Multivariate Markov switiching common factor models for the UK

Transcription

1 Loughborough Universiy Insiuional Reposiory Mulivariae Markov swiiching common facor models for he UK This iem was submied o Loughborough Universiy's Insiuional Reposiory by he/an auhor. Addiional Informaion: This is Business Cycle Volailiy and Economic Growh Research Paper No. /. Meadaa Record: hps://dspace.lboro.ac.uk/34/74 Publisher: c Loughborough Universiy Please cie he published version.

2 This iem was submied o Loughborough s Insiuional Reposiory by he auhor and is made available under he following Creaive Commons Licence condiions. For he full ex of his licence, please go o: hp://creaivecommons.org/licenses/by-nc-nd/.5/

3 Deparmen of Economics Business Cycle Volailiy and Economic Growh Research Paper No. / MULTIVARIATE MARKOV SWITCHING COMMON FACTOR MODELS FOR THE U.K. Terence C. Mills and Ping Wang May This paper forms par of he ESRC funded projec (Award No. L3853) Business Cycle Volailiy and Economic Growh: A Comparaive Time Series Sudy, which iself is par of he Undersanding he Evolving Macroeconomy Research programme. My hanks o Kae Morrison for providing excellen research assisance.

4 ABSTRACT We esimae a model ha incorporaes wo key feaures of business cycles, comovemen among economic variables and swiching beween regimes of boom and slump, o quarerly U.K. daa for he las four decades. A common facor, inerpreed as a composie indicaor of coinciden variables, and esimaes of urning poins from one regime o he oher, are exraced from he daa by using he Kalman filer and maximum likelihood esimaion. Boh comovemen and regime swiching are found o be imporan feaures of he U.K. business cycle. The composie indicaor produces a sensible represenaion of he cycle and he esimaed urning poins agree fairly well wih independenly deermined chronologies. These esimaes are sharper han hose produced by a univariae Markov swiching model of GDP alone. A fairly ypical sylised fac of business cycles is confirmed by his model - recessions are seeper and shorer han recoveries. Keywords: BUSINESS CYCLES, REGIME SWITCHING, MARKOV MODELS, COMOVEMENT. J.E.L. Classificaion: C5, E3

5 . INTRODUCTION The wo empirical regulariies of business cycles highlighed by Burns and Michell (946) - comovemen among economic variables hrough he cycle and asymmery in he evoluion of he cycle - have undergone a resurgence of ineres in recen years, promped by he developmen of new ime series echniques. Two of he mos influenial models of he business cycle are Sock and Wason s (989, 99, 993, 999) linear common facor model and Hamilon s (989) regime swiching model. Sock and Wason develop a linear dynamic facor model where business cycles are measured by comovemens in various componens of economic aciviy. Using several macroeconomic ime series, hey exrac a single unobserved variable and inerpre i as he sae of he economy. They hen compare his variable wih he U.S. Deparmen of Commerce (DOC) composie index, and find ha he similariy beween he wo is sriking, especially over he business-cycle horion. The disadvanage of heir model, however, is ha is lineariy canno capure business cycle asymmery, and forces expansions and conracions o have he same ampliude and duraion. To capure such asymmery, Hamilon (989) develops a regime swiching model in which oupu growh swiches beween wo saes according o a firs order Markov process. Expansions can herefore be gradual and drawn ou while recessions may be shorer and seeper - he 'sylised facs' of modern business cycles. Applying his model o he U.S., he shows ha shifs beween posiive and negaive oupu growh accord well wih he NBER s chronology of business cycle peaks and roughs. Being based on a single ime series, however, Hamilon s model canno capure he noion of economic flucuaions corresponding o comovemens of many aggregae and secoral variables. I may well be impossible for only one coinciden variable o capure all underlying business cycle informaion, which is he conclusion of boh Filardo (994) and Diebold and Rudebusch (996). Indeed, Diebold and Rudebusch provide boh empirical and heoreical suppor for combining hese wo key feaures of he business cycle, alhough hey do no fully esimae a model. Building on his research, however, several sudies do esimae hese wo feaures simulaneously wihin he regime swiching common facor model: for example, Chauve (998), Kim and Yoo (995), and Kim and Nelson (998). The common facor is defined o be an unobserved variable ha summarises he common cyclical movemens of a se of coinciden macroeconomic variables, as in Sock and Wason (99). However, i is also subjec o discree shifs so ha i can capure he asymmeric naure of business cycle phases, as in Hamilon (989). Wihin a mulivariae framework, all hree papers repor ha inferences abou he sae of he

