Markov-Switching Common Dynamic Factor Model with Mixed-Frequency Data

M arkov-SwitchingCommonD ynamicFactor

M odelwithM ixed-FrequencyD ata

KonstantinA . Kholodilin

kholodilin@ ires.ucl.ac.be

September18, 2001

In this paperweconsideracoincidenteconomicindicatormodelwith regime-switchingdynamics and with thetimeseries observed atdi¤ erent frequencies, forinstance, atmonthlyandquarterlyfrequencies. U ntilnow the only solution was todrop the lowerfrequency series and toestimate themodelbasedonlyon thehigherfrequencyseries. T his approach leads tothesigni…cantinformation losses. W eproposean approach allowingto overcomethis problem andtoestimateanonlineardynamiccommon fac-torwith themissingobservations takingadvantageofalltheinformation available.

Keywords: common dynamic factor, M arkov switching, mixed fre-quency data, Kalman …lter, compositeeconomicindicator.

JEL Classi…cation : C5, E3.

1 Introduction

T he estimation ofa coincidenteconomic indicator(CEI) atthe regionaland nationallevelplays averyimportantroleinmeasuringandpredictingthestate ofa¤ airs in a given region or country. T his indicator can then be used for the politicaland analyticalpurposes. T herefore this coincidenteconomic in-dicatorshould be readily available, reliable, and representative ofthe cyclical movements inthemain sectors oftheeconomy.

T heCEI shouldsatisfythetwode…ningconditions ofthebusiness cycleput forward byB urns and M itchelland stressed byD iebold and R udebusch (19 9 6) intheirsurveyofthemodernturningpoints modeling, namely: comovementof theindividualmacroeconomicseries within the cycleand asymmetricbusiness cycledynamics, whenthebehavioroftheeconomyduringexpansionsisdi¤ erent from thatin therecessions.

U ntilrecently there existed a separation between the modelcapturingthe commondynamicsofdi¤ erentmacroeconomicvariablesatthebusinesscyclefre-quencies, ontheonehand, andthenonlinearapproachtreatingdi¤ erentphases ofthebusiness cycleasymmetrically, on theotherhand. T he…rstapproach is,

amongothers, prominently represented by Stockand W atson (19 88, 19 89 and 19 9 2) whointroducedacoincidenteconomicindicatormodelaimingatextract-ingalatent, orunobserved, commondynamicmodel. T hesecondapproachwas greatlyadvanced byH amilton’s (19 89 ) breakthrough paper.

H owever, theneed foramodelsynthesizingthesetwoapproaches expressed byD ieboldand R udebusch (19 9 6) called toexistenceaM arkov-switchingcom-mon dynamicfactormodel. T his approachbecamefeasiblethanks tothetech-nique introduced in Kim (19 9 4), which permits toestimate M arkov-switching models putin a state-space form. A lmostimmediately quite a numberofap-plications ofthis technique to estimating nonlinearcommon dynamic models surged. Chauvet(19 9 8) andKim and N elson (19 9 9 ) applied ittomodelingthe commondynamicfactorwithM arkov-switchingdynamics, Kim andY oo(19 9 5) extended this modeltothetime-varyingtransition probabilities case.

T his modelallows to estimate simultaneously both the common factor(s), underlying common dynamics ofseveralmacroeconomic time series, and the probabilities of the recessions corresponding to this factor. In other words, thisapproachincorporatesnonlineardynamicsinthecommonfactorextraction by combiningtheunobserved componentmodelofStockand W atson with the M arkovregime-switchingmethodologyofH amilton.

H owever, thepracticalapplicationoftheseapproachis impededbythelack oftherelevantdatameasuredathigh(say, monthly) frequencies. A lotofvalu-ableinformation is lostbecausemany importanttimeseries areonly available atthequarterlyorannualfrequencies. Forinstance, theCEI estimatedwiththe monthlydatadoes nottakeintoaccounttheinformationcontainedintheG D P series which is available only atquarterly orlowerfrequencies. T his problem is especiallysevereattheregionallevel, sincetheregionalstatisticaldatabases aremuch morepoorthan thenationalones.

