LONG MEMORY AT THE LONG-RUN AND THE SEASONAL MONTHLY FREQUENCIES IN THE US MONEY STOCK. Guglielmo Maria Caporale. Brunel University, London

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1 LONG MEMORY AT THE LONG-RUN AND THE SEASONAL MONTHLY FREQUENCIES IN THE US MONEY STOCK Guglielmo Maria Caporale Brunel Universiy, London Luis A. Gil-Alana Universiy of Navarra Absrac In his paper we show ha he monhly srucure of he US money sock can be specified in erms of a long-memory process, wih roos a boh he zero and he seasonal monhly frequencies. We use a procedure ha enables us o es simulaneously for he roos a all hese frequencies. The resuls show ha he roo a he long-run or zero frequency plays a much more imporan role han he seasonal one, hough he laer should also be aken ino accoun. Keywords: Seasonaliy, Long Memory, Fracional Inegraion JEL Classificaion: C5; C Corresponding auhor: Professor Guglielmo Maria Caporale, Brunel Business School, Brunel Universiy, Uxbridge, Middlesex UB8 3PH, UK. Tel.: +44 (0) Fax: +44 (0) Guglielmo-Maria.Caporale@brunel.ac.uk The second named auhor graefully acknowledges financial suppor from he PIUNA Proec a he Universiy of Navarra, Pamplona, Spain.

2 . Inroducion Dickey, Hasza and Fuller (DHF, 984), Hylleberg, Engle, Granger and Yoo (HEGY, 990), Beaulieu and Miron (993), and Tam and Reinsel (998), amongs ohers, have proposed es saisics for seasonal uni roos in raw ime series. More precisely, if x is he ime series we observe, wih a changing seasonal paern, we can consider he model s ( L ) x u,,,..., () where s is he number of ime periods in a year, and u is an I(0) process, defined for he purposes of he presen paper as a covariance-saionary process wih specral densiy ha is posiive and finie a any frequency. Noe ha he polynomial in () can be decomposed ino ( L)( + L + L L s ) ( L) S( L). Tha is, he seasonal difference operaor can be wrien as he produc of he firs difference operaor and he moving-average filer S(L), conaining furher roos of modulus uniy. The roo a he long-run or zero frequency hen appears as a componen of he seasonal polynomial in (). However, here are many cases when his frequency plays a maor role, accouning no only for some of he seasonal behaviour bu also for he rending sochasic behaviour of he series. In his paper, we focus on monhly daa, (i.e. s ), and presen a version of he esing procedure of Robinson (994) ha enables us o consider simulaneously uni roos wih possibly fracional orders of inegraion a boh he zero and he seasonal frequencies. In paricular, we examine models such as: d ( L) ( L d ) x u,,,..., () for given real values d and d. Here, he firs fracional polynomial can be expressed in erms of is binomial expansion such ha, for all real values d, d ( ) d L ( ) d ( d ) L L d L , and similarly, for he seasonal componen,

3 d ( ) d L ( ) ( d ) L 4 L d L + 0 d.... Clearly, seing d and d 0 in () amouns o esing he classical uni roo model (Dickey and Fuller, 979; Phillips, 987; ec); if d 0 and d, we have seasonal uni roos (e.g., HEGY, 990), and if d d, we obain he airline model inroduced by Box and Jenkins (976).. The esing procedure We use a simple version of he ess of Robinson (994), specifically a Lagrange Muliplier (LM) es of he null hypohesis: H o : d o o o ( d, d )' ( d, d )' d, (3) in () for given real numbers d o and d o. The es saisic is given by: ˆ T ˆ' ˆ R a A aˆ, (4) 4 ˆ σ where T is he sample size, and aˆ π T * ψ ( ) g( ; ˆ) τ I( ) ; π ˆ σ σ ( ˆ) τ g( ; ˆ) τ I( ), T T Aˆ * * * * ψ ( ) ψ ( )' ψ ( ) ˆ( ε )' ˆ( ε ) ˆ( ε )' ˆ( ε ) ψ ( )' T ψ ( )' [ ψ ( ), ψ ( )]; ˆ ε( ) log ( ; ˆ g τ ) ; ψ ( ) log sin ; τ ψ ( ) log sin + log cos + log cos + π log cos cos 3 + π + log cos cos 3 + π log cos cos 6 + 5π log cos cos, 6

