working papers THE IMPACT OF PERSISTENT CYCLES ON ZERO FREQUENCY UNIT ROOT TESTS Tomás del Barrio Castro Paulo M.M. Rodrigues A.M.

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1 workig papers THE IMPACT OF PERSISTENT CYCLES ON ZERO FREQUENCY UNIT ROOT TESTS Tomás del Barrio Castro Paulo M.M. Rodrigues A.M. Robert Taylor September The aalyses, opiios ad fidigs of these papers represet the views of the authors, they are ot ecessarily those of the Baco de Portugal or the Eurosystem Please address correspodece to Paulo M.M. Rodrigues Baco de Portugal, Ecoomics ad Research Departmet Av. Almirate Reis, - Lisboa, Portugal; Tel.: , Paulo M.M. Rodrigues@bportugal.pt

2 BANCO DE PORTUGAL Av. Almirate Reis, - Lisboa Editio Ecoomics ad Research Departmet Pre-press ad Distributio Admiistrative Services Departmet Documetatio, Editig ad Museum Divisio Editig ad Publishig Uit Pritig Admiistrative Services Departmet Logistics Divisio Lisbo, September Number of copies ISBN ISSN 8- (prit) ISSN 8- (olie) Legal Deposit o. 3/83

3 The Impact of Persistet Cycles o Zero Frequecy Uit Root Tests Tomás del Barrio Castro a, Paulo M.M. Rodrigues b ad A.M. Robert Taylor c a Departmet of Applied Ecoomics, Uiversity of the Balearic Islads b Baco de Portugal, NOVA School of Busiess ad Ecoomics, Uiversidade Nova de Lisboa, CEFAGE c Grager Cetre for Time Series Ecoometrics, Uiversity of Nottigham Abstract I this paper we ivestigate the impact of o-statioary cycles o the asymptotic ad ite sample properties of stadard uit root tests. Results are preseted for the augmeted Dickey-Fuller ormalised bias ad t-ratio-based tests (Dickey ad Fuller, 99, ad Said ad Dickey, 98), the variace ratio uit root test of Breitug () ad the M class of uit-root tests itroduced by Stock (999) ad Perro ad Ng (99). The limitig distributios of these statistics are derived i the presece of o-statioary cycles. We show that while the ADF statistics remai pivotal (provided the test regressio is properly augmeted), this is ot the case for the other statistics cosidered ad show umerically that the size properties of the tests based o these statistics are too ureliable to be used i practice. We also show that the t-ratios associated with lags of the depedet variable of order greater tha two i the ADF regressio are asymptotically ormally distributed. This is a importat result as it implies that extat sequetial methods (see Hall, 99 ad Ng ad Perro, 99) used to determie the order of augmetatio i the ADF regressio remai valid i the presece of o-statioary cycles. Keywords: Nostatioary cycles; uit root tests; lag augmetatio order selectio. JEL classi catios: C, C

4 Itroductio Peaks at low o-zero frequecies imply the existece of cycles i time series, a feature preset i may macroecoomic, acial ad other time series; see, iter alia, Coway ad Frame (), Birgea ad Kilia () ad Priestley (98). The importace ad iterest i tred growth, cyclicality ad seasoal uctuatios i ecoomic ad acial time series dates back several decades; see Burs ad Mitchell (9). For istace, Caova (99) discusses the literature o Bayesia learig (Nyarko, 99) ad o oisy traders i acial markets (Campbell ad Kyle, 993) where models which geerate irregularly spaced but sigi cat cycles i ecoomic activity ad asset prices are proposed. The presece of cycles is also documeted i the political ecoomy literature (electoral cycles i govermet variables, Alesia ad Roubii, 99), ad aturally arises i the busiess cycle literature. Iterestigly, the existece of both complex ad real uit roots ca iduce growth cycles similar to those observed i ecoomic data; see Alle (99). Autoregressive (AR) processes with roots o the complex uit circle are o-statioary ad display persistet cyclical behavior similar to that of persistet busiess cycles (Paga, 999 ad Bieres, ). Bieres () ds evidece that busiess cycles may ideed be due to complex uit roots. Shibayama (8) studies ivetories ad moetary policy by estimatig VAR models ad also detects complex roots that geerate cycles of aroud to moths, which are close to busiess cycle legths. Give that cycles are a importat feature of ecoomic ad acial variables it is importat to evaluate their implicatios o the performace of pre-testig procedures, i particular o the limitig ull distributios ad ite sample properties of zero frequecy uit root test statistics. Cosequetly, followig the body of empirical evidece summarised above, i this paper we will focus o the case where the cyclical compoet is characterised by a secod order autoregressive compoet with complex roots i the eighbourhood of uity, although geeralisatios to higher-order o-statioary factors will also be discussed. The asymptotic distributios of the least-squares estimates of a AR() with complex uit roots has bee addressed i Ahtola ad Tiao (98), Gregoir () ad Taaka (8), the latter geeralises the results i Ahtola ad Tiao (98) by allowig the error term i the AR() model to follow a statioary process. For asymptotic results for geeral AR processes see, iter alia, Cha ad Wei (988). Cha ad Wei (988) cosidered the limitig distributios of the least squares estimate of a geeral ostatioary AR model of order p (AR(p)) with characteristic roots at di eret frequecies o or outside the uit circle, each of which may have di eret multiplicities (see also Nielse, ). This was the rst comprehesive treatmet of least squares estimates for a geeral ostatioary AR(p) model. Cha ad Wei (988) showed that the locatio of the roots of the time series played a importat role i characterizig the limitig distributios. Jegaatha (99) geeralized this idea to the ear-itegrated cotext where the limitig distributios of the least-squares estimators are expressed i terms of Orstei-Uhlebeck processes. Extesios to vector AR processes are provided i Tsay ad Tiao (99) ad to processes with determiistic treds i Cha (989). We will focus attetio o the covetioal augmeted Dickey-Fuller [ADF] tests (Dickey ad Fuller, 99, Said ad Dickey, 98, Hamilto, 99), the variace ratio uit root test of Breitug () ad the triity of so-called M uit-root tests itroduced by Stock (999) ad popularised by Perro ad Ng (99). The M tests have bee icreasigly popular i the uit root literature; ideed, i discussig the ADF tests Haldrup ad Jasso (,p.) argue that... practitioers ought to abado the use of these tests... i favour of the M tests because of... the excellet size properties ad early optimal power properties of the latter. The results we preset i this paper, however, suggest quite the opposite coclusio holds i cases where the time series process admits (ear-) uit roots at cyclical frequecies. I particular we show that while the limitig distributios of the two ADF statistics remai pivotal (provided the test regressio is properly augmeted), this is ot the case for the variace ratio or the three M statistics, all of which become too ureliable to be used i practice. We also show that the ormalized bias ad the t-ratio tests associated with the lags of the depedet variable (of order greater tha two) i the ADF regressio are asymptotically ormally distributed. This is a importat result as it implies that the stadard sequetial methods (see Hall, 99, ad Ng ad Perro, 99) used to determie the order of augmetatio i the ADF regressio remai valid whe cyclical (ear-) uit roots are preset i the data. The remaider of the paper is orgaized as follows. I sectio we outlie our referece time series model which allows for cyclical uit roots ad brie y outlie the ADF, variace ratio ad M uit root tests. I Sectio 3, i the case where cyclical uit roots are preset i the data, we establish the large sample behaviour of these tests together with those of the covetioal t-ratios for testig the sigi cace of lagged depedet variables i the ADF test regressio. Extesios to allow for ear uit roots (both

