TESTING OF SEASONAL FRACTIONAL INTEGRATION IN UK AND JAPANESE CONSUMPTION AND INCOME *

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1 TESTING OF SEASONAL FRACTIONAL INTEGRATION IN UK AND JAPANESE CONSUMPTION AND INCOME * by L A Gil-Alaña Humbold Universiy, Berlin, and Universiy of Navarre, Spain and P M Robinson London School of Economics and Poliical Science Conens: Absrac 1. Inroducion and summary. Tess for seasonal inegraion 3. Empirical applicaions 4. The UK case 5. The Japanese case 6. Concluding remarks Appendix References Tables 1 10 The Sunory Cenre Sunory and Toyoa Inernaional Cenres for Economics and Relaed Disciplines London School of Economics and Poliical Science Discussion paper Houghon Sree No. EM/00/40 London WCA AE November 000 Tel.: * Research suppored by ESRC Gran R The second auhor s research was also suppored by a Leverhulme Trus Personal Research Professorship.

2 Absrac The seasonal srucure of quarerly UK and Japanese consumpion and income is examined by means of fracionally-based ess proposed by Robinson (1994). These series were analysed from an auoregressive uni roo viewpoin by Hylleberg, Engle, Granger and Yoo (HEGY, 1990) and Hylleberg, Engle, Granger and Lee (HEGL, 1993). We find ha seasonal fracional inegraion, wih ampliudes possibly varying across frequencies, is an alernaive plausible way of modelling hese series. Keywords: Fracional inegraion; nonsaionariy; seasonaliy. JEL No.: C by he auhors. All righs reserved. Shor secions of ex, no o exceed wo paragraphs, may be quoed wihou explici permission, provided ha full credi, including noice, is given o he source.

3 1. Inroducion and summary Many macroeconomic ime series conain imporan seasonal componens. A simple model for a ime series y is a regression on dummy variables S i, y s 1 = m + m S + ε, ε ~ i i, (1) 0 i i d i = 1 where s is he number of ime periods in a year and he m i are unknown coefficiens. Sochasic processes have also been widely used in modelling seasonaliy, for example, he saionary seasonal ARMA s s Φ ( L ) y = Θ ( L ) ε, ε ~ i i d, () p q where Φ p (L s ) and Θ q (L s ) are polynomials in L s (he seasonal lag operaor) of orders p and q respecively, wih he zeros of Φ p (L s ) ouside he uni circle and he zeros of Θ q (L s ) ouside or on he uni circle. As an alernaive o (1) and (), i may be appropriae o allow for sochasic seasonal saionariy, as is implici in he pracice of seasonal differencing (see eg. Box and Jenkins, 1970) whereby he operaor (1 L s ) produces a saionary weakly dependen sequence. For example, for quarerly daa, ρ(l s ) = (1 L 4 ) can be facored as (1 L)(1 + L)(1 + L ), conaining four zeros of modulus uniy; one a zero frequency; one a wo cycles per year, corresponding o frequency π; and wo complex pairs a one cycle per year, corresponding o frequencies π/ and 3π/ (of a cycle π). A good deal of empirical work has followed his approach: Hylleberg, Engle, Granger and Yoo (1990) (henceforh HEGY) found evidence for seasonal uni roos in quarerly U.K. nondurable consumpion and disposable income, using a procedure ha allows ess for uni roos a some seasonal frequencies wihou mainaining heir presence a all such frequencies. Beaulieu and Miron (1993) exended he HEGY procedure o monhly daa and examined welve macroeconomic series in monhly and quarerly daa. By conras wih previous sudies, hey concluded ha evidence in favour of a seasonal uni roo was weak. These findings have been seriously quesioned by Hylleberg, Jorgesen and Sorensen (1993), who concluded ha seasonaliy is in many cases variable, no fixed. Hylleberg, Engle, Granger and Lee (1993) (henceforh 1

4 HEGL) performed he HEGY es on quarerly series of Japanese real consumpion and real disposable income, suggesing ha income is inegraed of order 1 (I(1)) a 0 and a all seasonal frequencies, π/, π and 3π/, and consumpion is I(1) a frequencies 0 and π, while some difficuly was found in separaing uni roos a frequency π/ (and 3π/) from a deerminisic seasonal paern. Osborn (1993) suggesed ha a nonsaionary periodic AR(1) or a periodically inegraed I(1) processes could be more useful. Seasonal uni roos can be viewed no only in an auoregressive framework bu also as a paricular case of seasonal fracionally inegraed processes. Consider he process s d ( 1 L ) y = u, (3) where d > 0 and u is an I(0) series, which is defined here as a covariance saionary process wih specral densiy bounded and bounded away from zero a all frequencies. Clearly, y has s roos of modulus uniy, all wih he same inegraion order d. (3) can be exended o presen differen inegraion orders for each seasonal frequency, whereas y is saionary if all orders are smaller han ½. We say ha y has seasonal long memory a a given frequency if he inegraion order a ha frequency is greaer han zero. Few empirical sudies have been carried ou in relaion o seasonal fracional models. The noion of fracional Gaussian noise wih seasonaliy was suggesed by Jonas (1981) and exended in a Bayesian framework by Carlin, Dempser and Jonas (1985) and Carlin and Dempser (1989). Porer-Hudak (1990) applied a seasonal fracionally inegraed model o quarerly U.S. moneary aggregae wih he conclusion ha a fracional ARMA model could be more appropriae han sandard ARIMAs. Advanages of seasonal fracionally differencing models for forecasing monhly daa are illusraed in Sucliffe (1994), and anoher empirical applicaion is found in Ray (1993). In he following secion we briefly describe some common ess for seasonal inegraion, and compare hem wih Robinson s (1994) ess for nonsaionary hypoheses which permi esing of seasonal fracional inegraion of any saionary or nonsaionary degree. Secion 3 describes models o be esed, using Robinson s (1994) approach, o macroeconomic daa

