O the Asymptotics of ADF Tests for Uit Roots Yoosoo Chag Departmet of Ecoomics Rice Uiversity ad Joo Y. Park School of Ecoomics Seoul Natioal Uiversity Abstract I this paper, we derive the asymptotic distributios of Augmeted- Dickey-Fuller (ADF) tests uder very mild coditios. The tests were origially proposed ad ivestigated by Said ad Dickey (984) for testig uit roots i ite-order ARMA models with iid iovatios, ad are based o a ite AR process of order icreasig with the sample size. Our coditios are sigi catly weaker tha theirs. I particular, we allow for geeral liear processes with martigale di erece iovatios, possibly havig coditioal heteroskedasticities. The liear processes drive by ARCH type iovatios are thus permitted. The rage for the permissible icreasig rates for the AR approximatio order is also much wider. For the usual t-type test, we oly require that it icrease at order o( = ) while they assume that it is of order o( ) for some satisfyig < =3. This versio: November, Key words ad phrases: ADF tests, uit roots, asymptotics, liear process, autoregressive approximatio. We wish to thak the Editor, a Associate Editor ad two aoymous referees for useful commets, which have greatly improved the expositio of a earlier versio of this paper. Park thaks the Departmet of Ecoomics at Rice Uiversity, where he is a Adjuct Professor, for its cotiuig hospitality ad secretarial support. This research was supported by Korea Research Foudatio.
. Itroductio The tests for uit roots i ARprocesses were rst proposed ad ivestigated bydickey ad Fuller (979, 98). The tests bydickey-fuller (DF) are based o ite-order AR models, the orders of which are assumed to be kow. They were later exteded by Said ad Dickey (984) to allow for ite ARMA processes of ukow order. They show that the tests, which are frequetly referred to as Augmeted-Dickey-Fuller (ADF) tests have the same limitig distributios as DF tests for ay ite-order ARMA processes with iid iovatios, if we icrease the order of the approximatig AR regressio models appropriately as the sample size grows. It seems that ADF tests are most commoly used by practitioers, alog with the tests by Phillips (987) ad Phillips ad Perro (988), which will be called PP tests i the paper. It is widely cojectured that ADF tests are valid for a class of models broader tha those cosidered by Said ad Dickey (984). I particular, it is routiely assumed that they have the same asymptotic distributios as DF tests, whe the uderlyig time series are geerated by geeral liear processes havig martigale di erece iovatios ad satisfyig mild coe ciet summability coditios. I much of the uit root literature, ADF tests are ideed cosidered uder the same coditios as PP tests, whose asymptotics have log bee established uder much weaker coditios. To the best of our kowledge, however, it has ot yet bee show rigorously that ADF tests have the same asymptotically ivariat distributios as DF tests for such a wide class of data geeratig processes. The rage of the icreasig rates for the AR approximatio order required i Said ad Dickey (984) is also believed to be uecessarily striget. They assume that it icreases at order o( ), for some satisfyig < =3, as the sample size gets large. I particular, the logarithmic rate of icrease i the order of approximatig AR regressios is ot permitted. This ca be a serious limitatio for the practical applicatios of the tests. I practice, we ofte use the order selectio rules such as AIC ad BIC, which set the AR order icreasig at a logarithmic rate. Ng ad Perro (995) ideed show that their results hold i the absece of the lower boud, ad thereby validate the use of such iformatio criteria. The purpose of this paper is to verify the validity of ADF tests uder a set of su ciet coditios that are exible eough to iclude most of the iterestig models used i practical ecoometrics. Our coditios allow for geeral liear processes drive by martigale di erece iovatios, which may well have coditioal heteroskedasticities as i ARCH processes. We oly assume a very mild summability coditio o their coe ciets. This cotrasts with Said ad Dickey (984), who cosider liear processes with geometrically decreasig coe ciets ad iid iovatios. Moreover, a much wider rage of the icreasig rates for the AR approximatio order is permitted. For the usual t-type test, we require that the rate icrease at order o( = ) i cotrast to o( =3 ) assumed i Said ad Dickey (984). The pla of the paper is as follows. Sectio itroduces the model ad assump- Patula (986, 988) exteded the tests to AR() models with martigale di erece errors ad to AR(p) models with ARCH errors.