5 at the zero ad cyclical frequecies) ad determiistic variables are discussed i sectio. Fiite sample simulatios are reported i sectio. Sectio cocludes. All proofs are collected i a mathematical appedix. I the followig bc deotes the iteger part of its argumet, ) deotes weak covergece, p! covergece i probability, ad x := y ( x =: y ) idicates that x is de ed by y (y is de ed by x). The Model ad Uit Root Tests. The Time Series Model We cosider a uivariate time series fx t g geerated accordig to the followig data geeratio process [DGP], (L) ( al)x t = " t ; " t IID(; ); t = ; ; : : : ; : () We assume throughout that the process is iitialised at x = x = x =, although weakeig this to allow these startig values to be of o p ( = ) would ot chage ay of the asymptotic results which follow. I () the autoregressive polyomial (L) = b cos()l + b L, with (; ), ad where L deotes the usual lag operator. Cosequetly, whe b = (jbj < ), (L) admits the complex cojugate pair of uit (stable) roots, exp (i) cos () i si (), at the spectral frequecy. We do ot assume that the value of is kow to the practitioer. The process additioally admits a zero frequecy uit root whe a =. I the case where a = b =, fx t g is therefore itegrated of order oe at both the zero ad spectral frequecies, deoted I () ad I (), respectively. I this case, it follows that z t := x t, where := ( L), will be I () but I (), while u t := x t, where := cos()l + L, will be I () but I (). Our focus i this paper is o testig the stadard zero frequecy uit root ull hypothesis that x t I (), H : a =, agaist the alterative that x t I (), H : jaj <, i the case where (L) admits the pair of complex uit roots at frequecy. Remark. The model i () is arguably quite simple. However, it ca be exteded to allow for: ear uit roots at the zero ad/or frequecies; determiistic compoets; weak depedece i f" t g, ad uit roots at other cyclical frequecies lyig i (; ) ad/or at the Nyquist () frequecy, without alterig the qualitative coclusios which ca be draw from the aalysis of (). For expositioal purposes we will therefore focus o () but we will discuss how our results geeralise to these cases. Remark. Notice that = cos()l + L geerates a (o-statioary) cycle of = periods. Cosequetly, i the case of data observed with a seasoal periodicity of S, = j=s; j f; ; : : : ; S g, where S := b(s )=c, geerates o-statioary seasoal cycles.. Zero Frequecy Uit Root Tests A large umber of procedures have bee proposed to test for zero frequecy uit roots; see, for example, Stock (99), Maddala ad Kim (998) ad Phillips ad Xiao (998) for excellet overviews. I this Sectio we review three popular classes of such tests. The rst tests we cosider are the augmeted Dickey-Fuller [ADF] ormalised bias ad t-ratio tests. These are computed from the auxiliary test regressio, x t = x t + kx j x t j + " k;t : () j= I (), k deotes the lag trucatio order chose to accout (parametrically) for (L) ad ay weak depedece i f" t g; i the simplest form of the DGP give i (), where " t is IID, k =. More geerally, where " t is a liear process satisfyig stadard summability ad momet coditios (see Chag ad Park, ), k eeds to be such that =k + k 3 =! as! ; see Said ad Dickey (98) ad Chag ad Park (). Based o OLS estimatio of (), the ADF t-ratio for testig H agaist H will be deoted P k t b := b=se (b) ad the associated ormalised bias statistic as Z b := b=( i= b i). The ADF tests remai the most popularly applied uit root tests due i part to their ease of costructio. At this poit we also outlie the sequetial lag selectio method due to Hall (99) ad Ng ad Perro (99). Here oe starts from a maximum lag legth, k max say i (), satisfyig the rate coditio above. Oe the rus the stadard t-test for the sigi cace of the termial lag usig critical values from the stadard ormal distributio. If the ull is rejected the ADF statistics are computed from () with lag legth 3

6 k max ; otherwise, the lag legth is reduced to k max ad the procedure repeated util the lag legth caot be reduced further, or if a pre-speci ed value for the miimum lag legth is attaied, at which poit the ADF statistics are computed for that lag legth. Secodly we will cosider the triity of so-called M uit root tests due to Stock (999) ad Perro ad Ng (99), iter alia: MSB := ^ MZ := X t= = X x t! (3) t= x t! x ^ () MZ t := MSB MZ () where ^ is a estimator of the log ru variace of f" t g. Followig Perro ad Ng (99) we ca cosider two alterative estimators for the log-ru variace. Firstly, a o-parametric kerel estimator based o the sample autocovariaces, ^ = s W A, with s W A := P h= +!(h=m)^ h, ^ h := P jhj t= ^" ;t^" ;t+jhj, where ^" ;t are the OLS residuals from regressig x t o x t, with kerel fuctio! () satisfyig e.g. the geeral coditios reported i Jasso (, Assumptio A3) ad the badwidth parameter m (; ) satisfyig =m+m =! as! (which correspods to Assumptio A of Jasso, ). Secodly, a parametric autoregressive spectral desity estimator, ^ := s AR ; of the form suggested by Berk (9), where s AR := P k ^ k= i= b i where ^ k := P be t ; b i ; i = ; :::; k; ad be t are obtaied from the OLS estimatio of () for a give value of k satisfyig the same coditios as give above i the cotext of the ADF tests. As oted i sectio, it has bee suggested by some authors (e.g. Haldrup ad Jasso, ) that the M tests, whe coupled with the modi ed AIC lag selectio method of Ng ad Perro (), are preferable to the stadard ADF tests outlied above due to their superior size properties, relative to the latter, i the presece of weak depedece i f" t g. Fially, we will cosider the variace ratio test (V RT ) proposed by Breitug (),! X X tx V RT A : () t= x t The variace ratio test has some appealig properties. First of all it requires o correctio, parametric or o-parametric, for serial correlatio from f" t g ad/or (L). Secod, by virtue of its lack of such a correctio factor, it has bee advocated by some authors (see, for example, Müller, 8) as a uit root test which avoids the criticisms of Faust (99) regardig the (theoretical) ucotrollability of the size of uit root tests based aroud (parametric or o-parametric) correctios for geeral weak depedece i f" t g. 3 Asymptotic Behaviour uder No-Statioary Cycles For the purpose of aalysig the impact of o-statioary cycles o the limit distributios of the zero frequecy uit root tests discussed i sectio. it will prove useful to rst cosider frequecy speci c orthogoal decompositios of x t ad x t. These results are collected together i Lemma. Lemma Let the time series process fx t g be geerated by () with a = b =. The for ay (; ) the followig decompositios hold: t= j= x j x t = S (t) + si [si ((t + ))S ;(t) cos ((t + )) S ; (t)] x t = + si [cos ( (t + )) S ;(t) + si ( (t + )) S ; (t)] + O p () () si [si ((t + ))S (t) cos ((t + )) S (t)] (8) where S (t) := P t j= " j, S ; (t) := P t j= " j cos (j), S ; (t) := P t j= " j si (j), := = ( cos ), := ( cos ()) = ( cos ), ad := si () = ( cos ).

