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Detection of the Industrial Business Cycle using SETAR
models
Dominique Guegan, Laurent Ferrara
To cite this version:
Dominique Guegan, Laurent Ferrara. Detection of the Industrial Business Cycle using SETAR models.
Journal of Business Cycle Measurement and Analysis, OECD Publishing, 2005, 2, pp.353-371.
�halshs-00201309�
using SETAR Models
Laurent Ferrara
and Dominique Guegan y
September13, 2005
Abstra t:
In this paper, we onsider a threshold time series model in order to take into a ount ertain stylized fa ts of the industrial business y le, su h as asymmetriesinthephasesofthe y le. Ouraimistopointoutsome thresh-olds under (over) whi h a signal of turningpoint ould be given. First, we introdu e the various threshold models and we dis uss both their statisti- altheoreti al and empiri alproperties. Espe ially, we review the lassi al te hniquesto estimatethe numberof regimes, thethreshold,the delayand the parameters of the model. Then, we apply these models to the Euro-zone industrial produ tion index to dete t, through a dynami simulation approa h,thedates ofpeaks and troughsinthebusiness y le.
Keywords:
E onomi y le, turning pointdete tion, Thresholdmodel,Euro-zoneIPI.
JEL Classi ation: C22,C51,E32.
Centre d'Observation E onomique de la CCIP and E.N.S. Ca han, MORA-IDHE, CNRSUMR8533. Adress: COE,27AvenuedeFriedland,75382ParisCedex08. E-mail: lferrara ip.fr
y
E.N.S. Ca han, Departement d'E onomie et Gestion, MORA-IDHE, CNRS UMR 8533, SeniorA ademi Fellow del'Institut Europla e de nan e (IEF). Adress: 61 Av-enuedu President Wilson,94235,Ca hanCedex, Fran e. E-mail: guegane ogest.ens- a han.fr
Re ently, we witnessedthedevelopmentof newtoolsinbusiness y le anal-ysis, mainly based on non-linear parametri modeling. Non-linear models have thegreat advantage to be exibleenoughto take into a ount ertain stylized fa ts of the e onomi business y le, su h as asymmetries in the phasesof the y le. In thisrespe t, emphasis has been pla edon the lass of non-linear dynami models that a ommodate the possibility of regime hanges. Espe ially, Markov-Swit hing models popularized by Hamilton (1989) have been extensively used in business y le analysis in order to des ribe the e onomi u tuations. Among the huge amount of empiri al studies,we an quote thepapers ofSi hel(1994), Lahiriand Wang (1994), Potter (1995), Chauvet and Piger (2003), Ferrara (2003), Clements and Krolzig(2003) orAnasandFerrara(2004b) asregardstheUS e onomyand thepapersofKrolzig (2001, 2004)orKrolzigand Toro(2001) onthe Euro-zone e onomy. Generally, the output of these appli ations is twofold. The authorsaim eitherat datingtheturningpointsof the y le orat dete ting inreal-timethe urrent regime ofthe e onomy.
However, datingand dete tingtheturningpointsof the y learequite dif-ferent obje tives. Datingisan ex post exer iseforwhi hseveralparametri and non-parametri methodologies are available. It turns out that simple non-parametri pro edures,su hasthefamousBryandBos han(1971) pro- edurestillusedbytheDatingCommitteeoftheNBER,aremore onvenient forthis kind of work (see Harding and Pagan, 2001, or Anas and Ferrara, 2004a, for a dis ussionon this issue). Real-time dete tion refers mainlyto short-term e onomi analysis, whi h is not an easy task for pra titioners. Indeed,several e onomi indi atorsarereleasedon aregularlymonthly ba-sis,oreven onadailybasisasregardsthe nan ialse tor, addingvolatility totheexistingvolatilityand thusleadingto anin ation oftheavailable in-formationset. Moreover, thedataareoftenstronglyrevisedandthediverse statisti almethods,su hasseasonaladjustmentor lteringte hniques,lead to edge-e e ts.