tios. The ADF tests ad their asymptotic results are preseted i Sectio 3. Sectio 4 cocludes the paper, ad Sectio 5 collect all mathematical proofs.. The Model ad Assumptios I this sectio, we preset the model ad assumptios. Let the time series (y t ) be give by y t = y t + u t () with (u t ) geerated as where " t ) is white oise, L is the usual lag operator ad u t = ¼(L)" t () ¼(z) = ¼ k z k k= The test of the uit root ull hypothesis = will be cosidered for (y t ) give as i (), agaist the alterative of statioarity j j <. The iitial value y of (y t ) does ot a ect our subsequet asymptotics as log as y = O p (), ad therefore, we set y = for expositioal brevity. We make the followig assumptios. Assumptio Let (" t ;F t ) be a martigale di erece sequece, with some ltratio (F t ), such that (a) E(" t ) = ¾, (b) (=) P " t p ¾ ad (c) Ej" t j r < K with r 4, where K is some costat depedig oly upo r. Assumptio Let ¼(z) 6= for all jzj, ad P k= jkj s j¼ k j < for some s. Our speci catio i () with coditios i Assumptios ad allows (u t ) to be geerated as a quite geeral liear process. Assumptio sets the iovatio sequece (" t ) to be martigale di ereces. With the coe ciet summability coditio i Assumptio, coditio (a) implies that (u t ) is weakly statioary. Give coditio (a), coditio (b) requires that the weak law of large umbers hold for the squared iovatios (" t ), for which it su ces to assume suitable mixig coditios for them. Uder Assumptio, we have due to the Marcikiewicz-Zygmud iequality i, e.g., Stout (974, Theorem 3.3.6), that the r-th momet of (u t ) exists ad is bouded uiformly i t. Note that secod order statioary ARCH ad GARCH models satisfy our coditios i Assumptio. The asymptotics developed i the literature for PPad other oparametric tests allow for ucoditioal, as well as coditioal, heterogeeity. For these tests, we commoly assume that (u t ) i () is geerated by martigale di erece iovatios (" t ) satisfyig E(" tjf t ) p ¾ (3)
3 ad for some costat K E(" 4 tjf t ) < K a:s: (4) which correspods to our Assumptio (i additio to the coe ciet summability coditio comparable to our Assumptio ) [see, e.g., Stock (994)]. Coditios i (3) ad (4) imply coditio (b) i Assumptio [see Hall ad Heyde (98, Theorem.3)]. Of course, coditio (4) aloe also implies coditio (c) i Assumptio. Therefore, besides ucoditioal homogeeity i coditio (a), Assumptio is weaker tha coditios (3) ad (4) take together. I place of Assumptio, we also cosider Assumptio Let (" t ;F t ) be a martigale di erece sequece, with some ltratio (F t ), such that (a) E(" tjf t ) = ¾ ad (b) Ej" t j r < K with r 4, where K is some costat depedig oly upo r. Assumptio is stroger tha Assumptio. Coditio (a) i Assumptio implies coditio (a) i Assumptio. Give coditio (b) i Assumptio, coditio (a) i Assumptio also implies coditio (b) i Assumptio [see agai Hall ad Heyde (98, Theorem.3)]. Obviously, coditio (3) holds uder coditio (a) i Assumptio, ad coditio (4) implies coditio (b) i Assumptio. Therefore, coditios (3) ad (4) are either ecessary or su ciet for Assumptio. Remark.: AR Approximatio Uder Assumptios ad, we may write with (L)u t = " t (z) = k z k k= ad approximate (u t ) i r-th mea by a ite order AR process with u t = u t + + p u t p + " p;t " p;t = " t + k=p+ k u t k It is well kow [see, e.g., Brilliger (975)] that coditio i Assumptio implies that P k= jkj s j k j <, ad we have P k=p+ j k j = o(p s ). Therefore, give the existece of the r-th momet of (u t ) implied by Assumptios ad, r Ej" p;t " t j r Eju t j r @ j k ja = o(p rs ) k=p+ The approximatio error thus becomes small as p gets large. Later, we will rely o the above AR approximatio for (u t ) with p icreasig as the sample size. To make it explicit that p is a fuctio of sample size, we will ofte write p = p.
