Working Paper A fractionally integrated exponential model for UK unemployment

Transcription

1 econsor Der Open-Access-Publikaionsserver der ZBW Leibniz-Informaionszenrum Wirschaf The Open Access Publicaion Server of he ZBW Leibniz Informaion Cenre for Economics Gil-Alaña, Luis A. Working Paper A fracionally inegraed exponenial model for UK unemploymen Discussion Papers, Inerdisciplinary Research Proec 373: Quanificaion and Simulaion of Economic Processes, No. 000,67 Provided in Cooperaion wih: Collaboraive Research Cener 373: Quanificaion and Simulaion of Economic Processes, Humbold Universiy Berlin Suggesed Ciaion: Gil-Alaña, Luis A. (000) : A fracionally inegraed exponenial model for UK unemploymen, Discussion Papers, Inerdisciplinary Research Proec 373: Quanificaion and Simulaion of Economic Processes, No. 000,67, hp://nbn-resolving.de/ urn:nbn:de:kobv: This Version is available a: hp://hdl.handle.ne/049/659 Nuzungsbedingungen: Die ZBW räum Ihnen als Nuzerin/Nuzer das unengelliche, räumlich unbeschränke und zeilich auf die Dauer des Schuzrechs beschränke einfache Rech ein, das ausgewähle Werk im Rahmen der uner hp:// nachzulesenden vollsändigen Nuzungsbedingungen zu vervielfäligen, mi denen die Nuzerin/der Nuzer sich durch die erse Nuzung einversanden erklär. Terms of use: The ZBW grans you, he user, he non-exclusive righ o use he seleced work free of charge, erriorially unresriced and wihin he ime limi of he erm of he propery righs according o he erms specified a hp:// By he firs use of he seleced work he user agrees and declares o comply wih hese erms of use. zbw Leibniz-Informaionszenrum Wirschaf Leibniz Informaion Cenre for Economics

2 A FRACTIONALLY INTEGRATED EXPONENTIAL MODEL FOR U.K. UNEMPLOYMENT L.A. GIL-ALANA Insiu für Saisik und Ökonomerie, Humbold Universiä zu Berlin, Berlin, Germany. Universiy of Navarre, Deparmen of Economics, Pamplona, Spain. ABSTRACT Fracionally inegraed models wih he disurbances following a Bloomfield (973) exponenial specral model are proposed in his aricle for modelling he U.K. unemploymen. This enables us a beer undersanding of he low-frequency dynamics affecing he series, wihou relying on any paricular ARMA specificaion for is shor-run componens, which, in general, requires many more parameers o esimae. The resuls indicae ha his exponenial model, confounded wih fracional inegraion, may be a feasible way of modelling unemploymen, also showing ha is order of inegraion is much higher han one and hus, leading o he conclusion ha he sandard pracice of aking firs differences may lead o erroneous resuls. Key words: Fracional inegraion; Long memory; Unemploymen JEL Classificaion: C

3 INTRODUCTION In his aricle, an alernaive way of modelling he U.K. unemploymen by means of combining fracional inegraion wih he Bloomfield (973) exponenial specral model for he disurbances is proposed. The main moivaion for using he fracional inegraion framework is ha i seems much more general han he radiional approaches based on deerminisic I(0) or uni roos I() models. We use he exponenial specral model of Bloomfield (973) because he usual ARMA represenaions require many more parameers o esimae compared wih his non-parameric approach, which also produces auocorrelaions decaying exponenially as in he AR processes. The aricle is organised as follows: Secion briefly describes he conceps of fracional inegraion and of he exponenial specral model of Bloomfield (973). Secion 3 shows he mehod of esimaing he parameers in his conex by using he While funcion, (which is an approximaion o he likelihood funcion), and we also presen a version of he ess of Robinson (994) for esing hese ype of models. In Secion 4 we esimae and es he models using differen measures of he U.K. unemploymen while Secion 5 conains some concluding remarks. AN EXPONENTIAL SPECTRAL FRACTIONALLY INTEGRATED MODEL Many economic ime series conain pleny of evidence of nonsaionariy and much conroversy in macroeconomics has revolved around he quesion of he suiabiliy of I() or uni roo models for describing raw ime series as opposed o he so-called rendsaionary models, where he raw series is described as an I(0) process plus a deerminisic rend. Uni roos and linear ime rends each consiue exremely specialised models for nonsaionariy, bu each has he advanage of concepual and compuaional simpliciy, and hey are naurally hough of as rival models because a uni roo, wih or wihou a drif,