5 of Unied Kingdom (Secion 4) and Japan (Secion 5) analyzed in HEGY (1990) and HEGL (1993) respecively. Secion 6 conains some concluding remarks.. Tess for seasonal inegraion 1 in We firs consider he Dickey, Hasza and Fuller (DHF) (1984) es of ρ s = s ( 1 ρ L ) y ε, ε ~ i i d (0, σ ). s = The es is based on he auxiliary regression s ( 1 L ) y = π y s + ε, (4) he es saisic being he -raio corresponding o π in (4). Due o he nonsandard asympoic disribuional properies of he -raios under he null hypohesis, DHF (1984) provide simulaed criical values for esing agains he alernaive π < 0. In order o whien he errors in (4), he auxiliary regression may be augmened by lagged (1 L s )y, and wih deerminisic componens, bu unforunaely his changes he disribuion of he es saisic. A limiaion in DHF (1984) is ha i oinly ess for roos a zero and seasonal frequencies, and herefore does no allow for uni roos a some bu no all seasonal frequencies. This defec is overcome by HEGY (1990) for he quarerly case. Their es is based on he auxiliary regression 4 ( 1 L ) y = π 1 y1 1 + π y 1 + π 3 y3 + π 4 y3 1 + ε, (5) where y 1 = (1+L+ L +L 3 )y removes he seasonal uni roos bu leaves in he zero frequency uni roo, y = -(1-L+ L -L 3 )y leaves he roo a π and y 3 = -(1- L )y leaves he roos a π/ and 3π/. The exisence of uni roos a 0, π, π/ (and 3π/) implies ha π 1 = 0, π = 0, and π 3 = π 4 = 0 respecively. The -raio for π 1 and π is shown by HEGY o have he familiar Dickey-Fuller disribuion (see Fuller, 1976) under he null of π 1 = 0 and π = 0 respecively, while he - raio for π 3, condiional on π 4 = 0 has he disribuion described by DHF (1984) for s =. Also a oin es of π 3 = π 4 = 0 is proposed based on he F-raio, and he criical values of he disribuion abulaed. A crucial fac in hese ess is ha 3

6 he same limi disribuions are obained when i is no known a priori ha some of he π s are zero: if he π s oher han he one o be esed are ruly nonzero, hen he process does no have uni roos a hese frequencies and he corresponding y s are saionary. If however some of he oher π s are zero, here are oher uni roos in he regression, bu he corresponding y s are now asympoically uncorrelaed and he null disribuion of he es saisic will no be affeced by he inclusion of a variable wih a zero coefficien which is orhogonal o he included variables. An exension of his procedure o allow oin HEGY-ype ess for he presence of uni roos a zero and all seasonal frequencies, and only for he seasonal frequencies, is given in Ghysels e al. (1994). I is shown ha he es saisics will have he same limiing disribuion as he sum of he corresponding squared -raios for π i (i = 1,,3,4) in he former, and π i (i =,3,4) in he laer es. All hese procedures es for a uni roo in he seasonal AR operaor and have sochasic nonsaionariy as he null hypohesis. Canova and Hansen (1995) seasonally exend he es of Kwiakowski e al. (199), and propose a Lagrange muliplier es (he CH es) based on he residuals from a regression exracing he seasonal and oher deerminisic componens, for esing he null of saionariy abou a deerminisic seasonal paern. Hylleberg (1995) compares small sample properies of HEGY and CH ess for seasonal uni roos in quarerly series, concluding ha boh ess complemen each oher. More recenly, Tam and Reinsel (1997) propose a es for a uni roo in he seasonal MA operaor, esing a deerminisic seasonal null agains a sochasic nonsaionary alernaive. They consider he (inegraed) SMA(1) model, y = µ + u, = 1 s,..., 0, (6) s s ( 1 L ) y = (1 α L ) u, = 1,,..., (7) where µ is a deerminisic seasonal mean, so ha µ - µ -s = 0, and u is iniially, a whie noise process. Thus, a es of α = 1 in (7) can be inerpreed as a es of deerminisic seasonaliy agains he alernaive α < 1 of sochasic inegraed seasonaliy. 4

7 The ess described above consider he possibiliy of only a single form of seasonal sochasic nonsaionary, in paricular, uni roos. We now describe he ess of Robinson (1994), which can es any ineger or fracional roo of any order on he uni circle in he complex plane. We observe {(y, z ), = 1,, n} where y = β ' z + x, = 1,..., (8) ρ ( L; θ ) x = u, = 1,,..., (9) x = 0, 0, (10) where β is a (kx1) vecor of unknown parameers and z is a (kx1) vecor of deerminisic variables ha migh include an inercep, a ime rend and/or seasonal dummies; ρ(l; θ), a prescribed funcion of L and he unknown (px1) parameer vecor θ, will depend on he model esed; u is an I(0) process wih parameric specral densiy σ f ( λ; τ ) = g( λ; τ ), π < λ π, π where he posiive scalar σ and he (qx1) vecor τ are unknown, bu g is of known form. In general we wish o es he null hypohesis H : θ = 0. (11) Under (11), he residuals are uˆ = ρ( L) y ˆ' β w, = 1,,, where o n n ρ ( L) = ρ( L;0); ˆ β = w w ' w ρ( L) y ; w = ρ( L) z. = 1 = 1 Unless g is compleely known funcion (eg. g 1, as when u is whie noise) we have o esimae he nuisance parameer vecor τ, for example by ˆ = arg min σ ( τ ), where Τ is a suiable subse of R q and τ τ Τ 1 σ ( τ ) n 1 n π 1 1 = g( λ ; ) I( ); I( ) uˆ τ λ λ = e n = 1 π n = 1 i λ ; λ = π. n The es saisic, derived from he Lagrange muliplier (LM) principle, is ˆ n ˆ' ˆ R = a A 1 aˆ = 4 σˆ rˆ' rˆ, (1) 5