4 Remark.: Beveridge-Nelso Represetatio We may write (u t ) as where ¹u t = u t = ¼()" t + (¹u t ¹u t ) (5) ¹¼ k " t k ; ¹¼ k = k= ¼ i i=k+ Uder our coditio i Assumptio, we have P k= j¹¼ k j < [see Phillips ad Solo (99)] ad therefore (¹u t ) is well de ed both i a.s. ad L r sese [see Brockwell ad Davis (99)]. Uder the uit root hypothesis, we may ow sum (5) o both sides gettig y t = ¼()w t + (¹u ¹u t ) (6) where w t = P t k= " k. Cosequetly, (y t ) behaves asymptotically as the costat ¼() multiple of (w t ). Note that (¹u t ) is stochastically of smaller order of magitude tha (w t ). The represetatios i (5) ad (6) were used origially by Beveridge ad Nelso (98) to decompose aggregate ecoomic time series ito permaet ad trasitory compoets. They are fully ad rigorously developed i Phillips ad Solo (99), ad used to obtai asymptotics for liear processes. Assumptio 3 Let p ad p = o( = ) as. Coditios i Assumptios ad 3 are sigi catly weaker tha the oes used by Said ad Dickey (984). They oly cosider (u t ) geerated by a ite order ARMA process ad thus e ectively look at the case that (¼ k ) decays geometrically. Their assumptio correspods to s = i our Assumptio. Also, they assume p = c for < =3. Therefore, for istace, the logarithmic rate for p is ot allowed i their result. This may restrict the use of the tests i some practical applicatios. 3 I additio to Assumptio 3, we also cosider Assumptio 3 Let p ad p = o((= log ) = ) as. Assumptio 3 Let p ad p = o( =3 ) as. I Assumptios 3 ad 3, we require p to icrease at slower rates. 3. ADF Tests ad Their Limitig Distributios The test of the uit root hypothesis for the time series (y t ) give by () ad () ca be based o the regressio y t = y t + px k 4y t k + " p;t (7) k= 3 Some statistical packages like SPLUS set by default the maximum lag legth to be log () for the order selectio criteria such as AIC ad BIC.
5 due to Remark. above. Note that uder the ull hypothesis we have = ad 4y t = u t. To itroduce the test statistics, we de e x p;t = (4y t ;:::;4y t p ) ad subsequetly let A = B = C = Now we have à à X y t " p;t y t x X à p;t x p;t x X p;t x p;t " p;t à à yt X y t x X à p;t x p;t x X p;t x p;t y t à à " X p;t " p;t x X à p;t x p;t x X p;t x p;t " p;t (8) (9) () ^ = A B ³ ^¾ = C A B s(^ ) = ^¾ B where ^ is the OLS estimator of, ^¾ is the usual error variace estimator, ad s(^ ) is the estimated stadard error for ^. We also let ^ () = px ^ p;k () where ^ p;k s are the OLS estimators of k s i regressio (7). The statistics that we will cosider i the paper are give by k= T = ^ s(^ ) S = (^ ) ^ () () (3) Note that T is the t-statistic for the uit root hypothesis, ad S is a ormalized uit root regressio coe ciet. The tests based o T ad S will be referred to respectively as the t-test ad the coe ciet test. They are the ADF tests which are extesios of the tests cosidered by Dickey ad Fuller (979, 98) for the AR() model. Said ad Dickey (984) looked at T oly. Now we derive the asymptotic ull distributios of the statistics T ad S de ed i () ad (3). I what follows, we assume that =. Also, we use the otatio k k to sigify the usual Euclidea orm. We de e kxk = (x + + x p) = for a p-vector x = (x i ), ad let kak = max x kaxk=kxk for a p p matrix A.
