Testing for Unit Roots in Semi-Annual Data

Transcription

1 Testing fo Unit Roots in Semi-Annual ata Sanda G. Feltham a and avid E. A. Giles b,* a Economic Analysis Banch, Ministy of Human Resouces, Victoia BC, Canada V8A4 b epatment of Economics, Univesity of Victoia, Victoia BC, Canada V8Y Revised, ecembe Abstact e conside the poblem of testing fo unit oots at the zeo and seasonal fequencies in timeseies data which ae ecoded semi-annually. The poposed methodology follows that of Hyllebeg et al. 99 and Beaulieu and Mion 993 fo quately and monthly data espectively. The non-standad asymptotic distibutions fo the single and joint tests ae deived, and vaious pecentiles of the finite-sample distibutions ae tabulated. Monte Calo simulation is used to investigate the powes of the tests, and we illustate thei application to seveal semiannual economic time-seies. JEL classifications: Keywods: C, C Unit oots; non-stationay data; seasonality; semi-annual data * Coesponding autho. dgiles@uvic.ca ; FAX: ; Voice:

2 . Intoduction Thee is now a well-established liteatue elating to the poblem of testing fo non-stationaity in seasonal economic time-seies data. Fo example, some of the ealie liteatue on this topic can be found in Hyllebeg 99, and a moe ecent oveview is given by Fanses 997. This issue is of consideable impotance as the distinction needs to be dawn between unit oots at the zeo fequency, and unit oots at some o all of the seasonal fequencies in the case of non-annual data. An incoect identification of the natue of such unit oots would lead to inappopiate filteing of the seies pio to its use in egession analysis, say, as well as inadequate testing fo possible cointegation between one such seies and anothe. A seasonal seies is one which has a spectum with distinct peaks at the seasonal fequencies, = j/s, j =,,..., s-, whee s is the numbe of seasons in the yea. The fequencies of inteest with quately data, fo example, ae at, ¼, ½, and ¾ cycles, o =, /,, and 3/. An integated seies is one that has an infinite mass in its spectum at some fequency,. In contast, the spectum of a stationay seies is finite, but non-zeo, at all fequencies. If a seasonal seies has a unit oot at all of its fequencies, it is said to be seasonally integated, o SIs, and it needs to be s-diffeenced to make it stationay. Ealy contibutions e.g., ickey et al., 984 to the poblem of testing fo non-stationaity in seasonal time-seies consideed only a null of SIs, and an altenative hypothesis of stationaity. That is, they did not conside the possibility of unit oots at only the seasonal non-zeo fequencies, and neithe did they allow fo unit oots at a sub-set of the seasonal fequencies. The moe ecent test poposed by Kunst 997 also falls into this categoy. In the case of quately data, this shotcoming was ectified patially by Osbone et al. 988, and moe fully with the tests poposed by Hyllebeg et al. 99 heeafte HEGY and Ghysels et al Fanses 99 and Beaulieu and Mion 993 povided a simila testing famewok fo monthly data. Economic time-seies data ecoded only on a semi-annual basis ae quite common. Fo example, many companies epot thei financial statements on this basis, at least on a povisional basis. The pupose of this pape is to investigate the popeties of a HEGY testing famewok in the case of semi-annual data, and to compae it with conventional ickey-fulle 98 testing fo unit oots at the zeo fequency in such time-seies. In the next section we descibe the testing pocedue, and section 3 discusses the asymptotic distibutions of ou test statistics. Section 4 deals with the finite-sample pecentiles fo these tests, and epots some small-sample powe esults. The