BesidesthewellknownMarkov-Swit hingapproa h,otherparametri mod-els have been proposed in the statisti al literature to allow for di erent regimes in business y le analysis. For instan e, probit and logit models have been used by Estrella and Mishkin (1998) to predi t US re essions. Thethresholdautoregressive(TAR)model,proposed rstbyTongandLim (1980),hasbeenusedtodes ribetheasymmetryobservedinthequartelyUS realGNPbydi erentauthors,su hasTiaoandTsay(1994), Potter(1995) andProietti(1998) forinstan e,andusingUSunemploymentmonthlydata by Hansen (1997). In the TAR model, the transition variable may be ei-theran exogenous variable,su hasa leadingindexforexample,ora linear
referredtoasaself-ex itingthresholdautoregressive(SETAR)model. This is the main di eren e with the Markov-Swit hing model whose generating pro essvariesa ordingtothestatesofthelatentMarkov hain. Thesetwo approa hes are omplementary be ause thenotion aptured under investi-gation is notexa tlythe same. Nevertheless, one of theinterest of SETAR pro esses lies on their predi tability, see for instan e De Goojier and De Bruin(1999) and Clementsand Smith(2001). When dealing with SETAR models, the transition is dis rete, but smooth transition is also hosen to studythebusiness y lebysome authors. Then,we gettheso- alledSTAR (smooth transition autoregressive) model, see for instan e Terasvirta and Anderson(1992) and vanDijk, Terasvirta and Franses (2002).
In this paper, we fo us on the dete tion of business y le turning points. Ouraim is ratherto point outsome thresholdsunder (over)whi ha signal ofturning point ould be given. We adopt theSETARapproa h be ausea thresholdmodelseems to be attra tive interms of business y le analysis. Here,we proposea prospe tive approa h to dete t the business y leasan alternative to othermore lassi alparametri approa hes.
Thispaperissplitintotwo parts. Ina rststep(se tiontwo),weintrodu e threshold models whi h allow to apture states in a time series, then, in se tionthree,wespe ifythemethodusedto estimatethedi erent parame-ters ofthethresholdmodels. Ina se ond step,we applydi erentthreshold modelsto the Euro-zone industrialprodu tionindex to dete t thedates of peaks and troughsinthebusiness y le (se tionfour). By usingadynami simulationapproa h,wealsoprovideameasureofperforman eofourmodel by omparisonto a ben hmarkdating hronology. Lastly, some on lusions and furtherresear h dire tions areproposed.
2 Des ription of models whi h apture states In this se tion, we spe ify some of the models whi h permit to take into a ounttheexisten eofvariousstatesinthedata. Forsakeofsimpli ity,we des ribethemodelsonlywithtworegimes,butthey anbeeasilygeneralised to more regimes. Fora review on erning these kindsof pro esses,werefer to Tong (1990), Fransesand van Dijk(2000)and Guegan(2003).
2.1 Threshold pro esses The ovarian e-stationary pro ess (Y
t
) follows a two-regimes threshold au-toregressivepro ess,denotedTAR(2,p
1 ,p
2
equa-Y t = (1 I(Z t d > ))( 1;0 + p1 X i=1 1;i Y t i + 1 " t ) + I(Z t d > )( 2;0 + p2 X i=1 2;i Y t i + 2 " t ); (1)
where is the threshold, d > 0 the delay, (" t
) a standardisedwhite noise pro ess, (Z
t
) the transition variable. Here, I(:) is the indi ator fun tion su h that I(Z
t d
> ) =1 if Z t d
> and zero otherwise. If, 8t, Z t
=Y t
, thepro essisreferredtoasself-ex itingTARpro ess(SETAR).Foragiven threshold and the position of the random variable Z
t d
with respe t to this threshold , the pro ess (Y
t
) follows here a parti ular AR(p) model. The model parameters are
j;i
, fori=0;:::;p j
and j =1;2, the standard varian e errors
1 and
2
, thethreshold and thedelayd. Thismodelhas beenintrodu ed rst byTong and Lim(1980).