4 wih π/t. I( ) is he periodogram of ˆ u d0 d o ( L) ( L ) x, and τˆ arg min τ T σ ( ), wih T * as a suiable subse of he R q Euclidean space. Finally, he funcion * τ g above is a known funcion coming from he specral densiy of u : f ( ; τ ) σ g( ; τ ), π π < π. Noe ha hese ess are purely parameric, and herefore require specific modelling assumpions abou he shor-memory specificaion of u. For insance, if u is a whie noise process, g, whils if u is an AR process of he form φ(l)u ε, hen g φ(e i ) -, wih σ V(ε ), he AR coefficiens being a funcion of τ. Based on H o (3), Robinson (994) esablished ha, under cerain regulariy condiions: Rˆ d χ, as T. (5) Thus, unlike oher procedures, we are in a classical large-sample esing siuaion for he reasons oulined by Robinson (994), who also showed ha he ess are efficien in he Piman sense agains local deparures from he null. A es of (3) will reec H o agains he alernaive χ, α χ, α χ H a : d d o if Rˆ >, where Prob ( > ) α. There exis oher versions of he ess of Robinson (994), esing, for example, only he roo a he long-run or zero frequency (e.g., Gil-Alana and Robinson, 997; Gil-Alana, 000), purely seasonal fracional models (Gil- Alana, 999, 00; Gil-Alana and Robinson, 00), or cyclical srucures (Gil-Alana, 00). However, a simulaneous es for he roos a boh he zero and he seasonal monhly componens has no been implemened ye. We carry ou such a es in he following secion. 3. Tesing for he orders of inegraion of he US monhly money sock The ime series analysed in his secion is he (seasonally unadused) US monhly money sock (billions of dollars), for he ime period 947m o 00m8, obained from he Federal Reserve Bank of S. Louis daabase. 3

5 Denoing he ime series x, we employ hroughou model (), esing H o (3) for values d o, d o 0, (0.5),, hereby including ess for a uni roo exclusively a he long-run or zero frequency (d o, d o 0), as well as ess for seasonal uni roos (d o 0, d o ), and uni and seasonal uni roos (d o d o ), in addiion o oher fracionally inegraed possibiliies. Iniially, we assume ha u is whie noise, bu hen we also allow for weakly paramerically auocorrelaed disurbances. In paricular, we consider AR() and seasonally monhly AR() processes for u. Only one non-reecion occurs for he hree ypes of disurbances, corresponding o (d o, d o ) (.5, 0.5). This suggess ha he order of inegraion a he longrun frequency has a much more imporan role han he one corresponding o he seasonal frequency. (Inser Figure abou here) In order o have a more precise view abou he non-reecion values, we performed again he ess of Robinson (994), bu his ime using incremens of 0.0 for d o and d o. Figure displays he non-reecion regions of (d o, d o ) for each ype of disurbances. I can be seen ha in all cases d o is higher han d o, highlighing once more he imporance of he roo a he zero frequency. In paricular, d o appears o be consrained beween. and.5, whils d o oscillaes around 0.5. Consequenly, shocks o he long-run componen will have permanen effecs, policy acions being required o bring he series back o is original rend. On he oher hand, seasonal shocks will be ransiory, mean reversion occurring a some poin in he fuure. Nex, we ry o esablish wha migh be he bes model specificaion for his series. For his purpose, we compue, for each model, he values of d o and d o producing he lowes saisics, which should be an approximaion of he maximum likelihood esimaes; his is because he procedure employed here is a Lagrange Muliplier es and is based on he While funcion, which is an approximaion o he likelihood funcion. The parameer values are displayed in Table. (Inser Table abou here) 4