3 application of the semi-annual HEGY tests is illustated in section 5, and some concluding emaks and ecommendations appea in section 6.. A testing famewok. Backgound In this section the famewok fo applying HEGY-type unit oot tests is outlined fo the case of semi-annual data. Recall the autoegessive model fo a stochastic seasonal pocess, x t : Bx t = t, whee t ~ iid,, B is a polynomial in the backshift opeato, B, and the oots of B= detemine whethe o not the seies is stationay. Fo the puposes of the following deivation the data geneating pocess of x t is assumed to be fee of any deteministic components. Following HEGY 99 and Beaulieu and Mion 993, B can be expessed in tems of elementay polynomials and a emainde: k B B * B kb B B k k whee k B = - / k B k =, ; B B / p k and *B is a emainde with oots outside the unit cicle k k k jk j k The k s ae the unit oots. In the case of semi-annual data we have = and = -, and so B = -B, B = +B, and B = -B. Substituting these into equation. yields: B B B B B * B B Next, let = - and = -, and substitute the ight hand side of.3 into the autoegession equation Bx t = t, so we have: B B xt B B xt B * B xt t 3

4 This can be ewitten in the fom of a egession equation on which the tests can be based: * B z z z 3t t t t whee: z t B xt xt xt z B x x x t t t t z3 t B xt xt xt t =,, 3,...,n.. Implementation of the tests To apply the unit oot tests with semi-annual data, equation.4 is estimated by Odinay Least Squaes OLS. Using a t-test, the null hypothesis of a unit oot at the zeo fequency = is tested against the one-sided altenative hypothesis <, which is equivalent to testing = vesus the altenative of stationaity, >. Similaly, the hypothesis of a unit oot at the fequency = is tested against the altenative <. In addition, the F-statistic fo = = may be used to test if unit oots exist at both fequencies simultaneously: that is, if the seies is seasonally integated. The asymptotic and finite-sample pecentiles of these non-standad t and F statistics ae consideed below, and vaious citical values ae given in Table. These tests ae evaluated unde the assumption that * B =. Howeve, as Beaulieu and Mion 993 point out, if B is of ode geate than S, then * B and the fitted model must be augmented with lagged values of the dependent vaiable as exta egessos in ode to whiten the eos. This follows fom the findings of Said and ickey 984 in the context of the familia augmented ickey-fulle AF tests fo unit oots at the zeo fequency. These augmentations will intoduce a finite-sample distotions in the null distibutions of the t and F tests that will die out asymptotically, povided the coect numbe of lags is added. In paticula, the numbe of such augmentation tems, q, must be allowed to incease as the sample size inceases. Beaulieu and Mion 99 investigated these finite-sample distotions in the case of monthly data, and found that the tue sizes significance levels of the individual unit oot t tests wee no highe than those implied by the asymptotic citical values. They theefoe suggest that this distotion may cause unde ejection of the unit oot hypotheses in small samples with such data. 4

5 3. Asymptotic null distibutions 3. HEGY-type tests fo semi-annual data This section pesents the asymptotic null distibutions fo the half-yealy data tests unde consideation in this pape. Fulle 976 poves that when = -, the limiting null distibutions of ˆ and the associated test statistics in a simple andom walk model, a andom walk with dift, and a andom walk with dift and tend, ae simply the mio image of the limiting distibutions when = ickey and Fulle, 979. Moeove, HEGY 99 show that the analysis of Chan and ei 988 can be used to extact the asymptotic distibution theoy fo the HEGY t tests fom the esults of ickey and Fulle 979 and Fulle 976. The asymptotic null distibutions fo ou two t statistics follow diectly fom the analyses of HEGY 99 and of Beaulieu and Mion 993. The latte povide the deivation of the asymptotic distibutions unde the assumption that t is a matingale diffeence 3 sequence with constant vaiance. A sequence of andom scalas, { t,} t=, is a matingale diffeence sequence if E t = fo all t and E t t-,, = fo t =, 3,...,n. Hamilton, 994, p.89. hen the unit oot tests ae applied in the context of a egession model without any deteministic tems the statistic t has the same limiting null distibution as the familia ickey-fulle F ttest statistic: [ ] {[ ] } L d t / / [ ] d { [ ] d} whee denotes a standad Bownian motion and is a stochastic integal Ghysels et al., 994; Hamilton, 994. Futhe, the asymptotic null distibution of t is the mio image of the usual F distibution; that is, the negative of the t has the asymptotic distibution in 3.. Essentially, these esults follow fom the fact that z t and z t in equation.5 ae asymptotically uncoelated. Ghysels, et al.994 show that this elationship with the usual F t-tests also holds in finite samples. By the same aguments, this will also be tue fo semi-annual data. Futhe, fom the esults of Engle, Gange, Hyllebeg and Lee 993, they pove that fo quately seasonal data 5