7 Remark 3. It is straightforward to show that the results stated i Lemma cotiue to hold uder weaker liear process coditios o f" t g provided Assumptios.-. of Gregoir (), adapted slightly to our situatio, are satis ed. Precisely, these coditios etail that " t = d(l)e t, where (" t ; F t ) is a martigale di erece sequece, with ltratio (F t ), such that E " t jf t = ad sup t E j" t j + jf t < a.s. for some >, ad where d(l) := + P j= d jz j is such that d (z) = for z = ad z =, ad P j= j jd jj <. Usig the decompositios provided i Lemma, we are ow i a positio to state the large sample behaviour of the elemets which form the uit root tests uder aalysis. These results are collected together i Lemma.

8 Lemma Let the coditios of Lemma hold. The, for ay (; ), the followig results hold as!, p x t ) W (r) + h i p si ((t + ))W (r) cos ((t + )) W si (r) t= + h i p cos ( (t + )) W (r) + si ( (t + )) W si (r) := b(r) (9) Z X x t ) ( cos ) W (r) dr Z W + ( cos ) si (r) h i + W (r) dr () X x t = X x t + o p () = X x t + o p () t= ) X x t x t ) t= t= si 8 si Z Z X x t x t ) cos + si X x t " t ) ( cos ) t= t= X t= + si x t " t ) si t= Z t= W (r) + h W (r) i dr () W (r) + h W (r) i dr () Z ( cos ()) Z Z W (r) + h W (r) i dr (3) W (r) dw (r) W (r) dw (r) Z W (r) dw (r) Z W (r) dw (r) W (r) dw (r) + W (r) dw (r) W (r) dw (r) X x t " t ) cos W (r) dw si (r) W (r) dw (r) Z W (r) dw (r) + W (r) dw (r) where, ad are as de ed i Lemma, ad W (r) ; W (r) ad W (r) are idepedet stadard Browia motio processes. Usig the results i Lemma, we are ow i a positio i Theorem to detail the asymptotic ull distributios of the uit root tests from sectio. whe the DGP cotais o-statioary cycles. Subsequetly, i Theorem, we will establish correspodig results for the t-ratios o the lagged depedet variables appearig i the ADF regressio (). Theorem 3 Let the coditios of Lemma hold. The for ay (; ), the followig results hold as!, ad for k i () () () () t b ) Z b ) V RT ) R W (r) dw (r) q R := DF () [W (r)] dr R W (r) dw (r) R [W (r)] := DF (8) dr R R W(r) dr + ( cos()) R r b (s) ds dr [W (r)] dr (9) ( cos()) si() R [W (r)] dr+ R

9 for ^ = s W A, MSB = O p [m] = ; MZ = O p (m) ; MZ t = O p [m] = () while for ^ = s AR MSB ) Z W (r) dr + ( Z cos ()) W si () (r) Z i dr + hw! = (r) dr () MZ ) MZ t ) b () R W(r) dr + ( cos()) p [ cos ()] b () [ cos ()] R [W (r)] dr+ R r R R W(r) dr [W + (r)] dr+ R ( cos()) [W (r)] dr () ( cos()) si() p [ cos ()] ( cos()) si() [W (r)] dr (3) where m is the badwidth used to compute s W A, ad where W (r) ; W (r) ad W (r) ad b (r) are as de ed i Lemma. Remark. I the case where jbj < i (L) i (), so that o o-statioary cycles are preset, it is well kow that both t b ad MZ t weakly coverge to DF, while Z b ad MZ weakly coverge to R DF. Moreover, i this case MSB ) W (r) = R =: MSB ad V RT ) [ R r W(s)ds] dr R =: VRT. W(r) dr Comparig these represetatios with those give i Theorem, it is see that oly the two ADF statistics, t b ad Z^, computed from () retai their usual pivotal limitig ull distributios i the presece of o-statioary cycles. I cotrast, the limitig ull distributios of V RT ad the triity of M statistics with autoregressive spectral desity estimators of the log-ru variace are o-pivotal, their fuctioal forms depedig i a complicated way o both o-stochastic ad stochastic fuctios, while the results i () show that the triity of M statistics computed usig a kerel-based estimator of the log-ru variace have degeerate limitig ull distributios i the presece of o-statioary cycles. This result obtais because x t is o-statioary which causes the kerel-based log-ru variace estimator to diverge to + at rate O p (m); see Taylor (3). Sice both x ad P t= x t are of O p (), the divergece of ^ W A to + implies that both the MZ ad MZ t statistics will diverge to, while MSB will coverge i probability to zero, i each case at the rates stated i (). Cosequetly, the three M tests based o these statistics will therefore all have asymptotic size of uity i the presece of o-statioary cycles. The impact o the ite sample size of the uit root tests based o each of the statistics discussed i Theorem whe usig the stadard asymptotic critical values (appropriate to the case where o-statioary cycles are ot preset) will be ivestigated i sectio. Remark. I this paper we have ot icluded the Z ad Z t uit root tests of Phillips (98) ad Phillips ad Perro (988) i the set of tests uder discussio. However, otig from expressios (.) ad (.) of Perro ad Ng (99,p.3) that Z = MZ (=)(b ) ad Z t = MZ t :( P t= x t =s W A )= (b ), i each case for the versio of the M test usig the kerel-based log ru variace estimator s W A, we see immediately from the discussio i Remark that both Z ad Z t will also diverge to at the same rates as are give for MZ ad MZ t, respectively, i (). Remark. The result i () for t b has previously bee give i Nielse (), ad is also show to hold for case of seasoally itegrated data i Ghysels et al. (99). Remark. It is straightforward but tedious, usig Lemma of Bieres (), to show that the results give i (), (8) ad () of Theorem will ot alter if we allow for weak depedece i f" t g of the form give i Remark 3. The limitig ull distributios for V RT ad the triity of M statistics with autoregressive spectral desity estimators of the log-ru variace will ow deped o additioal uisace parameters arisig from the MA parameters, fd j g j=. Moreover, the results i Theorem are qualitatively uchaged if we allow (L) to be a pth order polyomial cotaiig additioal uit roots with frequecies i the rage (; ], provided k p i (). Speci cally, i this case the results i () ad (8) will cotiue to hold, as will the order results i (), while the limitig ull distributios for V RT