6 Lemma 3. Uder Assumptios, ad 3, we have for large (a) y t " p;t = ¼() w t " t + o p () (b) yt = ¼() X wt + o p () (c) " p;t = " t + o p (p s ) Lemma 3. Uder Assumptios, ad 3, we have for large à (a) x p;t x p;t = O p() (b) x p;t y t = O p (p = ) (c) x p;t " p;t = o p (p = ) The results i Lemma 3. are well expected from Remarks. ad.. If we let A ;B ad C be de ed as i (8) (), the it follows from Lemmas 3. ad 3. that sice A = B = C = ¼() w t " t + o p () ¼() X wt + o p () " t + o p () à à X y t x X à p;t x p;t x X p;t x p;t " p;t à y t x X p;t x p;t xp;t x p;t " p;t = o p () due to Lemma 3.. Uder give assumptios, the parameter estimates ^¾ ad ^ () used to de e the statistics T ad S are cosistet. Moreover, if we let ^ p = (^ p; ;:::; ^ p;p ) ; p = ( ;:::; p ) the ^ p is also cosistet for p uder suitable coditios.
7 Lemma 3.3 Uder Assumptios, ad 3, we have ^¾ p ¾ as. Lemma 3.4 Uder Assumptios, ad 3, we have k^ p pk = o p (p = ) ad ^ () = () + o p () for large. Lemma 3.5 Let Assumptios, ad 3 hold. We have ^ p = p + O p ((log =) = ) + o(p s ) uiformly for large. Moreover, it follows that ^ () = () + O p (p(log =) = ) + o(p s ) for large. Cosequetly, we have uder Assumptios, ad 3 T = w t " t à = + o p () ¾ wt Moreover, we have uder Assumptios, ad 3 or uder Assumptios, ad 3 S = w t " t wt + o p () Note that ¼() = = () ad () 6=. Therefore, if we de e ^¼ () = =^ (), the we have ^¼ () p ¼() if ad oly if ^ () p (). The asymptotic ull distributios of T ad S ca ow be easily obtaied usig the results i Phillips (987) or Cha ad Wei (988). They are give i the followig theorem. Note that, uder Assumptio, the required ivariace priciple holds for the partial sum process costructed from (" t ). See Hall ad Heyde (98, Theorem 4.). We let W be the stadard Browia motio. Theorem 3.6 (Limitig Distributios of ADF Tests) Uder Assumptios, ad 3, we have T d Z µz W t dw t = Wt dt
8 as. Moreover, uder Assumptios, ad 3 or uder Assumptios, ad 3, we have as. S d Z Z W t dw t W t dt The asymptotic ull distributios of T ad S are thus idetical to those of the correspodig statistics studied i Dickey ad Fuller (979, 98). They are tabulated i Fuller (995). Our result for T oly requires that p = o( = ), cotrastigly with the more striget coditio p = o( =3 ) used i Berk (974). This relaxatio is possible, sice T does ot ivolve ay estimates of the coe ciets ( k ) of the lagged di erece terms. Our coditio p = o( = ) i Assumptio 3 is ot su ciet for the cosistecy of these estimates. This ca be a serious limitatio, sice the lag legth is ofte selected usig some hypothesis tests o these coe ciets. To validate such procedures, we must have slower icreasig rates for p as we specify i Assumptios 3 or 3. The rate p = o((= log ) = ) is su ciet for the models with homogeeous martigale di erece iovatios, while we must have p = o( =3 ) as i Berk (974) for more geeralmodels with possiblyheterogeous martigale di erece iovatios. Ideed, these are the coditios that we impose to get the asymptotics of S, which icludes the estimate of (). Remark 3.: Models with Determiistic Treds The models with determiistic treds ca be aalyzed similarly. If the time series (z t ) is give by z t = ¹ + y t or z t = ¹ + t + y t (4) ad (y t ) is geerated as i (), the uit root hypothesis ca be tested i regressio (7) usig residuals obtaied from the prelimiary regressio (4). Their distributios are give similarly as those i Theorem 3.4, respectively with demeaed ad detreded Browia motios W ¹ t = W t Z W s ds; W t = W t + (6t 4) Z W s ds (t 6) Z sw s ds i place of stadard Browia motio W. Though we do ot report the details, our results here ca easily be exteded to obtai the asymptotic theory for the uit root tests i models ivolvig determiistic treds. Remark 3.: Near Uit Root Models Asymptotics for the ear uit root models ca also easily be obtaied. If we use local-to-uity formulatio for ad let = c
9 with some costat c >, the the limitig distributios of ADF tests are give as the same as those i Theorem 3.4 with the stadard Browia motio W replaced by Orstei-Uhlebeck process W c, which is give by W c (t) = Z t exp[ c(t s)]dw(s) This ca also be show similarly give our assumptios. 4. Coclusio I this paper, we provide rigorous derivatios of the asymptotics for ADF tests of uit roots. The required coditios are exible eough to iclude a wide class of uit root models geerated by very geeral time series models: liear processes drive by martigale di ereces with coe ciets decayig at polyomial orders. Moreover, our results are obtaied uder a miimal assumptio o the icreasig order for the approximatig autoregressios. 5. Mathematical Proofs Proof of Lemma 3. For Part (a), write y t " p;t = y t " t + y t (" p;t " t ) We have X y t " t = ¼() w t " t + ¹u " t ¹u t " t = ¼() w t " t + O p ( = ) It therefore su ces to show that y t (" p;t " t ) = ¼() X w t (" p;t " t ) + ¹u (" p;t " t ) ¹u t (" p;t " t ) = R + R + R 3 = o p () (5) to deduce the stated result. To show (5), we rst write " p;t " t = k=p+ k u t k = k=p+ ¼ p;k " t k
where k=p+ ¼ p;k c k=p+ k = o(p s ) as i Berk (974, Proof of Lemma, p49). Also, deote by ± ij the usual Kroecker delta. To show that R = o p (), we write Xt X w t (" p;t " t ) = " i ¼ p;j " t j We have Moreover, we have B @E 4 which is bouded by = i= j=p+ = ¾ X X k=p+ j=p+ j=p+ j=p+ t X X ¼ p;j " t i " t j k=p+ i= ( k)¼ p;k + ( k)¼ p;k X k=p+ j=p+ t X X ¼ p;j (" t i " t j ¾ ± ij ) i= j¼ p;k j = o(p s ) 3 = XXt ¼ p;j (" t i " t j ¾ ± ij ) 5 C A i= " # Xt X j¼ p;j j @E (" t i " t j ¾ ± ij ) A ck = @ i= j=p+ j¼ p;j ja = o(p s ) for some costat c. It ow follows immediately that R = o p (). To deduce that R = o p (), we simply ote that (" p;t " t ) = k=p+ X ¼ p;k " t k = o p ( = p s ) which follows similarly as above. Fially, to show that R 3 = o p (), we write ¹u t (" p;t " t ) = i=j=p+ = ¾ X + k=p+ X ¹¼ i ¼ p;j " t i " t j i=j=p+ ¹¼ k ¼ p;k = X ¹¼ i ¼ p;j (" t i " t j ¾ ± i+;j )