6 the asymptotic null distibutions fo F statistics fo testing eithe i that the seies is seasonally integated; o ii that thee ae unit oots at all seasonal fequencies; ae the same as the limiting distibutions of the sum of the appopiate squaed t-statistics. Similaly, the asymptotic null distibution of ou F statistic, fo seasonally integated half-yealy data, follows diectly fom the limiting distibutions of the sum of the squaes of t and t. So, in the case of no dift, no tend, and no seasonal dummy vaiable in the fitted egession we have: L [ ] d [ ] d F [ ] d [ ] d whee i fo i =, denote independent standad Bownian motions Ghysels, et al., 994. Next, conside the asymptotic null distibutions of the tests when deteministic tems ae added to the fitted egession equation,.4. The geneal effects of the vaious deteministic tems ae discussed and then the actual distibutions ae pesented following the notation of Beaulieu and Mion 993. The addition of dift o tend tems affects the asymptotic null distibution of t because these components have thei spectal mass at the zeo fequency HEGY, 99. Howeve, the asymptotic distibution of t is independent of the dift and tend tems as z t is asymptotically othogonal to tems that ae not peiodic Beaulieu and Mion, 993. Recall that the asymptotic distibution of the F statistic is elated to those of the squaed t and squaed t statistics. Thus, the asymptotic null distibution of the F statistic will also change with the addition of deteministic tems, as the distibution of t changes when a dift and/o tend is added to the fitted egession. In the case of the t statistics, the effect of adding a seasonal dummy vaiable into the HEGY egession is the evese of that of adding constant and tend components. Once a constant tem is included in the egession, the addition of a seasonal dummy vaiable does not affect the asymptotic null distibution of t any futhe. The subsequent addition of a seasonal dummy vaiable does, howeve, change the asymptotic null distibution of t, and so the asymptotic popeties of F also change in this case. Again, this all follows fo easons analogous to those in the case of quately data. Some futhe insights into these, and othe elationships between the tests, in case of quately data ae given by Smith and Taylo

7 The foms of the vaious asymptotic distibutions can now be summaized. Let the distibutions of t and t when deteministic tems ae excluded fom the egession equation be denoted as: L N t L N ; t Following the notation of Beaulieu and Mion 993 N x and N x epesent the pat of the x numeato that diffes fom that in the above t and t distibutions, espectively. Similaly and x epesent the pat of the denominato that diffes. The fou vaiations of deteministic tems in the fitted egession model ae epesented by x =,,, and, coesponding to egessions with a constant, a constant and tend, a constant and seasonal dummy, and finally a constant, tend and seasonal dummy. Thus, the geneal fom of the limiting null distibutions fo the t-statistics can be witten as: L [ ] d N t X [ ] d X / L [ ] d N t X / [ ] d X The five possible asymptotic null distibutions fo each of the test statistics ae summaized in Table. Only the pats of the numeato and denominato that diffe fom the no-dift, no-tend model, ae given fo the t statistics. This is done to povide visual epesentation of the distibutions that ae the same. The full asymptotic null distibution of the F statistic is given in tems of these components. [Table About Hee] 3.3 ickey-fulle tests and semi-annual data Seveal popeties of the ickey-fulle t-test have been analyzed in the context of seasonal data. Fo example, HEGY 99 showed it to have asymptotic equivalence to thei t statistic in the no dift/no tend case. Ghysels, et al. 994 demonstated that when unit oots exist at seasonal fequencies the ickey-fulle t-test is still a valid test fo unit oots at the zeo fequency, povided the usual F egession is appopiately augmented with l s- lagged values of the 7