10 ad the triity of M statistics with autoregressive spectral desity estimators of the log-ru variace will ow deped o stochastic ad o-stochastic fuctios relatig to these frequecies but will remai o-degeerate. I Theorem we ow establish the large sample behaviour of the t-ratios o the lagged depedet variables i (). Theorem Let the coditios of Lemma hold ad de e the vector of parameters from () as := [; ; ; 3 k ] =: [; ; ; ]. For k i (), the as!, A cos () + B (^ ) ) V (A + B ) A (^ ) ) V (A + B ) p ^ ) N ; H H () () () where Z A := V Z B := V V := Z W (r) dw (r) + W (r) dw (r) + V (A + B ) W (r) dw (r) W (r)dr; A := Z W (r) dw (r) si () V (A + B ) W (r) dr; B := Z h W (r) i dr where W (r) ; W (r) ad W (r) are as de ed i Lemma, ad where H ad ad (A.), respectively, i the Appedix. Moreover, for < i k, are de ed i (A.9) t bi := (b i i )=s:e:(b i ) ) N(; ): () Remark 8. The results i (A.) ad (A.) imply that ^ ad ^, like b, are super-cosistet. From (), it is see that the parameters o the lagged di erece terms from lag three owards are root cosistet asymptotically ormal [CAN]. Uder the coditios of Theorem, =, sice " t IID(; ); however, the stated results also hold whe " t is a statioary AR(k ) process provided k k. Moreover, where " t displays the geeral weak depedece of the form give i Remark 3 the foregoig results still remai valid although here, for a give lag trucatio k, the parameters j, j 3, take the role of pseudo-parameters i the same sese as i, for example, Chag ad Park () ad Ng ad Perro (99). Remark 9. The key result i Theorem is that give i () which establishes that stadard t-tests for the sigi cace of the lagged depedet variables of order three ad above ca be coducted usig stadard ormal critical values. This is a importat result i that it implies that the sequetial lag speci catio methods of Hall (99) ad Ng ad Perro (99), as outlied i sectio., remai valid i the presece of o-statioary cycles. I cotrast, it follows straightforwardly from the represetatios i (A.) ad (A.) that the t-ratios associated with ^ ad ^ have o-stadard limitig distributios which are fuctioals of the idepedet stadard Browia motio processes, W (r) ; W (r) ad W (r). I the sceario cosidered i Remark where (L) is a pth order polyomial of o-statioary factors the it is straightforward but tedious to show that the OLS estimators associated with the rst p lagged depedet variables i () will have o-stadard limitig distributios (ow beig fuctioals of p idepedet stadard Browia motio processes) while those for p + owards will agai be root- CAN with their associated t-ratios havig stadard ormal limitig ull distributios. Extesios to Near-Itegratio ad Determiistics. Near-Itegratio I this sectio we geeralise the results give i sectio 3 to the case where the data are geerated accordig to the ear-itegrated process, ( ' L) cos()' L + ' L x t = " t ; " t IID(; ); t = ; ; : : : ; (8) 8

11 where ' := exp c ' + c ; ad (; ) : This process is ear-itegrated at the zero ad frequecies with a commo o-cetrality parameter c. Uder this settig we ca establish the followig Lemma. Lemma. Let the time series process fx t g be geerated by (8) with x = x = x =. The for ay (; ) the followig results hold: where p x o t = J c;; (t=) + si([t si() p + ])Jc;;(t=) cos([t + ])J c;; (t=) + si () p cos ( [t + ]) J c;; (t=) + si ( [t + ]) Jc;; (t=) + O p = p (9) J c;; (x) := J c;; (x) := ad where, as! ; bxc p X p bxc p X ' bxc j " j j= p ' bxc j " j cos (j) ; J c;; (x) := J c;; (t=) J c;; (t=) J c;; (t=) A J c; (r) J c; (r) J c; (r) A p bxc X ' bxc j= j " j si (j) where J c; (r), J c; (r) ad J c; (r) are idepedet stadard Orstei-Uhlebeck processes such that dj c; (r) = cj c; (r) dr + dw (r) ad dj j c; (r) = cj j c; (r) dr + dw j (r), j = ;, with W (r), Wk (r) ad W k (r) the stadard Browia motios de ed i Lemma. Usig the results i Lemma 3, it is the straightforward to show that the results give i Theorems ad carry over to this cotext, substitutig W (r), W k (r) ad W k (r) by J c;(r), J c; (r) ad J c; (r); respectively, throughout. The commets i Remark agai apply i this case. Remark. The assumptio of a commo o-cetrality parameter to the zero ad frequecies, as embodied i (8), ca be relaxed. To that ed, cosider the case where the DGP admits a di eret o-cetrality parameter at the zero ad frequecies; viz, ' L cos()' L + ' L x t = " t ; " t IID(; ); t = ; ; : : : ; (3) where ' := exp c ' + c ad ' := exp c c ' + : For data geerated by (8) rather tha (3), the result give i Lemma 3 still hold, provided, J c;; (x) ; Jc;; (x) ad J c;; (x) are replaced by J c;; (x) := P bxc p j= ' bxc j "j, Jc ;; (x) := p P bxc p j= ' bxc j "j cos (j) ad J c ;; (x) := p P bxc p j= ' bxc j "j si (j), respectively, ad J c; (r), Jc; (r) ad J c; (r) are similarly replaced by J c;(r), Jc ; (r) ad J c ; (r), respectively, where dj c ; (r) = c J c; (r) dr + dw (r) ad dj j c ; (r) = c J j c ; (r) dr + dw j (r), j = ;. The results give i Theorems ad agai carry over, mutatis mutadis, as do the commets i Remarks ad 9.. Determiistic Compoets Where determiistic compoets are preset i the DGP, the previous results ca be exteded i a straightforward fashio. More speci cally, we cosider the cases where () is costructed from de-treded data, deoted ^x t, obtaied as the OLS residuals from the regressio of x t oto either a costat (demeaed data) or a costat ad liear tred (liear de-treded data); i.e., ^x t = ^x t + kx j ^x t j + ^" k;t ; j= 9