6 economy obained from he model exhibi significanly higher correlaions wih he NBER reference daes han if jus a single variable, such as oupu growh, was used. The basic idea behind hese sudies is ha informaion abou business cycles can be exraced from a group of series raher han a single series, so ha esimaed business cycles reflec informaion from various economic secors. Furhermore, he exraced facor can be compared wih, for example, he DOC coincidence index, and more imporanly, i can be used for real ime assessmen of he economy. Previous research using hese models has ypically used daa from he U.S., and few sudies of oher economies have been underaken. In he U.K., research on he asymmery of business cycles have been based on he univariae Hamilon regime swiching model, for example, Krolig and Sensier () and Simpson, Osborn and Sensier (). However, none of hese have aemped o combine asymmery wih a common facor derived from a se of indicaor series. We hink such an exension is imporan for wo relaed reasons. Firs, if a se of indicaors can correcly provide signals of changes in aggregae economic aciviy, hen his would be helpful o any business or governmen in heir decision making, as hey are ypically affeced by economic expansions and conracions. Second, in sudying aggregae flucuaions like business cycles, i is useful o be able o analyse a group of imporan economic ime series. Individual series measure only one aspec of economic aciviy, so hey canno capure he idea of cyclical flucuaions corresponding o comovemens of many aggregae and secoral variables. Knowledge of hese feaures for he U.K. economy is herefore imporan for policy makers and forecasers.. MODEL SPECIFICATION As saed in he inroducion, our model combines he common facor model wih regime swiching. Suppose ha Yi is (he logarihm of) a macroeconomic variable ha moves conemporaneously wih overall economic condiions. I can be modelled as consising of wo sochasic auoregressive processes - a single unobserved componen, which corresponds o he common facor, and an idiosyncraic componen. Defining y = ( Y Y ) i i i i i i, he model can be wrien as follows, y = λ c +, i =,, n, () ( L) c = µ S + v φ, v ~ i. i. d. N(,), () ψ ( L) = ε, i. i. d. N(, σ ) i i i ε (3) i ~ i 3

7 c is he growh rae of he common facor, which is dependen on wheher he economy is in expansion or recession, and i eners each of he n equaions wih a differen weigh λ i, which measures he sensiiviy of he ih variable o he business cycle. The variables i are idiosyncraic erms having an AR represenaion. Their innovaions ε i can be hough of as measuremen errors and v is he innovaion o he common facor. The funcions ψ i (L) and φ (L) are polynomials in he lag operaor, where L is he lag operaor and = L. To incorporae he asymmery of business cycles, he common facor is assumed o be generaed by a Markov swiching process of he ype proposed by Hamilon (989), so ha ( S ) S µ = µ + µ S (4) where S is an unobservable sae variable ha swiches beween sae (recession) and sae (expansion) wih ransiion probabiliies governed by he Markov process [ S = i S = i] pii [ S = j S = i] = pii P = P < p ii <, i, j =, In he absence of equaion (4), we have he Sock and Wason dynamic facor model. For he idenificaion of he model, i is assumed ha he variance of v is uniy. The innovaions v and ε i are assumed o be independen for all and i. Wih he availabiliy of he esimaion mehod developed by Kim (994), he model can be esimaed by maximising he likelihood funcion. Inferences abou he unobserved nonlinear facor and he laen Markov sae can hen be obained a he same ime. The mehod consiss of a combinaion of Hamilon s algorihm and he nonlinear discree version of he Kalman filer: we refer o Kim (994) for echnical deails. To faciliae esimaion, he model can be expressed in sae-space represenaion. Wih AR() processes for boh he common facor and idiosyncraic erm, and wih n = 4 (as in he applicaion below), he model can be expressed as he measuremen and ransiion equaions 4

8 5 = 4, 4 3, 3,, e e e e e e e e c c y y y y λ λ λ λ (5) and + + = 4 3 4, 4, 3, 3,,,,, 4 4 4, 4 3, 3,, S v c c c c ε ε ε ε ψ ψ ψ ψ φ φ µ (6) 3. DATA AND RESULTS We chose four ime series ha are represenaive coinciden economic indicaors: oupu, income, sales and employmen. These series are GDP a facor cos, real household disposable income, reail sales, and employee jobs. All series are seasonally adjused quarerly observaions and logarihms are used. The sample period is from 959Q o Q. Graphs of he four series are shown in Figure. We firs es wheher he four series are individually inegraed and, if hey are, wheher hey are coinegraed. 3 We find ha we canno rejec he hypohesis ha each of he series is inegraed, and neiher can we rejec he hypohesis of no coinegraion Excep for he reail sales series, aken from Daasream, all oher daa are from he Office of Naional Saisics. The series codes are YBHH, NRJR, UKRETTOTG and BCAJ, respecively. We also ried workforce raher han employees, producing resuls similar o hose repored here. Monhly income is only available afer 986Q. 3 Resuls are available upon reques.

9 among hese variables. Therefore, we use he firs differences of he variables (muliplied by one hundred) as is implied by he model se ou in equaions () o (6). As in he model, all series are demeaned by subracing he sample mean from each difference. As in equaion (6), we began wih second order auoregressive specificaions for boh he common componen and he four idiosyncraic componens, producing he resuls presened in Table. The esimaed model suggess ha he common facor and GDP are, in fac, whie noise. Resricing he appropriae four parameers o ero produced he esimaes shown in Table. As he likelihood raio saisic from he wo models has a value of only.3, he resriced model is aken as our preferred one. Table Esimaes of he dynamic facor model wih Markov swiching Common facor φ φ µ µ p p (.477).48 (.38) (.464).358 (.9).876 (.46).947 (.57) Idiosyncraic componen y -.8 (.786) ψ i ψ i σ i -. (.83).83 (.593) λi.59 (.857) y (.874) -.38 (.53).363 (.838).67 (.3) y (.5) -.86 (.367).7993 (.).7664 (.6) y (.85).45 (.8).394 (.43). (.375) Log-likelihood y Noe: The order of he variables in i is GDP, income, sales and employmen. Sandard deviaions are in parenheses 6

10 Table Esimaes of dynamic facor model wih Markov swiching wih resricions Common facor φ φ µ µ p p (.363).3448 (.84).83 (.49).9457 (.466) Idiosyncraic componen ψ i ψ i σ i y (.587) λi.56 (.833) y (.869) -.34 (.54).38 (.834).675 (.74) y (.4) (.37).7974 (.99).7667 (.48) y (.8).48 (.8).3943 (.4).4 (.365) Log-likelihood value y Noe: The order of he variables in i is GDP, income, sales and employmen. Sandard deviaions are in parenheses The esimaed model seems successful in exracing informaion abou flucuaions in economic aciviy. The resuls suppor he presence of asymmeric business cycles ha swich beween wo differen saes, wih sae having a significanly negaive mean and sae a significanly posiive mean. The ransiion probabiliies associaed wih hese wo regimes of recession and expansion are.8 and.946 respecively. These esimaes imply ha he average duraion of he expansionary regime is ( p ) 8. 4 = quarers, which may be conrased wih ( p ) 5.6 quarers for he average duraion of he recessionary regime. The = esimaes of he mean growh raes of he business cycle common facor are. 7 and.34. Therefore, recessions on average are boh seeper and shorer, boh by a facor of approximaely hree, han expansions, which is consisen wih he findings of Kim and Nelson (998) for he U.S. Figure plos he exraced Markov swiching 7

11 common facor in boh levels and firs differences. 4 The levels of his series, which may be inerpreed as an index of he business cycle, accuraely reproduce he sylised facs of he U.K. experience, while he differences show clearly he volailiy of he 97s and he relaive sabiliy of he 99s. Moving o he idiosyncraic componen, our esimaes show ha sales has he highes weighing on he common facor, suggesing ha his series is he mos sensiive coinciden variable. This is consisen wih he common observaion ha sales respond immediaely o changes in economic condiions. The nex mos sensiive series is income, followed by GDP and employmen. Using monhly daa from he U.S., Kim and Nelson (998) found ha indusrial producion had he highes weighing, followed by income, sales and, finally, employmen. We suspec ha our differen ordering is because indusrial producion responds more swifly han GDP o economic condiions, especially when he economy is close o urning poins. Employmen is he leas sensiive o business cycle movemens and also has he smalles innovaion variance among he four variables. The negaive coefficiens of ψ i and ψ i for income and sales indicae ha he idiosyncraic componens of hese series exhibi negaive serial correlaion, while he employmen series behaves differenly wih posiive idiosyncraic auocorrelaion. Figure 3 plos he probabiliy ha he economy is in a recesssion: panel (a) shows he filered probabiliy condiional on informaion available hrough, Pr [ S Ψ ], (,,, T ) =, while panel (b) shows he smoohed probabiliy based = = T. The wo on he complee se of informaion up o T, Pr[ Ψ ], (,,, ) S T = graphs are very similar and clearly pick ou and dae correcly he hree major recessions ha he U.K. economy has experienced during he las four decades. Apar from hese, he plos idenify several brief recessions during he 96s, ypically relaed o an overvalued exchange rae and consequen balance of paymens deficis. Unforunaely, here is no official U.K. business cycle chronology ha we may relae our resuls o. We have hus compared our inferred probabiliies of recessions wih he chronology provided by Aris, Konolemis and Osborn (997), where hey use boh heir own procedure and one provided by Bry and Boschan (97) 5. Their daing is based on monhly indusrial producion and finishes in 993, and so can only be used for rough comparisons. We find ha our recession probabiliies are more closely relaed o Bry and Boschan's daing, which are shown as shaded areas on he plos of Figure 3. Almos all of heir recessions are picked up by our model, alhough he duraions of each recession are somewha differen. 4 The deails of how o obain he levels of he common facor are described in Sock and Wason (99). 5 Boh chronologies are presened in Table D of Aris, Konolemis and Osborn (997) 8

12 We also esimae a univariae Markov swiching model using GDP daa only, where he growh rae of GDP is assumed o follow an AR(4) process, as in Hamilon (989). The resuls are shown in Table 3. I is ineresing o find ha he ransiion probabiliies associaed wih he wo regimes of expansion and recession are.957 and.7 respecively, and he associaed average duraions are 3.4 and 3.35 quarers for expansions and recessions respecively. These esimaes hus suppor previous findings ha univariae Markov swiching models end o find lower probabiliies of recession and hence shorer average recession duraions. Noe ha he mean growh of GDP in recessions is 3.% per annum, while i is.6% per annum during expansions. Figure 4 plos he filered and smoohed recession probabiliies from he univariae Markov swiching model, again wih he Bry and Boschan business cycle daing. As before, he hree major recessions are idenified, bu he model fails o accoun for several of he recessions during he 96s. Table 3 Esimaes from he univariae Markov swiching model Parameer Esimaes µ (.688) µ.666 (.575) p.7 (.333) p.9566 (.36) σ.8654 (.56) φ (.893) φ (.954) φ.65 (.73) 3 φ -.45 (.887) 4 Log-likelihood value

13 4. CONCLUSIONS We have esimaed a model ha incorporaes wo key feaures of business cycles, comovemen among economic variables and swiching beween regimes of boom and slump, o quarerly U.K. daa for he las four decades. A common facor, inerpreed as a composie indicaor of coinciden variables, and esimaes of urning poins from one regime o he oher, were exraced from he daa by using he Kalman filer and maximum likelihood esimaion approach of Kim (994). Boh comovemen and regime swiching are found o be imporan feaures of he U.K. business cycle. The composie indicaor produces a sensible represenaion of he cycle and he esimaed urning poins agree fairly well wih independenly deermined chronologies. These esimaes are sharper han hose produced by a univariae Markov swiching model of GDP alone. A fairly ypical sylised fac of business cycles is confirmed by his model - recessions are seeper and shorer han recoveries. REFERENCES Aris, M.J., Konolemis, Z.G., and Osborn, D.R. (997), Business cycles for G7 and European counries, Journal of Business, 7, Bry, G. and Boschan, C. (97), Cyclical Analysis of Time Series: Seleced Procedures and Compuer Programs, Technical Paper No., New York: NBER. Burns, A.F. and Michell, W.A. (946), Measuring Business Cycles, NewYork: NBER. Chauve, M. (998), An economeric characeriaion of business cycle dynamics wih facor srucure and regime swiching, Inernaional Economic Review, 39, Diebold, F.X. and Rudebusch, G.D. (996), Measuring business cycles: A modern perspecive, Review of Economics and Saisics, 78, Filardo, A.J. (994), Business cycle phases and heir ransiional dynamics, Journal of Business and Economic Saisics,, Hamilon, J.D. (989), A new approach o he economic analysis of nonsaionary ime series and he business cycle, Economerica, 57, Kim, C.-J. (994), Dynamic linear models wih Markov swiching, Journal of Economerics, 6, -.

14 Kim, C.-J. and Nelson, C.R. (998), Business cycle urning poins, a new coinciden index, and ess for duraion dependence based on a Markov-swiching model of he business cycle, Review of Economics and Saisics, 8, 88-. Kim, M.-J. and Yoo, J.-S. (995), New index of coinciden indicaors: A mulivariae Markov swiching facor model approach, Journal of Moneary Economics, 36, Krolig, H.-M. and Sensier, M. (), A disggregaed Markov-swiching model of he UK business cycles, Mancheser School, 68, Simpson, P.W., Osborn, D.R. and Sensier, M. (), Modelling business cycle movemens in he UK economy, Economica, forhcoming. Sock, J.H. and Wason, M.W. (989), New indexes of coinciden and leading indicaors, in O.J. Blanchard and S. Fischer (ediors), MBER Macroeconomics Annual, , Cambridge: MIT Press. Sock, J.H. and Wason, M.W. (99), A probabiliy model of he coinciden economic indicaors, in K. Lahiri and G.H. Moore (ediors), Leading Economic Indicaors: New Approaches and Forecasing Records, 63-95, New York: Cambridge Universiy Press. Sock, J.H. and Wason, M.W. (993), A procedure for predicing recessions wih leading indicaors: Economeric issues and recen experiences, J.H. Sock and M.W. Wason (ediors), Business Cycles, Indicaors, and Forecasing, 95-56, Chicago, Universiy of Chicago Press. Sock, J.H. and Wason, M.W. (999), Business cycle flucuaions in U.S. macroeconomic ime series, in J. Taylor and M. Woodford (ediors), Handbook of Macroeconomics, 3-64, Amserdam: Elsevier.

15 Figure. Time series of he four coinciden variables (a) Logarihm of GDP (b) Logarihm of income

16 (c) Logarihm of sales (d) Logarihm of employee jobs 3

17 Figure. Exraced Markov swiching common facor (a) Common facor, c (b) Growh rae of common facor, c 4

18 Figure 3. Filered and smoohed recession probabiliies from Markov swiching and common facor model (a) Filered probabiliies (b) Smoohed probabiliies 5

19 Figure 4. Filered and smoohed recession probabiliies from univariae model of GDP (a) Filered probabiliies (b) Smoohed probabiliies 6