8 diffeenced seies. They use Monte Calo simulations to see how the existence of seasonal unit oots influences the finite-sample size and powe of the test. Thei esults show that, in the case of quately data, when l < s- the size of the F test is seveely distoted. hen l s- the size is close to its nominal size, povided the data geneating pocess is fee of negative moving aveage components. Although we have not explicitly exploed these chaacteistics of the F t-test in the context of semi-annual data hee, intuitively it is clea that coesponding esults to those above will hold with s =. 3.3 Moving-aveage eos Although the HEGY test povides fo identification of oots at diffeent fequencies, it has been found to suffe fom size distotions fom negative MA components in the data geneating pocess. In paticula, fo the data geneating pocess: Y t = d Y t-d + u t, whee u t = t + t- + 4 t-4, and = -.9, the nea cancellation of - B in the autoegessive component of u t makes the unit oot at the zeo fequency difficult to detect. This also explains why the ickey-fulle test suffes fom size distotions with negative MA components, because it has the same distibution as the HEGY t test when l s-. This is not a tivial poblem as such situations aise fequently in pactice. A lage bias in size o vey low powe was also found to occu when seasonal dummies ae in the data geneating pocess, but ae not included in the egession. This povides justification fo including deteministic tems o additional lags even though they may be ielevant and thus may educe the powe of the test. 4. Finite-sample esults 4. Citical Values e have investigated the small-sample distibutions of ou thee semi-annual unit oot tests unde both the null and altenative hypotheses using Monte Calo simulations. All of these simulations 8

9 wee implemented using SHAZAM 997 on a EC Alpha 3 wokstation. The data geneating pocess used was x t = x t- + t whee the t values wee geneated by the nomal andom numbe geneato in SHAZAM. Five diffeent vaiations of the basic HEGY auxiliay egession wee fitted to the geneated data: dift, seasonal dummy vaiable, and tend,s,t; dift, seasonal dummy, and no tend,s,nt; dift, no seasonal dummy vaiable, and tend,ns,t; dift, no seasonal dummy vaiable, and no tend,ns,nt; and no dift, no seasonal dummy vaiable, and no tend N, NS, NT The initial value fo y t was set to zeo and then the fist obsevations wee dopped to wash out the effect of this initial value. All simulations wee based on, epetitions. Citical values associated with the st, 5 th and th pecentiles of the undelying distibutions wee geneated fo the t test of H : = and fo the t test of H : =. Coesponding citical values wee calculated fo the F test of H : = = by detemining the 9 th, 95 th, and 99 th pecentiles of the empiical distibutions. These esults ae pesented in Table. [Table About Hee] Thee is no discenable change in the t citical values when a dift and/o tend ae added to the fitted egessions, as was indicated by the asymptotic distibution theoy in the pevious section. In contast, the citical values fo t change when eithe a dift o tend is added to the fitted model on which the tests ae based. The subsequent addition of a seasonal dummy vaiable does not then change the finite sample distibution of t, but it does change that of t. As pedicted by the asymptotic distibution theoy, the F statistic citical values change with each vaiation to the deteministic components in the fitted egession, as its distibution is the same as the distibution of the sum of the squaes of t and t. It is also woth noting the vey close accodance between the no-dift, no-tend values fo t in Table, the coesponding ickey-fulle F citical values of MacKinnon 99, and the HEGY 99 values. Fo example, with a % significance level and n =, the citical values ae in Table, fom MacKinnon, and -.6 fom HEGY. The citical values fo t i.e., fo testing H : = in Table can be compaed with the HEGY 99 citical values fo the fitted model with no deteministic tems. Fo the % level and n = the two statistics ae.6 and.6 espectively. 9