8 1/ * n 1/ where = ˆ π rˆ A aˆ; σˆ = σ ( ˆ); τ aˆ = ψ ( λ ) g( λ ; ˆ) τ σˆ n 1 I( λ ), Aˆ n n 1 1 n 1 n 1 n 1 ( ) ( )' ( ) ˆ( )' ˆ( ) ˆ( )' ˆ( ) ( )' ; ψ λ ψ λ ψ λ ε λ x ε λ ε λ x ε λ ψ λ 1 = 1 = 1 = 1 = = i λ ψ ( λ ) = Re log ρ( e ;0) ; ˆ( ε λ ) = log g( λ ; ˆ) τ θ τ where he sum on * is over λ such ha -π < λ < π. λ (ρ l - λ 1, ρ l + λ 1 ), l = 1,,, s, such ha ρ l l = 1,,, s < are he disinc poles of ρ(l). Noe ha Rˆ is a funcion of he hypohesized differenced series which has shor memory under (11), and hus we mus specify he frequencies and inegraion orders of any seasonal roos. Robinson (1994) esablished under regulariy condiions ha ˆ R as n, d χ p and also he Piman efficiency propery of LM in sandard problems. If p = 1, an approximae one-sided 100α% level es of (11) agains alernaives H : θ 0. (13) 1 > reecs H o if rˆ > z α, where he probabiliy ha a sandard normal variae exceeds z α is α, and conversely, a es of (11) agains alernaives H : θ 0. (14) 1 < reecs H o if rˆ < -z α. A es agains he wo-sided alernaive θ 0, for any p, reecs if Rˆ exceeds he upper criical value of he χ disribuion. We can compare Robinson s (1994) ess wih hose in HEGY (1990). Exending (5) o allow augmenaions of he dependen variable o render he errors whie noise, and deerminisic pahs, he auxiliary regression in HEGY (1990) is 4 φ L)(1 L ) y = π y + π y + π y + π y + η + ε, (15) ( where φ(l) is a saionary lag polynomial and η is a deerminisic process ha migh include an inercep, a ime rend and/or seasonal dummies. If we p 6

9 canno reec he null hypohesis π 1 = 0 agains he alernaive π 1 < 0 in (15), he process will have a uni roo a zero frequency wheher or no oher (seasonal) roos are presen in he model. In Robinson s (1994) ess, aking (9) wih d + θ ρ L ; θ ) = (1 L) ( (16) wih d = 1, (11) implies a single uni roo a zero frequency. However, we could have insead or alernaively or d + θ ρ L ; θ ) = (1 L ) ( (17) 3 d + θ ρ ( L ; θ ) = (1 L + L L ) 4 + θ ρ ( (18) d L ; θ ) = (1 L ). (19) If again d = 1, under (11), x displays uni roos a frequencies zero and π in (17); zero and wo complex ones corresponding o frequencies π/ and 3π/ in (18), or all of hem in (19). Using HEGY s (1990) ess, he non-reecion of he null π = 0 in (15) will imply a uni roo a frequency π independenly of oher possible roos, and his can be consisen wih (8) (10) oinly wih (17) or (19) among oher possibiliies covered by Robinson s (1994) ess. Furhermore, esing sequenially, (or oinly as in Ghysels e al., 1994), he differen null hypoheses in (15), if we canno reec ha π i = 0 for i = 1,, 3 and 4, he overall null hypohesized model in HEGY (1990) becomes 4 φ ( L)(1 L ) y = η + ε, = 1,,..., (0) and we can compare i wih he se-up in Robinson (1994), using (8) (10) and (19) wih φ(l)u = ε, = 1,,, which, wih d = 1, under he null (11) becomes 4 4 φ ( L)(1 L ) y = φ( L) β '(1 L ) z + ε, = 1,,... (1) Clearly, if we do no include explanaory variables in (8) and (15), (i.e., η = z 0), (1) becomes (0), and including regressors, he difference beween he wo models will be due purely o deerminisic componens. Robinson s (1994) ess also allow esing differen inegraion orders for each of he seasonal frequencies. Thus, insead of (19) we could consider for insance, + θ ρ ( d 1 θ1 θ ) 3 3 ; θ ) (1 ) + d + = (1 + ) (1 + d L L L L () 7

10 and es he null θ = (θ 1, θ, θ 3 ) = 0 for differen values of d 1, d and d 3. This possibiliy is also ruled ou in HEGY (1990) and he oher ess presened above, which us concenrae on he uni roo siuaions. We can also compare he ess of Robinson (1994) wih hose in Tam and Reinsel (1997). They considered (6) and (7), where u is now a saionary and inverible ARMA process and esed H : α = 1 (3) o in (7) agains he alernaive α < 1. The non-reecion of (3) in (6) and (7) would imply ha y follows a deerminisic seasonal paern plus a saionary sochasic process, while is reecion would be evidence of seasonal inegraion. We can ake fracional operaors insead of he AR and MA ones in (7): s d s γ ( 1 L ) y = (1 L ) u, = 1,,..., (4) wih d > 0, and given he common facors appearing in boh sides in (4), calling δ = γ - d, he model can be rewrien as (6) wih s δ ( 1 L ) y = u, = 1,,..., (5) and we can es H o δ = 0 agains he alernaive δ > 0. Thus, (7) and (5) are idenical under he null. The null and alernaive versions of (5) are covered by Robinson s (1994) seing, wih β z in (8) replaced by µ, and s = 4, d = 0 and θ = δ in (19). The null χ limi disribuion of Robinson s (1994) ess is consan across specificaions of ρ(l; θ) and z and hus does no require case by case evaluaion of a nonsandard disribuion, unlike of he oher ess described. Ooms (1997) proposes Wald ess based on Robinson s (1994) model in (8) (10), which have he same limi behaviour as LM ess of Robinson (1994), bu require efficien esimaes of he fracional differencing parameers. He suggess a modified periodogram regression esimaion procedure of Hassler (1994), whose disribuion is evaluaed under simulaion. Also Hosoya (1997) esablishes limi heory for long memory processes wih singulariies no resriced o zero frequency and proposes a se of quasi-log-likelihood raio saisics o be applied o raw ime series. Robinson s (1994) ess were applied 8