6 I can be seen ha, if u is whie noise, d.7 and d 0.4; he corresponding values if u is AR() are d.0 and d 0.09; finally, if u is modelled as a seasonal AR() process, d. and d 0.9. Thus, in all hree cases he order of inegraion a he zero frequency is higher han, whils he seasonal one is slighly above 0. Several diagnosic ess were hen carried ou on he residuals of he esimaed models, indicaing ha he bes model is he one wih seasonal AR disurbances. Our preferred specificaion is herefore he following: (. L) 0.9 ( L ) x u,,,... u 0.43u + ε,,,... wih whie noise ε. Clearly, he sandard approach of aking firs differences or firs seasonal differences would no appropriae here, since he former would resul in a series sill exhibiing a long-memory componen, whils he laer would enail overdifferencing. 4. Conclusions The sochasic behaviour of he US money sock has been examined in his paper using a procedure ha enables us o consider simulaneously roos wih fracional orders of inegraion boh a he zero and he seasonal frequencies. The resuls sugges ha he roo a he zero frequency should be considered independenly of he seasonal frequency, hough he laer should also be aken ino accoun, exhibiing an order of inegraion of abou 0.5. Finally, he fac ha he roo a he long-run frequency is higher han, while he one affecing he seasonal srucure is smaller han, has some implicaions in erms of policy acion and forecasing. In paricular, whils shocks affecing he seasonal srucure appear o be mean revering, hose affecing he long run end o persis forever, requiring policy-makers o ake appropriae acions o resore equilibrium. 5

7 References Beaulieu, J.J. and J.A. Miron, 993, Seasonal uni roos in aggregae U.S. daa, Journal of Economerics 55, Box, G.E.P. and G.M. Jenkins, 976, Time Series Analysis: Forecasing and Conrol, ( nd eds.) San Francisco, C.A.: Holden-Day. Dickey, D. A., D. P. Hasza and W. A. Fuller, 984, Tesing for uni roos in seasonal ime series, Journal of he American Saisical Associaion 79, Gil-Alana, L.A., 999, Tesing fracional inegraion wih monhly daa, Economic Modelling 6, Gil-Alana, L.A., 000, Mean reversion in he real exchange raes, Economics Leers 69, Gil-Alana, L.A., 00, Tesing sochasic cycles in macroeconomic ime series, Journal of Time Series Analysis, Gil-Alana, L.A., 00, Seasonal long memory in he aggregae oupu, Economics Leers 74, Gil-Alana, L.A. and P.M. Robinson, 997, Tesing of uni roos and oher nonsaionary hypoheses in macroeconomic ime series, Journal of Economerics 80, Gil-Alana, L.A. and P.M. Robinson, 00, Tesing seasonal fracional inegraion in he UK and Japanese consumpion and income, Journal of Applied Economerics 6, Hylleberg, S., R.F. Engle, C.W.J. Granger and B.S. Yoo, 990, Seasonal inegraion and coinegraion, Journal of Economerics 44, Porer-Hudak, S., 990, An applicaion of he seasonal fracionally differenced model o he moneary aggregaes, Journal of he American Saisical Associaion 85, Robinson, P.M., 994, Efficien ess of nonsaionary hypoheses, Journal of he American Saisical Associaion 89, Tam, W. and G. C. Reimsel, 997, Tess for seasonal moving average uni roo in ARIMA models, Journal of he American Saisical Associaion 9,

8 FIGURE Region of values of d o and d o where H o (3) canno be reeced a he 95% significance level i) wih whie noise disurbances: 0,75 d 0,5 0,5 0,,,3,4,5 d ii) wih AR() disurbances: 0,75 d 0,5 0,5 0,,,3,4,5 d iii) wih seasonal AR() disurbances: 0,75 d 0,5 0,5 0,,,3,4,5 d 7

9 TABLE Model specificaions according o he lowes saisics in Figure u d d AR coeff. Seasonal AR coeff. Whie noise AR () Seasonal AR()