4 implies a consan or linear rend funcion, he disincion hen being in he disurbance erms. However, in he las few years, an increasing amoun of lieraure sudying he source of nonsaionariy in macroeconomic ime series in erms of fracionally differenced ime series, has appeared. We can consider a process like d ( L) x = v, =,,... () where v is an I(0) covariance saionary process wih specral densiy which is posiive and finie a zero frequency. Clearly, if d = 0, x = v and he process is weakly dependen as opposed o he srong dependence case when d > 0. The macroeconomic lieraure sresses he cases d = 0 and d = bu we can define () for any real d by he expansion ( L ) d = + Γ ( d + ) ( L). = Γ ( d + ) Γ ( + ) If 0 < d < 0.5 in () hen, x is a covariance saionary process, having auocovariances which decay much more slowly han hose of an ARMA process, in fac so slowly as o be nonsummable. Models such as () provide a ype of flexibiliy in modelling low-frequency dynamics no achieved by non-fracional ARIMA models, and saionary fracional models have been shown by Granger (980), Robinson (978), o arise from aggregaion of ARMA series wih randomly varying coefficiens. On he oher hand, AR modelling of he I(0) process v is very convenional, bu many oher ypes of I(0) process exis, including ones ouside he saionary and inverible ARMA case. We propose in his paper he use of he exponenial specral model of Bloomfield (973), in which v is defined exclusively in erms of is specral densiy funcion, given by m σ f ( λ ; τ ) = exp cos( ). τ r λ r () π r =

5 Suppose ha v follows an ARMA process of form p v = φ v + ε r r r = r = q θ r ε r, where ε is a whie noise process and all zeros of φ(l) lying ouside he uni circle and all zeros of θ(l) lying ouside or on he uni circle. Clearly, he specral densiy funcion of his process is hen q i r λ θ r e σ r = f ( λ ; ϕ) =, (3) p π i r λ φ e where ϕ corresponds o all he AR and MA coefficiens and σ is he variance of ε. Bloomfield (973) showed ha he logarihm of an esimaed specral densiy funcion is ofen found o be a fairly well-behaved funcion and can hus be approximaed by a runcaed Fourier series. He showed ha () approximaes (3) well where p and q are of small values, which usually happens in economics. Like he saionary AR(p) model, his has exponenially decaying auocorrelaions and hus, using his specificaion, we do no need o rely on so many parameers as in he ARMA processes, which always resuls edious in erms of esimaion, esing and model specificaion. The Bloomfield model for I(0) processes, confounded wih he fracional model () has no been used very much in previous economeric applicaions (hough he Bloomfield model iself is a well-known model in oher disciplines, eg. Beran, 993), and one byproduc of his work is is emergence as a credible alernaive o he fracional ARIMAs which have become convenional in parameric modelling of long memory. Among he few examples found in he lieraure are Gil-Alana and Robinson (997) and Velasco and Robinson (999). The following secion shows he mehod of esimaing and esing he r = r 3

6 parameers in his conex of fracionally inegraed models wih Bloomfield (973) exponenial specral disurbances. ESTIMATION AND TESTING IN THE FREQUENCY DOMAIN Given a covariance saionary process {x, =,, }, where x is given by () (wih d < 0.5) and v is an I(0) process wih specral densiy funcion of form as in (), we are firsly concerned wih he esimaion of he parameers of he model, ha is, d and hose appearing in (). Based on parameric approaches, d is esimaed oinly wih all he oher parameers ha specify he model. Since v is defined in erms of is specral densiy funcion, he esimaion mus be carried ou in he frequency domain. Fox and Taqqu (986) assumed Gaussianiy, and minimized he While funcion (an approximaion o he exac likelihood funcion) of a covariance saionary process wih I(0) disurbances of a very general form (and hus, including he Bloomfield (973) exponenial specral model). Calling ψ he parameer vecor o be esimaed, hey minimized 4π π π log f ( λ ; ψ ) + I ( λ) d λ f λ ψ ( ; ) (4) where I(λ) is he periodogram of he process x, defined as I ( λ ) = π T T = x e i λ. The esimae was shown o be consisen and asympoically normal under appropriae condiions, which are saisfied by he fracional model () wih 0 < d < 0.5. Furhermore, Velasco and Robinson (999) show a way of esimaing d for nonsaionary series wih 0.5 d <, and even for any degree of nonsaionary (d 0.5) by means of apering. Anoher 4

