Long memory in US real output per capita. Guglielmo Maria Caporale and Luis A. Gil-Alana. January Department of Economics and Finance

Transcription

1 Deparmen of Economics and Finance Working Paper No Economics and Finance Working Paper Series Guglielmo Maria Caporale and Luis A. Gil-Alana Long memory in US real oupu per capia January 009 hp://

2 LONG MEMORY IN US REAL OUPU PER CAPIA Guglielmo Maria Caporale Brunel Universiy, London and Luis A. Gil-Alana Universiy of Navarra January 009 Absrac his paper analyses he long memory properies of quarerly real oupu per capia in he US (948Q 008Q3) using non-parameric, semi-parameric and parameric echniques. he resuls vary subsanially depending on he mehodology employed. Evidence of mean reversion is obained in a parameric conex if he underlying disurbances are weakly auocorrelaed. We also examine he possibiliy of a srucural break in he daa and he resuls indicae ha here is a sligh reducion in he degree of persisence afer he break ha is found o occur in he second quarer of 978. JEL Classificaion: C, O40 Keywords: Fracional Inegraion, Long Memory, Convergence Corresponding auhor: Professor Guglielmo Maria Caporale, Cenre for Empirical Finance, Brunel Universiy, Wes London, UB8 3PH, UK. el.: +44 (0) Fax: +44 (0) Guglielmo-Maria.Caporale@brunel.ac.uk he second-named auhor graefully acknowledges financial suppor from he Miniserio de Ciencia y ecnologia (ECO ECON Y FINANZAS, Spain)..

3 . Inroducion Following he seminal work of Barro (99) and Barro and Sala-i-Marin (99, 99, 995), in he las couple of decades a vas lieraure on convergence has been produced. his is a key issue o assess he empirical relevance of neoclassical versus endogenous growh models. he sandard Solow (956) model implies ha all counries should converge o a level of income ha is deermined by heir respecive saving raes and populaion growh raes, and ha over ime poor counries or regions end o grow faser han rich ones. herefore, evidence of convergence has radiionally been inerpreed as supporing he neoclassical model. However, Barro and Sala-i-Marin (99, 99) argued ha he esimaed rae of convergence is only consisen wih he neoclassical model if diminishing reurns o capial se in very slowly, and ha endogenous growh models wih consan reurns and gradual diffusion of echnology also fi he daa (see also Mankiw e al., 99, Quah, 993, and Sala-i-Marin, 996). heir approach is based on esing for β-convergence in a regression of he firs difference of per capia oupu agains an exogenous rae of echnical progress, as well as lagged seady-sae and acual per capia oupu, where β is he slope coefficien. heir conclusion is ha he rae of convergence o is seady sae value of per capia oupu is exponenial and approximaely equal o % for mos economies. More recenly, hough, Michelacci and Zaffaroni (000) have poined ou ha his finding canno be reconciled wih he oher sylized facs of uni roos in oupu (see Nelson and Plosser, 98), and a fairly smooh rend of oupu per capia in he OECD economies (see Jones, 995). hey show ha exending he sandard Solow model o allow for cross-secional heerogeneiy in he adusmen speed resuls in oupu exhibiing long memory, and ha per capia oupu is well represened by a meanrevering long memory process wih 0.5 d <, where d is he fracional inegraion

4 parameer in oher words, i is no covariance saionary, bu sill mean-revering, wih he implicaion ha sandard uni roo ess will no reec he null of non-saionariy even when convergence akes place. heir inerpreaion is ha, given he long-memory properies of he oupu series, convergence does ake place, bu a a hyperbolic very slow rae raher han an exponenial one. However, heir analysis has been criicised by Silverberg and Verspagen (000), who have argued ha i canno really shed ligh on he ime series properies of oupu per capia (and herefore ake forward he convergence debae), as i relies on quesionable filering of he daa, and on using he semiparameric Geweke and Porer- Hudak (GPH, 983) mehod as modified by Robinson (995a), which has been shown o be biased in small samples. Insead, Silverberg and Verspagen (000) use Beran s (994) FGN esimaor and Sowell s (99) parameric maximum likelihood esimaor, which are no affeced by small-sample bias, and show ha he evidence of fracional inegraion in he range [0.5, ), which is a key resul in he paper by Michelacci and Zaffaroni (000), disappears when hese more appropriae mehods are used. In anoher recen paper Mayoral (006) examined annual real GNP and GNP per capia in he US for he ime period , using several parameric (Sowell, 99, Mayoral, 004, Velasco and Robinson, 000) and semi-parameric (Geweke and Porer- Hudak, 983 and everovsky and aqqu, 997) long-memory mehods. Her resuls, hough slighly differen depending on he echnique used, provide evidence ha he orders of inegraion lie in he inerval [0.5, ), implying nonsaionariy, high persisence and mean-revering behaviour. he presen paper makes he following conribuions: firs, we invesigae if mean reversion akes place in quarerly per capia real oupu in he US ( ) by using a variey of non-parameric, semi-parameric and fully parameric echniques

