power relaive o he homoskedasic case. In all cases, as expeced, power increases as T increases, and as d 0 moves away from. Tables 4 7 abou here 5

16 So far we have been reaing he value of d as if i is pre-speci ed. Now we use he GMD esimaion procedure described earlier o esimae d 0. Using he rimming rule de ned in (7) ; he resuling esimae is hen used o replace d wih ^d : Under homoskedasiciy, DGM show via Mone Carlo experimens ha replacing he value of d wih ^d has very lile impac on he size and power properies of he FD-F es wihou Whie s correcion. We now check wheher hese resuls coninue o hold under heeroskedasiciy. Tables 9 o presen he sizeand power properies of he FD-F ess wih and wihou Whie s correcion. The ^d and W ^d ess are calculaed in he same way as before, bu using he inpu ^d as de ned in (7) : Following DGM, we se c = 0:0 and hus ^d 0:98: By comparing he resuls in Tables 8 - wih he corresponding resuls in Tables -7, he rejecion frequencies when d is replaced wih ^d are broadly similar o hose when d is speci ed a priori. Thus, he esimaion of d using he GMDesimaor under uncondiional heeroskedasiciy will no a ec he performance of he W ^d ess. 6 Conclusion Tables 9 abou here We have shown ha, via Mone Carlo simulaions, he OLS -saisics su er from subsanial size disorions when he errors are heeroskedasic. Thus, he FD-F -saisics compued using OLS sandard errors derived under he assumpion of homoskedasiciy will give misleading saisical inferences. We sugges FD-F -saisic ha uses Whie heeroskedasiciy robus sandard errors o accoun for he presence of uncondiional heeroskedasiciy of unknown form. We demonsrae ha Whie s version of he FD-F saisics has a sandard limiing null disribuion una eced by uncondiional heeroskedasiciy. A Mone Carlo sudy shows ha he proposed mehod is e ecive in reducing he size disorion. In he absence of heeroskedasiciy, he power loss due o he use of Whie sandard errors insead of homoskedasic sandard errors urns ou o be very small. References [] Anderson, T.W. (97) The saisical analysis of ime series. Wiley New York. [] Beare, B. (004) Robusifying uni roo es o permanen changes in innovaion variance. Yale Universiy, mimeographed. [3] Beran, J. (995) Maximum likelihood esimaion of he di erencing parameer for inverible and shor and long memory auoregressive inegraed moving average models. Journal of he Royal Saisical Sociey, 57, [4] Bollerslev, T. (986) Generalized auoregressive condiional heeroskedasiciy. Journal of Economerics, 3,

17 [5] Busei, F. and A.C. Harvey (00) Tesing for he presence of a random walk in series wih srucural breaks. Journal of Time Series Analysis,, [6] Busei, F. and A.M.R. Taylor (003) Variance shifs, srucural breaks, and saionariy ess. Journal of Business and Economic Saisics,, [7] Busei, F. and A.M.R. Taylor (004) Tess of saionariy agains a change in persisence. Journal of Economerics, 3, [8] Cavaliere, G. (004) Uni roo ess under ime-varying variances. Economeric Reviews, 3, [9] Cavaliere, G. and A.M.R. Taylor (005) Saionariy ess under ime-varying second momens. Economeric Theory,, -9. [0] Cavaliere, G. and A.M.R. Taylor (006) Tesing for uni roos in ime series models wih non-saionary volailiy. Journal of Economerics, forhcoming. [] Cavaliere, G. and A.M.R. Taylor (006) Tesing for a change in persisence in he presence of a volailiy shif. Oxford Bullein of Economics and Saisics, 68, [] Davidson, J. (994) Sochasic limi heory: an inroducion for economericians. Oxford Universiy Press. [3] Dolado, J., Gonzalo, J. and L. Mayoral (00) A fracional Dickey-Fuller es for uni roos. Economerica, 70, [4] Engle, R. F. (98) Auoregressive condiional heeroskedasiciy wih esimaes of variance of U.K. in aion. Economerica, 50, [5] Hamori, S. and A. Tokihisa (997) Tesing for a uni roo in he presence of a variance shif. Economics Leers, 57, [6] Harris D. and H. Kew (007) Roo-T consisen esimaor of fracionally inegraed model wih uncondiional heeroskedasic errors. Melbourne Universiy, mimeographed. [7] Hosking, J.R.M. (98) Fracional di erencing, Biomerica, 68, [8] Kim, J. (000) Deecion of change in persisence of a linear ime series. Journal of Economerics, 95, [9] Kim, J., J. Belaire Franch and R. Badillo Amador (00) Corrigendum o "Deecion of change in persisence of a linear ime series". Journal of Economerics, 09, [0] Kim, T.H., S. Leybourne and P. Newbold (00) Uni roo ess wih a break in innovaion variance. Journal of Economerics, 09, [] Mayoral M. (007) Minimum disance esimaion of saionary and non-saionary ARFIMA processes. Economerics Journal, 0,

