A Single-Index Dynamic Factor Model for Current-Quarter. Estimates of Economic Activity in Hong Kong

Transcription

1 A Sngle-Inex Dynam Faor Moel for Crren-Qarer Esmaes of Eonom Avy n Hong Kong Sefan Gerlah Hong Kong Moneary Ahory Hong Kong Inse for Moneary Researh Unversy of Basel an CEPR an Mahew S. Y Hong Kong Inse for Moneary Researh Prelmnary Draf Sepember Prelmnary Absra Ths paper apples he sngle-nex ynam faor moel, evelope by Sok an Wason (99), o proe rren-qarer esmaes of eonom avy n Hong Kong. The Hang Seng nex, a resenal propery pre nex, real sales an oal expors are se as onen naors n he moel. Prnpal Componen Analyss s frs se o oban an mpresson of he ommon omponen of he naor seres. Ths omponen an he ynam faor enfe by he Sok-Wason mehoology are srongly orrelae an seem o aon for eonom flaons n Hong Kong reasonably well. JEL: E E7 Key wors: Bsness yles, ynam faor moel, Kalman flerng The vews expresse n he paper are solely hose of he ahors an o no neessarly refle hose of he HKMA, he HKIMR, s Conl of Avsers or Boar of Dreors. Cona nformaon: Sefan Gerlah, HKMA, Garen Roa, h floor, Cenral, Hong Kong, e-mal: sefan_gerlah@hkma.gov.hk; Mahew S. Y, HKIMR, Garen Roa, 8 h floor, Cenral, Hong Kong, e-mal: mahew_sf_y@hkma.gov.hk.

2 . Inroon To on eonom poly, some egree of foresgh s neee, as many poly aons reqre a long lag o have an effe. In lgh of hs nee, governmen agenes, enral banks an eonom researh nses aross he worl are ronely prong maro eonom foreass for poly prposes. Mh researh has been evoe o he sse of how o onsr more arae an more mely eonom foreass. In parlarly, one approah, sng he fa ha eonom avy evolves n yles, aemps o fn relable foreasng ools for bsness yle rnng pons. Ths approah has le o an exensve lerare evelopng, from he early lanmark sy by Brns an Mhell (96) o he mh more formalse approah of Sok an Wason (989, 99 an 99). Brns an Mhell (96) evelop a ls of ompose leang, onen, an laggng nexes of bsness yles, sng a large nmbers of eonom varables (so-alle naor varables). These nexes have playe an mporan role for smmarsng he sae of maroeonom avy n he Une Saes. Brns an Mhell s efnon of bsness yles has wo key feares. The frs s he omovemens among nval naor varables. The seon s her vson of bsness yles no separae phases or regmes. Usng Brns an Mhell s noon of bsness yles, Sok an Wason (989, 99 an 998) formlae a moern sasal framework o sy bsness flaons. They assme ha he omovemens have a ommon elemen ha an be apre by a sngle nerlyng, nobservable varable an ha hs nobservable varable represens he general sae of he eonomy. They all her moel, whh proves a formal efnon of he nobservable sae of he eonomy, he sngle-nex moel. Usng hs moel, hey ompe a ompose nex of onen naors. Sne he semnal work of Sok an Wason, he sngle-nex moel, also alle he one-faor moel, has been wely se by many oher researhers. Reen lerare nles Camba-Menez () an Gara-Ferrer an Ponela (), who mofy he moel o foreas GDP growh for Eropean onres. For Germany n parlar, Banholz an Fnke () se he moel o evelop leang naors of eonom avy n he onry. In Asa, Fka an Onoera () an Chen an Ln ()