5 based on fracional inegraion and long-memory processes. he resuls indicae ha he behaviour of US per capia real oupu is capured well by a linear rend model wih saionary long-memory behaviour, implying ha shocks affecing he series will rever o is rend someime in he fuure. Moreover, he possibiliy of srucural change is also invesigaed, and i is found ha here has been a decrease in he degree of persisence of he series during he las hree decades. he layou of he paper is as follows. Secion discusses he relevance of fracional inegraion o es for convergence. Secion 3 oulines he mehods used here. Secion 4 presens he empirical resuls. Secion 5 focuses on long memory and srucural change. Secion 6 summarises he main findings and offers some concluding remarks.. Fracional inegraion and economic growh Given a covariance saionary process {x, 0, ±, }, wih auocovariance funcion E(x Ex )(x - -Ex ) γ, according o McLeod and Hipel (978), x displays he propery of long memory if lim γ is infinie. An alernaive definiion, based on he frequency domain is as follows. Suppose ha x has an absoluely coninuous specral disribuion, so ha i has a specral densiy funcion, denoed by f(), and defined as f ( ) γ cos, π < π. π hen, x displays long memory if he specral densiy funcion has a pole a some frequency in he inerval [0, π]. Mos of he empirical lieraure has concenraed on 3

6 he case where he singulariy or pole in he specrum occurs a he zero frequency. his is he sandard case of I(d) models of he form: d ( L) x u, 0, ±,..., () x 0, 0, where L is he lag operaor (Lx x - ) and u is I(0) defined as a covariance saionary process wih specral densiy funcion ha is posiive and bounded a any frequency. hus, he process u could iself be a saionary and inverible ARMA sequence, when is auocovariances decay exponenially; however, i could decay a a much slower rae han exponenially. When d 0 in (), x u, and x is said o be weakly auocorrelaed as opposed o he case of srongly auocorrelaed if d > 0. Moreover, if 0 < d < 0.5, x is sill covariance saionary, bu is lag- auocovariance γ decreases very slowly, a he rae of d- as, and so he γ are non-summable. We say hen ha x has long memory given ha f() is unbounded a he origin. Also, as d in () increases beyond 0.5 and hrough (he uni roo case), x can be viewed as becoming more nonsaionary in he sense, for example, ha he variance of he parial sums increases in magniude. Processes of he form given by () wih posiive non-ineger d are called fracionally inegraed, and when u is ARMA(p, q) x has been called a fracionally ARIMA (or ARFIMA) model. his ype of model provides a higher degree of flexibiliy in modelling low frequency dynamics which is no achieved by nonfracional ARIMA models. here are several heoreical argumens ha can be pu forward o usify long memory (and fracional inegraion) in aggregae ime series (Robinson, 978; Granger, 980; aqqu e al., 997; Chambers, 998; Parke, 999; ec.). Robinson (978) and During he 960s, Granger (966) and Adelman (965) poined ou ha mos aggregae economic ime series have a ypical shape wih he specral densiy increasing dramaically as he frequency approaches zero and ha differencing he daa leads o overdifferencing a he zero frequency. 4

7 Granger (980) showed ha if he individual series follow heerogeneous AR() processes of he form: hen he aggregae series xi, α i xi, + ui,, i,,..., N, N i, x x i,,..., can exhibi long memory if, for example, he α i are drawn from a Bea B(p, q) disribuion for cerain values p and q. Wih sligh variaions, his is also he argumen employed in Michelacci and Zaffaroni (000), Silverberg and Verspagen (000), Mayoral (006) and ohers when using long range dependence in aggregae oupu series. A crucial issue here is hen o deermine he appropriae order of he series o disinguish beween permanen and ransiory changes in oupu. hus, for example, if i is I(0) saionary, shocks affecing he series will be ransiory and he degree of decay will depend on he srucure describing he shor-run dynamics, being, for example, exponenial if hey are auoregressive. On he oher hand, if he series is I(), shocks will be permanen and hus persising forever. By allowing for fracional inegraion, we permi a much richer degree of flexibiliy in he dynamic behaviour of he series o analyse he persisence of shocks: if d (0, 0.5), he series is covariance saionary and mean revering, wih he effecs of shocks disappearing in he long-run, hough a a slower (hyperbolic) rae han in he I(0) case; if d [0.5,), he series is no longer covariance saionary bu is sill mean revering, while d implies ha he series is nonsaionary and non-mean-revering. In he case of d in he inerval [0.5, ) some auhors argue ha mean reversion is a misnomer given he nonsaionariy naure of he process (Phillips and Xiao, 999). 5