10 The esults in Table ae based on a data geneating pocess d.g.p. of x t = x t- + t. That is, unit oots ae pesent at both the zeo and fequencies the semi-annual seies is seasonally integated. An altenative appoach is to conside a data geneating pocess that has a unit oot at only one of these fequencies, but then still base the unit oots tests on the HEGY-type integating egession as in equation.4. Citical values wee also geneated fo these tests. In the test fo a unit oot only at the zeo fequency, a simple andom walk was used as the d.g.p.; in the second case the d.g.p. had a unit oot at the fequency but not at the zeo fequency. Fo completeness, these esults appea in Table 3. [Table 3 About Hee] 4. Powes of the tests The Monte Calo expeiment was also used to simulate the size-adjusted powes of the test statistics developed in this pape, fo the following thee fitted egessions: dift, seasonal dummy vaiable, and tend, S, T; dift, seasonal dummy vaiable, and no tend, S, NT; and dift, no seasonal dummy, with tend, NS, T. Using the d.g.p. of Table, namely x t = x t- + t, the size-adjusted powes of the tests of =, = and = =, wee simulated fo = [...] and vaious sample sizes n. Each powe cuve was geneated using, epetitions. The esults of these empiical powe simulations fo the t, t, and F tests ae given in Tables 4-6, fo the case whee the d.g.p. is as in Table ; and in Tables 7 and 8 fo the case whee the d.g.p. is as in Table 3. Illustative plots appea in Figues -3 and Figues 4 and 5 espectively. [Tables 4 to 8 About Hee] These esults show that the powe cuves fo lage sample sizes have the basic expected shape. That is, they move towads % as moves away fom. Thee is easonable powe when n =, but vey low size-adjusted powe when the sample size is small. This is no supise given othe esults fo simila such unit oot tests. The pattens in the esults as the sample size is inceased eflect the consistency of the tests. Changing the significance level of the test yields the usual tadeoff between the size and powe. As expected, the tests based on auxiliay egessions

11 without a dift o tend have moe powe than thei countepats with deteministic tems, because the data geneating pocess does not include deteministic tems. [Figues to 5 About Hee] 5. Some applications The application of the testing pocedues discussed in this pape is illustated hee fo fou quite diffeent semi-annual economic time-seies. In each case, we compae the esults obtained when the possibility of seasonal unit oots is entetained, with those obtained when the AF tests ae used to test fo unit oots only at the zeo fequency. The seies ae published only in semiannual fom, and ae: i New Zealand total knitted fabic sales tonnes, June 965 to ecembe 997, seies SEPH.SATT Statistics New Zealand, 998; ii Canadian pe-cast concete pice index, 98=, 977 to 99, matix 4 Statistics Canada, 998; iii Canadian poduction of maketable gas millions of cubic metes, oil-equivalent, June 979 to ecembe 989 Petoleum Monitoing Agency Canada, vaious yeas; iv U.S.A. six-monthly incease in numbe of cellula phone subscibes, ecembe 984 to ecembe 995 ateman, 998. These seies ae shown 4 in Figue 6, and Table 9 pesents the esults of testing fo a unit oot only at the zeo fequency, using the AF tests. e allowed fo dift and/o tend in the AF egessions, and used the stategy of olado et al. 99 to detemine thei final inclusion 5. The augmentation level, q, fo the egessions was chosen as the minimum equied to obtain clean autocoelation and patial autocoelation functions fo the associated esiduals 6 ; and a seasonal dummy vaiable was included with any dift tem 7. As can be seen, the AF tests suggest in each case that the seies ae I. Accodingly, one would infe that the seies can be made stationay via simple fist-diffeencing. In Table we show the esults of applying the semi-annual HEGY tests outlined in this pape. In the cases i, iii and iv we find that in fact the seies ae SI that is, they each have unit oots at both the zeo and fequencies. So, in fact two-peiod diffeencing is the appopiate filte needed to make these seies stationay. The ealie conclusion that seies ii has a unit oot