7 5 esimae wih he same asympoic behaviour is obained if (4) is replaced by a sum over he Fourier frequencies, i.e. minimizing, ) ; ( ) ( ) ; ( log = + T f I f T ψ λ λ ψ λ (5) wih λ = π/t. In he model given by () and (), ψ = (d, τ ), and. ) ( cos exp ) ; ( = = m r r d i r e f λ τ π σ ψ λ λ (6) Subsiuing now (6) in (5), he minimum in (5) can be easily carried ou hrough a compuer programme. We nex describe a esing procedure suggesed by Robinson (994) o es he order of inegraion in raw ime series in his conex of exponenial specral disurbances. Suppose we observe {y, =,, T}, where T x z y...,,,, ' = + = β (7) and z is a (kx) vecor of exogenous regressors, (like z = (,) o include, for insance, he case of a linear ime rend); x is described by () wih he disurbances following a specral densiy funcion as in (). In general, we wish o es he null hypohesis : o, o d d H = (8) for a given real number d o. When z is nonempy, we form, ) ( d z L w o = aking 0., 0 = z Based on he null model, he leas-squares esimae of β and residuals are, ) ( ' ˆ T d T y L w w w o = = = β

8 and is periodogram is d ˆ o v = ( L) y ˆ' β w, =,,.... P ( λ ) vˆ = π T T = vˆ e i λ. Unless f in () is a compleely known funcion, (e.g., as when v is whie noise), we have o esimae he nuisance parameer vecor τ. The esimae mus be a Gaussian one, ha is, i mus have he same limi disribuion as he efficien maximum likelihood esimae based on he assumpion ha v, v,, v T, is Gaussian. One such esimae, which fis naurally ino he frequency domain seing, is ˆ τ = arg min τ σ ( τ ), where he minimizaion is carried ou over a suiable subse of R m, and σ ( τ ) T m π = Pˆ ( ) exp v λ τ r T = r = cos( λ ) r. Nex we form m( λ ) = log sin λ and he es saisic is T aˆ s ˆ =, (9) ˆ b σˆ where ˆ σ = σ ( ˆ), τ aˆ T m π = m ( λ ) Pv ( λ ) exp τ r T = r = ˆ cos ( λ r) and 6

9 bˆ = π 6 m = = m + =. Under he null hypohesis (8), Robinson (994) esablished under regulariy condiions ha sˆ d N(0, ) as T. (0) The condiions on v in (0) are far more general han Gaussianiy, wih a momen condiion only of order required. An approximae one-sided 00α%-level es of (8) agains alernaives : d d o () H > is given by he rule: Reec H o if s ˆ > zα ", () where he probabiliy ha a sandard normal variae exceeds z α is α. Conversely, an approximae one sided 00α% level es of (8) agains alernaives is given by he rule: : d d o (3) H < Reec H o if sˆ < z ". (4) As hese rules indicae, we are in a classical large sample esing siuaion for reasons described by Robinson (994), who also showed ha he above ess are efficien in he Piman sense ha agains local alernaives H : d = d o + δt -/ for δ 0, he es has an asympoic normal disribuion wih variance and mean which canno (when v is Gaussian) be exceeded in absolue value by ha of any rival regular saisic. In he following secion, a fracionally inegraed Bloomfield (973) model is esimaed and esed using differen measures of he U.K. unemploymen. All calculaions α 7