8 3. Mehodology In his secion we briefly describe he mehods employed for he empirical analysis in Secion 4, which are based on non-parameric, semi-parameric and parameric echniques for modelling long-range dependence. I is well known ha he findings on persisence and long memory can vary subsanially depending on he mehod used, and herefore a robusness check is crucial. 3.. Non-parameric approaches he wo mehods presened here es he null hypohesis of shor memory (i.e. d 0 in ()) agains long memory (d > 0) and/or ani-persisence (d < 0). 3 Firs we describe a procedure developed by Lo (99). he modified R/S saisic (Lo, 99) is: Q ( q) k k max ( ) min ( ), ˆ ( ) k x x k x x σ q where q ˆ σ ( q) ˆ σ x + ω ( q) ˆ γ, and ω ( q), <, q + and x is a saionary series (-0.5 < d < 0.5) of sample size, wih sample mean x, sample variance ˆ σ x, and sample auocovariance a lag given by γ. his saisic was furher normalized as: V Q q q ( ) ( ). () An advanage of he modified R/S saisic is ha i allows us o obain a simple formula for he fracional differencing parameer d, since ˆ d Log ( Q ( q)). Log ( ) 3 A process is said o be ani-persisen if i reverses iself more ofen han a random series would (see Mandelbro, 977). 6

9 he null hypohesis of I(0) includes ARMA models, hough, as poined ou by Haubrich and Lo (00), i does no conain a rend-saionary model. he limi disribuion of V (q) is derived in Lo (99) and he 95% confidence inerval wih equal probabiliies in boh ails is [0.809,.86]. Several Mone Carlo experimens conduced by everovsky e al. (999) and Willinger e al. (999) show ha his mehod is biased in favour of acceping he null of no long memory as he bandwidh parameer q increases. herefore, hese auhors warned abou using Lo s modified mehod in isolaion. 4 Lee and Schmid (996) showed ha he KPSS es proposed by Kwiakowski, Phillips, Schmid and Shin (99) has he same power as Lo s saisic agains long memory alernaives. Giraiis, Kokoszka, Leipus and eyssiere (003) proposed a cenering of he KPSS saisic based on he parial sum of he deviaions from he mean. hey called his mehod he rescaled-variance V/S saisic, which is given by M ( q) * * * Var ( S, S,..., S ), (3) ˆ σ ( q) k * * * * * where S ( x x), and Var ( S, S,..., S ) ( S S k * ) is heir sample variance. According o Giraiis e al. (003) he V/S es is more suiable for series ha exhibi high volailiy, and various Mone Carlo experimens conduced by hese auhors show ha he V/S es is less sensiive o he choice of he bandwidh number q. hese auhors showed ha he asympoic disribuion of M (q) is given by F k (πc / ), where F k is he limi disribuion of he sandard Kolmogorov saisic and c is he criical value a a chosen significance level (e.g. he 5% level). 4 Oher papers dealing wih he small sample disribuion of he R/S saisic are Harrison and reacy (997) and Izzeldin and Murphy (000). 7

10 As for he choice of he opimal bandwidh parameer q (q * ) in he wo mehods described above, q 0 corresponds o he classic Hurs-Mandelbro R/S saisic. Haubrich and Lo (00) suggesed using Andrew s (99) daa-dependen procedure o deermine he opimal bandwidh, which is given by * 3 * 3 a q, (4) ˆ * 4 ρ wih a, where ρˆ is he firs order AR coefficien. ( ˆ ρ ) 3. Semi-parameric approaches he main advanage of he semi-parameric mehods is ha hey only specify he I(d) srucure, wihou modelling he d-differenced process, which is merely described as an I(0) process. In doing so hey avoid he problem of poenial misspecificaion in he shor-run dynamics of he series. here exis several procedures for esimaing he fracional differencing parameer in semiparameric conexs. Of hese, he log-periodogram regression esimae proposed by Geweke and Porer-Hudak (983) has been he mos widely used. his mehod was laer modified by Künsch (986) and Robinson (995a) and has been analysed, among ohers, by Hurvich and Ray (995), Velasco (999a, 000) and Shimosu and Phillips (00). Based on he While funcion, Robinson (995b) proposed anoher esimaor, which is essenially a local While esimaor in he frequency domain, using a band of frequencies ha degeneraes o zero. he esimaor is implicily defined by: ˆ m d arg min log ( ) log d C d d s, (5) m s 8

