A state space approach to calculating the Beveridge Nelson decomposition

Transcription

1 Eonomis Leers 75 (00) loae/ eonbase A sae spae approah o alulaing he Beveridge Nelson deomposiion James C. Morley* Deparmen of Eonomis, Washingon Universiy, Campus Box 08, Brookings Drive, S. Louis, MO , USA Reeived 7 February 00; aeped Oober 00 Absra A sae spae approah provides a general unified framework for alulaion of he Beveridge Nelson deomposiion for a wide variey of ime series models, inluding all univariae and veor ARIMA models. 00 Elsevier Siene B.V. All righs reserved. Keywords: Beveridge Nelson deomposiion; Sae spae models; Kalman filer JEL lassifiaion: C; C3. Inroduion The deomposiion mehod inrodued by Beveridge and Nelson (98) provides a onvenien way o esimae he permanen and ransiory omponens of an inegraed ime series. Given a foreasing model for he firs-differenes of he series, he Beveridge Nelson (BN) rend is he long-run foreas of he level of he series (minus any deerminisi drif) and he BN yle is he gap beween he presen level of he series and is long-run foreas. In praie, alulaion of he exa BN rend and yle is ofen ompliaed by he presene of infinie sums in he long-run foreas. This paper poins ou, however, ha exa alulaion of he BN rend and yle is relaively sraighforward if he foreasing model an be as ino sae spae form. Examples inlude all univariae and veor ARIMA models. Thus, he sae spae approah provides a general unified framework for BN rend/yle alulaion for all of he ases disussed in he previous lieraure, inluding Cuddingon and Winers (987), Miller (988), Newbold (990), and Arino and Newbold (998). *Tel.: ; fax: address: morley@wueon.wusl.edu (J.C. Morley) / 0/ $ see fron maer 00 Elsevier Siene B.V. All righs reserved. PII: S (0)0058-X

2 4 J.C. Morley / Eonomis Leers 75 (00) 3 7. Moivaion To moivae he sae spae approah, firs onsider an inegraed ime series hy ha an be mos auraely foreas using a saionary univariae AR() model for is firs differenes: ` (DY m) 5 f(dy m) e () u u where e i.i.d.n(0, s ), f,, and DY ; Y Y. By onsidering he implied Wold form from he AR() model, i is sraighforward o show ha, under he assumpion of normaliy, he minimum mean squared error (MSE) -period-ahead foreas of he firs differene is: E [(DY m)] 5 f (Dy m) () where lower-ase Dy denoes he realized value of he random variable DY. The BN rend, denoed, is defined as he minimum MSE foreas of he long-run level of he series (minus he deerminisi drif) or, equivalenly, he presen level of he series plus he infinie sum of he minimum MSE -period-ahead firs differene foreass: J ;lim E [Y J? m] 5 y lim O E [(DY m)] (3) J J ` J ` 5 Thus, subsiuing () ino (3), he BN rend of observaion y for he AR() ase is: f 5 y ]] (Dy m) (4) f Tha is, he rend is he presen level plus he long-run impa of he ransiory momenum in he series implied by he deviaion of Dy from is seady-sae level m ; E[DY ]. Meanwhile, he BN yle, denoed, for he AR() ase is: f 5 ]] (Dy m) (4) f Noe ha he yle is defined in he onvenional way as he deviaions from he rend (i.e. ; y ). Then, as disussed in Morley e al. (00), he BN rend and yle provide esimaes of he permanen and ransiory omponens of y. 3. A sae spae approah Given he BN rend/ yle deomposiion for he AR() ase, i is sraighforward o generalize o any ase where he firs differenes of hy ` an be mos auraely foreas by a model ha an be as ino sae spae form, inluding all univariae and veor ARMA models. In pariular, suppose (DY m) is a linear ombinaion of he elemens of a kx sae veor X : Beveridge and Nelson (98) define he BN yle as he rend minus he level.