11 o non-seasonal daa by Gil-Alaña and Robinson (1997), and given he vas amoun of empirical work based on AR srucures, an empirical sudy of fracional-based ess for seasonal daa seems overdue. 3. Empirical applicaions The relaionship beween consumpion and income is arguably one of he mos imporan in macroeconomics. The mos influenial and perhaps mos widely esed view of his relaionship is he permanen income hypohesis (see Hall, 1989). We concenrae on he univariae reamen of hese wo variables, and apply differen versions of Robinson s (1994) ess o some seasonally unadused, quarerly daa for Unied Kingdom and Japan, using he same daase as in HEGY (1990) and HEGL (1993) respecively. For boh counries we follow he same procedure. We es (11) in a version of (8), y = β + β + β S + β S + β S + x, 1,,... (6) = wih (9) and (10), where S 1, S and S 3 are seasonal dummies. We es in a sequenial way. Since he daa are quarerly, we sar by assuming ha x in (6) has four roos and ake ρ(l; θ) as in (19). Given ha θ is scalar, we es H o (11) agains he one-sided alernaives (13) and (14). In order o allow differen inegraion orders a differen frequencies we also consider ρ + ( d1 θ1 θ ; θ ) (1 ) + (1 + d L = L L ), (7) and more generally, (). Therefore, θ = (θ 1, θ ) under (7) and (θ 1, θ, θ 3 ) under () and we es here (11) agains he wo-sided alernaive θ 0. Clearly, when deparures are acually of he specialized form (19), a es of (11) direced agains (19) will have greaer power han ones direced agains (7) or (), bu he ess have power agains a wider range of alernaives. Following his sequenial way of esing we nex assume x displays only hree roos: wo of hem complex, corresponding o frequencies π/ and 3π/, and one real ha migh be eiher a zero or a frequency π. Thus, we perform he ess in case of (18) and 3 + ρ ( d θ L ; θ ) = (1 + L + L + L ), (8) 9

12 and exending now he ess o allow differen inegraion orders a he complex and a he real roos, we also consider wo-sided ess where and + θ ρ ( d 1 θ1 ) ; θ ) = (1 ) + (1 + d L L L (9) ρ + ( d1 θ1 θ ; θ ) (1 ) + (1 + d L = + L L ). (30) In a furher group of ess, we assume ha he hypohesized model conains only wo roos, one a zero frequency and he oher a π. Again we look firs a one-sided ess agains (17) and hen a wo-sided ess agains ρ + ( d1 θ1 θ ; θ ) (1 ) + (1 + d L = L L). (31) Finally we consider he possibiliy of a single roo (or perhaps wo complex ones), and herefore look a (16) as well as and finally, + θ ρ ( d L ; θ ) = (1 + L), (3) + θ ρ ( d L ; θ ) = (1 + L ). (33) The form of  for hese various choices of ρ is derived in he appendix. I is found ha Â, ineresingly, does no vary wih he null hypohesized inegraion order d or inegraion orders d i, clearly faciliaing he compuaions. In all hese cases he ess will be performed for differen model specificaions in (6). Firs, we assume ha β i 0 a priori; nex β i = 0, i, (including an inercep); nex β i = 0, i 3, (a ime rend); nex β = 0, (an inercep and dummy variables); finally ha all β i are unknown. In all cases we consider a wide range of null hypohesized d (and d i s when p > 1), from 0.50 hrough.5 wih 0.5 incremens, and whie noise u, hough in some cases we exend o I(0) parameric auocorrelaion in u. Clearly, non-reecions of (11) when d (and he d i s) equal 1 imply uni roos, and non-reecions wih d = 0 will sugges deerminisic models of form advocaed by Tam and Reinsel (1997). 4. The U.K. case We analyze he quarerly Unied Kingdom daase used in HEGY (1990). c is log consumpion expendiure on non-durables and y is log 10

13 personal disposable income, from hrough The conclusions of HEGY (1990) were ha c could be I(1) a each of he frequencies 0, π/ (and 3π/) and π; y may conain only wo roos, a zero and π; c y can have four uni roos if dummies are no inroduced, bu wo uni roos of he same form as in c if hey are. Table 1 repors resuls for he one-sided saisic ˆr, when ρ(l; θ) in (9) is (19). Firs, in Table 1(i), we ake u as a whie noise process, and observe ha for he wo individual series (c and y ), he null is never reeced when d = 0.75 and d = 1. Also, d = 1.5 is no reeced when we include as regresors an inercep and dummies. For he differenced series (c y ), he values of d where H o is no reeced are slighly smaller (d = 0.50 and d = 0.75), and he null hypohesis is clearly reeced in all cases, in favour of less nonsaionary alernaives, suggesing ha if he wo individual series were in fac I(1), a degree of fracional inegraion may exis for a given coinegraing vecor (1, - 1), using a simplisic version of he permanen income hypohesis heory as discussed by Davidson e al. (1978) for example. The fac ha he uni roo null is never reeced for c is consisen wih HEGY (1990), bu his hypohesis is no reeced for y, while HEGY (1990) found evidence of only wo uni roos (a frequencies 0 and π) in his series. Various ess of his hypohesis will be performed laer in a furher group of ess. Also, HEGY (1990) inroduced augmenaions, including lagged values of he series. Thus, we also performed he ess wih AR u. In Tables 1(ii) and (iii) we give resuls for AR(1) and AR() u respecively. Tess allowing higher order AR u were also performed, yielding similar resuls. The non-reecion values are now d = 0.50 and d = 0.75, and in hose cases where he former is reeced, his is always in favour of saionary alernaives. The lower inegraion orders observed in hese wo ables compared wih Table 1(i) can in large par be due o he fac ha he AR esimaes are Yule-Walker ones, enailing roos ha canno exceed one in absolue value bu can be arbirarily close o i, so hey pick up par of he nonsaionary componen. 11

14 Table repors resuls of he wo-sided ess Rˆ in (1) when θ is a (x1) vecor. ρ(l; θ) is now given by (7) and herefore we allow differen inegraion orders for he real and complex roos, leing d 1 and d ake each of he values 0.50 (0.5) We concenrae on he cases of no regressors, of an inercep, and of boh, an inercep and a ime rend, and presen he resuls only for hose cases where we observe a leas one non-reecion value for each (d 1, d ) combinaion across he series. If here are no regressors, H o is reeced in all cases for he individual series, while for c y we observe several nonreecions when d 1 = 0.50, 0.75, 1.00 and 1.5 and d = 0.50 and Including an inercep or a linear ime rend, he resuls are similar in boh cases, wih mos of he non-reecions occurring when d is smaller han d 1, and also observing smaller orders for c y han for he individual series. In Table 3 we exend hese ess o allow differen inegraion orders a zero and π, and hus ρ(l; θ) is in (). Again we only presen he resuls for hose cases where we observe a leas one non-reecion value. The resuls are consisen wih he previous ones: in fac, when here are no regressors, he null is always reeced for c and y, while for c y here are some nonreecions, wih he lowes value achieved a d 1 = 1 and d = d 3 = Including a consan or a ime rend, he resuls seem o emphasize he imporance of he roo a zero frequency over he ohers, given is greaer inegraion order. Following his sequenial way of esing we nex assume x can be modelled wih hree roos, and hus remove from (19) he roo a zero frequency (in which case ρ(l; θ) adops he forms (8) or (30)), or a π (i.e., ρ(l; θ) as in (18) or (9)). Though we do no presen he resuls, hey show ha H o is reeced in all series and across all cases, indicaing he imporance of hese wo roos, as suggesed by HEGY (1990). In he nex group of ables we suppose x has only wo roos, a zero and π. Firs we ake ρ(l; θ) as in (17), so he same inegraion order is assumed a boh frequencies. This way of specifying he model is ineresing in view of he resuls in HEGY (1990), who suggesed ha only wo uni roos a hese frequencies were presen in y, and in some cases for c y. Resuls for whie 1