11 C( d) m m d π s m I( s ) s, s, s 0, where I( s ) is he periodogram of he raw ime series, x, given by: I( ) s π x e i s, and d (-0.5, 0.5). Under finieness of he fourh momen and oher mild condiions, Robinson (995b) proved ha: m ( dˆ do ) d N(0, / 4) as, where d o is he rue value of d. his esimaor is robus o a cerain degree of condiional heeroskedasiciy (Robinson and Henry, 999) and is more efficien han oher semiparameric compeiors. Alhough here exis furher refinemens of his procedure, (Velasco, 999b, Velasco and Robinson, 000; Phillips and Shimosu, 004, 005; ec.), hese mehods require addiional user-chosen parameers, and he esimaes of d may be very sensiive o he choice of hese parameers. In his respec, he mehod of Robinson (995b) seems compuaionally simpler. 3.3 Parameric approaches Esimaing d paramerically along wih he oher model parameers can be done in he frequency domain or in he ime domain. In he ime domain, Sowell (99) analysed he exac maximum likelihood esimaor of he parameers of he ARFIMA model, using a recursive procedure ha allows quick evaluaion of he likelihood funcion, which is given by: (π ) n / Σ / exp X ' n Σ X n, 9

12 where X ( ) n x, x,..., xn and ~ N( 0,Σ) X n. 5 Oher parameric mehods of esimaing d based on he frequency domain were proposed, among ohers, by Fox and aqqu (986) and Dahlhaus (989). Small sample properies of hese and oher esimaors were examined in Smih e al. (997) and Hauser (999). In he firs of hese aricles, several semi-parameric procedures were compared wih Sowell s maximum likelihood esimaion mehod, finding ha Sowell s (99) procedure ouperforms he semiparameric ones in erms of bias and mean square errors. Hauser (999) also compares Sowell s (99) procedure wih ohers based on he exac and he While likelihood funcion in he ime and he frequency domain, and shows ha Sowell s procedure dominaes he ohers in he case of fracionally inegraed models. In his paper we will employ a mehod suggesed by Robinson (994). here are several reasons for using his mehod. Firs, i allows us o include deerminisic erms such as an inercep or a linear ime rend unlike oher mehods such as Lo s (99) non-parameric approach. Anoher advanage of his mehod is ha i is valid for any real value of d, herefore encompassing saionary (d < 0.5) and nonsaionary (d 0.5) hypoheses, unlike he mehods described above ha require firs differencing o render he series saionary prior o he esimaion of d. Moreover, he limi disribuion is sandard normal and is he mos efficien mehod under Gaussianiy of he error erm. We employ here he following model, y β + β + x,,,... (6) 0 d ( L) x u,,,..., (7) where u is assumed o be I(0), and given he parameric naure of his mehod, u has o be specified in a parameric form, ha may be a whie noise process, or more generally, 5 See also Beran (995), anaka (999), Dolado, Gonzalo and Mayoral (003) for oher parameric mehods in he ime domain. 0

13 some ype of weak auocorrelaion (i.e, ARMA) srucure. In his approach we es he null hypohesis: H : d do, o for any real value d o in (6) and (7), and he limi disribuion is N(0, ). he funcional form of he es saisic is presened in Appendix A Empirical resuls he ime series daa analysed in his secion is real per capia US GDP, quarerly, for he ime period 948Q 008Q3. We will presen resuls based on boh he original ime series and is (firs-difference) log-ransformaion. Real GDP daa (GDPC96) were obained from he US Deparmen of Commerce_ Bureau of Economic Analysis (hp:// while hose of populaion were rerieved from he Civilian Noninsiuional Populaion (CNP60V) obained from he Deparmen of Labor, Bureau of Labor Saisics (hp:// [INSER FIGURE ABOU HERE] Figure displays plos of he ime series and heir firs differences. Boh series appear o be nonsaionary in levels, increasing over ime. On he oher hand, heir firs differences have a saionary appearance. Firs, we perform he non-parameric mehods described in Secion 3.. he resuls based on Lo s (99) modified R/S saisic and Giraiis e al. s (003) procedure are displayed in able. 6 Empirical applicaions based on his procedure can be found for example, in Gil-Alana and Robinson, (997), Gil-Alana (000, 005), ec.