3 apply he moel o mprove foreass of he Japanese eonomy an o enfy rnng pons an bsness yles n Tawan respevely. In he ase of Hong Kong, real GDP fgres are pblshe almos hree monhs afer he referene qarer. Ths arrangemen may no be promp enogh o monor he eonomy on an ongong bass. Ths, an naon of he sae of he eonomy appears warrane. The APEC Sy Cenre of he Hong Kong Unversy has evelope a hgh freqeny maroeonom foreasng moel for Hong Kong n ollaboraon wh Lawrene Klen of he Wharon Shool of he Unversy of Pennsylvana. Sne he frs qarer of, he APEC Sy Cenre has been reglarly prong qarerly foreass of he growh of Hong Kong GDP sng a large nmber of hgh freqeny fnanal an maroeonom varables, Prnpal Componen Analyss (PCA) an he brge eqaon approah. In hs paper, we apply Sok an Wason s sngle-nex moel o onsr a onen naor of eonom avy n Hong Kong. For ompaonal effeny, we only se for monhly naor seres wo fnanal seres an wo maroeonom seres for he analyss. Applyng he sngle-nex moel on he selee nor seres an sng he Kalman fler for he esmaon of moel parameers an sae veors, we generae esmaes of he growh rae of he real GDP n he same qarer before s pblshe. Ths paper proees as follows. Seon esrbes he aa se n he empral ses n hs paper. Seon presens he resls of applyng PCA on obanng an mpresson of he nobservable omovemen omponen. Seon esrbes he sngle-nex moel. Seon 5 presens he empral resls from sng he moel an an nex of onen naors of eonom avy n Hong Kong. The las seon onles. Hgh freqeny maroeonom foreasng moels for Hong Kong, APEC Sy Cenre, Hong Kong Inse of Eonom an Bsness Sraegy Unversy of Hong Kong. The foreass are avalable on he followng webse lnk: hp://

4 . The Daa for Empral Analyss The prpose of hs seon s o explan he hoe of naor seres se n he snglenex moel esmae below. A large nmber of monhly fnanal an maroeonom varables may onan sefl nformaon abo real eonom avy n Hong Kong. Gven he range of possble naor seres, we frs ree he mensonaly of he problem by seleng a sbse of he naor seres: he Hang Seng nex, he -monh Hong Kong nerbank offer rae, a resenal propery pre nex, oal expors, reane mpors, real sales, ors arrvals an elery onsmpon. The sample pero s from Janary 99 o Deember. Or hoe of hese egh seres s base on he assmpon ha hey may be more relevan o be se as onemporaneos naors. The egh naor varables, exep he HIBOR seres whh s n perenage pons, are shown n logarhm sale n Fgre. Frhermore, he fve maroeonom varables have been seasonally ajse by he X- meho. The fgre sggess ha all varables an be haraerse as havng a sohas ren, exep he neres rae seres. For beer ompaonal effeny, wol be esrable o se a sbse of hese egh seres. In orer o sele he seres ha are more nformave for real avy, we frs look a he ross orrelaon beween her growh raes an he seasonally ajse qarerly growh raes of real GDP. Fgre shows he ross orrelaes beween he growh raes of eah of he egh seres (for neres raes we se he level of he seres) an he qarerly real growh of Hong Kong from lea 8 qarers o lag 8 qarers. I s lear ha all of hem exep he elery onsmpon splay onserable orrelaon wh GDP. The Hang Seng Inex, propery pre nex, oal expors, real sales an reane mpors seres have srong posve orrelaon wh he GDP, whereas he HIBOR seres shows negave orrelaon as expee. Bease he frs for seres have he larges onemporaneos orrelaon, we sele hem as onen naors for he empral work. The aa an he Hong Kong real GDP growh ome from he CEIC Daa L aabase. The ross-orrelaon fnon of GDP an naor seres (I) s esmae by he followng eqaon: ρ GDP k, I () = ( GDP GDP)( I I) ( GDP GDP) ( I I) k

5 Regarng he me seres properes, he selee for naor seres all have a n roo, onfrme by he agmene Dkey-Fller es. Followng he Sok an Wason s mehoology, we assme ha he for naor seres are no onegrae. Dfferen moellng sraeges are neee, however, for nobservable-omponen moels wh onegrae varables.. Prnpal Componens Analyss To oban an mpresson of he nobservable omovemen omponen of he for naor seres, we frs apply Prnpal Componens Analyss (PCA) o exra he frs prnpal omponen from he aa. I s well known ha he general objeves of PCA are aa reon an nerpreaon. 5 I explans he varane-ovarane srre of he aa sng lnear ombnaons of he orgnal varables. Fgre shows he monhly growh raes of he for naor seres. We fn ha f we apply PCA o hese seres, he frs an seon prnpal omponens (PCs) all work srongly orrelae wh he qarerly GDP growh. I s bease he prnpal omponens exrae from he monhly growh raes wll lose he orhogonal propery when hey are onvere no qarerly freqeny for he allaons of he orrelaon wh he qarerly GDP growh. However, f we frs onver he monhly growh raes no qarerly raes an hen apply PCA on hem, he frs PC s hghly orrelae wh he qarerly real GDP growh. Table naes ha he frs an seon PCs apre abo 5% an % of he varane for he aa. The orrelaon beween he frs PC an he qarerly GDP growh rae s.59, mh larger han ha,.9, of he seon PC. The hr an forh PCs are even less orrelae wh he qarerly GDP growh rae. One nerpreaon of hs fnng ha here s an nobservable nerlyng omponen among he qarerly growh raes of he Harvey, Fernanez-Maho, an Sok (987) sss moellng sraeges for nobservable-omponen moels wh onegrae varables. 5 For a goo nroon o he heory an he applaons of Prnpal Componen Analyss, see Jollffe (986).