Oe may easily see that the rst term is of order o(p s ) ad that the secod term is of order o p ( = p s ), usig the same argumets as above. The result i Part (b) follows immediately from Phillips ad Solo (99). Moreover, the result stated i Part (c) ca easily be deduced from the iequality à = " p;t à " " = t # = (" p;t " t ) ad the fact that E " # (" p;t " t ) 6 = E 4 A3 @ 7 ¼ p;k " t k 5 k=p+ = ¾ X ¼ p;k = o(p s ) k=p+ Note that (=) P " t = O p (). Proof of Lemma 3. We use various results i Berk (974) i the proof. Though his results were derived uder the iid assumptio for (" t ), the results cited here hold uder our coditios (a) ad (c) i Assumptio ad Assumptio, which imply secod-order statioarity ad uiform boudedess of the fourth momets for (u t ) ad (" t ). To show the result i Part (a), we let k = E(u t u t k ) be the autocovariace fuctio of (u t ), ad de e pp = ( i j ) p i;j= The it follows from Berk (974, Proof of Lemma 3, p493) that à E x p;t x p;t pp c p for some costatc. Therefore, à x p;t x p;t pp = O p( = p) Moreover, as is well kow [see, e.g., Berk (974, Equatio(.4), p493)], = O() pp for allp. The result stated i Part (a) ow follows readily, sice à à x p;t x p;t pp x p;t x p;t pp
The proof for Part (a) is therefore complete. To show the result i Part (b), we let ( k ) be the autocovariace fuctio of (u t ) ad use the fact E " X (u t i u t j i j )# = O() which holds uiformly i i ad j. See, e.g., Berk (974, Equatios (.) ad (.), page 49). I what follows, we let y t = for all t by covetio. Let j p ad write where y t u t j = R = y t u t + R y t u t j y t u t We will show that R = O p () uiformly i j, j p. First ote that we have for each j = ;:::;p y t u t = ad we may rewrite R as say. We have R = y t j u t j + (y t y t j )u t j R = jx @ u t i Au t j i= i= jx = @ i j A + t= j+ t= j+ y t u t y t u t = R R " jx X # (u t i u t j i j ) i= = O() + O p ( = p) uiformly i j, j p. Moreover, if we write R = = t= j+ Ã t X u t i u t i= j t= j+i= X u t u t i + t X i= j+t= j+ u t u t i = R a + Rb
3 the it follows that j R a X = j @ ad that R b = (j ) i i= t X i= j+ A + i + t= j+ t= j+ 3 j X 4 (u t u t i i ) 5 = O(p) + O p ( = p) i= 4 t X i= j+ 3 (u t u t i i ) 5 = O(p) + O p (p 3= ) uiformly i j, j p. The stated result follows immediately, sice P y t u t = O p (). Part (c) readily follows from Berk (974). Note that x p;t " p;t x p;t (" p;t " t ) + x p;t " t It follows from Berk (974, Equatio (.3), p49) that E x p;t (" p;t " t ) c p @ k k=p+ A = o( p s ) Moreover, due to Berk (974, above Equatio (.7), p493), we have E x p;t " t = p¾ Eu t = O(p) To complete the proof, ote that p s ; p = o( p ) sice p, p = o( = ) ad s uder our assumptios. Proof of Lemma 3.3 We have ^¾ = " t + o p () from Lemmas 3., 3. ad the subsequet discussios, ad the cosistecy of ^¾ follows directly from (b) i Assumptio. This completes the proof. Proof of Lemma 3.4 We write à X ^ p p = x p;t x X p;t x p;t " p;t
4 ad the stated result for ^ p follows immediately from à X ^ p p x p;t xp;t x p;t " p;t due to parts (a) ad (c) of Lemma 3.. To prove the result for ^ (), we let p be thep-dimesioal vector of oes. It follows that ^ () = p^ p Furthermore, () = @ p p + k k=p+ A = ³ p p + o(p s ) as we metioed i Remark.. Therefore, j^ () ()j p^ p p p + o(p s ) p = ^ p p + o(p s ) from which the stated result for ^ () follows immediately. Proof of Lemma 3.5 We rst write (u t ) as u t = p; u t + + p;p u t p + e p;t where the coe ciets ( p;k ) are de ed so that (e p;t ) are ucorrelated with (u t k ) for k = ;:::;p. Moreover, we de e u t = ~ p; u t + + ~ p;p u t p + ~" p;t It follows from Haa ad Kavalieris (986, Theorem.) ad Bühlma (995, Proof of Theorem 3.) that max j~ p;k p;k j = O((log =) = ) a:s: (6) k p px j p;k k j c j k j = o(p s ) (7) k= where c is some costat. Sice it follows that k=p+ j~ p;k k j j~ p;k p;k j + j p;k k j max j~ p;k k j = O((log =) = ) + o(p s ) a:s: k p