12 ad similarly the variace ratio ad M uit root tests are costructed usig the de-treded data, although i the de itio of the MZ statistic the term ^x eeds to be added to the umerator of (); see, for example, Müller ad Elliott (3). I both these cases the results i Theorems ad (together with the correspodig results i sectio. for the ear-itegrated case) remai valid provided the stadard Browia motio process, W (r), ad the stadard Orstei-Uhlebeck process, J c;, i () ad (8) are replaced by their OLS de-treded couterparts; e.g., W f R (r) := W (r) W (s) ds, for the OLS de-meaed case, ad W c (r) := W (r) ( r) R W (s) ds (r ) R sw (s) ds, for the OLS liear de-treded case. Fially ote also, that for the correspodig uit root tests based o local GLS de-tredig (see, iter alia, Elliott, Rotheberg ad Stock, 99, ad Ng ad Perro, ) the the previous results agai hold but ow replacig the stadard Browia motio, W (r), ad stadard Orstei-Uhlebeck processes, J c; by their local GLS de-meaed or liear de-treded couterparts; see Elliott et al. (99,pp 8-8) for precise details. Mote Carlo Experimets I this sectio, we use simulatio methods to ivestigate the ite sample properties of the uit root tests discussed i sectio. whe (ear-) o-statioary cycles are preset i the data. All results reported i this sectio are based o ; Mote Carlo replicatios usig the RNDN fuctio of Gauss 9.. Uless otherwise stated, results are preseted for both de-meaed ad liear de-treded data. Our rst set of experimets, reported i Tables a (OLS de-meaed data) ad Table b (liear OLS de-treded data), relate to data geerated accordig to the DGP + c L cos() + c L + + c L x t = t NIID(; ) (3) with c =, so that the uit root ull hypothesis holds, ad with f=; =; =; =; =g ad c f; ; g, i each case for a sample size of =, iitialised at x = x = x =. Although the frequecy = would ot be cosidered a low frequecy compoet, it is oetheless a seasoal frequecy compoet for ay case where the umber of seasos is eve (e.g. mothly or quarterly data) ad therefore seems worth icludig. For comparative purposes, results for the covetioal radom walk, ( L)x t = t NIID(; ); iitialised at x =, are also provided i the rows labelled. The results i Tables a ad b report the empirical (ull) rejectio frequecies of the uit root tests from sectio. i each case for a omial % sigi cace level usig the asymptotic critical values appropriate to the case where (ear-) o-statioary cycles are ot preset i the data; that is, from DF, DF, MSB or VRT, as appropriate. I Table, the ADF t b ad Z^ tests were computed from the ADF regressio () for the true lag legth, k =. I the cotext of the triity of M tests, the superscript used i the omeclature of Tables a ad b deotes the log ru variace estimator used; the subscripts b ad q idicate that ^ = s W A with the Barlett ad quadratic spectral kerels, respectively, while the subscript AR idicates that ^ := s AR. For ^ = s W A, results are reported for the Bartlett ad quadratic spectral kerel, usig the data-depedet badwidth formulatios for these kerels suggested i Newey ad West (99, equatios (3.8) to (3.) ad Table ). For ^ := s AR, we agai set k =. Also reported i Tables a ad b are the empirical rejectio frequecies for the covetioal t-ratio tests o the lagged depedet variables x t ad x t from (), deoted t^ ad t^, i each case compared to the. level critical values from the stadard ormal distributio (a % rule). Isert Tables a ad b about here As predicted by the asymptotic distributio theory i Theorem ad sectio., it is oly the ADF t b ad Z^ tests which display ite sample size properties which are robust to the presece of (ear-) o-statioary cycles i the data. The size properties of the t b test are somewhat better tha those of Z^ which is a little over-sized, most otably so i the case of liear de-treded data, but both show o sigi cat variatios i size from the radom walk base case uder o-statioary or ear-o-statioary cycles for all of the frequecies cosidered. Agai i lie with the predictios from the asymptotic theory, we see that this is ot the case for the other tests cosidered. Agai as predicted by the results i Theorem ad sectio., the empirical sizes of the V RT test vary cosiderably across both ad c. As might be expected, for a give frequecy, the size distortios i V RT decrease as c icreases; this is because the cyclical compoet at frequecy moves further away from the o-statioarity boudary as c icreases. The size distortios for the three M tests with the autoregressive spectral desity estimator show eve greater variatio across tha the V RT test. Overall though, eve though the V RT ad M