15 noise u are given in Table 4 and he non-reecion values occur when d = 0.75 and 1 for c and y, and when d = 0.50 for c y, suggesing again he possibiliy of a fracional coinegraing relaionship a hese wo frequencies wih coinegraing vecor (1, -1). The hypohesis of wo uni roos is always reeced for c if we include regressors. These reecions are in line wih HEGY (1990), who indicaed ha complex uni roos should be included. For y we observe ha d = 1 is no reeced in 3 of he 5 possible specificaions in (6), which is also consisen wih HEGY (1990). If we allow inegraion orders o differ beween zero and π frequencies, (i.e., ρ(l; θ) as in (31)), he only non-reecion values occur when d 1 = 0.75 and d = 0.50 for y wih an inercep and wih a linear ime rend. Finally we assume x has only wo complex roos, a π/ and 3π/, or a single one eiher a π or zero. Thus ρ(l; θ) akes he form given in (33), (3) and (16) respecively. As expeced, H o is always reeced in he firs wo cases, indicaing he imporance of he roo a zero frequency o describe rending behaviour. Table 5 gives resuls of rˆ for whie noise u and ρ(l; θ) as in (16), and we observe ha if here are no regressors he I(1) null is no reeced for c and y, bu is srongly reeced for c y. There are few non-reecions in his able and hey correspond o values of d ranging beween 0.50 and 1 for he individual series. For c y, he only wo non-reecion cases occur a d = 0.50 if dummies are included, bu for he remaining specificaions his null is srongly reeced in favour of saionary alernaives. The fac ha he uni roo is reeced in his able for all series when some regressors are included in (6) is consisen wih HEGY (1990), who suggesed he need of a leas one seasonal uni roo. Summarizing now he main resuls obained in he U.K. case, we can say ha if x in (6) is I(d) wih four roos of he same order and u is whie noise, he values of d where he null is no reeced range beween 0.75 and 1 for he individual series and are slighly smaller for he difference c y. If u is AR, d ranges beween 0.50 and 0.75 for he hree series considered. Allowing differen inegraion orders a each frequency, we observe ha he roo a zero 13

16 frequency seems more imporan han he seasonal ones, a he seasonal roo a π appears also more imporan han he wo complex ones a π/ and 3π/. If we ake x as I(d) wih wo real roos, he model seems more appropriae for y han for c or c y, which is in line wih resuls in HEGY (1990). Finally, modelling x as fracionally inegraed wih a single roo a zero frequency, he range of d where H o is no reeced goes from 0.50 o 1 for he individual series bu close o saionariy for c y, bu using a single seasonal roo a frequency π or a pair of complex ones a frequencies π/ and 3π/ seems inappropriae in view of he grea proporion of reecions. 5. The Japanese case We analyze here he log of oal real consumpion (c ), he log of real disposable income (y ), and he difference beween hem (c y ) in Japan from o in 1980 prices. These series have been analyzed in HEGL (1993) o es he presence of seasonal inegraion and coinegraion. In his work (and in an earlier version, HEGL, 1991), hey apply he HEGY (1990) ess o hese daa and heir conclusions can be summarized as follows: for c, a uni roo is observed a all frequencies 0, π/, 3π/ and π if here are no regresssors in he model or if only a ime rend is included; however, if dummies are also included, only wo uni roos are observed, one a zero frequency and one a frequency π. For y, uni roos are no reeced a any frequency when here are no regressors or when a ime rend and/or dummies are inroduced, bu if only an inercep is included he uni roo a zero frequency is reeced. Finally, for c y, uni roo nulls are no reeced a any frequency, independenly of he regressors used. Table 6 is analogous o Table 1, showing he one-sided es saisic rˆ when ρ(l; θ) in (9) akes he form (19). Table 6 (i) repors resuls for whie noise u, and he firs hing ha we observe is ha if β i 0 in (6), we canno reec (11) for d = 0.75 and d = 1 in case of eiher c or y, while for c - y, hese wo cases are also no reeced, along wih d = Including regressors, he uni roo hypohesis is reeced in boh series in favour of more nonsaionary 14

17 alernaives, while he nulls d = 0.75 and d = 1 are never reeced for c - y. Thus, if ρ(l; θ) = 1 L 4 and u is whie noise, he wo individual series are clearly nonsaionary wih d greaer han 1 in mos cases; however, heir difference seems less nonsaionary, suggesing ha some fracional coinegraion could exis beween boh series, wih coinegraing vecor (1, -1). The fac ha d = 1 is no reeced for c and y when here are no regressors, and for c - y independenly of he regressors used in (6), is consisen wih he resuls in HEGL (1993) hough hey allow AR srucure in he differenced series. Therefore in Tables 6 (ii) and (iii) we suppose ha u in (9) is an AR(q) wih q = 1 or. The range of non-reecion values of d goes from 0.50 hrough 1 for c and c - y, and from 0.50 hrough 1.5 for y. As we explained before for he U.K. case, his smaller degree in he inegraion order of he series (compared wih Table 6 (i)), could be in large par due o compeiion beween inegraion order and AR parameers in describing he nonsaionary componen. If we concenrae on he AR(1), we see ha he uni roo is no reeced for y bu is for c when dummy variables are included in he model, again in line wih HEGL (1993). So far we have assumed ha he four roos in x mus have he same inegraion order. In he following ables we allow inegraion orders o differ beween complex roos and real ones. Table 7 corresponds o wo-sided ess when ρ(l; θ) in (9) akes he form given in (7). When here are no regressors, he null is reeced in all cases for boh c and y, while for c - y we observe some non-reecions when d 1 = d = 0.50, 0.75 and 1. These hree possibiliies were no reeced in Table 6 (i) when we employed he one-sided ess. Including an inercep or a ime rend, we observe now some non-reecions for c and y. Saring wih c, H o is no reeced when d 1 = 1.5 or 1.50 and d = 0.50, 0.75 and 1, observing herefore a greaer degree of inegraion a zero and π frequencies han a π/ and 3π/. Similarly, for y, all non-reecions occur when d 1 is slighly greaer han d, and for c - y, he lowes saisics are obained a d 1 = d = The null hypohesis of a uni roo a all frequencies is 15