18 [] McConnell, M.M. and G. Perez-Quiros (000) Oupu ucuaions in he Unied Saes: wha has changed since he early 980s? American Economic Review, 90, [3] Nicholls, D.F. and A.R. Pagan (983) Heeroskedasiciy in models wih lagged dependen variables. Economerica, 5, [4] Perron, P. (989) The grea crash, he oil price shock, and he uni roo hypohesis. Economerica, 57, [5] Perron, P. (990) Tesing for a uni roo in a ime series wih a changing mean. Journal of Business and Economic Saisics, 8, [6] Phillips, P.C.B. and K.-L. Xu (006) Inference in auoregression under heeroskedasiciy. Journal of Time Series Analysis, 7, [7] Robinson, P.M. and D. Marinucci (00) Narrow-band analysis of nonsaionary processes. Annals of Saisics, 9, [8] Sowell, F.B. (990) The fracional uni roo disribuion. Economerica, 58, [9] Sock, J.H., (999) A class of ess for inegraion and coinegraion. In: Engle, R.F., Whie, H. (Eds.), Coinegraion, causaliy and forecasing. A Fesschrif in honour of Clive W.J. Granger. Oxford Univerisy Press, Oxford, pp [30] Tanaka, K. (999) The nonsaionary fracional uni roo. Economeric Theory, 5, [3] van Dijk, D., D.R. Osborn and M. Sensier (00) Changes in variabiliy of he business cycle in he G7 counries. Economeric Insiue Repor EI 00-8, Eramus Universiy Roerdam. [3] Whie, H. (980) A heeroskedasiciy-consisen covariance marix esimaor and a direc es for heeroskedasiciy. Economerica, 48, [33] Whie, H. (984) Asympoic heory for economericians. Academic Press. 8

19 Appendix: Proofs Lemmas 3 and 4 below give he weak law of large numbers and he Cenral Limi Theorem for fracional inegraed processes under Assumpions V and E, respecively. Lemma 3 Le fe g be a sequence of random variables generaed according o () wih and f" g saisfy Assumpions V and E respecively. Consider he following fracionally inegraed processes: z = e ; < = and x = e ; < =: Then (i) sup T E jz j r < for r 4; (ii) 0 < B T P T E z ; and (iii) T P T (x mz n E (x m z n )) p! 0 for m; n = 0; : I seems useful o sae he following resuls here. The sochasic sequence T P T z does no converge o zero in probabiliy. To see his, we wrie T P T z = T T X z E z + T TX E z : () The rs erm on he righ-hand side of he above equaion converges in probabiliy o zero by Lemma 3(iii) wih x = z : The second erm is bounded away from zero uniformly in T by Lemma 3(ii). Lemma 4 Le fz g and fe g be de ned as in Lemma 3. If he condiions of Lemma 3 are sais ed, hen as T! ; T T X E z e! = X T T = d z e! N (0; ) : Proof of Lemma 3 Par (i) Under he Type model of fracional inegraion, he series z can be wrien as X z = e (>0) = j ( ) e j (3) j=0 9