13 tests with ^ = s W A are ot asymptotically degeerate uder (ear-) o-statioary cycles the results i Tables a ad b suggest that oe of these tests could reliably be used i practice. The degeeracy of the kerel-based M tests is clearly re ected i Tables a ad b, although agai as with the V RT test there is some amelioratio of this i the case of the MSB test for a give value of as c is icreased. Fially, we observe that the empirical rejectio frequecy of the t^ test with a % rule is uity throughout (except i the radom walk case where it is essetially correctly sized), implyig that i the presece of (ear-) o-statioary cycles the sequetial method of Hall (99) ad Ng ad Perro () will always retai both lagged depedet variables i (), as would be hoped. Ideed, we obtaied the same outcome whe a tighter % rule was used. More geerally, i ureported simulatios we foud that i the case outlied i the latter part of Remark, where (L) is a pth order polyomial of o-statioary factors, the same dig holds for the t-test o the pth lag, so that p lagged depedet variables will always be icluded i the ADF regressio. Isert Tables about here I a secod experimet, we ow ivestigate i detail the impact of a umber of covetioal lag selectio methods o the empirical size ad power properties of the t b test (agai ru at the omial asymptotic % level). Correspodig results for Z^ are available o request. Results are reported for the sequetial method (deoted SQ) outlied i sectio. usig a % rule (as is commoly doe i practice), the stadard AIC rule, ad also the modi ed AIC (deoted MAIC) rule of Ng ad Perro (). For all of these methods, results are reported for maximum lag legths of k max := b[ ]= c ad k max := b[ ]= c; with the subscript or o each method deotig which of these maximum lag legths was used. No miimum lag legth was set for ay of the selectio methods. The data were agai geerated accordig to (3) with the parameter settigs as were cosidered for the results i Tables a ad b, but augmeted to iclude results for c f; ; ; 3; ; ; 3g, rather tha just c =. For the case of c = (Table ) results are preseted oly for c =. To esure comparability with the other cases of, the results reported for = i Tables ad 3 pertai to the DGP + c L ( :L )x t = t NIID(; ) iitialised at x = x = x = ; this process, like the cases cosidered where =, phas a true lag legth of k = i (), but here the cyclical pair of roots are stable (both have modulus : ad lie at frequecy =). Fially, for comparative purposes, results are preseted both for tests based o OLS de-tredig ad the correspodig tests based o local GLS de-tredig. Table reports the empirical size (c = c = ) obtaied with the three lag-order selectio methods ad also reports the average order of lag augmetatio selected by each approach. The results i Table idicate that the empirical size of the ADF test with OLS de-tredig is reasoably close to the omial level throughout for both SEQ ad AIC, although both methods yield slightly over-sized tests i the case of k max with liear de-tredig. As regards the tests which are based o the MAIC rule, here a degree of uder-sizig is see throughout; this is perhaps ot uexpected give that the pealty fuctio o which the MAIC rule is based is i fact misspeci ed whe o-statioary cycles are preset i the DGP, as is the case here. Similar commets apply whe local GLS de-tredig is used, although here it is oteworthy that the tests based o MAIC ca be very badly udersized i the case of liear de-tredig; ideed all of the tests display a tedecy to udersize here. As regards the average lag legth chose, it is see that SQ, AIC, AIC ad MAIC get reasoably close to the true order (recall that this is two throughout), while both MAIC ad SQ over- t the lag order, most otably so i the case of SQ. Such over- ttig will of course ecessarily lead to power losses uder the alterative, as will be see i the results Tables 3-. I the case of SQ it should be oted that this result is ot attributable to the presece of o-statioary cycles because it happes to the same degree i the = case where ostatioary cycles are ot preset i the data. I cotrast for MAIC (ad to a lesser extet MAIC ) we see that the degree of over- ttig is higher whe o-statioary cycles are preset relative to the case where they are ot. Table 3 (c = ), Table (c = :) ad Table (c = ) report the empirical power of the t b test for c = ; ; 3; ; ; 3 uder the various lag selectio rules. Results i these tables are reported oly for the case of liear de-treded data; qualitatively similar results were see for the case of de-meaed data ad may be obtaied from the authors o request. Overall, for a give value of c, the results are qualitatively very similar regardless of the frequecy at which the o-statioary cyclical roots occur. The best power performace for both OLS ad local GLS de-tredig is obtaied whe either SQ or AIC is used to specify the lag augmetatio legth, cosistet with the digs i Table o average lag legth tted by the various rules. The rami catios of the over- ttig see i Table for the MAIC ad SQ rules is clearly see i Tables 3- with the tests based o these rules showig cosiderably lower power throughout tha the tests based o the other lag selectio methods, other thigs beig equal. Aother iterestig aspect of the power results is see most clearly i the pure o-statioary cycles

14 case i Table 3. Here we see that the ite sample power of the OLS de-treded tests are, for a give lag selectio rule, fairly isesitive to frequecy at which the o-statioary cycle occurs, ad ideed as to whether a o-statioary cycle occurs or ot. The same caot be said for the local GLS de-treded tests. To illustrate, i the = case, where o o-statioary cycles are preset, the power of the local GLS de-treded tests are clearly superior to those of the correspodig OLS de-treded tests; for example, with SQ ad c = the local GLS test has power. %, while the OLS test has power. %. However, for = the coverse teds to be the case with the OLS de-treded tests ow the more powerful; for example, for = =(=) usig SQ, for c = 3 the OLS test ow has power of.% (.%) ad the local GLS test.9% (33.9%). This re ects the fact that the local GLS de-tredig method is based o the assumptio that, aside from the possible zero frequecy uit root, the process is statioary, ad it is clear that where this assumptio is violated the ite sample power of the local GLS de-treded tests su ers cosiderably relative to their OLS de-treded couterparts. This is most likely attributable to the fact that the local GLS estimates of the parameters characterisig the determiistic tred compoet will be highly ie ciet, relative to the correspodig OLS estimates, i this case. As the cyclical compoet becomes less persistet (i.e. as c icreases away from zero) the so we would expect the ite sample power of the local GLS de-treded tests to recover, ad a compariso of the results i Tables ad with those i Table 3 shows that this is ideed the case; i the foregoig example whe c = the local GLS test has power.% ad 3.% for = ad =, respectively, while the OLS test has power.3% ad.% for = ad =, respectively. Coclusios I this paper we have show that amog popularly applied uit root test statistics, oly the ADF t-ratio ad ormalised bias statistics have pivotal limitig ull distributios i the presece of (ear-) ostatioary cycles i the data. Other commoly employed uit root test statistics, such as the variace ratio statistic of Breitug () ad the triity of M statistics due to Stock (999) ad Perro ad Ng (99), were show either to admit o-pivotal limitig ull distributios or to have o-degeerate limitig ull distributios, i the latter case yieldig tests with a asymptotic size of oe, whe (ear-) o-statioary cycles are preset. Additioally, we have show that the t-ratios o the lagged depedet variables withi the ADF test regressio also retai stadard ormal limitig ull distributios such that sequetial lag speci catio also remais valid uder (ear-) o-statioary cycles. Cosequetly, we strogly recommed the use of ADF uit root tests coupled with sequetial lag selectio i cases where it is suspected that (ear-) o-statioary cycles may be preset i the data. I such cases our results also suggest that the ite sample power advatages of local GLS de-treded ADF-type tests over their OLS de-treded couterparts see whe ay cyclical behaviour is statioary are likely to be overtured whe o-statioary cycles are preset. Refereces Ahtola, J.A., ad G.C. Tiao (98). Distributio of the least squares estimators of autoregressive parameters for a process with complex roots o the uit circle. Joural of Time Series Aalysis 8, -. Alesia, A., ad N. Roubii (99). Politics cycles i OECD coutries. Review of Ecoomic Studies, 9, Alle, D.S. (99). Filterig permaet cycles with complex uit roots. Workig Paper 99-A, Federal Reserve Bak of St. Louis ( Berk, K.N. (9). Cosistet autoregressive spectral estimates. The Aals of Statistics, Bieres, H.J. (). Complex uit roots ad busiess cycles: are they real? Separate appedix Bieres, H.J. (). Complex uit roots ad busiess cycles: are they real? Ecoometric Theory, Birgea, L., ad L. Kilia (). Data-drive oparametric spectral desity estimators For ecoomic time series: a Mote Carlo study. Ecoometric Reviews, 9-.