10 were carried ou using Forran. A diskee conaining he codes for he esimaion and esing programmes is available from he auhor on reques. AN EMPIRICAL APPLICATION TO THE U.K. UNEMPLOYMENT Four differen measures of unemploymen were considered. Firsly, we looked a he number of people claiming unemploymen benefis. This measure is known as he claiman coun (CC) and is available monhly. We look a his series (U ) and also a is logarihmic ransformaion (log U ). Anoher measure, which is relaed o he unemploymen rae, is he CC series as a percenage of he workforce. We also look a his series, (u ), as well as is logisic ransformaion: u * = u log u. All hese monhly series sar in January 97 and end in Augus 998. These series have been invesigaed in a number of papers by Gil-Alana (999a,b,c), sudying heir orders of inegraion in erms of non-parameric and parameric (ARFIMA) models. This aricle is herefore a complemenary work in ha direcion. In all hese previous works, he order of inegraion of he U.K. unemploymen was found o be much higher han one and hus, reecing he hypohesis of a uni roo. In his aricle we wan o invesigae he order of inegraion of he series when he disurbances follow a Bloomfield (973) exponenial specral model. Across Tables -4 we presen he esimaed values of d when x is given by () and v follows he Bloomfield (973) model () wih m = 0 (i.e., v is whie noise); ; and 3. These esimaes were found minimizing (5) using a grid search over he range [-5, 5] for τ and [-0.5, 0.5) for d of lengh 0.0 wih x based on he second differences. In all hese ables we also display he esimaed values of d when v follows an AR(p) process wih p = 0; ; and 3. The 8

11 esimaion in hese cases was carried ou using he Sowell s (99) procedure of esimaing by maximum likelihood in he ime domain. Thus, he difference observed in he esimaed values of d when p and m are boh equal o zero is clearly due o he differen mehod of esimaion used. Saring wih U, we observe in Table ha if v is whie noise, he esimaed value of d is.66 when using he ime domain esimaion procedure, and.65 when using he frequency domain approach. Allowing v o be weakly paramerically auocorrelaed, he order of inegraion of U seems o be higher, ranging beween.83 and.05 when modelling v wih auoregressions bu slighly greaer, and ranging beween.9 and.0 when v follows he Bloomfield (973) exponenial specral model. Thus, we observe ha he order of inegraion of his series is much higher han one, flucuaing around when he disurbances are weakly auocorrelaed. (Tables and abou here) Table displays he resuls for log U. Again all he values are higher han. If v is whie noise, he esimaed value of d is now.63 in he ime domain and.6 in he frequency domain. If v is an AR process, d oscillaes beween.7 and.84, and modelling v wih Bloomfield (973), he values of d oscillae beween.73 and.0. Thus, we again observe a higher value when using he exponenial specral model. Comparing hese resuls wih hose in Table we observe ha using he logarihmic ransformaion, he orders of inegraion are slighly smaller hough sill grealy above one. Taking u as he measure of unemploymen, he resuls are given in Table 3. If v is whie noise, he esimaed value of d is in boh cases around.50. Tha means ha if we ake firs differences, he differenced series behaves as in he boundary case beween saionariy and nonsaionariy. Allowing v o be weakly auocorrelaed, he values range 9

12 beween.70 and.9 when using auoregressions, and beween.87 and.08 wih he exponenial specral disurbances. (Tables 3 and 4 abou here) Finally, using he logisic ransformaion of u, (u * ) as he measure for unemploymen, we again observe ha if v is whie noise, he esimaed value of d is around.50 bu allowing weak dependence in he disurbances, he values of d are slighly higher, ranging beween.64 and.8 when v is AR and beween.69 and.0 wih he Bloomfield (973) model. We can conclude he analysis of hese four ables by saying ha when esimaing he order of inegraion of he U.K. unemploymen, he value of d seems o be much higher han one. Given ha boh mehods of esimaion are based on maximum likelihood, convenional ess based on he saisic ( dˆ d) / SE( dˆ ) were performed, reecing he uni roo null (ie., d = ) in all cases across all series. If he disurbances are whie noise, he order of inegraion is slighly higher han.60 for U and log U, and is around.50 for u and u *. If we allow he disurbances o be weakly paramerically auocorrelaed, he orders of inegraion are even higher, flucuaing beween.70 and.0 in all cases. We also observe higher values when using he Bloomfield (973) exponenial specral model raher han he auoregressions, hough he difference beween hem is in all cases smaller han 0.5. Across Tables 5 8 we es he order of inegraion of he series using he ess of Robinson (994) described in Secion 3. Denoing any of he measures of unemploymen y, we employ hroughou he model () and (7) wih z = (,),, z = (0,0) oherwise, so y = α + β + x, =,,... (5) d ( L) x = v, =,,..., (6) 0