3 J.C. Morley / Eonomis Leers 75 (00) DY m 5fh h??? h gx k (5) where h, i 5,,...,k, is he weigh of he ih elemen of X in deermining (DY m). Furhermore, i suppose sae veor X evolves aording o he following firs-order sohasi differene equaion: v (6) N(0, V) and he eigenvalues of F are less one in modulus. Then, i is sraighforward o show ha he minimum MSE -period-ahead foreas of he firs differene DY is: f kg E [DY ] 5 h h??? h F E[X ] (7) Noe ha, sine he sae veor may onain unobserved elemens (see, for example, he ARIMA(,,) example onsidered below), he expeaion E [X ] may have o be obained prior o alulaing (7). Forunaely, he Kalman filer, whih an be employed o obain exa maximum likelihood esimaes for sae spae models wih unobserved elemens, provides his expeaion. Thus, denoing he Kalman filered alulaion of he expeed sae veor X u; E [X ], he BN rend of observaion y for he general ase is: f kg u 5 y h h??? h F(I F ) X (8) Meanwhile, he BN yle of y for he general ase is: f kg u 5h h??? h F(I F ) X (9) Again, he BN rend and yle provide esimaes of he permanen and ransiory omponens of y. 4. Two examples To illusrae he general usefulness of his approah, wo examples are provided. The firs example is a bivariae veor error orreion model (VECM) as used for aggregae inome and onsumpion in Cohrane (994). The seond example is a univariae ARIMA(,,) model as used for real GDP in Morley e al. (00). 4.. A bivariae VECM Cohrane (994) employs a speial ase of he approah proposed here o alulae he BN rend and yle of observed aggregae inome ( y ) given a VECM foreasing model of aggregae inome hy ` and onsumpion hy `. A slighly simplified form of his model has he following sae spae 3 represenaion: See Harvey (990) for he full deails of he Kalman filer and sae spae models. Tehnially, he Kalman filer alulaes he minimum MSE linear proeion of he sae veor on he observable daa. This is equal o he expeed value under a Normaliy assumpion. 3 Cohrane (994) onsiders seond-order dynamis for he firs differenes of oupu and onsumpion.

4 6 J.C. Morley / Eonomis Leers 75 (00) 3 7 DY m g g g3 DY m vy DY m 5 g g g3 DY m v (0) Y Y a g g g g g g Y Y a v v y or, more ompaly, v (09) N(0, V) and he eigenvalues of F are less one in modulus, whih orresponds o oinegraion of aggregae inome and onsumpion wih oinegraing veor [ ]. Then, noing ha (DY m) 5f00X g, he BN rend of observed y is: f g x 5 y 00F(I F ) () where he Xu erm in (8) is se o he realized value of he sae veor x sine, in his example, is elemens are all observable a ime. The BN yle of y is: 500F(I f g F ) x () 4.. A univariae ARIMA(,,) Morley e al. (00) employ he sae spae approah proposed here o alulae he exa BN rend and yle of observed real GDP ( y ) given a redued form ARMA(,) foreasing model of he firs differenes of hy `. I is imporan o noe ha he alulaion of he exa BN rend and yle is nonrivial in his ase due o he presene of unobservable moving average erms in he foreasing equaion. Also, i should be noed ha here are muliple possible sae spae represenaions for he model. However, he ompanion form represenaion for an ARMA(,) is onvenien sine i has (DY m) as he firs elemen of he sae veor X : DY m f f u u DY m e DY m DY m 0 e e e (3) e e or, more ompaly: v (39) N(0, V) and he eigenvalues of F are less one in modulus (equivalenly, he roos of ( f z f z ) 5 0 lie ouside he uni irle). Then, he BN rend of observed y is: f g u 5 y F(I F ) X (4) Meanwhile, he BN yle is: 5 f 0 0 0gF(I F ) X (5) u

5 J.C. Morley / Eonomis Leers 75 (00) Again, he problem of unobserved moving average erms in he foreasing equaion is solved by using he Kalman filer alulaion of X 5 E [X ]. u 5. Conlusion A sae spae approah provides a sraighforward and general unified framework for BN rend/ yle alulaion for a wide variey of models, inluding all univariae and veor ARIMA models. Aknowledgemens I have reeived helpful ommens from Charles Nelson and Eri Zivo. Referenes Arino, M.A., Newbold, P., 998. Compuaion of he Beveridge Nelson deomposiion for mulivariae eonomi ime series. Eonomis Leers 6, Beveridge, S., Nelson, C.R., 98. A new approah o he deomposiion of eonomi ime series ino permanen and ransiory omponens wih pariular aenion o measuremen of he business yle. Journal of Moneary Eonomis 7, Cohrane, J.H., 994. Permanen and ransiory omponens of GNP and sok pries. Quarerly Journal of Eonomis 04, Cuddingon, J.T., Winers, L.A., 987. The Beveridge Nelson deomposiion of eonomi ime series: A quik ompuaional mehod. Journal of Moneary Eonomis 9, 5 7. Harvey, A.C., 990. The Eonomeri Analysis of Time Series, nd Ediion. MIT Press, Cambridge, MA. Miller, S.M., 988. The Beveridge Nelson deomposiion of eonomi ime series: Anoher eonomial ompuaion mehod. Journal of Moneary Eonomis 6, Morley, J.C., Nelson, C.R., Zivo, E., 00. Why are unobserved omponen and Beveridge Nelson rend-yle deomposiions of GDP so differen. Washingon Universiy and Universiy of Washingon, Manusrip. Newbold, P., 990. Preise and effiien ompuaion of he Beveridge Nelson deomposiion of eonomi ime series. Journal of Moneary Eonomis 6,