20 and since < =; he coe cien j ( ) is square summable. Choosing r = 4; we have E z 4 (4)! 4 = E P j ( ) e j = P Firs observe ha j=0 i=0 j=0 k=0 l=0 i ( ) j ( ) k ( ) l ( ) E (e i e j e k e l ) : E (e i e j e k e l ) = E (e i e j ) E (e k e l ) + E (e i e k ) E (e j e l ) +E (e i e l ) E (e j e k ) + ( i; j; k; l) (5) where (:; :; :; :) denoes he join fourh order cumulans of e. I follows ha equaion (4) can be rewrien as E z 4 (6) = 3 P i=0 j=0 k=0 l=0 + P i=0 j=0 k=0 l=0 i ( ) j ( ) k ( ) l ( ) E (e i e j ) E (e k e l ) i ( ) j ( ) k ( ) l ( ) ( i; j; k; l) : We will show ha each erm on he righ hand side of equaion (6) is uniformly bounded in T. Since fe g are independen and E (e ) = 0; non-zero expecaions arise in he following hree pairs: (i) i = j and k = l; (ii) i = k and j = l; (iii) i = l and j = k: The rs erm on he righ side of equaion (6) is uniformly bounded in T since 3 P i=0 k=0 i ( ) k ( ) E e i E e = 3 P P i ( ) k ( ) i k i=0 k=0 3 P P i ( ) k ( ) 4 i=0 k=0 3 4 P i ( i=0 k k=0 ) P k ( ) < : (7) Regarding he second erm on he righ hand side of equaion (6) ; he independence assumpion for e implies ha he fourh order cumulans are zero excep when i = j = 0

21 k = l: To see his, consider he case where i = j 6= k = l: Then, equaion (5) becomes ( i; i; k; k) = E e ie k E e i E e k = E e ie k E e i E e k = 0: Similar argumens follow for cases where i = k 6= j = l and i = l 6= j = k: However, if i = j = k = l; equaion (5) becomes ( i; i; i; i) = E e 4 i 3E e i : In his case, he fourh order cumulans are zero only when he process fe g is Gaussian because E e 4 i = 3 4 i : Therefore, he second erm in (6) is uniformly bounded in + m T under Assumpion V since = X 4 i ( ) j ( i; i; i; i)j i=0 X 4 i ( ) E e 4 i=0 X i=0 X i=0 i 3E e i 4 i ( ) E e 4 i i 4 i ( ) sup E e X sup E e Combining equaions (7) and (8) yields i=0 sup E z 4 B < : T Then (i) follows direcly from he Liapunov inequaliy. Par (ii) To show (), we wrie T E z = T TX 4 i ( ) < : (8) E e X T X = T j ( ) j j= X T X T j ( ) : j=

22 Since > 0, i is only required o show ha In order o show his, we wrie T 0 < B lim T! T X T X j= j ( ) TX X j ( ) < : j= X T TX TX TX = T j ( ) T j ( ) = = j= T T j= X X T TX j ( ) T j ( ) j= TX j ( ) j= T j= TX X T TX j ( ) T j ( ) : j= j= As T! ; he rs erm approaches a nie posiive limiing value since he sequence j ( ) is square summable, ha is TX 0 < B lim j ( ) < : T! j= This hus implies ha he second erm converges o zero. The hird erm converges o zero by Cesaro summaion. This complees he proof for par (iii). Par (iii) We show ha he sequence fx z E (x z )g sais es he condiions of he Chebyshev Law of Large Numbers (see Davidson 000 pg 4). The rs condiion, which requires he sequence has zero mean, is sais ed rivially. The second condiion requires ha lim E T T! TX (x z E (x z ))! = 0: (9) Like he series z in (3) ; x can be wrien as X x = e (>0) = j ( ) e j : (30) In order o simplify noaion, we will rewrie he coe cien i ( ) in (30) as i and he coe cien j ( ) in (3) as i : Under he condiion of Lemma 3 ( < = and < =), he coe ciens i and i are square summable. j=0

23 To show (9) ; we wrie TX E T (x z E (x z ))! T T = T s=0 s=0 s=0 je ((x z E (x z )) (x s z s E (x s z s )))j! P i j (e i e j E (e i e j )) i=0 j=0 E Ps Ps k l (e s k e s l E (e s k e s l )) k=0 l=0! P i j (e i e j E (e i e j )) i=0 j= E! P : (3) k s l s (e k e l E (e k e l )) k=s l=s+ Because of independence propery of e ; he following hree pairs have non-zero expeced values: (a) i = j and k = l; (b) i = k and j = l; and (c) i = l and j = k: In each of hese cases, we will show ha he erm converges o zero as T! : Case (a). The erm in equaion (3) for which i = j and k = l simpli es as follows: T T P s=0 j= l=s+ +T s=0 l=s+ P P s=0 j= l=s+;l6=j j j l s l s E e j E e j e l E e l j l j l jl s j l s E e l l (3) j j j j j l s j l s E e j E e j e l E e l : The inequaliy is riangle inequaliy. The second erm is zero since for l 6= j E e j E e j e l E e l = E e j E e j E e l E e l = 0: 3