12 only at the zeo fequency is upheld in Table, but these examples cetainly illustate the need fo cae when applying standad AF tests to semi-annual data. [Figue 6 About Hee] [Tables 9 and About Hee] 6. Conclusions In this pape we have discussed a famewok fo testing fo unit oots in time-seies data that ae epoted at a maximum fequency of twice a yea. In such cases, the possibility of unit oots at the zeo and/o fequencies aises, and the testing stategies developed by Hyllebeg et al. 99 and Beaulieu and Mion 993 povide a natual famewok to exploit. The asymptotic null distibutions of the tests we popose ae deived, discussed and summaized, and in cetain cases they can be linked to existing esults associated with the well-known tests of ickey and Fulle 979, 98. Simulated pecentiles fo the finite-sample null distibutions of the test statistics ae tabulated fo vaious sample sizes and combinations of deteministic tems in the fitted egessions. The powes of the tests ae simulated unde a simila ange of situations, and thei application is illustated with seveal actual semi-annual time-seies. The analysis in this pape may be extended in seveal diections. The obvious one is to investigate the popeties of coesponding tests fo cointegation. Othe topics of inteest include the size-obustness of the tests to autocoelation in the data-geneating pocess; to the method used to select an appopiate augmentation level ; and to stuctual beaks in the data. These emain mattes fo futue eseach. Acknowledgements The authos ae gateful to Robet aeseke, Judith Giles, Nilanjana Roy, Gugsa ekneh and paticipants in the Univesity of Victoia Econometics Colloquium, fo thei helpful comments on an ealie vesion of this wok.

13 3 Table Effects of deteministic tems on asymptotic distibutions of semi-annual unit oot tests Model t t F N, NT, NS N N N N N, NT, NS d N N / ] [ ] [ N d N d d, T, NS d d d d d d N N / ] [ ] [ N d N d 4 d d d d, NT, S N N d N / ] [ ] [ d N d d / ] [ ] [ d N d, T, S N N N N / ] [ ] [ d N d / ] [ ] [ d N d

14 Table Citical values fo the HEGY-type unit oot tests. d.g.p.: B x t = t Model = = = =, S, T, S, NT, NS, T, NS, NT N, NS, NT n % % % % 99% 9 9%

15 Table 3 Citical values fo the individual seasonal unit oot tests d.g.p.: - Bx t = t d.g.p.: + Bx t = t Model = = N % % % %, S, T, S, NT, NS, T, NS, NT N, NS, NT

16 Table 4 Powe of the t test. d.g.p.: B x t = t , S, T % n= n= n= n= n= n= % n= n= n= , S, NT % n= n= n= n= n= n= % n= n= n= , NS, T % n= n= n= n= n= n= % n= n= n=

17 Table 5 Powe of the t test. d.g.p.: B x t = t , S, T % n= n= n= n= n= n= % n= n= n= , S, NT % n= n= n= n= n= n= % n= n= n= , NS, T % n= n= n= n= n= n= % n= n= n=

18 Table 6 Powe of the F test. d.g.p.: B x t = t , S, T % n= n= n= n= n= n= % n= n= n= , S, NT % n= n= n= n= n= n= % n= n= n= , NS, T % n= n= n= n= n= n= % n= n= n=

19 Table 7 Powe of the t test. d.g.p.: Bx t = t , S, T % n= n= n= n= n= n= % n= n= n= , S, NT % n= n= n= n= n= n= % n= n= n= , NS, T % n= n= n= n= n= n= % n= n= n=

20 Table 8 Powe of the t test. d.g.p.: + Bx t = t , S, T % n= n= n= n= n= n= % n= n= n= , S, NT % n= n= n= n= n= n= % n= n= n= , NS, T % n= n= n= n= n= n= % n= n= n=

21 Table 9 Results of AF tests Seies i Seies ii Seies iii Seies iv Fitted Model Test Statistic Outcome Statistic Outcome Statistic Outcome Statistic Outcome n q , S, T t dt I F ut , S, NT t d I.989 n.a. F ud n.a. N, NS, NT t.837 I n.a..848 I n.a. Notes: n is the sample size; q is the augmentation level fo the tests;, S, T denotes dift, seasonal dummy, tend ;, S, NT denotes dift, seasonal dummy, no tend ; N, NS, NT denotes no dift, no seasonal dummy, no tend. The othe notation is defined in the text.