18 no reeced in his series, which is again consisen wih Table 6 (i) and wih resuls of HEGL (1993). In Table 8 we are slighly more general in he specificaion of ρ(l; θ) in (9), and a differen inegraion order is allowed a each frequency. Therefore, ρ(l; θ) akes he form (). Similarly o Table 7, when here are no regressors he null is always reeced for he individual series, while for c - y, here are non-reecions a some alernaives, wih d 1 greaer han d or d 3. Including an inercep or a ime rend, he resuls emphasize he imporance of he roo a zero frequency over he ohers for he hree series. Performing he ess under he assumpion ha ρ(l; θ) is of forms (18) or (8) - (30), we always reeced. Thus, following his sequenial way of performing he ess, we nex assume ha x has only wo roos, one a zero frequency and he oher a π. Firs we ake ρ(l; θ) as in (17), so θ consiss of a single parameer. Table 9 gives resuls for one-sided ess wih whie noise u. We observe ha he resuls are quie variable across he differen specificaions of (6), and while he orders of inegraion range beween 0.50 and 1.5 for he individual series, for he difference c - y he only non-reecions occur when d = 0.50 wih seasonal dummies. The resuls for he uni roo case are consisen wih hose in HEGL (1993). In fac, he uni roo null is no reeced for c when dummies are included, bu is nearly always reeced for y and c - y, due perhaps o exclusion of uni roos a frequencies π/ and 3π/, as was suggesed by hese auhors. Exending he ess o allow differen inegraion orders a he same wo frequencies, we observed us a single case where he null was no reeced and i corresponded o c wih no regressors and d 1 = 1.5 and d = Finally, we examine he case of x conaining a single roo, and concenrae on he case when his roo is a zero frequency, i.e. (16). Table 10 shows resuls merely for whie noise u, and we observe ha he uni roo null is no reeced for c and y when here are no regressors, bu srongly reeced for c - y. In fac, in he laer series he null is reeced in favour of saionary 16

19 alernaives for he whole variey of specificaions in (6), suggesing ha a his zero frequency, a cerain degree of fracional coinegraion migh also occur, wih reference again o he permanen income hypohesis. Modelling x wih a single roo a frequency π (i.e. (3)) or as an I(d) process wih wo complex roos corresponding o frequencies π/ and 3/ (i.e., (33)), produced reecions for all cases and across all series. In conclusion we can summarize he main resuls obained for he Japanese case by saying ha if x is I(d) wih four seasonal roos of he same order d, and u is whie noise, he values of d where he null is no reeced are a leas one for c and y, and less han or equal o one for c - y. If u is AR, d ranges in mos cases from 0.50 o 1 for he hree series, and, allowing differen inegraion orders for he differen frequencies, he mos noiceable fac is he relaive imporance of he roo a zero frequency over he ohers. Taking x as I(d) wih wo roos, a zero and a frequency π, he null is no reeced for c when d ranges beween 0.75 and 1.5 while for y and c - y, he non-reecion cases correspond o d < 1. Finally, if we assume ha x has a single roo a zero frequency or a frequency π (or wo complex ones corresponding o frequencies π/ and 3π/), he uni roo hypohesis will be reeced in pracically all cases in favour of less nonsaionary alernaives. 6. Concluding remarks Our approach, based on Robinson (1994), has he advanage over sandard auoregressive-based mehods of allowing for fracional componens, differen memory parameers across seasonal frequencies, and of sandard null limi disribuion heory and Piman efficiency agains local alernaives. Our Lagrange muliplier esing avoids esimaion of parameers under he alernaive hypohesis, unlike Wald and likelihood-raio ype ess, while, possessing he same null and local limi behaviour as such ess. In he empirical work, we selec a wide range of null hypoheses, wih respec o memory parameer, insead of esimaing hem, and he resuls may give some impression of he local power performance of he ess. 17

20 We have presened a variey of model specificaions for quarerly consumpion and income daa in Japan and U.K.. Given he variey of possibiliies covered by Robinson s (1994) ess, one canno expec o draw unambiguous conclusions abou he very bes way of modelling hese series. In fac, using hese ess, he null hypohesized model will permi differen deerminisic pahs; differen lagged srucures allowing roos a some or all seasonal frequencies, each of hem wih a possibly differen inegraion order; and differen ways of modelling he I(0) disurbances u. Looking a he resuls presened above as a whole, some common feaures are observed for all series in boh counries, however, and hey can be summarized as follows: Firs, modelling x as a quarerly I(d) process, we observe ha inegraion orders are slighly smaller if u is AR raher han whie noise, due perhaps o he AR componen picking up par of he nonsaionary componen. The resuls emphasize he imporance of real roos over complex ones, given he greaer inegraion order observed for he former, and his is even clearer when we allow differen inegraion orders for each frequency. Excluding one real roo resuls in reecing he null in pracically all siuaions. If ρ(l; θ) is given by (17), we observe several non-reecions, and separaing he roos a zero and a π, he resuls emphasize he imporance of he roo a zero. Modelling he series, however, as a simple I(d) process wih a single roo does no seem appropriae in mos of he cases. Anoher common feaure observed across all he ables is he fac ha inegraion orders for he individual series seem o range beween 0.50 (or 0.75) and 1.5, independenly of he lag funcion ρ(l; θ) used when modelling x in (9) and he inclusion or no of deerminisic pahs in (6), indicaing clearly he nonsaionary naure of hese series. In fac, hough his was no shown in he ables, he null was pracically always reeced when d ranged beween 0 and 0.50, and herefore we found conclusive evidence agains deerminisic paerns of he form proposed by Tam and Reinsel (1997); however, c y seems less inegraed in pracically all cases. Therefore, if we consider ha he series are well modelled by a given funcion ρ(l; θ), a cerain degree of fracional coinegraion would exis beween consumpion and income for a 18