5 ad that j~ () ()j px j~ p;k p;k j + k= px j p;k k j + k= = O(p(log =) = ) + o(p s ) a.s. k=p+ j k j from (6) ad (7). However, from the applicatio of simple least squares algebra, we have B @ ^ p;. ^ p;p C A = B @ ad we may easily deduce ~ p;. ~ p;p C A (^ ) à X x p;t x p;t ^ () = ~ () + O p ( p) à X x p;t y t Note that we have from parts (a) ad (b) of Lemma 3. that à X à p x p;t x X p;t x p;t y t à X k p k x p;t xp;t x p;t y t = O p (p) where p is the p-vector of oes, ad ^ = O p ( ). The stated results ow follow immediately. Proof of Theorem 3.6 Obvious ad omitted. Refereces Berk, K.N. (974). Cosistet autoregressive spectral estimates, Aals of Statistics, 489-5. Beveridge, S. ad C.R. Nelso (98). A ew approach to decompositio of ecoomic time series ito permaet ad trasitory compoets with particular attetio to measuremet of the busiess cycle, Joural of Moetary Ecoomics 7, 5-74. Brilliger, D.R. (975). Time Series: Data Aalysis ad Theory. Holt, Riehart ad Wisto: New York. Bühlma, P. (995). Movig-average represetatio of autoregressive approximatios, Stochastic Processes ad their Applicatios, 6, 33-34.
6 Cha, N.H. ad C.Z. Wei (988). Limitig distributios of least squares estimates of ustable autoregressive processes, Aals of Statistics 6, 367-4. Dickey, D. A. ad W. A. Fuller (979). Distributio of estimators for autoregressive time series with a uit root, Joural of the America Statistical Associatio 74, 47-43. Dickey, D.A. ad W.A. Fuller (98). Likelihood ratio statistics for autoregressive time series with a uit root, Ecoometrica 49, 57-7. Fuller, W.A. (995). Itroductio to Statistical Time Series, d ed. Wiley: New York. Hall, P. ad C.C. Heyde (98). Academic Press: New York. Martigale Limit Theory ad Its Applicatio. Haa, E.J. ad L. Kavalieris (986). Regressio, autoregressio models, Joural of Time Series Aalysis, 7, 7-49. Ng, S. ad P. Perro (995). Uit root tests i ARMA models with data depedet methods for selectio of the trucatio lag, Joural of the America Statistical Associatio 9, 68-8. Patula, S.G. (986). O asymptotic properties of the least squares estimators for autoregressive time series with a uit root, Sakhyã, Series A, 48, 8-8. Patula, S.G. (988). Estimatio of autoregressive models with ARCH errors, Sakhyã, Series B, 5, 9-38. Phillips, P.C.B. (987). Time series regressio with a uit root, Ecoometrica 55, 77-3. Phillips, P.C.B. ad P. Perro (988). Testig for a uit root i time series regressio, Biometrika 75, 335-346. Phillips, P.C.B. ad V. Solo (99). Asymptotics for liear processes, Aals of Statistics, 97-. Said, S.E. ad D.A. Dickey (984). Testig for uit roots i autoregressive-movig average models of ukow order, Biometrika 7, 599-68. Stock, J. H. (994). Uit roots, structural breaks ad treds, i R.F. Egle ad D.L. McFadde (eds), Hadbook of Ecoometrics, Vol. IV, pp. 739 84, Elsevier, Amsterdam. Stout, W.F. (974). Almost Sure Covergece. Academic Press: New York.