24 The erm (3) converges o zero as T! since T = T s=0 l=s+ l= j l j l T P T j l j l l= T P T j l j l l= = sup E e sup E e = sup E e j l j l j l s j l s E e l l lp j s j s E e 4 l s= lp s= lp 4 l j s j s E e 4 l + 4 l j s j s s= T P T l= T P T = sup E e j l j l lp j s j s s= P P P l l= P P T l l= l l=0 l : l=0 s s= P s s=0 The rs inequaliy follows from he riangle inequaliy, he second inequaliy follows from equaion (3); he hird inequaliy holds because he coe ciens l (= l ( )) and l (= l ( )) are square summable. Case (b). The erm in equaion (3) for which i = k and j = l simpli es as follows: T T T s=0 k=s l=s+ s=0 k=s s=0 k=s j k j j k sup E e 4 s P k k=0 = sup E e 4 s P k k=0 j k j l j k s j l s E e k e l s j l=s+ j k j j k s j P l=s P P l l=0 l l=0 q l l s j l j l s rsup E e 4 sup T T = s=0 s=0 s TP P s P k k=s E e 4 k E e 4 l k k=s P E e 4 P l l=s l l=s The rs inequaliy follows from Cauchy-Schwarz inequaliy, he second inequaliy follows from equaion (3) and he hird inequaliy holds since he coe ciens k and l are square summable. From he las equaliy, he erms P P k=s k and l=s l go o zero as s!. Thus by Cesaro summaion, as T! ; i follows ha s P P T s=0 k k=s 4 l l=s! 0: :

25 Case (c). The case when i = l and j = k; disappears similarly as in case (b). This complees he proof for par (iii). Proof of Lemma 4 De ne where X T = s T z e s T = TX E z e : The sequence fx T g is a maringale di erence sequence, since E X T jf T ( ) = E s T z e jf T ( ) = s T z E e jf T ( ) = s T z E (e ) = 0 a:s: Theorem 6..3 of Davidson gives condiions under which he process fx T g obeys he cenral limi heorem for maringale di erences; ha is TX X T = T = P T z e q T P T E! d N (0; ) : z e The rs condiion requires he square sequence X T obeys he weak law of large numbers or TX p! (33) and he second condiion requires Regarding he rs condiion, noe ha X T max jx p T j! 0: (34) T P TX E XT T = E z P T E = z e and hus equaion (33) can be re-wrien as e TX X T E X T p! 0: (35) Even hough X T is a maringale di erence sequence, he squared sequence, XT ; is no a maringale di erence sequence. Consequenly, he WLLN for maringale di erence sequence 5

26 canno be used o imply ha equaion (35) holds. We proceed by rewriing he squared sequence as TX X T E XT = T P T z e E z e T s T = T s X T T T z e T T X z E z! : Thus i su ces o show ha equaion (35) is rue if: T T X z e z p! 0; (36) and and T T X z E z p! 0 (37) 0 < B T s T B < : (38) To show (36) ; we noe ha z e z is a maringale di erence sequence as E z e z jf = E z e jf E z jf = z E e jf z = z z = 0 a:s: By Minkowski inequaliy, Lemma 3(i), and (3), we have E z e z = q E z 4 q e4 + E z 4 4 q q E z 4 E e 4 + E z 4 r sup T E z r! 4 sup E e 4 + sup E z 4 < : T Thus by WLLN for maringale di erences (Theorem 6.. of Davidson) equaion (36) holds. For he consisency in (37) o be valid, we have o check ha he sequence z E z sais es he condiions of he Chebyshev Law of Large Numbers (see Davidson 000 pg 4). 6