Fortunately, the problem of discrepancy in the frequency ofobservations seems tobesolved. T hesolutionwas recentlyproposed byM arianoand M ura-sawa (2000). T hey considera modelwhere di¤ erentfrequencies, say monthly and quarterly, fordi¤ erentvariables entering the modelare allowed. T his is especially usefulifwe wantourcoincidentindicator to be a proxy forsome aggregateobservable variable, e.g. G D P. A s arule the G D P dataarereleased atmuchlowerfrequencythanindividualseries characterizingspeci…csectors of theeconomy. T heM urasawaandM ariano’smodelenablesustotakeadvantage ofthevaluableinformation containedin thelower-frequencytimeseries.

O uridea is to apply this approach to the M arkov-switching common dy-namic factormodelso thatto be able to estimate CEI which considers both thecomovementofthemacroeconomicvariables and theasymmetryofthedif-ferentbusiness cyclephases withoutlosingtheimportantinformation which is otherwisewasted becauseofthediscrepancies in theobservation spacing.

T herestofthepaperis structuredasfollows. Inthenextsectionwediscuss the technicaldetails ofconstruction and estimation ofthe M arkov-switching common factormodels. In the section three we consider application ofthis methodologytotherealdata. Sectionfourconcludes thepaper. A llthegraphs and tables areputintotheA ppendixfollowingthelistofreferences.

2 T hemodelandits estimation

T hemodelofthecommon factorwith nonlinear(M arkov-switching) dynamics as theoneestimated byKim and N elson can beexpressed as follows:

¢ yt=±+ °(L )¢ ct+ ut (1)

where¢ ytis then£1 vectorofthe …rstdi¤ erences ofthe observed time series in logs;¢ ctis …rstdi¤ erenceoftheunobserved common factorhavinga regime-switchingdynamics;utisthen£1 vectorofthespeci…c, oridiosyncratic, components characterizingtheindividualdynamics ofeach oftheobserved se-ries, and°(L )is thelagpolynomialin thefactorloadings.

T hecommon dynamicfactoris modeled as:

Á(L )¢ ct=¹(st)+ "t (2) whereÁ(L )istheA R (p)lagpolynomial;¹(st)isthecommonfactorintercept dependingonthestatevariablestfollowinga…rst-orderM arkov-chainprocess, and"t»N ID(0;¾2(st))¡thus the variance ofthe common factorshockmay alsobe state-dependent. In amore generalspeci…cation the coe¢cients ofthe autoregressivepolynomialÁ(L )maydepend on thestatetoo.

T hevectoroftheidiosyncraticcomponents canberepresented as follows:

Ã(L )ut =´t (3)

where´t »N ID(0;§)and both the lag polynomialÃ(L )and variance-covariance matrix§have adiagonalstructure. Each idiosyncraticcomponent is modelled asA R(qi)wherei= 1;:::;n:In principle, the autoregressive order maybedi¤ erentacross thespeci…ccomponents and maybeequaltozero.

N ow assume that di¤ erent series1 _{are observable at di¤ erentfrequencies.}

Supposethatn1 time series (y1t) areobserved atthe lowerfrequencyf, while the restofthe seriesn2 =n¡n1 (y2t) are measured ata higherfrequency, which wemaynormalizeto1:D enotebyy¤

1ttheunobserved values ofthe…rst

n1 measuredatthehigherfrequency. T hentheobservedseriescanbeexpressed

in terms oftheunobserved as follows:

y1t= 1 f f¡1 X i=0 L iy1¤t (4) H enceaftertakingthe…rstdi¤ erenceoftheobservablelowerfrequencyseries, thegrowthrates oftheseseries would beas:

(1 _¡L f_)y 1t= 1 f( f¡1 X i=0 L i₎2₍₁ ¡L )y¤_1t (5) where(Pf_i=0¡1 L i₎2₌P2f¡1 i=0 (f+ 1 ¡ji¡fj)L iorsimpler ( f¡1 X i=0 L i₎2_{= 1 + L + 2L} 2_{+ 3L} 3_{+ :::+ 3L} 2f¡4_{+ 2L} 2f¡3_{+ L} 2f¡2_{+ L} 2f¡1 ₍₆₎