21 given coinegraed vecor (1, -1), using a very simplisic version of he permanen income hypohesis. We can finally compare hese resuls wih hose obained in HEGL (1993) and HEGY (1990) for uni roo siuaions. Resuls in HEGL (1993) for Japanese daa indicaed he presence of uni roos a all frequencies for y and c - y, and he same conclusions hold for c if dummies are excluded, hough only wo real uni roos would be presen if hese dummy variables are included. Looking now a our ables, we observe ha he uni roo null is no reeced for y in any specificaion in (6) when ρ(l; θ) adops he form (19) wih AR u. Similarly, for c - y we canno reec he uni roo null for he same ρ(l; θ) and whie noise u. For c, he null of four uni roos is no reeced when here are no dummies, bu if hese are included non-reecions will occur when ρ(l; θ) akes he form (17). For he U.K. case, resuls in HEGY (1990) suggesed ha four uni roos could be presen for c, and for c - y if dummies are excluded, and wo real uni roos for y and for c - y if hey are included. Our resuls again show a cerain consisency wih heirs, given ha he uni roo null is no reeced for consumpion if ρ(l; θ) is (19) wih whie noise u, and for income his hypohesis is no reeced if ρ(l; θ) akes he form (17). Appendix In his appendix we analyze he marix  in Rˆ in (1) when ρ(l; θ) in (9) adops he form in () and u is whie noise, so ha Aˆ = n * ψ ( λ ) ψ ( λ )', where ψ(λ ) = (ψ 1 (λ ), ψ (λ ), ψ 3 (λ )) for λ π, wih λ cos r λ ( λ ) = Re, 1 r ψ λ [ ] i log (1 e ) = log sin = = [ ] i λ r cos r λ ( λ ) = Re log (1 + e ) = logcos = ( 1), r = 1 r ψ λ r cos r λ 3 ( λ ) = Re ( 1). r ψ λ [ ] i log (1 + e ) = log cos λ = = r r

22 Then  can be approximaed in large samples by π ~ 1 A = ψ ( λ) ψ ( λ)' d λ = π π ~ ( ), A i where ~ A ~ A ~ ~ 11 = A = A33 = r = 1.644, r = 1 6 ~ ~ ~ 13 = A31 = A3 = A3 = ( 1) r r 0.411, r = 1 ~ A ~ 1 = A1 = ( 1) r r 0.8. r = 1  in (1) approximaes n imes he expeced value of he second derivaive marix of he Gaussian log-likelihood wih respec o he (px1) parameer vecor θ. (See, Robinson, 1994, page 1433). Thus, given he non-diagonaliy of Â, we rule ou he possibiliy of esing, as in HEGY (1990), for he presence of roos independenly of he exisence of oher roos a any oher frequencies in he process. 1 For he remaining specificaions of ρ(l; θ), A ~ can be easily obained from he above expressions. Thus, if ρ(l; θ) is given by (19), ψ(λ) = ψ 1 (λ) + ψ (λ) + ψ 3 (λ) and A ~ = 1.64; under (7), ψ(λ) = [ψ 1 (λ) + ψ (λ), ψ 3 (λ)] and he (x) marix A ~ = [(1.64, -0.8) ; (-0.8, 1.64) ]; under (18), ψ(λ) = ψ 1 (λ) + ψ 3 (λ) and A ~ =.46; under (8), ψ(λ) = ψ (λ) + ψ 3 (λ) and A ~ =.46; under (9), ψ(λ) = [ψ 1 (λ), ψ 3 (λ)] and A ~ = [(1.64, -0.41) ; (-0.41, 1.64) ]; under (30), ψ(λ) = [ψ (λ), ψ 3 (λ)] and A ~ = [(1.64, -0.41) ; (-0.41, 1.64) ]; under (17), ψ(λ) = ψ 1 (λ) + ψ (λ) and A ~ = 1.64; under (31), ψ(λ) = [ψ 1 (λ), ψ (λ)] and A ~ = [(1.64, -0.8) ; (-0.8, 1.64) ]; under (16), (3) or (33), ψ(λ) = ψ 1 (λ), ψ (λ) or ψ 3 (λ) respecively, wih A ~ = 1.64 in each case. Allowing AR(q) u, g(λ; τ) below (10) akes he form π 1 q = 1 i e λ τ, and  will be given by he expression below (1), wih he l h ˆ ε ( λ) given by elemen of 0

23 ˆ ε ( λ) l q cosl λ ˆ τ cos( l ) = g( λ; ˆ). τ = 1 A diskee conaining he FORTRAN codes for he ess can be obained from he firs auhor upon reques and i is also available in Gil-Alaña (1997), pages and on he JAE web sie. 1

24 References Bealieu, J.J. and J.A. Miron (1993), Seasonal uni roos in aggregae U.S. daa, Journal of Economerics, 55, Box, G.E.P. and G.M. Jenkins (1970), Time series analysis: Forecasing and Conrol, San Francisco: Holden-Day. Canova, F. and B.E. Hansen (1995), Are seasonal paerns consan over ime? A es for seasonal sabiliy, Journal of Business and Economic Saisics, 13, Carlin, J.B. and A.P. Dempser (1989), Sensiiviy analysis of seasonal adusmens: empirical case sudies, Journal of he American Saisical Associaion, 84, 6-0. Carlin, J.B., A.P. Dempser and A.B. Jonas (1985), On mehods and models for Bayesian ime series analysis, Journal of Economerics, 30, Davidson, J.E., D.F. Hendry, F. Srba and S. Yeo (1978), Economeric modelling of aggregae ime series relaionships beween consumer s expendiure and income in he U.K., Economic Journal, 91, Dickey, D.A., D.P. Hasza and W.A. Fuller (1984), Tesing for uni roos in seasonal ime series, Journal of he American Saisical Associaion, 79, Fuller, W.A. (1976), Inroducion o saisical ime series, Willey Series in Probabiliy and Mahemaical Saisics, Willey, New York, NY. Ghysels, E., H.S. Lee and J. Noh (1994), Tesing for uni roos in seasonal ime series: some heoreical exensions and a Mone Carlo invesigaion, Journal of Economerics, 6, Gil-Alaña, L.A. (1997), Tesing fracional inegraion in macroeconomic ime series, Ph. D. hesis, London School of Economics, Universiy of London, London. Gil-Alaña, L.A. and P.M. Robinson (1997), Tesing of uni roos and oher nonsaionary hypoheses in macroeconomic ime series, Journal of Economerics, 80, Hall, R.E. (1989), Consumpion, modern business cycle heory, ed. R.J. Barro, Cambridge, Harvard Universiy Press. Hassler, U. (1994), Misspecificaion of long memory seasonal ime series, Journal of Time Series Analysis, 15,