14 [INSER ABLE ABOU HERE] Saring wih real per capia GDP (in firs differences), we canno reec he null of I(0) saionariy for any bandwidh parameer when using Lo s (99) modified R/S saisic. Moreover, he esimaed values of d are smaller han 0.0 in all cases. Using he V/S saisic of Giraiis e al. (003) he same evidence agains long memory is obained for all bandwidh parameers. Performing he same ype of analysis on he growh rae series, evidence of shor memory is obained in all cases. hus, using hese non-parameric procedures, here is no evidence of long memory in he firs differences of US real oupu (boh he raw series and he one in logs). [INSER FIGURE ABOU HERE] Nex we compue he esimaes of d based on he semi-parameric mehod of Robinson (995b). he resuls displayed in Figure refer o dˆ in (5) for he whole range of values of he bandwidh number m,,, / (on he horizonal axis). 7 Figure a refers o real per capia GDP while Figure b displays he esimaes for he growh rae, and, in boh cases, we presen he 95% confidence inervals corresponding o he I() and he I(0) hypohesis respecively. he resuls are consisen in boh cases. Evidence of I() behaviour in real oupu (or, alernaively, I(0) in he growh rae) is found if he bandwidh parameer is small. However, if m is higher han /4, his hypohesis is reeced in favour of higher orders of inegraion. If we focus on m () / 6, he I() hypohesis canno be reeced for he original series and he I(0) one for he growh rae. Overall here is srong evidence agains mean reversion (d < ) for real 7 he choice of he bandwidh is crucial in view of he rade-off beween bias and variance: he asympoic variance is decreasing wih m while he bias is growing wih m.

15 per capia GDP, and we found some evidence of long memory (d > 0) in he growh rae series if he bandwidh is large, a resul ha migh be spurious and reflec he negleced ARMA srucure for he d-differenced process. However, a limiaion of he above approaches is ha hey do no allow he inclusion of deerminisic erms such as inerceps and linear rends. hus, in wha follows, we assume ha he series have an inercep and/or a linear ime rend, and ha i is he demeaned (derended) series ha exhibis long memory. For his purpose we employ he parameric approach of Robinson (994) described in Secion 3.3, assuming ha he disurbances are whie noise and also auocorrelaed. In paricular, we consider he se-up in (6) and (7), esing H o (8) for d o -values from o.500 and 0.00 incremens in he real per capia oupu, and from o in he growh rae. In oher words, he esed (null) model is: y d 0 β + β + x, ( L) o x u,,,..., wih I(0) u. Performing he saisic rˆ as given in Appendix A we should expec a monoonic decrease in he value of he es saisic wih respec o he values of d o. Such monooniciy is a consequence of he one-sided alernaives employed in his procedure. hus, for example, we would expec ha if H o (8) is reeced wih d o 0.50 agains he alernaive H a : d > 0.50, an even sronger reecion occurs when esing H o wih d o [INSER FIGURE 3 ABOU HERE] Figure 3 shows he values of he es saisic for he wo cases of an inercep and a linear ime rend under whie noise and AR() disurbances, as well as he confidence bands for he non-reecion cases. I can be seen ha, in he wo cases of 3

16 whie noise disurbances, here is a monoonic decrease, and he values of d o for which H o canno be reeced (displayed in able ) range, for real per capia GDP, beween.6 and.333 wih an inercep and beween.9 and.334 wih a linear rend. For he growh rae series, he non-reecion values are consrained beween 0.5 and wih an inercep, and beween 0.46 and wih a linear rend respecively. hus, alhough here are sligh differences beween he raw and he logged series, he implicaions are he same in he wo cases, wih values above for he wo series in levels. In he case of auocorrelaed errors, we observe a lack of monooniciy in he values of he es saisic wih respec o d: we obain non-reecion values when d is close o 0 and bu reecions for values in beween. his may be explained by he low power of his mehod if he roos of he AR polynomials are close o he uni circle. In fac, his happens wih all parameric procedures due o he compeiion beween he fracional differencing parameer and he AR parameers in describing he ime dependence. When employing higher AR orders we obain essenially he same resuls. [INSER ABLE ABOU HERE] We repor in able he values of d ha produce he lowes saisics in absolue value. hese values should be approximaions o he maximum likelihood esimaes since Robinson s (994) mehod is based on he While funcion, which is an approximaion o he likelihood funcion. 8 hus, we choose he values of d where he es saisic crosses he 0-axis in he plos in Figure 3. For real per capia GDP he mos ineresing case in view of he LR ess is he one wih a linear ime rend and AR() u, 8 We also compued Sowell s (99) and Beran s (994) saisics, and he resuls were raher similar o hose repored here. 4