T herefore the vectorofthe growth rates ofthe observed series may be de-composed as: µ (1 _¡L f_)y 1t (1 _¡L )y2t ¶ =±+ °(L ) Ã 1 f( Pf¡1 i=0 L i)2 1 ! (1 _¡L )ct+ Ã 1 f( Pf¡1 i=0 L i)2 1 ! ut (7 ) In orderto be estimated, this modelcan be expressed in the state-space form. T hemeasurementequation: ¢ yt=A st+ wt (8) T ransition equation: st=¹t+ C st¡1+ vt (9 ) where¢ yt=¡(1 ¡L f)y1t (1 ¡L )y2t ¢0

is then_£1 vectorofobserved variables indi¤ erences;

st=¡¢ c¤t ut ¢0isthem£1 statevectorcontainingthecommondynamic factorvector¢ c¤ t= ¡ ¢ ct¡1 ¢ ct¡2 ::: ¢ ct¡r ¢0_{, with} r= max_fp;2f_¡1_g;

and thespeci…ccomponents vector

ut=

¡

u1t ::: u1t¡l ::: unt ::: unt¡qn

¢0

;withl= max_fq1;2f¡1g; ¹_t=¡¹(st) 0 ::: 0 ¢0is thevectorofintercepts;and …nally

vt=¡"t 0 ::: ´1t ::: ´nt ::: 0

¢0_{is thevectorofdisturbances}

:

T hedimension ofthestatevector,m, is determined as:

m=r+ n1 ¤l+ Pn_{i=n1+ 1}qi

T hesystem matrices havethefollowingstructure: T hemeasurementn_£mmatrix:

A = 0 B B B @ °₁i(f)¤ 0 °₂ or¡1 i(f)¤ ... °n iqn 1 C C C A

where¤is the1£(2f¡1 )vectorofcoe¢cients ofthe(Pf_i=0¡1 L i₎2_;o

kis the

k£1 vectorofzeros, andikis the …rstrowofthek£kidentity matrix, and

i(f)is theindicatorfunction: i(f)= ½ _{0, if} t= 1h;:::;(f_¡1 )h 1, otherwise whereh= 1;2;3;::: T hem£mtransition matrix: C = 0 B B B B B B B B B @ © or 0 Ir¡1 ª1 ol Il¡1 ... ªn oqn 0 Iqn¡1 1 C C C C C C C C C A

where©andªi (i = 1;:::;n) are the row vectors of the autoregressive coe¢cients;Ikis thek£kidentitymatrix.

T hen_£nvariance-covariancematrixofthedisturbancestothemeasurement equation: R = µ I(f) O O O ¶

whereI(f)is thediagonaln1£n1 matrixwiththeindicatorfunctions,i(f),

on themain diagonal. T hem_£mvariance-covariancematrixofthedisturbances tothetransition equation: Q = 0 BB B B B B @ ¾2_(s t) 0 ... ¾2 1 ... 0 ¾2 n 1 CC C C C C A

W e introduce three identifyingassumptions in this speci…cation ofmodel. First, thevariance-covariancematrixQ isdiagonal. Secondly, wemayseteither

°1 = 1 or¾2(st= 1 )= 1:W echosethe…rstoption.

T heunobservedvaluesofthelower-frequencytimeseriesaretreatedasmiss-ing. A s M ariano and M urasawa (2000) have shown, they can be replaced by any random variableas soon as itis notcorrelated with theparameters ofthe modelwe aregoingtoestimate. In particular, these missingobservations may besubstitutedbyzeros. T hus, the…rstn1 observedvariableswillbeconstructed

as follows:

y1t=

½ _{0, if}

t= 1h;:::;(f_¡1 )h

Inprinciple, wecandothiskindofsubstitutionnotonlyfortheobservations betweentheobservedvalues ofthelower-frequencytimeseries, butalsoincase ofthe series which are shorterthan the others. In the generalcase we may de…netheindicatorfunction as:

i(f)=

½ _{1, if} t2¥ 0, otherwise

where¥is thesetofdates forwhich theshortesttimeseries is observable. Forinstance, when thet1 initialobservations are missing, the set¥willbe

de…ned as:

¥=ftjt> t1g

Inthecasewhenthesamevariableis alsotheonewhichis measured atthe lowerfrequency, thede…nition of¥willbeas:

¥=ftjt> t1 andt= 16 h;:::;(f¡1 )hg, whereh2Z

W eestimatethemodelusingmaximum likelihood method. Forthederiva-tion of the approximate likelihood funcForthederiva-tion for the common dynamic factor models with M arkovswitchingwereferourreadertoKim and N elson (19 9 9 ).