15 Breitug, J. (). Noparametric tests for uit roots ad coitegratio. Joural of Ecoometrics 8, Burs, A.F., ad W.C. Mitchell (9). Measurig Busiess Cycles. NBER. N.Y. Campbell, J., ad A. Kyle (993). Smart moey, oise tradig ad stock price behaviour. Review of Ecoomics Studies, 3. Caova, F. (99). Three tests for the existece of cycles i time series. Ricerche Ecoomiche, 3 Cha, N.H. (989). Asymptotic Iferece for ustable autoregressive time series with drifts. Joural of Statistical Plaig ad Iferece 3, 3-3. Cha, N.H. (). Iferece for time series ad stochastic processes. Statistica Siica, Cha, N.H., ad C.Z. Wei (988). Limitig distributio of least squares estimates of ustable autoregressive processes. The Aals of Statistics,, 3-. Chag, Y., ad J.Y. Park (). O the asymptotics of ADF tests for uit roots. Ecoometric Reviews, 3-8. Choi, I. (993) Asymptotic ormality of the least squares estimates for higher order autoregressive itegrated processes with some applicatios. Ecoometric Theory 9, 3-8. Coway, P., ad D. Frame (). A spectral aalysis of New Zealad output gaps usig Fourier ad wavelet techiques. Reserve Bak of New Zealad Discussio Paper Series DP/, Reserve Bak of New Zealad. Dickey, D. A., ad W.A. Fuller (99). Distributio of the estimators for autoregressive time series with a uit root. Joural of the America Statistical Associatio, -3. Elliott, G., T.J. Rotheberg ad J.H. Stock (99). Ecoometrica, E ciet tests for a autoregressive uit root. Faust, J. (99). Near observatioal equivalece ad theoretical size problems with uit root tests. Ecoometric Theory, 3. Ghysels, E., H.S. Lee ad J. Noh (99). Testig for uit roots i seasoal time series: some theoretical extesios ad a Mote Carlo ivestigatio. Joural of Ecoometrics, -. Gregoir, S. (999). Multivariate time series with various hidde uit roots, Part. Ecoometric Theory, 3-8. Gregoir, S. (). E ciet tests for the presece of a pair of complex cojugate uit roots i real time series. Joural of Ecoometrics 3, -. Haldrup, N., ad M. Jasso (). Improvig Power ad Size i Uit Root Testig. Palgrave Hadbooks of Ecoometrics: Vol. Ecoometric Theory, Chapter. T. C. Mills ad K. Patterso (eds.). Palgrave MacMilla, Basigstoke. Hall, A.R. (99). Testig for a uit root i time series with pretest data-based model selectio. Joural of Busiess ad Ecoomic Statistics, -. Hamilto, J.D. (99). Time Series Aalysis. Priceto Uiversity Press. Jasso, M. (). Cosistet covariace matrix estimatio for liear processes. Ecoometric Theory 8, 9 9. Jegaatha, P. (99). O the asymptotic behavior of least-squares estimators i AR time series with roots ear the uit circle. Ecoometric Theory, 9 3. Maddala, G.S., ad I. Kim (998). Uit Roots, Coitegratio, ad Structural Chage. Cambridge Uiversity Press. Müller, U. (8). The impossibility of cosistet discrimiatio betwee I() ad I() processes. Ecoometric Theory, 3. 3

16 Müller, U.K., ad G. Elliott (3). Tests for uit roots ad the iitial coditio. Ecoometrica, 9 8. Nyarko, Y. (99). Learig i misspeci ed models ad the possibility of cycles. Joural of Ecoomic Theory,. Newey, W.K., ad K.D. West (99). Automatic lag selectio i covariace matrix estimatio. Review of Ecoomic Studies, 3-3. Ng, S., ad P. Perro (99). Uit root tests i ARMA models with data depedet methods for selectio of the trucatio lag. Joural of the America Statistical Associatio 9, 8-8. Ng, S., ad P. Perro (). Lag legth selectio ad the costructio of uit root tests with good size ad power. Ecoometrica 9, 9. Nielse, B. (). The asymptotic distributio of uit root tests of ustable autoregressive processes. Ecoometrica 9, -9. Paga, A. (999). Some uses of simulatio i ecoometrics. Mathematics ad Computers i Simulatio 8, Perro P., ad Ng, S. (99). Useful modi catios to some uit root tests with depedet errors ad their local asymptotic properties. Review of Ecoomic Studies 3, 3 3. Phillips, P.C.B. (98), Time series regressio with a uit root, Ecoometrica, 3. Phillips, P.C.B., ad P. Perro (988). Testig for a uit root i time series regressio. Biometrika, Phillips, P.C.B., ad Z. Xiao (998). A primer o uit root testig. Joural of Ecoomic Surveys, 3-. Priestley, M.B. (98). Spectral Aalysis ad Time Series. New York, NY: Academic Press. Said, S.E., ad D.A. Dickey (98). Testig for uit roots i autoregressive-movig average models of ukow order. Biometrika, 99-. Shibayama, K. (8). O the periodicity of ivetories. Departmet of Ecoomics, Uiversity of Ket at Caterbury. Stock, J. (99). Uit roots, structural breaks ad treds. Hadbook of Ecoometrics, Vol IV, Chap.. Stock, J.H. (999), A class of tests for itegratio ad coitegratio. Egle, R.F. ad White, H. (eds.), Coitegratio, Causality ad Forecastig. A Festschrift i Hoour of Clive W.J. Grager, Oxford: Oxford Uiversity Press, 3. Taaka, K. (8). Aalysis of models with complex roots o the uit circle. Joural of the Japaese Statistical Society 38. Taylor, A.M.R. (3). Robust statioarity tests i seasoal time series processes. Joural of Busiess ad Ecoomic Statistics, -3. Tsay, R.S., ad G.C. Tiao (99). Asymptotic properties of multivariate ostatioary processes with applicatios to autoregressios. Aals of Statistics, 3-3.

17 A Appedix Before presetig the proofs of the mai text, we ote the followig trigoometric idetities which will be used i the sequel: where i each case (; ): si t cos () si ( (t + )) si () cos ( (t + )) (A.) cos t cos () cos ( (t + )) + si () si ( (t + )) (A.) si (t ) cos () si ( (t + )) cos () si () cos ( (t + )) si () si ( (t + )) cos (t ) cos () cos ( (t + )) + cos () si () si ( (t + )) si () cos ( (t + )) (A.3) (A.) cos () cos () si () (A.) Moreover, for the DGP i (), ad usig the represetatio of the partial sum of a AR() process with complex uit roots give i Equatio () of Bieres (,p.93), we ca write the followig spectral decompositios of x t, x t ad x t, ad x t = tx si [ (t + si j= j)] " j = si [si ( (t + )) S (t) cos ( (t + )) S (t)] ; (A.) tx si [ (t + j)] x t = L " j si = j= si [si (t) S (t ) cos (t) S (t )] = cos si [si ( (t + )) S (t ) cos ( (t + )) S (t )] si si [cos ( (t + )) S (t ) + si ( (t + )) S (t )] (A.) tx x t = L si [ (t + si = = j= j)] " j si [si ( (t )) S (t ) cos ( (t )) S (t )] cos () si [si ( (t + )) S (t ) cos ( (t + )) S (t )] cos si [cos ( (t + )) S (t ) si ( (t + )) S (t )] (A.8) si where we have de ed the -frequecy partial sum processes, S (t) := P t j= " j cos (j) ad S (t) := P t j= " j si (j). Proof of Lemma : From Property. i Gregoir (999,p.), which makes uses of the idetity = ( cos ()) cos()l + L ( cos() + L) + ( L) (A.9) ( cos ()) see Gregoir (999,p.), it follows that if {x t } is geerated by () the, x t = ( cos ) tx j= " j + cos + L ( cos ) tx si [ (t + si j= j)] " j (A.) =: C t + C t : (A.)