17 wih an esimaed value of d of 0.3. In case of he growh rae series we observe ha he esimaed value of d is also very sensiive o he choice of he error erm. We obain values around 0.6 for he whie noise case (and hus wih values significanly above for he undifferenced log-series), and ani-persisence (d < 0) in case of auocorrelaed disurbances wih values of -0.6 (wih an inercep) and (wih a linear rend). hus, according o hese wo specificaions, he log-series are mean-revering wih values of d equal o (wih an inercep) and (wih a linear ime rend). Based on he -ess on he deerminisic erms and LR ess on he specificaion of he auocorrelaed srucure, we choose as poenial models, for he real per capia GDP, he following specificaion: y x, ( L) x u, u 0.93u (4.86) (89.85) + ε (-values in parenhesis), and for he growh series, 0.6 ( L)log y x, ( L) x u, u 0.887u (90.66) + ε, implying he laer equaion an order of inegraion for he log-real per capia oupu of abou hus, evidence of mean reversion is obained in he wo cases, he unlogged and he logged versions of real per capia US oupu. 9 Overall, our findings are parially consisen wih hose of Mayoral (006), who using a long span of US daa on boh GNP and GNP per capia covering 33 years reaches he conclusion ha hese series are highly persisen bu mean-revering series ha can be modelled as fracionally inegraed processes, a resul which is found o be robus even when allowing for breaks in he deerminisic componen of he model. As 9 We also consider AR(k) processes wih k, 3 and 4 for he I(0) error erm u, and he resuls were in all cases similar o he AR() case. Moreover, he orders of inegraion were in all cases in he inerval (0, 0.5) for he wo series in levels. We repor in he paper he values for he AR() case since i produced he lowes saisics. 5

18 Mayoral (006) poins ou, his evidence is inconsisen wih endogenous growh models, in he conex of which permanen policy changes should have permanen effecs on economic growh. here is an imporan difference beween our findings and hose of Mayoral (006), namely our esimaed orders of inegraion are all in he range (0, 0.5) while in Mayoral (006) hey are in he inerval [0.5, ). An explanaion for his may be he differen daa frequency employed. Since Mayoral s daa are annual hey are characerised by a higher degree of aggregaion which may induce a higher degree of persisence in he daa. 5. Long memory and srucural change In his secion we ake ino accoun he possibiliy of a srucural break in he daa. his is a relevan issue in he conex of fracional inegraion since i has been argued by many auhors ha fracional inegraion migh be an arificial arefac generaed by he exisence of breaks in he daa (see, e.g., Cheung, 993; Diebold and Inoue, 00; Giraiis e al., 00; Mikosch and Sarica, 004; Granger and Hyung, 004; ec.). [INSER ABLE 3 ABOU HERE] able 3 displays for he wo series he esimaes of he break daes and he deerminisic erms along wih he fracional differencing parameers using he procedure developed by Gil-Alana (008). 0 We employ models wih inercep and inerceps wih linear rends combined wih whie noise and AR() disurbances. Employing higher AR orders leads essenially o he same resuls for boh break daes and fracional differencing parameers. Saring wih he real per capia GDP series (in 0 his mehod is briefly described in Appendix B. 6

19 able 3a) we observe ha he break dae is found o occur a 978Q in he four cases examined, and he same break dae is found for he growh rae series (see able 3b) if he disurbances are auocorrelaed. We repor in he able in bold he esimaes corresponding o he seleced model for each series. hus, for real per capia GDP he seleced model is * y x, ( L) x u, u 0.774u ε,,..., (9.437) (0.553) and 0.34 * y x, ( L) x u, u 0.875u ε, +,..., (0.749) (6.546) and for he growh rae series, * y x, ( L) x u, u 0.74u (9.68) ε, *,..., and * y x, ( L) x u, u 0.876u ε, *,..., (6.793) wih * y ( - L)log(y ), and * 978Q for he wo series. hus, we observe a reducion in he degree of persisence (as measured by he fracional differencing parameers) in he wo cases, from o 0.34 in case of he raw series, and from o in he log of GDP. [INSER FIGURE 4 ABOU HERE] As in he previous secion he models were seleced according o he -values for he deerminisic erms along wih LR ess. 7