27 The rs condiion, which requires he sequence has zero mean, is sais ed rivially. The second condiion requires ha he variance of he sum ends o zero as T! ; ha is lim E T T! TX z E z! = 0: (39) Unlike he above proof, his ime he sequence z E z is no a maringale di erence sequence and herefore he variance of he sum of erms is no equal o he sum of he variances. In order o simplify he noaion, we will express z as z = = = e ( >0)! X i ( ) e i i=0 X X i j e i e j ; i=0 j=0 where i = i ( ) : To show (39), we wrie TX E T z E z! T s=0 4 T E s=0 s E z E z z s E z s s P k=0 i=0 j=0 s P l=0 i j (e i e j E (e i e j )) k l (e s k e s l E (e s k e s l ))! : By leing i = i and k = k ; he above equaion ends o zero by argumens similar o hose used in proving Lemma 3(i). Thus by he Chebyshev Law of Large Numbers, T T X as required. To show (38), we rs show ha z E z p! 0; X T 0 < B T E z e : (40) 7

28 To see his, we wrie T T X E z e = T TX = T T X = E e E z j= X j ( ) j TX TX j j ( ) T +j j= TX 4 j= = j j ( ) : (4) T ( ) is con- Since he coe cien j ( ) is square summable, which means ha P T vergen, Lemma 8.3. of Anderson (97) implies ha 0 < B lim T! TX j= j j ( ) = T X j ( ) < ; j=0 j=0 j where P j=0 j ( ) > 0. Given ha 4 is a sricly posiive consan, equaion (4) is bounded away from zero for all T : Nex for (38) we show T T X Following from he argumens in equaion (4) ; we wrie E z e B < : (4) T T X E z e X T X 4 T j ( ) j= X T X 4 T j ( ) = j=0 X = 4 j ( ) = B < : (43) j=0 Thus (38) has been proved. Equaions (38) ; (36) and (37) imply ha he rs condiion of he cenral limi heorem saed in (33) holds. Regarding he second condiion for he cenral limi heorem (see equaion (34)), we noe ha for any > 0 and for some > ; P max jx T j > T TX P (jx T j > ) 8 TX E jx T j :

29 Se = 4 and noe ha fe g are independen, we obain TX E XT 4 = s 4 T TX E z 4 e 4 T P T = E z4 e4 T P T E z e = T P T E z4 E e 4 T P T E z e T sup T E z 4 sup E e 4 T P T E! 0 z e as T! : This follows because Lemma 3(i) and equaion (3) uniformly bound he numeraor and (38) uniformly bounds he denominaor away from zero and in niy. Proof of Theorem Par (a) Asympoic Disribuion. In order o simplify noaion we wrie z for d y : Following from he discussion in secion 3., z I ( d ) under he uni roo null hypohesis. The values of ( d ) will lie in he inerval (0; =) since d (=; ). Using his and noing ha y = e, he W -raio in (6) can be re-wrien as W (d ) = = TP z s TP s z e ^e T z T 0 TP e E (z e ) T P T z T P T ^e C E (z e ) A : Since = < d < ; he series fz g is asympoically saionary and by Lemma 4, i follows ha TP s T z T e E (z e )! d! N (0; ) : 9

30 I remains o show ha or equivalenly T T P T z T P T ^e E (z e ) p! z ^e E (z e ) T E (z e ) p! 0: (44) Equaion (38) boh bounds he denominaor uniformly away from zero and ensures ha i is nie. I su ces o show ha he numeraor converges in probabiliy o zero as T! : We noe ha he residuals ^e can be wrien as ^e = y ^ d y = e ^z. Using his, we wrie he numeraor in (44) as T = T = T z e ^z E z e z e ^z e + ^ z z e E z ^T E z z 3 e + ^ T P T z 4 (45) The aim is o show ha all he hree erms in (45) converge in probabiliy o zero. The rs erm converges in probabiliy o zero by equaions (36) and (37). As for he second and hird erms in (45), we will rs show ha ^! p 0: Under he null hypohesis, ^ in (6) can be wrien as ^ = T P T z e T P T z The numeraor T P T z e converges o zero in probabiliy by Lemma 3(iii) wih = 0: As for he denominaor, we have already shown in equaion () ha T P T : z does no converge o zero in probabiliy. Therefore ^! p 0: Now, coming back o he second erm in (45) ; since ^! p 0, i will converge in probabiliy o zero if T z 3 e = O p () : (46) Before proving equaion (46), we sae he Holder s inequaliy. For any random variables X and Y and for a > 0; if E jxj a < and E jy j a=(a ) < hen E jxy j (E jxj a ) =a E jy j a=(a ) (a )=a : 30