3 A pplication

3.1 Simulatedexample

First, inordertocheckourmodel, wehavesimulatedasimplecommondynamic factormodelwith aM arkov-switchingdynamics. T heparameters used tosim-ulatethe arti…cialtimeseries arepresented in thesecond column oftheT able 1 ofA ppendix. T herewere…vetimeseries with540 observations ineachgener-ated. T henthe…rstseries was chosentobethelow-frequencyseries. T herefore there were ”quarterly” observations calculated as the means overeach ”quar-ter”. T hus, forthis timeseries we may observeonly the dataaggregated over each three observations, while the remainingfourtime series are observed at the”monthly” frequency.

W eestimated anonlineardynamiccommon factormodelwith di¤ erentob-servation frequencies, whose structure replicates the D G P ofthesimulated se-ries. T heestimatesoftheparameterstogetherwiththecorrespondingstandard errors, and p-values, arecontained in the columns 3 through 5 ofthe T able1. T he estimated parameters, save forthe variance ofthe idiosyncratic compo-nentcorrespondingtothe quarterly observed series, are very close tothe true parameters.

Figure 1 comparingthe true and estimated common componentas wellas the true regime with the smoothed conditionalprobabilities ofthe regime 2 (recession), also shows strikingsimilarity between the true and estimated se-ries. T he conditionalregime probabilities sometimes miss the recession when its duration is veryshort.

T hus, ourmodel, when itcorresponds tothe D G P ofthe series, estimates theunknown parameters oftheprocess su¢cientlywell.

3.2 R ealexample

H avingtested the performance ofourmodelon the arti…cialdata, we applied ittotheactualdata. T hedatausedarethesameas inM arianoandM urasawa (2000) study. T hese arethe quarterly U S realG D P series from the …rstquar-terof19 59 tillthe lastquar…rstquar-terof19 9 8 and fourmonthly U S macroeconomic timeseries stretchingfrom January19 59 toD ecember19 9 8, namely: employees on nonagriculturalpayrolls; personalincome less transferpayments; index of industrialproduction;and manufacturingand tradeseries2_.

T oselectthelagorder, we applied A kaikeinformation criterion (A IC) and Schwarz B ayesianinformation criterion (SB IC) computedas follows:

A kaikeinformation criterion:

A IC = 2 log[L (µ)]¡2[n1p+n2q] (10)

whereL (µ)is thelikelihood function valueatmaximum;n1 numberofthe

low-frequencyseries (in this casewehaveonlyonesuch timeseries - quarterly G D P );n2 is thenumberofthehigh-frequencyseries;pandqaretheorders of

theA R polynomials ofthelow- and high-frequencyseries, respectively. Schwarz B ayesian information criterion:

S B IC = 2 log[L (µ)]¡[n1p+ n2q]log(T) (11)

whereT is thenumberofobservations.

T he values ofthe log-likelihoods forthe various autoregressive ordercom-binations(p;q)as wellas thetwoinformation criteriaarepresented in T able2 oftheA ppendix. T heA IC chooses (3,3) whileSB IC selects (1,2) combination as theoptimalone. W earegoingtousethelattercombination as amorepar-simonious. T his is the samecombination which was suggested by the SB IC in thelinearcase(seeM arianoand M urasawa(2000)).

W e representthe estimates ofthe parameters ofthe linearcommon factor model(takenfrom M arianoandM urasawa(2000)) andourownestimatesofthe common factormodelwith M arkov switchingin the T able 3 ofthe A ppendix. T heestimatedparameters forthelinearand nonlinearmodels areverysimilar, with the exception ofthe autoregressive parameterofthe common dynamic factorwhich is slightlysmallerwhen theM arkovswitchingis introduced.