20 Figure 4 displays he impulse response funcions and heir corresponding 95% confidence bands for he wo series in he wo subsamples. he resuls are similar for he wo series, a rapid decrease being observed in he pos-978 period. [INSER FIGURE 5 ABOU HERE] Figure 5 displays he esimaed ime rends for he wo subsamples in he original daa and he wo esimaed inerceps in case of he growh rae series. We observe ha in he original series he esimaed rends fi he daa relaively well, and here is some persisence in he deviaions of he raw daa from he rends ha is modelled hrough a saionary long-memory process. 6. Conclusions his paper analyses he long memory properies of quarerly per capia real oupu in he US using non-parameric, semi-parameric and parameric echniques. he resuls vary subsanially depending on he mehodology employed. Evidence of fracional inegraion wih mean-revering behaviour is obained in a parameric conex if he underlying disurbances are weakly auocorrelaed: shocks affecing he series rever o he original rends hough a a very slow hyperbolic rae. We also examine he possibiliy of a srucural break in he daa and he resuls indicae ha here is a sligh reducion in he degree of persisence afer he break ha is found o occur in he second quarer of 978. Noe ha he approach employed in his aricle does no direcly es he hypohesis of fracional inegraion versus srucural breaks as insead in Mayoral (006), who allows for breaks in he deerminisic erms imposing he same degree of 8

21 inegraion before and afer he break(s). he same happens wih oher mehods such as hose used by Robinson (994), Hidalgo and Robinson (996) and Lazarova (005), which allow breaks in he deerminisic componens wih long-memory innovaions. Unlike hese approaches, we allow differen orders of inegraion before and afer he break dae. Oher mehods such as Ohanissian e al. (008) can also be employed, hough his mehod explois he invariance propery of he long-memory parameer for emporal aggregaion. However, he presen paper does no deal wih emporal aggregaion since i focuses exclusively on quarerly daa. Moreover, Ohanissian e al. s (008) mehod is semi-parameric (and based on Geweke and Porer-Hudak s 983 approach) while we employ purely parameric specificaions in he presence of a break. An ineresing exension of our analysis would be o esimae a more flexible model for he break, for insance incorporaing Markov swiching along wih fracional inegraion. However, a presen he necessary heory is sill missing and no procedure is available o oinly esimae he parameers in such a model. Work in his direcion is now in progress. 9

22 0 Appendix A he LM es of Robinson (994) for esing H o : d d o, in (6) and (7) is a A r ˆ ˆ ˆ ˆ / / σ, where is he sample size and ); ( ˆ) ; ( ( ˆ) ˆ ); ( ˆ) ; ( ) ( ˆ I g I g a τ π τ σ σ τ ψ π ) ( ) ˆ( )' ˆ( ) ˆ( )' ˆ( ) ( ) ( ˆ A ψ ε ε ε ε ψ ψ ). ( arg min ˆ ; ˆ); ; ( log ) ˆ( ; sin log ) ( σ τ τ π τ τ ε ψ g â and  in he above expressions are obained hrough he firs and second derivaives of he log-likelihood funcion wih respec o d (see Robinson, 994, page 4, for furher deails). I( ) is he periodogram of u evaluaed under he null, i.e.: ; ' ˆ ) ( ˆ do w y L u β, ) ( ; ) ( ' ˆ do do z L w y L w w w β z (, ), and g is a known funcion relaed o he specral densiy funcion of u :. ), ; ( ) ; ; ( π π τ π σ τ σ < g f Appendix B We examine a model of he form: b d u x L x y,,..., ) ( ; + + β α,,,...,, ) ( ; u x L x y b d β α where he α's and he β's are he coefficiens corresponding respecively o he inerceps and he linear rends; d and d may be real values, u is I(0), and b is he ime of a break ha is supposed o be unknown. his model can also be wrien as:

23 ~ ~ ( L) d y α (d) + β (d) + u,,..., b, ~ ~ ( L) y α (d ) + β (d ) + u, b d + ~ i i ~ i d d where (d ) ( L), and (d ) ( L), i,. i,...,, he procedure is based on he leas square principle. Firs we choose a grid for he values of he fracionally differencing parameers d and d, for example, d io 0, 0.0, 0.0,,, i,. hen, for a given pariion { b } and given iniial d, d -values, () () (d o, d o ), we esimae he α's and he β's by minimising he sum of squared residuals, min b () d o ~ () ~ () ( L) y α (do ) β (do ) w.r..{ α, α, β, β} + () d o ~ () ~ () ( L) y α (do) β (do) b +. ) ˆ b o o ˆ ) b o o () ( () ( Le α ( ; d, d ) and β ( ; d,d ) denoe he resuling esimaes for pariion { b } and iniial values () d o and d () o. Subsiuing hese esimaed values ino he obecive funcion, we have RSS( b ; () d o, d () o ), and minimising his expression for all (i) b; o values of d o and d o in he grid we obain RSS( b ) arg min RSS( d, { i, } d ( ) o ). hen, he esimaed break dae, ˆ k, is such ha ˆ k arg min i,...,m RSS(i ), where he minimisaion is done over all pariions,,, m, such ha i - i- ε. hen, he regression parameer esimaes are he associaed leas-squares esimaes of he esimaed k-pariion, i.e., α ˆ ˆ ({ˆ }), βˆ ˆ ({ˆ }), and heir corresponding i α i k i β i differencing parameers, dˆ i dˆ i ({ˆ k}), for i and. k