13 reaing separaely he cases α = β = 0 a priori, (i.e., including no regressors in he undifferenced regression); α unknown and β = 0 a priori, (i.e., including an inercep); and finally, α and β unknown (i.e., including a linear ime rend). We model he I(0) process v o be boh whie noise (m = 0) and weakly auocorrelaed wih he Bloomfield (973) model of orders, and 3. Clearly, if v is whie noise, when d =, he differences ( L) y behave, for >, like a random walk when β = 0, and a random walk wih a drif when β 0. However, we repor es saisics no merely for he null d o = in (8) bu for d o =.0;.40;.50;.60;.80 and. The es saisic repored in Table 5 (and also in Tables 6-8) is he one-sided one given by (9), so ha significanly posiive values of his, see (), are consisen wih () whereas significanly negaive ones, see (4), are consisen wih (3). A noable feaure observed across he ables is ha ŝ is in all he cases monoonically decreasing wih respec o d o. This is somehing ha we should expec of any reasonable saisic, given correc specificaion and adequae sample size, because, for example, we would wish ha if d =.0 is reeced agains d >.0, an even more significan resul in his direcion would be obained when d = is esed. Saring wih U in Table 5, he firs hing we observe is ha he nulls d = and d =.0 are boh reeced in all cases in favour of alernaives wih d >.0. Also, d =.40 is always reeced excep for whie noise v and α = β = 0. If we do no include regressors and v follows a Bloomfield (973) model, he values of d where H o (8) is no reeced are.50 and.60 when m = ;.50,.60 and.80 when m =, and all hese values along wih.00 when m = 3. If we include an inercep or a linear ime rend in (7) he resuls seem more conclusive: d =.60 is he only non-reecion case for whie noise v ; d =.80 and are no reeced wih Bloomfield (973) disurbances and m = and ; and d = is he only nonreecion case wih m = 3. These resuls are clearly consisen wih hose given in Table,

14 where he order of inegraion was found o be around.60 for whie noise disurbances and ranging beween.80 and for he exponenial specral model of Bloomfield (973). (Tables 5 and 6 abou here) Table 6 repors he resuls for he log U. The mos sriking poin we observed here is ha he null d = is no reeced in any case when α = β = 0 a priori, however, including regressors, his hypohesis is srongly reeced in favour of alernaives wih d equal o or greaer han.50. This migh relae o he fac ha (-L) d ends o zero for all posiive d smaller han and his is faser as d aproximaes, becoming exacly zero when d =. Thus, i migh be he case ha when esing for a uni roo, he model should include an inercep raher han imposing α = β = 0. We see ha including an inercep, H o (8) is no reeced for d =.60 wih m = 0 and ranges beween.50 and wih he Bloomfield (973) disurbances. (Tables 7 and 8 abou here) Tables 7 and 8 correspond respecively o u and u *. Saring wih u, we again observe ha he values of d flucuae beween.40 and. If v is whie noise, d =.50 is he only case where H o is no reeced, which is compleely in line wih he esimaion carried ou in Table 3. Similarly, if v is weakly auocorrelaed and we include regressors, he nonreecion values are d =.80 when m = ; d = when m = ; and d =.60,.80 and when m = 3, which is once more consisen wih Table 3. Finally, measuring unemploymen in erms of u *, he resuls are very similar o hose given in Table 6 (for log U ). If we do no include regressors, he uni roo null hypohesis is no reeced along wih oher fracionally hypoheses wih d >. However, including an inercep or a linear ime rend, all he nonreecions occur when d.50, clearly showing he nonsaionary characer of he series. We can conclude by saying ha for all hese measures of unemploymen, he orders of inegraion seem higher han one, and his is observed wheher or no we include