B ased on the parameterestimates ofthe nonlinearcommon factormodel withdi¤ erentobservationfrequencies, wecalculatedtheestimateofthecommon factorin thesamewayas itis donein Kim and N elson (19 9 9 ), thatis,

2_{T hedataweredemeanedandnormalizedtohaveunitvariance. T heywerekindlyprovided} tous by Y .M urasawa.

ct=ct¡1 + ¢ ct+± (12)

where±isthemeanofthecommonfactorcomputedasinStockandW atson

(19 88).

Figure 2 shows the evolution ofthecommon factorand theconditionalre-cession probabilities obtainedfrom theestimation ofourmodelplottedagainst the N B ER recession dates, where the latterare represented by the shading. T he smoothed recession probabilities exactly correspond to the N B ER reces-sion chronology, the only di¤ erence beingthe recesreces-sion detected by ourmodel in theverybeginningofthesampleand missed bytheN B ER .

4 Summary

In this paperweintroduceaM arkov-switchingcommon dynamicfactormodel with missingobservations. U ntilnowonly thedataofthesamefrequency and with the same length were used toestimate the latentcommon factormodels with M arkov-switchingdynamics. B uildingon theextension ofthelinearcom-monfactormodeltothecaseofthedatawith di¤ erentobservation frequencies proposed byM arianoand M urasawa(2000), weo¤ erasolution totheproblem ofmissingobservations in thenonlinearcase.

T his would allowto preventthe losses ofvaluable information concerning the evolution ofthe common dynamic factorwhich may be contained in the lower-frequency timeseries and, in general, in thetimeseries with anytypeof missingvalues.

R eferences

[1]Chauvet M . (19 9 8) ”A n Econometric Characterization of B usiness Cycle D ynamics withFactorStructureandR egimeSwitching”

InternationalEco-nomicR eview39 ,9 69-9 6.

[2]ChauvetM ., P otterS. (2000) ”Coincidentand L eading Indicators ofthe StockM arket”JournalofEmpiricalFinance7 , 87 -111.

[3]D ieboldF.X ., R udebuschG .D .(19 9 6) ”M easuringB usinessCycles: A M od-ern P erspective”T he R eviewofEconomics andStatistics7 8, 67-7 7 . [4]H amiltonJ.D . (19 9 4)T imeSeries A nalysis. N ewJersey: P rinceton U

niver-sityP ress.

[5]Kim C.-J.(19 9 4) ”D ynamicL inearM odelswithM arkov-Switching”Journal

ofEconometrics60, 1-22.

[6]Kim C.-J., N elson C.R . (19 9 9 )State-Space M odels with R egime Switching:

Classical and G ibbs-Sampling A pproaches with A pplications. Cambridge:

[7 ]Kim M .-J., Y ooJ.-S. (19 9 5) ”N ewIndex ofCoincidentIndicators: A M ul-tivariate M arkov SwitchingFactorM odelA pproach”JournalofM onetary

Economics36, 607 -30.

[8]M ariano R .S., M urasawa Y . (2000) ”A N ewCoincidentIndex ofB usiness Cycle B ased on M onthly and Q uarterly Series” Institute ofEconomicR e-search (KyotoU niversity) discussion paper518.

[9 ]Stock J.H ., W atson M .W . (19 88) ”A P robability M odelofthe Coincident EconomicIndicators”, N B ER workingpaper27 7 2.