6 naor for seres ha s hghly orrelae wh he overall eonom onons n Hong Kong.. The Sngle-Inex Dynam Faor Moel In hs seon we presen he sngle-nex moel we se o onsr a onen nex for he Hong Kong eonomy. We frs olne he moel, show how an be wren n sae-spae form an sss how o esmae sng Kalman flerng.. Spefaon The sngle-nex moel we se follows Sok an Wason (99). I an be formlae n erms of he frs fferenes of he for naor varables, as follows: () Y = D + γ C +, =,..., () ( C δ ) = φ ( C δ ) +... φ ( C p p δ ) + η, η ~... N(, σ η ) () = q q +, ~... N(, σ ) an =,..., where Y enoes he logarhm of seres. In he above moel, Y onsss of wo sohas omponens: an nobservable ommon omponen, C an an osynra omponen,. Boh of hese omponens are moelle as aoregressve sohas proesses, AR(p) an AR(q), respevely. For a normalsaon, he sale of C, s enfe by seng σ η o ny. In aon, all shoks are assme o be nepenen The man enfyng assmpon n he above moel s ha he omovemens n he naor seres arse from he sngle sore fferen weghs, γ, =,..., C,.e. C eners eah naors wh a. Ths s mae prese by assmng ha an mally norrelae a all leas an lags for all seres. C are 5

7 6 Noe ha, as he parameers D an δ are no separaely enfe, Sok an Wason (99) sgges wrng he moel n evaon from means, hs onenrang he D an δ γ erms o of he lkelhoo fnon:,..., ) (,... ~,... (6) (,)... ~,... (5),...,, () = = + + = = + = an N N y q q p p σ η η φ φ γ where Y Y y = an δ = C. As he sngle-nex moel n evaon from means s lnear n he nobservable omponens, we an se he Kalman fler o onsr he Gassan lkelhoo fnon an o esmae he nknown parameers of he moel by maxmsng he lkelhoo fnon. However, o se he Kalman fler, we have o ransform he above hree eqaons no sae spae form. 6. Sae-Spae Represenaon The sae spae form of he sysem s omprse of a measremen eqaon an a ranson (or alle sae) eqaon. The measremen eqaon, whh relaes he observe varables o he elemens of he sae veor, s gven by: = (7) C y y y y γ γ γ γ 6 For a ssson of sae-spae moels an he Kalman fler, see Harvey (989, 99) or Hamlon (99).

8 7 The ranson eqaon, whh esrbes he evolon of he nobservable sae veor, whh n or ase onans an an her lags an be wren: + = (8) C C η φ φ φ where [ ] η s he veor of srbanes, whh we assme s agonal. Ths assmpon mples ha shoks o he nobservable ommon omponen an he osynra omponens are mally norrelae a all leas an lags.. Esmaon As menone before, o esmae he moel, whh s gven by he measremen eqaon an he ranson eqaon, we form he lkelhoo fnon: = T T T v F v F π) ( T L log log log 9) (, where T, v an F enoe he sample sze, he preon errors an he mean sqare marx of he preon errors, respevely. Esmaes of he moel an hen be obane by nmerally maxmsng he lkelhoo fnon, sng he Kalman fler. Sne he naor varables are non-saonary, n esmang he moel we follow he sggesons of Harvey (989) an assme ha he pror sae veor s a ranom varable an has a ffse srbon, ha s, we assme ha s ovarane marx s gven by I κ wh κ. Ths s anamon o assmng ha nohng s known abo he nal sae.