18 Cosequetly, otig that the -frequecy compoet, C t ; of (A.) is, C t = cos ( cos ) si (si ((t + ))S (t) cos ((t + ))S (t)) + ( cos ) si (si (t) S (t ) cos (t) S (t )) it follows from (A.) that, after simpli catio, C t = si [si ((t + ))S (t) cos ((t + ))S (t)] + si [cos ( (t + )) S (t) + si ( (t + )) S (t)] + O p () where := cos ( cos ) ad := si ( cos ) : As a cosequece, we therefore have, o de ig the zero-frequecy partial sum process S (t) := P t j= " j, that x t = ( cos ) S (t) + si [si ((t + ))S (t) cos ((t + )) S (t)] + si [cos ( (t + )) S (t) + si ( (t + )) S (t)] + O p () (A.) which establishes the result i (). The result i (8) follows directly from (A.). Before provig the results i Lemma, we rst provide some additioal results i a preparatory lemma, relevat to the computatio of the uit root statistics from sectio.. Lemma A. Let the coditios of Lemma hold. The i) ii) X x t = t= X x t = t= ( cos ) X S(t) t= X + ( cos ) si t= S(t ) + S(t ) + o p ( ) X S(t) + S (t) si + o p ( ) t= v) vi) iii) iv) X x t = t= X x t = t= X x t x t = t= X t= x t x t = cos + si X S(t ) + S (t ) si + o p ( ) t= X S(t ) + S (t ) si + o p ( ) t= X si S(t ) + S(t ) + o p ( ) t= X S(t ) + S(t ) + o p ( ): t= Proof of Lemma A.: The decompositios i (i)-(vi) obtai usig the results i Lemma, the trigoometric idetities i (A.)-(A.), ad ivokig the result that the di eret partial sums that compose the momets expressed i (i)-(vi) are asymptotically ucorrelated (see Cha ad Wei, 988, Theorem 3.., p.393). Note that the asymptotic ucorrelatedess of these compoets also holds i the ear itegrated cotext cosidered i Sectio ; see Jegaatha (99, Propositio, p.8).

19 Proof of Lemma : First de e W (t=) := p brc p X j= p brc W (t=) := p X j= cos (j) " j = si (j) " j = p p S (r) p p S (r) ad W (t=) := P brc p j= " j. Usig these de itios, we the have from Lemma, ad otig that x t = S (t) = P t j= " j, that h i p x t = W ; (t=) + p si ((t + ))W ; (t=) cos ((t + )) W si ; (t=) + i p hcos # ( (t + )) W; (t=) + si ( (t + )) W si ; (t=) + o p () ) W (r) + h p si ((t + ))W (r) si cos ((t + )) W (r) i + h i p cos ( (t + )) W (r) + si ( (t + )) W si (r) := b(r) (A.3) p x t = ) h i p si ((t + ))W; (t=) cos ((t + )) W si ; (t=) h i p si ((t + ))W (r) cos ((t + )) W si (r) (A.) p x t = W ; (t=) ) W (r) : (A.) Next observe that X x t " t = t= ad, hece, ( cos ) X t= ( cos ()) S (t )" t si X S (t t= X S (t )S (t) + t= X S (t t= X )S (t) S (t! t= )S (t) + o p () )S (t)! X x t " t = t= ) ( cos ) + si X W ((t t= ) =) W (t=) X W ((t ) =) dw (t=) X W ((t t= t= ) =) dw (t=)! X X W ((t ) =) dw (t=) + W ( cos ()) ((t ) =) dw (t=) + o p () t= t= Z Z Z W (r) dw (r) + W (r) dw ( cos ) si (r) W (r) dw (r) Z Z W (r) dw (r) + W ( cos ()) (r) dw (r) : The proof of the results for P t= x t " t ad P t= x t " t follow alog similar lies ad are therefore omitted. Furthermore, the stated covergece results for P t= x t ; P t= x t ; P t= x t x t ad P t= x t x t follow straightforwardly from (i), (ii), (v) ad (vi) of Lemma A., respectively, ad applicatios of the CMT.!

20 Proof of Theorems ad : First we de e z t := [x t x t x t x t 3... x t k ] ad := (; ; ; 3 ; :::; k ) : Hece, to aalyse the covergece of the OLS parameter estimates from () we will make use of the followig expressio for the OLS estimatio error " # " X # X b = z t z t z t " t : (A.) t= Notice that uder the coditios of Theorems ad, the true values of the parameters are give by = (; cos ; ; ; :::; ). Usig a similar approach to Choi (993), we de e the (k + ) (k + ) matrix t= cos cos A := cos cos 3 =: [A A ] (A.) where A is a (k + ) 3 matrix ad A a (k + ) (k ) matrix. Notice that A is a ltratio matrix sice, A z t = ( x t ; x t ; :::; x t k ). Usig the matrix A, ad itroducig the scalig matrix := diag f ; g where := diagf; ; g ad := diagf p ; :::; p g; the latter a (k ) (k ) matrix, we ca rewrite the scaled estimator from (A.) as 8 X X 3 9 X 3 >< A z t z t A A z t z t A >= A z t " t b = A t= t= X X t= >: A z t z t A A z t z X t A >; A z t " t t= t= t= 8 >< = A >: X t= A z t z t A o p () o p () X t= A z t z t A 3 9 > = >; X A z t t= X A z t t= " t " t 3 : (A.8) We ow establish covergece results for the elemets i (A.8). First we observe that X X X x t x t x t x t x t t= t= t= X A z t z t A = X X X x t= t x t (x t ) x t x t t= t= t= X X X x t x t x t x t (x t ) which, as! ; will coverge, usig results i Lemma, to t= t= t= 3 where Z (r)dr := X Z A z t z t A ) (r)dr t= V (A + +B ) ( cos ) ( cos ) si (A +B ) 8 si (cos + )(A +B ) si (A +B ) 8 si (A +B ) si cos (A +B ) si (cos + )(A +B ) si cos (A +B ) si (A +B ) si 3 (A.9) 8