15 deerminisic regressors like an inercep and/or a linear ime rend in he model. Thus, hough we do no sress in his aricle any paricular specialized model for any series, hese resuls, as a whole, show ha he sandard pracice of aking firs differences when modelling he U.K. unemploymen may sill lead o series wih a componen of long memory behaviour. CONCLUSIONS A fracionally inegraed model wih he disurbances following a Bloomfield (973) exponenial specral model has been proposed in his aricle for modelling he U.K. unemploymen. This ype of model can be considered as an alernaive o he moscommonly used fracionally ARIMA (ARFIMA) ones, wih he Bloomfield (973) srucure describing he shor-run dynamics wihou need of esimaing so many parameers as in he ARMA case. A mehod based on he While funcion for esimaing by maximum likelihood in he frequency domain, along wih a procedure suggesed by Robinson (994) for esing hese ype of models were performed using four differen measures of unemploymen. These measures were: he number of people claiming unemploymen benefis, (U ); is logarihmic ransformaion, (log U ); he number of people claiming benefis as a percenage of he workforce, (u ); and is logisic ransformaion, (u * ). Using an esimaion procedure based on he frequency domain, he orders of inegraion were found o be around.60 for U and log U, and around.50 for u and u *, if he disurbances were whie noise. Similar resuls were obained when esimaing hrough he ime domain. Allowing weakly auocorrelaed disurbances, eiher hrough auoregressions or hrough he Bloomfield (973) exponenial specral model, he orders of inegraion were found o be higher, ranging in all cases beween.70 and.0. 3

16 Performing he ess of Robinson (994) on hese series, he resuls lead o he same conclusions, wih he orders of inegraion ranging around.50 when modelling he disurbances as whie noise, bu obaining higher values when allowing weak dependence on he disurbances. We also observed ha in many cases more han one non-reecion case appeared. This is however no a all surprising, noing ha when fracional hypoheses are enerained, some evidence supporing hem may appear, because his migh happen even when he uni roo model is highly suiable. On he oher hand, ofen he bulk of hese hypoheses are reeced, suggesing ha he opimal local power properies of he ess, shown by Robinson (994), may be suppored by reasonable performance agains non-local alernaives. The frequency domain approach used in his paper seems o be very unpopular amongs economericians, and hough here exis ime domain versions of he Robinson s (994) ess (cf, Robinson, 99), he preference here for he frequency domain se-up is moivaed by he somewha greaer elegance of formulae i affords when he Bloomfield model is used. We should finally menion ha we have no inended in his paper o invesigae any model specificaion for he U.K. unemploymen, raher o show ha fracionally inegraed models wih Bloomfield (973) exponenial disurbances are feasible alernaives. The resuls indicae ha all hese series are nonsaionary, wih he orders of inegraion much higher han. In fac, he above resuls show ha d is in pracically all cases higher han.50 and hus, he sandard approach of aking firs differences sill produces nonsaionary series, which may hen lead, when esimaing by leas squares, o erroneous conclusions. Several oher lines of research can be developed which should prove relevan o he analysis of hese and oher macroeconomic daa. In paricular, i would be worh o proceed o build up confidence inervals for he fracional differencing parameer, especially in he 4

17 Bloomfield (973) case. Also, he quesion of how bes o exend his model o a mulivariae se-up remains o be invesigaed. 5

18 REFERENCES Beran, J., Fiing long-memory models by generalized linear regression, Biomerika, 80, (993), Bloomfield, P., An exponenial model for he specrum of a scalar ime series, Biomerika, 60, (973), 7-6. Fox, R. and Taqqu, M.S., Large-sample properies of parameer esimaes for srong dependen saionary Gaussian ime series, Annals of Saisics, 4, (986), Gil-Alana, L.A. and Robinson, P.M., Tesing of uni roo and oher nonsaionary hypoheses in macroeconomic ime series, Journal of Economerics, 80, (997), Gil-Alana, L.A., Esimaion of fracional ARIMA models for he U.K. unemploymen, Cenre for Economic Forecasing, London Business School, Discussion Paper 05-99, (999a). Gil-Alana, L.A., Tesing he order of inegraion in he U.K. unemploymen, Cenre for Economic Forecasing, London Business School, Discussion Paper 06-99, (999b). Gil-Alana, L.A., Semi-parameric esimaion of he fracional differencing parameer in he U.K. unemploymen, Preprin. (999c). Granger, C.W.J., Long memory relaionships and aggregaion of dynamic models, Journal of Economerics, 4, (980), Robinson, P.M., Saisical inference for a random coefficien auoregressive model, Scandinavian Journal of Saisics, 5, (978), Robinson, P.M., Tesing for srong serial correlaion and dynamic condiional heeroskedasiciy in muliple regression, Journal of Economerics, 47, (99), Robinson, P.M., Efficien ess of nonsaionary hypoheses, Journal of he American Saisical Associaion, 89, (994),