9 5. Empral Resls Table presens he esmaes of φ, γ an he varanes of he srbanes n he sngle-nex moel on he for naor seres (n he orer of he Hang Seng nex, he resenal propery pre nex, real sales an oal expors). All parameers are onssen wh ha pree by heory an sgnfan a he 5 peren level, exep γ 7. Wh regar o he esmae aoregressve oeffens, he roos of φ (B) all le ose he n rle, wh wo of hem omplex onjgaes. Ths, he esmae AR() proess of s saonary an exhbs a ylal paern. Regarng he osynra omponens, we anno reje he hypohess ha an eqal zero. Ths, he growh raes of he wo fnanal naor seres are an AR() proess pls whe nose. Fgres shows he esmae nobservable ommon omponen,, agans he sanarse growh raes of eah of he for naor seres. The nobservable ommon omponen looks smlar o he wo fnanal seres bease he faor loangs, γ an γ, are large. Alhogh he faor loangs n he wo maroeonom seres are smaller, hey also play an mporan role n generang he nobservable ommon omponen. As ssse above, f we sar he oal expors seres, he esmae wll be orrelae mh less wh he qarerly GDP growh. We onver he no qarerly freqeny an, wh an ajsmen on s sale, plo agans qarerly real GDP growh n Fgre 5, wh s wo sanar error onfene bans. The onemporaneos orrelaon beween he wo seres s reasonably hgh, a aron.58. Or moel, however, somewha neresmae he GDP growh n. The neresmaon may e o a few reasons. Frs, he propery marke an real sales were sll epresse by he lak of onfene n he Hong Kong eonomy an 7 We have also re anoher moel who he forh seres (oal expors). For ha moel, alhogh all he esmae parameers are sgnfan a he 5 peren level, he esmae seres,, has a mh smaller orrelaon wh real GDP growh han ha of he orgnal moel. As or man prpose s o prove an esmae of he rren sae of he eonomy, we ee o keep he forh naor seres an se he orgnal moel, espe ha γ s only sgnfan a aron he peren level. 8

10 he hgh nemploymen rae. Seon, he sok marke performe mh less sasfaorly n he las qarer of bease of he h of Sepember nen. Lasly, he growh rebon n was manly e o he goo expor performane of he Hong Kong eonomy. In orer o onsr an nex of onen naors of eonom avy n Hong Kong, we nee o esmae he mean growh rae for he omovemen omponen C. Ths mean s allae as a weghe average of he growh raes of he naor seres. The weghs are hose mplly se o onsr from he naor varables an an be esmae from he Kalman Fler algorhm. In or moel, he esmae mean growh rae for C s.98 perenage pons. Fgre 6 shows he nex of onen naors of eonom avy n Hong Kong, onsre by applyng he above snglenex ynam faor moel on he for naor seres. 6. Conlson We have shown ha he sngle-nex ynam faor moel s sefl n assessng rren-qarer real eonom avy n Hong Kong. As he Hong Kong GDP fgres are pblshe wh a me lag of wo o hree monhs, or approah an proe esmaes of he GDP growh of he same qarer, parlarly on showng any rnng pons before he ssemnaon. We have also onsre an nex of onen naors of eonom avy n Hong Kong from he sngle-nex ynam moel. Wh he avalably of hs nex, we an aop he mehoology of sae-spae moels wh regme swhng 8, evelope by Km (99), o enfy he bsness yle flaons n Hong Kong n fre work. 8 For he eals of he mehoology, see Km an Nelson (999). 9

11 Referenes Banholz, H. an M. Fnke (), In Searh of Leang Inaors of Eonom Avy n Germany, forh omng n Jornal of Foreasng. Brns, A. F., an W. A. Mhell (96), Measrng Bsness Cyles, New York: NBER. Camba-Menez, G., R. J. Smh, G. Kapeanos, an M. R. Weale (), An Aoma Leang Inaor of Eonom Avy: Foreasng GDP Growh for Eropean Conres, Eonomers Jornal, : Chen, S. an J. Ln (), Ienfyng Trnng Pons an Bsness Cyles n Tawan: a Mlvarae Dynam Markov-Swhng Faor Moel Approah, Aaem Eonom Papers, 8: 89-. Debol, F. X., an G. D. Rebsh (996), Measrng Bsness Cyles: A Moern Perspeve, The Revew of Eonoms an Sass, 78: Fka, S. an T. Onoera (), A New Compose Inex of Conen Eonom Inaors n Japan: How Can We Improve Foreas Performanes?, Inernaonal Jornal of Foreasng, 7: Gara-Ferrer, A. an P. Ponela (), Foreasng Eropean GNP Daa hrogh Common Faor Moels an oher Proeres, Jornal of Foreasng, : 5-. Hamlon, J. D. (99), Tme Seres Analyss, Prneon Unversy Press, Prneon, NJ. Harvey, A. C. (989), Foreasng, Srre Tme Seres Moels an he Kalman Fler, Cambrge Unversy Press, Cambrge. Harvey, A. C. (99), The Eonomer Analyss of Tme Seres, e. MIT Press, Cambrge.