22 Table Results of seasonal unit oot tests Seies i Seies ii Seies iii Seies iv Fitted Model Test Statistic Outcome Statistic Outcome Statistic Outcome Statistic Outcome, S, T q t Zeo oot Zeo oot Zeo oot Zeo oot t oot.54 oot oot F SI SI SI, S, NT q t Zeo oot Zeo oot Zeo oot Zeo oot t oot oot oot F SI SI SI, NS, T q t Zeo oot Zeo oot Zeo oot Zeo oot t oot oot oot F SI.49 SI SI, NS, NT q t Zeo oot Zeo oot Zeo oot Zeo oot t oot oot oot F SI.967 SI SI N, NS, NT q t Zeo oot Zeo oot Zeo oot Zeo oot t oot oot oot F SI.78 SI SI

23 Figue Powe of the t test. ift, seasonal dummy and tend in egession. % size. d.g.p.: B x t = t Powe % 8 6 n= n=5 n= 4 size=% Rho Figue Powe of the t test. ift, seasonal dummy and tend in egession. % size. d.g.p.: B x t = t Powe % 8 6 n= n=5 n= 4 size=% Rho 3

24 Figue 3 Powe of the F test. ift, seasonal dummy and tend in egession. % size. d.g.p.: B x t = t Powe % 8 6 n= n=5 n= 4 size=% Rho Figue 4 Powe of the t test. ift, seasonal dummy and tend in egession. % size. d.g.p.: Bx t = t Powe % 8 6 n= n=5 n= 4 size=% Rho 4

25 Figue 5 Powe of the t test. ift, seasonal dummy and tend in egession. % size. d.g.p.: + Bx t = t Powe % 8 6 n= n=5 n= 4 size=% Rho 5

26 Figue 6b: Canadian Pecast Concete Pice Index 98= ate Pice Index Figue 6a: New Zealand Knitted Fabic Sales ate Figue 6c: Canadian Gas Poduction ate Figue 6d: Numbeof U.S. Cell Phones ate Tonnes Millions Cubic Metes Millions

27 Refeences Beaulieu, J. J., Mion, J. A., 993. Seasonal unit oots in aggegate U.S. data. Jounal of Econometics 55, Chan, N.H., ei, C. Z., 988. Limiting distibutions of least squaes estimates of unstable autoegessive pocesses. Annals of Statistics, 6, ickey,. A., Fulle,. A., 979. istibution of the estimatos fo autoegessive time seies with a unit oot. Jounal of the Ameican Statistical Association, 74, ickey,. A., Fulle,. A., 98. Likelihood atio statistics fo autoegessive time seies with a unit oot. Econometica, 49, ickey,. A., Hasza,. P., Fulle,. A., 984. Testing fo unit oots in seasonal time seies. Jounal of the Ameican Statistical Association 79, ods, J.L., Giles,.E.A., 995. Altenative stategies fo augmenting the ickey-fulle test: sizeobustness in the face of pe-testing. Jounal of Statistical Computation and Simulation, 53, olado, J.J., Jenkinson, T., Sosvilla-Riveo, S., 99, Cointegation and unit oots. Jounal of Economic Suveys, 4, Fanses, P. H., 99. Seasonality, nonstationaity and the foecasting of monthly time seies. Intenational Jounal of Foecasting 7, Fanses, P. H., 998. Time Seies Models fo Business and Economic Foecasting. Cambidge Univesity Pess, Cambidge. Fanses, P. H., Hobijn, B., 997. Citical values fo unit oot tests in seasonal time seies. Jounal of Applied Statistics 4, Fulle,. A., 976. Intoduction to Statistical Time Seies. iley, New Yok. Ghysels, E., Lee, H. S., Noh, J., 994. Testing fo unit oots in seasonal time seies: some theoetical extensions and a Monte Calo investigation. Jounal of Econometics 6, Hyllebeg, S., 99. Modelling Seasonality. Oxfod Univesity Pess, Oxfod. Hyllebeg, S., Engle, R. F., Gange, C.. J., Yoo, B. S., 99. Seasonal integation and cointegation. Jounal of Econometics 44, Kunst, R. M.,997. Testing fo cyclical non-stationaity in autoegessive pocesses. Jounal of Time Seies Analysis 8,