19 Sowell, F., Maximum likelihood esimaion of saionary univariae fracionally inegraed ime series models, Journal of Economerics, 53, (99), Velasco, C. and Robinson, P.M., While Pseudo-Maximum likelihood esimaion of nonsaionary ime series, Preprin. (999). 7

20 TABLE * Esimaion of he fracionally differencing parameer d for U Fracionally inegraed wih AR u Fracionally inegraed wih Bloomfield u AR(p) Value of d Bloomfield (m) Value of d p = 0.66 m = 0.65 p =.83 m =.9 p =.9 m =.09 p = 3.05 m = 3.0 *: U corresponds o he number of people claiming unemploymen benefis. TABLE * Esimaion of he fracionally differencing parameer d for log U Fracionally inegraed wih AR u Fracionally inegraed wih Bloomfield u AR(p) Value of d Bloomfield (m) Value of d p = 0.63 m = 0.6 p =.7 m =.73 p =.74 m =.80 p = 3.84 m = 3.0 *: log U is he log ransformaion of he number of people claiming unemploymen benefis. TABLE 3 * Esimaion of he fracionally differencing parameer d for u Fracionally inegraed wih AR u Fracionally inegraed wih Bloomfield u AR(p) Value of d Bloomfield (m) Value of d p = 0.50 m = 0.49 p =.70 m =.87 p =.88 m =.08 p = 3.9 m = 3.06 *: u is he number of people claiming unemploymen benefis as a percenage of he workforce. TABLE 4 * Esimaion of he fracionally differencing parameer d for u * Fracionally inegraed wih AR u Fracionally inegraed wih Bloomfield u AR(p) Value of d Bloomfield (m) Value of d p = 0.49 m = 0.49 p =.64 m =.69 p =.7 m =.83 p = 3.8 m = 3.0 *: u * is he logisic ransformaion of u. 8

21 TABLE 5 * Tesing d = d o for U wih Bloomfield (973) exponenial specral disurbances Values of d o m z No regressors m = 0 An inercep A ime rend No regressors m = An inercep A ime rend No regressors m = An inercep A ime rend No regressors m = 3 An inercep A ime rend *: U is he number of people claiming unemploymen benefis. In bold: The non-reecion values a he 95% significance level. TABLE 6 * Tesing d = d o for log U wih Bloomfield (973) exponenial specral disurbances Values of d o m z No regressors m = 0 An inercep A ime rend No regressors m = An inercep A ime rend No regressors m = An inercep A ime rend No regressors m = 3 An inercep A ime rend *: log U is he log ransformaion of he number of people claiming unemploymen benefis. In bold: The nonreecion values a he 95% significance level. 9

22 TABLE 7 * Tesing d = d o for u wih Bloomfield (973) exponenial specral disurbances Values of d o m z No regressors m = 0 An inercep A ime rend No regressors m = An inercep A ime rend No regressors m = An inercep A ime rend No regressors m = 3 An inercep A ime rend *: u is he number of people claiming unemploymen benefis as a percenage of he workforce. In bold: The non-reecion values a he 95% significance level. TABLE 8 * Tesing d = d o for u * wih Bloomfield (973) exponenial specral disurbances Values of d o m z No regressors m = 0 An inercep A ime rend No regressors m = An inercep A ime rend No regressors m = An inercep A ime rend No regressors m = 3 An inercep A ime rend : u * is he logisic ransformaion of u. In bold: The non-reecion values a he 95% significance level. 0

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