12 Harvey, A.C, F.J. Fernanez-Maho, an J.H.Sok (987), Foreasng an Inerpolaon Usng Veor Aoregressons wh Common Trens, Annales Eonome e e Sasqe, No.6-7: Jollffe, I. T. (986), Prnpal Componen Analyss, Sprnger Verlag, New York. Km, C., (99), Dynam Lnear Moels wh Markov-Swhng, Jornal of Eonomers, 6:-. Km, C., an C. R. Nelson (999), Sae-Spae Moels wh Regme Swhng, The MIT Press, Cambrge, Massahses. Sok, J. H., an M. W. Wason (989), New Inexes of Conen an Leang Eonom Inaors, n: O. Blanhar, an Fsher, S. (es.), NBER maroeonom annal 989, MIT Press, Cambrge, MA, pp Sok J. H., an M. W. Wason (99), A Probably Moel of he Conen Eonom Inaors, n: Lahr, K., an Moore, G. (es), Leang Eonom Inaors New Approahes an Foreasng Reors, Cambrge Unversy Press, New York, pp Sok J. H., an M. W. Wason (99), A Proere for Preng Reesson wh Leang Inaors: Eonomer Isses an Reen Experene, n: J.H. Sok, an M.W. Wason,. (es), Bsness Cyles, Inaors an Foreasng, Unversy of Chago Press for NBER, pp Sok, J. H., an M. W. Wason (998), Bsness Cyle Flaons n U.S. Maroeonom Tmes Seres, NBER Workng Paper No. 658, Cambrge, MA.

13 frs prnpal omponen Table. PC Analyss Seon Prnpal Componen Thr Prnpal Componen Forh Prnpal omponen Egenvale Varane Proporon Cmlave Proporon Egenveor Real Sales Propery Pre Hang Seng Inex Toal Expors

14 Table Maxmm Lkelhoo Esmaes (Sample pero: Janary 99 o Deember ) Parameers Esmaes S. Error Asympo -vales φ φ φ , , γ γ γ.7..5 γ.7.7. σ.87 σ.56 σ.7 σ.66 Log Lkelhoo -8.

15 Fgre : Inaor Seres (n logarhm sale, exep neres rae). Hang Seng Inex Three Monh Inerbank Rae Toal Expors (SA) Real Sales (SA) Tors Arrvals (SA) Propery Pre Inex Elery Consmpon (SA) Reane Impors (SA)

16 Fgre : Cross Correlaon beween Qarerly Real GDP Growh an Inaor Seres 9 Hang Seng Inex Three Monh Inerbank Rae & Toal Expors Real Sales Tors Arrvals Elery Consmpon Propery Pre Reane Impors The ross-orrelaon fnon of GDP an naor seres (I) s esmae by he followng eqaon: ρ GDP k, I () = ( GDP GDP)( I I) ( GDP GDP) ( I I) k 5

17 Fgre : Inaors (Sanarse Growh Rae) e R a h w o Gr se ar n a S - - Hang Seng Inex e R a h w o Gr se ar n a S Propery Pre Inex e R a h w o Gr e s ar n a S Real Sales e R a h w o Gr e s ar n a S Toal Expors

18 Fgre : Inaor Seres an Comovemen Componen 6 Sanarse Growh Rae - - Hang Seng Inex Sngle-Inex Moel Sanarse Growh Rae - - Propery Pre Inex Sngle-Inex Moel Sanarse Growh Rae Real Sales Sngle-Inex Moel Sanarse Growh Rae - - Toal Expors Sngle-Inex Moel

19 Fgre 5: Comovemen Componen an GDP Growh 8 Perenage Pons - Sngle-Inex Moel GDP (QoQ, sa) Fgre 6: Inex of Conen Inaors n Hong Kong 6 Jan 99 = 5 Inex