5 A ppendix

T able1. Simulated example: trueand estimatedparameters P arameter T rue Estimated St. error p-value

p1 1 0.9 5 0.9 3 0.02 0.0 p22 0.84 0.87 0.03 0.0 ¹1 0.4 0.43 0.03 0.0 ¹2 -0.6 -0.68 0.05 0.0 °2 0.5 0.44 0.05 0.0 °3 0.8 0.81 0.02 0.0 °4 2.0 2.01 0.06 0.0 °5 1.7 1.7 3 0.05 0.0 Á 0.6 0.56 0.03 0.0 Ã1 -0.5 -0.61 0.16 0.0 Ã2 0.6 0.59 0.04 0.0 Ã3 -0.1 -0.06 0.05 0.12 Ã4 -0.2 -0.17 0.06 0.0 Ã5 -0.8 -0.84 0.02 0.0 ¾2 1 0.25 0.9 9 0.16 0.0 ¾2 2 0.36 0.38 0.02 0.0 ¾2 3 0.16 0.16 0.01 0.0 ¾2 4 0.49 0.53 0.05 0.0 ¾2 5 0.81 0.81 0.06 0.0 ¾2 c 0.16 0.15 0.02 0.0

T able2. L agselection analysis

(p,q) L ogL ik A IC SB IC (0,0) -1643.52 -3287 .04 -3287 .04 (0,1) -1605.68 -3221.36 -3242.82 (0,2) -1565.85 -3151.7 -319 4.62 (0,3) -1555.9 5 -3141.9 -3204.48 (1,0) -1626.37 -3254.7 4 -3259 .03 (1,1) -1589 .89 -319 1.7 8 -3217 .53 (1,2) -1550.31 -3122.62 -3169.83 (1,3) -1539 .81 -3111.62 -317 8.37 (2,0) -1625.38 -3254.7 6 -3263.34 (2,1) -1589 .3 -319 2.60 -3222.64 (2,2) -1549 .68 -3123.36 -317 4.86 (2,3) -1539 .81 -3113.62 -3184.54 (3,0) -1625.29 -3256.58 -3269.10 (3,1) -1589 .26 -319 4.52 -3227 .89 (3,2) -1545.7 3 -3117 .46 -317 1.69 (3,3) -1534.41 -3104.82 -317 9 .9 1

T able3. R esults ofestimation oflinearand M arkov-switchingmodels

P arameter L inear¤ _{N onlinear}

Coe¢cient Coe¢cient St. error p-value

p1 1 - 0.9 7 0.01 0.0 p22 - 0.83 0.13 0.0 ¹1 - 0.06 0.02 0.0 ¹2 - -0.39 0.11 0.0 °₂ 0.48 0.49 0.04 0.0 °3 0.83 0.83 0.06 0.0 °4 2.10 2.10 0.13 0.0 °5 1.7 1 1.7 2 0.11 0.0 Á 0.56 0.35 0.07 0.0 Ã1 1 -0.02 -0.04 0.10 0.36 Ã1 2 -0.7 8 -0.7 9 0.11 0.0 Ã21 0.11 0.10 0.05 0.01 Ã22 0.45 0.45 0.05 0.0 Ã31 -0.04 -0.05 0.05 0.17 Ã32 0.02 0.02 0.08 0.43 Ã41 -0.03 -0.02 0.07 0.41 Ã42 -0.06 -0.06 0.07 0.21 Ã51 -0.44 -0.44 0.05 0.0 Ã52 -0.22 -0.22 0.05 0.0 ¾2 1 0.19 0.19 0.04 0.0 ¾2 2 0.02 0.02 0.00 0.0 ¾2 3 0.10 0.10 0.01 0.0 ¾2 4 0.26 0.27 0.03 0.0 ¾2 5 0.60 0.60 0.04 0.0 ¾2 c 0.08 0.06 0.01 0.0

* T heestimates oftheparameters forthelinearmodelaretakenfrom M ar-ianoand M urasawa(2000).

True and estimated common dynamic factor

1 41 81 121 161 201 241 281 321 361 401 441 481 521 0 50 100 150 200 250 300

Smoothed recession probabilities vs. true low regime

1 41 81 121 161 201 241 281 321 361 401 441 481 521 0.00 0.25 0.50 0.75 1.00 Figure1:

US coincident indicator, 1959:1-1998:12

1959 1962 1965 1968 1971 1974 1977 1980 1983 1986 1989 1992 1995 1998 0 20 40 60 80 100 120

Smoothed recession probabilities vs. NBER dates

1959 1962 1965 1968 1971 1974 1977 1980 1983 1986 1989 1992 1995 1998 0.00 0.25 0.50 0.75 1.00 Figure2: