Mixed-frequency VAR models with Markov-switching dynamics

Transcription

1 Mixed-frequency VAR odels wih Markov-swiching dynaics Maxio Caacho * Universidad de Murcia Absrac: This paper exends he Markov-swiching vecor auoregressive odels o accoodae boh he ypical lack of synchroniciy ha characerizes he real-ie daily flow of acroeconoic inforaion and econoic indicaors sapled a differen frequencies. The resuls of he epirical applicaion sugges ha he odel is able o capure he feaures of he NBER business cycle chronology very accuraely. JEL Classificaion: E32, C22, E27. Keywords: Business Cycles, Oupu Growh, Tie Series. * M. Caacho, Deparaeno de Méodos Cuaniaivos para la Econoía y la Epresa. Faculad de Econoía y Epresa. Capus de Espinardo, 31 Universidad de Murcia, Spain. Phone: Fax: E-ail: caacho@u.es 1

2 1. Inroducion Early proposed by Sis (198), he Vecor Auoregression (VAR) specificaion is one of he os successful, flexible and easy o use odels for he analysis of ulivariae ie series. Since he influenial work of Hailon (1989) any auhors have used Markov-swiching exensions of hese odels o capure he business cycle regie shifs ypically observed in econoic daa. Soe exaples of Markov- Swiching VAR (MS-VAR) odels are Krolzig (1997), Caacho and Perez Quiros (22) and Paap, Sergers and van Dik (29). Alhough econoic daa are rarely colleced a he sae insances in ie, hese sandard MS-VAR applicaions are resriced o use econoic indicaors ha us be available a he sae frequency. In addiion, hese applicaions also rely on he unrealisic assupion ha he real-ie daa flow of all he variables involved in he epirical analyses occurs a he sae ie. To overcoe hese drawbacks, his paper develops a odel ha exends o a Markov-swiching conex he Mixed Frequency VAR (MF-VAR) odel proposed by Mariano and Murasawa (211). I call his odel Markov-Swiching Mixed Frequency VAR (MS-MF-VAR). The exension offers he ineresing addiional inforaion of convering he business cycle signals provided by several quarerly and onhly econoic indicaors ino recession probabiliies. I apply he ehod o US coinciden indicaors o copue inferences of he US business cycles. The coponen indicaors, used by Mariano and Murasawa (211) in a linear MF-VAR odel o consruc a onhly index of he econoic aciviy, are quarerly real GDP and he four onhly coinciden indicaors ha ake up he coinciden indicaor currenly released by he Conference Board. Using hese indicaors in a Markov-swiching MF-VAR odel, he inferred onhly probabiliies of recession sugges ha he nonlinear exension of he odel is capable of idenifying he US business cycles wih very high accuracy. 2. MF-VAR odels wih Markov-swiching 2.1. Specificaion Le X, be a vecor of econoic indicaors ha ay include 1 quarerly indicaors, X,1, and 2 onhly indicaors, X,2. Le e assue ha ln X, conains a uni roo. To sar wih, le e assue ha all he indicaors are observed every onh, alhough I will relax his assupion soon. If he saple ean of he wihin quarer aciviy can be well approxiaed by he geoeric ean, Mariano and Murasawa (23) show ha he quarer-over-quarer q growh rae of quarerly indicaors copued a each onh of he saple, Y,1, can be expressed as he averaged su of previous onh-over-onh growh raes q Y,1 = Y,1 + Y 1,1 + Y 2,1 + Y 3,1 + Y 4,1. (1)

3 The onh-over-onh growhs of he econoic indicaors, Y ( Y,1 ', Y,2 ) = ' ', are hen assued o follow a Markov-swiching VAR(p) process. Therefore, he consan ers, he auoregressive coefficiens and he covariance arix are driven by an unobservable wo-sae variable s where ~ i (, ) s Y = η + φ Y φ Y + ε, (2) s 1, s 1 p, s p ε Σ. The sae variable is assued o evolve according o an irreducible 2-sae Markov chain whose ransiion probabiliies are defined by (,,..., ) p s = s = i s = h ϕ = p s = s = i = p, (3) i where i,=,1, and ϕ refers o he inforaion se up o period. In shor, his odel endogenously peris he odel paraeers o swich as he dae and regie changes. The MF-VAR specificaion his odel can be easily saed in sae space represenaion and esiaed by using he Kalan filer. Le e assue ha p 5, for q Y = Y ', Y ' ', he easureen equaion, Y = H β + E, insance p=1. 1 Calling (,1,2 ) where ~ (, ) E i R, can be defined as s Y Y Y 1 I I I 1 I 1 1 I 1 1 = 3, 3,, 3, 3 Y 2 I, (4) I 2 I 2 I 2 I 2 2 Y 3 Y 4 where E = and R=. The ransiion equaion, β = µ + F β 1 + V, wih V ~ i, Q, can be saed as s s s Y ηs φ1, s Y 1 ε Y 1 I Y 2 Y = 2 + I Y +, (5) 3 Y 3 I Y 4 Y 4 I Y 5 where 1 Modifying he expressions for higher lag orders is sraighforward. 3

4 Q s Σs =. (6) 2.2. Esiaion and signal exracion The Kalan filer can be used o esiae odel s paraeers and o infer unobserved coponens. Saring he algorih wih iniial values β and P, he predicion equaions when s -1 =i and s = are ( i, ) i β = µ + F β, (7) ( i, ) P = F P F + Q, (8) i ', where β i 1 is an inference on β condiional on inforaion up o -1 given he saes s - 1=i and s =, ( i, ) i P 1 is is covariance arix, 1 1 β is an inference on β 1 based on i inforaion up o -1 given he sae s -1 =i and 1 1 is is covariance arix. These expressions can be used o copue predicion errors and heir covariance arix ( i, ) ( i, ) v = Y H β, (9) P 1 1 ( i, ) ( i, ) Ω = H P H, (1) 1 1 ' which can be used o evaluae he log likelihood funcion ln ( 2 i, i,, ) 1, 1 1 ' i i l = π Ω + v Ω 1 v 1 = 1 i= 1 2. (11) A each ieraion, one can use he nonlinear filer proposed by Hailon (1989), o p s =, s = i ϕ and he filered probabiliies copue he oin probabiliies ( 1 ) p( s = ϕ ). Once he observed variables are realized a he end of ie, in each ieraion he sae vecor and is covariance arix are updaed as follows 1 ' 1 β = β + P H Ω v (12) i, i, i, i, i, P = P P H ' Ω H P. (13) i, i, i, i, i, As noed by Ki (1994), he above Kalan filer produces 2-fold increases in he oal nuber of cases o consider. Thus, if he filer does no reduce he nuber of ers a each ie, i becoes copuaionally unfeasible. In line wih his auhor, I propose o collapse he Kalan filer o a single poserior a each and subsiue (12) and (13) by β = 2 i= 1 ( i, =, = ϕ β ) p s s i 1 ( = ϕ ) p s, (14) 4

5 P = 2 i= 1 ( = ϕ ) ( i, ) ( i, ) i, p s, = s 1 = i ϕ P + β β β β ', (15) p s respecively. Once paraeers are esiaed, he soohed probabiliies, which are based on all he inforaion available in he saple, can be esiaed ieraively fro he filered probabiliies as follows 2 p( s + 1 = k ϕ T ) p( s = ϕ ) p ( s + 1 = k s = ) p( s = ϕ T ) =. (16) p s = k ϕ k = The soohed probabiliies are very useful o evaluae he in-saple accuracy of a Markov-swiching odel o capure he business cycles Dealing wih issing observaions So far, we have assued ha all he variables included in he odel are always available a onhly frequencies for all ie periods. However, his assupion is quie unrealisic when using he odel o copue real-ie inferences of he business cycles for wo reasons. The firs reason has o do wih ixing quarerly and onhly frequencies. Since quarerly daa is only observed in he hird onh of he respecive quarer, quarerly indicaors exhibi wo issing observaions in he firs wo onhs of each quarer. The second reason has o do wih he flow of real-ie daa, which iplies ha he publicaion lag of he indicaors is differen. Therefore, he daa vinages ypically exhibi issing daa a he end of he saple of he indicaors wih larger publicaion delays. As described in Mariano and Murasawa (23), he syse of equaions reains valid wih issing daa afer a suble ransforaion. These auhors propose replacing he issing observaions wih rando draws r, whose disribuion canno depend on he paraeer space ha characerizes he Kalan filer. 2 Then, he easureen equaion are ransfored convenienly in order o allow he Kalan filer o skip he issing observaions when updaing. Le Y i be he i-h eleen of he vecor Y and R ii be is variance. Le His be he i-h row of he arix Hs, which has 5 coluns, and le 5 be a row vecor of 5 zeroes. The easureen equaion can be replaced by he following expressions Y Yi if Yi is observable =, (17) r oherwise + i His if Y is observable i H + is =, (18) oherwise E 5 Ei if Yi is observable =, (19) r oherwise + i 2 2 We assue ha ~ (, r ) r σ for convenience bu replaceens by consans would also be valid. 5

6 and R Riis if Y is observable i =, (2) σ oherwise + is µ + s, which is a colun vecor wih he drifs 2 r η + s in he firs 5 cells and 4 zeroes elsewhere, where η is if Y is observable i η + is =. (21) oherwise This subsiuion leads o a ie-varying sae space odel wih no issing observaions so he Kalan filer can be direcly applied o Y + i, µ +, H +, E + i, R +. s is is 3. Applicaion The purpose of his secion is o show how he regie-swiching MF-VAR described in he previous secion works in epirical applicaions. Toward his end, I exend he epirical analysis of Mariano and Murasawa (21), who consruc a coinciden index of he US econoic aciviy fro a linear MF-VAR, o addiionally copue inferences of he US business cycles fro a MS-MF-VAR. Following hese auhors, he coponen indicaors are quarerly growh raes of real GDP and he four onhly coinciden indicaors ha ake up he Coinciden Indicaor released by he Conference Board: Eployees on non-agriculural payrolls, Personal incoe less ransfer payens, Index of indusrial producion and Manufacuring and rade sales, all of he in onhly growh raes. The daa vinage was downloaded on July, 18h 213 and he saple period is fro January 196 o June 212. I is worh noing ha any issing daa appear in his daa vinage. According o he release calendar of he indicaors used in he analysis, several ouliers appear a he end of he ie series. In paricular, he laes figures of he indicaors are March 213 for GDP, March 213 for sales, May 213 for incoe and June 213 for indusrial producion and eployen. In addiion, he quarerly GDP is observable every hird period only. Following Mariano and Murasawa (21), I selec a lag lengh p=1. Following Hailon (1989) and Chauve (1998) who consider ha he shifs do no depend on he dynaics of he auoregressive process or he covariance arices, only he drifs are allowed o swich. Figure 1 copares he onhly esiaes of he US econoic aciviy ha are obained fro he linear MF-VAR odel and fro y Markovswiching exension. Boh indicaors are in consonance wih he NBER-referenced business cycles, which are ploed as shaded areas. The posiive growh raes are soeies inerruped by broad changes of direcion ha see o ark quie well he US recessions. In spie of he siilar perforance of he linear and nonlinear approaches o consruc onhly indicaors of he US econoic aciviy, he laer approach offers he ineresing addiional inforaion of convering he business cycle signals provided by he econoic indicaors ino recession probabiliies. To asser he accuracy of he odel o accoun for he business cycles, Figure 2 plos he values of he soohed recession probabiliies of sae s =1. According o he reasonable aching beween he quarers of high probabiliies of sae 1 and he NBER recessions, i is easy o inerpre sae 1 as recession and he series ploed in his figure as probabiliies of being in recession. The 6

7 probabiliies are close o eiher zero or one, suggesing ha he odel is capuring well he underlying paern of he dichooous shifs beween expansions and recessions. In spie of he high correlaion beween he probabiliies of recession and he NBER referenced recessions, he quesion is wheher or no y new exension ouperfors exising odels. The naural copeior is he MS-VAR odel of Krolzig (1997), Caacho and Perez Quiros (22) and Paap, Sergers and van Dik (29). In addiion, I check he relaive perforance of he odel wih he Markov-Swiching Dynaic Facor (MS-DF) odel proposed by Ki and Yoo (1995) and Chauve (1998). 3 To quanify he relaive abiliy of hese odels o deec he acual sae of he business cycle, I copue Quadraic Probabiliy Score (QPS), which is he ean squared deviaion of he differen probabiliies of recession fro a NBER-referenced recessionary duy. Table 1 shows ha MF-MS-VAR ouperfors MS-VAR and MS-DF, alhough he iproveens wih respec o MS-VAR are larger. The p-values of he null of differen accuracy (Diebold and Mariano, 1995) show ha he iproveens are saisically significan in he case of MF-MS-VAR versus MS-VAR, bu one canno reec he null of equal predicive accuracy of MF-MS-VAR and MS-DF a any reasonable significance level. However, his analysis is in-saple and ois he effec of he asynchronous releases ha characerizes he real-ie flow of acroeconoic inforaion. To perfor a ore realisic assessen of he acual epirical reliabiliy of MF- MS-VAR, I develop a pseudo real-ie analysis as suggesed by, aong ohers, Giannone, Reichlin and Sall (28). Towards his end, I use he laes available daase o consruc daa vinages ha are recursively updaed onhly in he iddle of each onh. The firs daa vinage of our experien refers o Sepeber 15, 1976 and he las daa vinage refers o July 15, 213. Since he publicaions daes of he indicaors exhibi relaively sable calendars, he successive vinages can easily replicae he publicaion lags ha characerize realie analyses. I iplies ha he pseudo real-ie probabiliies inferred fro MS-VAR and MS-DF us be copued as wo-onh-ahead forecass since sales exhibi publicaion delays of abou wo onhs. 4 According o Table 1, MF-MS-VAR ouperfors boh MS-VAR and MS-DF since i exhibis lower QPS han hese odels. According o he Diebold-Mariano ess, he gains are saisically significan in boh cases, alhough a significance levels greaer han.5 in he laes case. 4. Conclusion Nowadays, here has been a grea deal of ineres in odeling he business cycle feaures of ulivariae ie series. The MS-VAR odels used in he epirical analyses are only of liied usefulness in pracice since hey are resriced o use econoic indicaors ha us be sapled a he sae frequencies and ha canno exhibi publicaion delays. The MF-VAR odels are resriced o linear analysis and do no provide recession probabiliies. To overcoe hese drawbacks, his paper exends he MS-VAR odels o deal wih he proble of ixed sapling frequencies and o accoun for asynchronously published econoic indicaors and he MF-VAR odels o accoun for regie swiching business cycle asyeries. 3 In conras o hese auhors, I do no ipose a single-index facor srucure in he odel. 4 All he odel use paraeers esiaed fro he laes available saple. Re-esiaing he odels is unfeasible since i would iply esiaing large nubers of paraeers wih shor saples. 7

8 Acknowledgeens Errors are y own responsibiliy. I hank he edior and an anonyous referee for heir coens and he financial suppor of MICINN (ECO ). Sofware (wrien in GAUSS) used in his paper can be obained fro he auhor s web page. References Caacho, M., Perez Quiros, G., 27. This is wha he leading indicaors lead. Journal of Applied Econoerics 17: Chauve, M An econoic characerizaion of business cycle dynaics wih facor srucure and regie swiches. Inernaional Econoic Review 39: Diebold, F., and Mariano, R Coparing predicive accuracy. Journal of Business and Econoic Saisics 13: Giannone, D., Reichlin, L., and Sall, D. 28. Nowcasing: The real-ie inforaional conen of acroeconoic daa. Journal of Moneary Econoics 55: Hailon, J., A new approach o he econoic analysis of nonsaionary ie series and he business cycles. Econoerica. 57, Ki, C Dynaic linear odels wih Markov swiching. Journal of Econoerics 6: Ki, M., and Yoo, J New index of coinciden indicaors: A ulivariae Markov swiching facor odel approach. Journal of Moneary Econoics 36: Krolzig, H-M Markov swiching vecor auoregression: Modeling, saisical inference and applicaion o business cycle analysis. Lecure Noes in econoics and aheaical syse, 454. Springer-Verlag, Berlin. Mariano, R., and Murasawa, Y. 23. A new coinciden index of business cycles based on onhly and quarerly series. Journal of Applied Econoerics 18: Mariano, R., and Murasawa, Y A coinciden index, coon facors, and onhly real GDP. Oxford Bullein of Econoics and Saisics 72: Paap, R., Segers, R., and van Dik, D. 29. Do leading indicaors lead peaks ore han roughs? Journal of Business and Econoic Saisics 27:

9 Table 1. Analysis of he relaive perforance In-saple Pseudo real-ie Quadraic probabiliy score MS-MF-VAR.8.6 MS-VAR MS-DF.9.8 p-value of no differen accuracy es MS-MF-VAR vs MS-VAR.. MS-MF-VAR vs MS-DF.36.5 Noes. Quadraic probabiliy score easures he ean squared deviaion of he differen ypes of inferences fro a recessionary duy ha is consruced fro he NBER business cycles. The es of no differen accuracy is he Diebold and Mariano (1995) es. 9

10 Figure 1. Monhly indicaors of quarerly growh rae Noes. Doed (sraigh) line refers o he onhly index of quarerly GDP growh rae fro Markov-swiching (lineal) ixed-frequency VAR odel. Shaded areas correspond o recessions as docuened by he NBER. The saple goes fro o Figure 2. Soohed probabiliies of sae Noes. Shaded areas correspond o recessions as docuened by he NBER. The saple goes fro o