25 Hosoya, Y. (1997), A limi heory for long-range dependence and saisical inference on relaed models, Annals of Saisics, 5, Hylleberg, S. (1995), Tess for seasonal uni roos. General o specific or specific o general, Journal of Economerics, 69, 5-5. Hylleberg, S., R.F. Engle, C.W.J. Granger and H.S. Lee (1991), Seasonal coinegraion. The Japanese consumpion funcion, , Discussion Paper, Universiy of California, San Diego, C.A. Hylleberg, S., R.F. Engle, C.W.J. Granger and H.S. Lee (1993), Seasonal coinegraion. The Japanese consumpion funcion, Journal of Economerics, 55, Hylleberg, S., R.F. Engle, C.W.J. Granger and B.S. Yoo (1990), Seasonal inegraion and coinegraion, Journal of Economerics, 44, Hylleberg, S., C. Jorgensen and N.K. Sorensen (1993), Seasonaliy in macroeconomic ime series, Empirical Economics, 18, Jonas, A.B. (1981), Long memory self similar ime series models, unpublished manuscrip, Harvard Universiy, Dep. of Saisics. Kwiakowski, D., P.C.B. Phillips, P. Schmid and Y. Shin (199), Tesing he null hypohesis of saionary agains he alernaive of a uni roo, Journal of Economerics, 54, Ooms, M. (1997), Flexible seasonal long memory and economic ime series, Preprin. Osborn, D.R. (1993), Discussion of Engle e al., 1993, Journal of Economerics, 55, Porer-Hudak, S. (1990), An applicaion of he seasonal fracionally differenced model o he moneary aggregae, Journal of he American Saisical Associaion, 85, Ray, B.K. (1993), Long range forecasing of IBM produc revenues using a seasonal fracionally differencing ARMA model, Inernaional Journal of Forecasing, 9, Robinson, P.M. (1994), Efficien ess of nonsaionary hypoheses Journal of he American Saisical Associaion, 89, Sucliffe, A. (1994), Time series forecasing using fracional differencing, Journal of Forecasing, 13, Tam, W. and G.C. Reinsel (1997), Tess for seasonal moving average uni roo in ARIMA models, Journal of he American Saisical Associaion, 9,

26 4

27 rˆ in (1) wih TABLE 1 d + θ ρ L ; θ ) = (1 L ) ( 4 for he U.K. daa i): Wih whie noise u Series z / d c Y c - y I I,T I,S I,T,S, I I,T I,S I,T,S, I I,T I,S I,T,S, ii): Wih AR(1) u Series z / d C Y c - y I I,T I,S I,T,S, I I,T I,S I,T,S, I I,T I,S I,T,S, iii): Wih AR() u Series z / d c y c y I I,T I,S I,T,S, I I,T I,S I,T,S, I I,T I,S I,T,S, : Non-reecion values for he null hypohesis (11) a 95% significance level; --: No inercep, no ime rend and no seasonal dummies; I: Inercep; I,T: Inercep and ime rend; I,S: Inercep and seasonal dummies; I,T,S: Inercep, ime rend and seasonal dummies. 5

28 TABLE d1 θ1 + θ Rˆ in (1) wih ρ ( ; θ ) = (1 ) + (1 + d L L L ) and whie noise u for he U.K. daa No inercep and no rend Inercep Inercep and a ime rend d 1 d c y c - y c y c - y c y c - y : Non-reecion values for he null hypohesis (11) a 95% significance level. TABLE 3 d Rˆ in (1) wih 1 θ + θ ( ; ) (1 ) (1 ) (1 ) + θ ρ θ + d = + + d L L L L and whie noise u for he U.K. daa No inercep and no rend Inercep Inercep and a ime rend d 1 d d 3 c y c - y c y c - y c y c - y : Non-reecion values for he null hypohesis (11) a 95% significance level. 6

29 rˆ in (1) wih TABLE 4 d + θ ρ L ; θ ) = (1 L ) ( and whie noise u for he U.K. daa Series z / d I c I,T I,S I,T,S I y I,T I,S I,T,S I c - y I,T I,S I,T,S : Non-reecion values for he null hypohesis (11) a 95% significance level. --: No inercep, no ime rend and no seasonal dummies; I: An inercep; I,T: An inercep and a ime rend; I,S: An inercep and seasonal dummies; I,S,T: An inercep, a ime rend and seasonal dummies. rˆ in (1) wih TABLE 5 d + θ ρ L ; θ ) = (1 L) ( and whie noise u for he U.K. daa Series z / d I c I,T I,S I,T,S I y I,T I,S I,T,S I c - y I,T I,S I,T,S : Non-reecion values for he null hypohesis (11) a 95% significance level. --: No inercep, no ime rend and no seasonal dummies; I: An inercep; I,T: An inercep and a ime rend; I,S: An inercep and seasonal dummies; I,S,T: An inercep, a ime rend and seasonal dummie 7

30 rˆ in (1) wih TABLE 6 d + θ ρ L ; θ ) = (1 L ) ( 4 for he Japanese daa i): Wih whie noise u Series z / d c y c - y I I,T I,S I,T,S, I I,T I,S I,T,S, I I,T I,S I,T,S, ii): Wih AR(1) u Series z / d c Y c - y I I,T I,S I,T,S, I I,T I,S I,T,S, I I,T I,S I,T,S, iii): Wih AR() u Series z / d c y c y I I,T I,S I,T,S, I I,T I,S I,T,S, I I,T I,S I,T,S, : Non-reecion values for he null hypohesis (11) a 95% significance level; --: No inercep, no ime rend and no seasonal dummies; I: Inercep; I,T: Inercep and ime rend; I,S: Inercep and seasonal dummies; I,T,S: Inercep, ime rend and seasonal dummies. 8

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