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31 Figures and ables Figure : ime series plos Real GDP per capia Log of real GDP per capia 0,06-0,05 -,5 0,04-3 0,03 0,0-3,5 0, Q 008Q3-4,5 948Q 008Q3 Firs differences of real GDP per capia Real GDP per capia growh 0,00 0,05 0,0008 0,03 0,0004 0,0 0-0,0004-0,0-0,0008-0,03-0,00 948Q 008Q3-0,05 948Q 008Q3 9

32 able : Non-parameric saisics for long memory a) Real per capia GDP (in firs differences) Lo s mod. (99) Giraiis e al. (003) Lo s mod. (99) Giraiis e al. (003) q *.64 (0.0870).436 (0.0659).333 (0.050).63 (0.045).300 (0.0377).8 (0.0870).73 (0.0373).5494 (0.0797).7049 (0.097).8 (0.0870) a) Real per capia GDP growh q *.435 (0.065).457 (0.0400).4 (0.04).0933 (0.06).0764 (0.034).0779 (0.036).54 (0.099).440 (0.0669).530 (0.0754).085 (0.048) In Lo s (99) modified R/S saisic, he 95% confidence inerval wih equal probabiliies in boh ails is [0.809,.86]. he values in parenheses refer o he esimaes of he d s. In Giraiis e al. (003), he criical value a he 5% level is

33 Figure : While esimaes of d based on a semiparameric mehod (Robinson, 995) a) Real per capia GDP,6, 0,8 0, b) Real per capia GDP growh 0,8 0,4 0-0,4-0, he horizonal axis refers o he bandwidh parameer while he verical one displays he esimaes of d. 3

34 Figure 3: Esimaes of d wih Robinson (994) for a range of values of d a) Real per capia GDP Whie noise wih an inercep Whie noise wih a linear ime rend AR() wih an inercep AR() wih a linear ime rend b) Real per capia GDP growh Whie noise wih an inercep Whie noise wih a linear ime rend AR() wih an inercep AR() wih a linear ime rend he horizonal axis refers o he range of values of d under H o. he verical one displays he values of he es saisic, and he bold lines refer o he 95% non-reecion bands. 3

35 able : Esimaes of d based on a parameric approach (Robinson, 994) a) Real per capia GDP Whie noise AR () Inercep. (.6,.333) (-0.356, -0.56) (0.757,.43) (0.956,.43) b) Real per capia GDP growh Linear ime rend.5 (.9,.334) 0.3 (0.96, 0.489) Whie noise AR () Inercep 0.60 (0.5, 0.394) -0.6 (-0.8, ) Linear ime rend 0.58 (0.46, 0.396) (-0.696, ) In parenhesis he 95% confidence band for he values of d. In bold he bes model specificaion for each series. 33

36 able 3: Esimaes of he coefficiens in a model wih a single break (Gil-Alana, 008) a) Real per capia GDP Break dae In + WN 978Q.38 In + AR 978Q.36 + WN 978Q.69 + AR 978Q Break dae In + WN 965Q In + AR 978Q WN 98Q AR 978Q Firs Subsample Second subsample d α β ρ d α β ρ (66.689) (.787) (63.348) (.668) (9.437) (0.553) b) Real per capia GDP growh Firs Subsample (7.09) (36.44) (.003) (0.749) (.537) (6.546) Second subsample d α β ρ d α β ρ (.67) (9.68) (.09) (6.078) (-.9) (-.063) (.355) (6.793) (3.90) (.75) (-.343) (-0.057) In inercep; ime rend; WN Whie noise and AR Auoregression. In bold he seleced model for each series. -values in parenheses. 34

37 Figure 4: Impulse response funcions and 95% confidence inervals Real GDP per capia ( s subsample) Real GDP per capia ( nd subsample),6,6,, 0,8 0,8 0,4 0, Growh rae series ( s subsample) Growh rae series ( nd subsample),6,6, 0,8 0, , 0,8 0, he hin lines refer o he 95% confidence bands for he impulse responses. 35

38 Figure 5: ime series plos and heir esimaed deerminisic erms Real GDP per capia 0,06 0,05 0,04 0,03 0,0 0, Q 978Q 008Q3 Real GDP per capia growh 0,035 0,05 0,05 0,005-0,005-0,05-0,05-0, Q 978Q 008Q3 36