28 MacKinnon, J.G., 99. Citical values fo cointegation tests, in R.F. Engle and C..J. Gange, eds., Long-Run Economic Relationships. Oxfod Univesity Pess, Oxfod, Osbon,. R., Rodigues, P. M. M., 998. The asymptotic distibutions of seasonal unit oot tests: a unifying appoach. iscussion Pape 98, School of Economic Studies, Univesity of Mancheste. Osbon,. R., Chui, A. P. L., Smith, J. P., Bichenhall, C. R., 988. Seasonality and the ode of integation of consumption. Oxfod Bulletin of Economics and Statistics 5, Petoleum Monitoing Agency Canada, vaious yeas. Canadian Petoleum Industy Monitoing Repots. Phillips, P. C. B., 987. Time seies egession with a unit oot. Econometica 55, Said, S. E., ickey,. A.,984. Testing fo unit oots in autoegessive-moving aveage models of unknown ode. Biometika 7, SHAZAM, 997. SHAZAM Econometics Compute Pogam, Use s Refeence Manual, Vesion 8.. McGaw-Hill, New Yok. Smith, R. J., Taylo, A. M. R., 998. Additional citical values and asymptotic epesentations fo seasonal unit oot tests. Jounal of Econometics, 85, Statistics Canada, 998. Cansim Main Index. Statistics New Zealand, 998. INFOS atabank. Stock, J. H., 988. A class of tests fo integation and cointegation. Mimeo., Kennedy School of Govenment, Havad Univesity, Cambidge, MA. ateman, R., 998. <www-stat.whaton.upenn.edu/~wateman/fsw/datasets/txt/cellula.txt>. 8

29 Footnotes. See also Fanses and Hobijn 997. Smith and Taylo 998 povide a ecent elegant discussion of the HEGY tests fo quately data.. See also Phillips 987 and Stock The assumption of a matingale diffeence sequence is stonge than that of seially uncoelated eos, but weake than independence Hamilton, 994, p In the case of seies iv the plotted seies is fo the numbe of cell phones, athe than the six-monthly incease in this numbe. The latte seies is the one that is analyzed. 5. In Table 9, t dt denotes the AF unit oot t-test with dift and tend tems included in the fitted egession; F ut is the coesponding AF F-test fo a unit oot and zeo tend; t d is the unit oot t-test with a dift but no tend in the fitted egession; F ud is the coesponding F-test fo a unit oot and a zeo dift; and t is the AF unit oot test when the fitted egession has no dift o tend tem included. Finite-sample citical values fo ou t-tests and F-tests come fom MacKinnon 99, and fom ickey and Fulle 979, 98, espectively. 6. The simulation esults of ods and Giles 995 favou this appoach, in tems of low pe-test size distotion, fo sample sizes such as ous. In fact the esults ae not sensitive to the method used to choose this augmentation level. 6. The need to include this dummy vaiable follows fom the esults of Ghysels et al Thee was no evidence that any of the seies othe than iv may be I. 7. 9