Using some algebrai notations, the model (1), with p 1
= p 2
= p, an be rewritten as a regression model. Denote I
d ( ) I(Z t d > ), 1 = [ 1;0 ; ; 1;p ℄ 0 , 2 =[ 2;0 ; ; 2;p ℄ 0 and Y 0 t 1 =[1;Y t 1 ; ;Y t p ℄, then, we getthefollowingrepresentationfrom (1):
Y t =(1 I d ( ))Y 0 t 1 1 +I d ( )Y 0 t 1 2 +((1 I d ( )) 1 +I d ( ) 2 )" t : (2) Ifwedenote theun onditionalstationarydistributionofthepro ess (Y
t ), to get its analyti al form is a non-trivial problem. However an impli it solutionisalways availableif thestationary pro ess (Y
t
) an be onsidered asan ergodi Markov hainoverR
n
. It isgiven foranyevent A,by: (A)=
Z 1
1
P(Ajx)(dx); where denotes the limiting distribution of (Y
t
). For SETAR pro esses introdu edin(1), di erentnumeri alsolutionshave beenproposedto solve thisproblem,seeJones (1978) andPemberton (1985). Inreality, weobtain anapproximationofthisdistribution, omputingempiri allytheper entage of points belonging to the rst regime or to the se ond one. This method gives an estimation of the un onditionalprobability (
1 or
2
) to be in a givenregime.
On ea h state, it is possible to propose more omplex stationary models like ARMA(p,q) pro esses, GARCH(p,q) pro esses (see for instan e Za-koian,1994)orlongmemorypro esses(seeDufrenot, Gueganand Peiguin-Feissolle,2005a and2005b). Notealsothattheregimes anbe hara terized by hanges in the varian e of the noise pro ess (see Pfann, S hotman and T hernig, 1996).
Insteadof usingasharp transitionbetweenthetwostates, hara terisedby theindi atorfun tionI(:),we anuseasmoothtransition. Thisisthebasi ideaof the smoothtransition autoregressive (STAR)pro ess. Inthat ase, byusingthepreviousnotations,thetwo-regimesSTAR(2,p
1 ,p 2 )pro ess(Y t ) followsthe re ursive s heme:
Y t = (1 G(Z t d ; ; ))( 1;0 + p1 X i=1 1;i Y t i + 1 " t ) + G(Z t d ; ; )( 2;0 + p2 X i=1 2;i Y t i + 2 " t ); (3)
whereGis some ontinuous fun tion,forinstan e thelogisti one:
G(Z t d ; ; )= 1 1+exp( (Z t d )) : (4)
Note that the transition fun tion G is bounded between 0 and 1. The parameter anbeinterpretedasthethresholdbetweenthetworegimesin thesensethatthe logisti fun tion hanges monotoni allyfrom 0to 1 with respe tto the value of thelagged exogeneous orendogenous variable Z
t d . Theparameter determinesthesmoothnessofthe hangeinthevalueofthe logisti fun tion,andthus,thesmoothnessofthetransitionofoneregimeto theother. As be omesvery large,thelogisti fun tion(4)approa hesthe indi ator fun tion I(Z
t d
> ). Consequently, the hange of G(Y t d
; ; ) from0to1be omesinstantaneousatZ
t d
= . Thenwe ndtheTARmodel asaparti ular aseofthisSTARmodel. When !0,thelogisti fun tion approa hes a onstant (equal to 0.5) and when = 0, the STAR model redu estoalinearARmodel. Thismodelhasbeendes ribedbyTerasvirta andAnderson(1992). OthergeneralisationsoftheSTARpro esshavebeen proposed, forinstan e byrepla ingthe logisti fun tionbythe exponential fun tion or by using long memory dynami s in ea h regime (see van Dijk, Fransesand Paap,2002).
3 Estimation for SETAR models
Inthefollowing, weusetheSETARpro essinordertomodelthee onomi business y le using the Euro-zone industrial produ tion index. We now des ribetheestimationpro edureweuse inse tionfour.
TheTAR modelintrodu edintheeighties'hasnotbeenwidelyusedin ap-pli ationsuntilre ently,primarilybe auseitwashardinpra ti etoidentify thethresholdvariableand to estimatetheasso iatedvalues, and,se ondly,
authors have proposed di erent ways to bypass this problem. We present inthisse tiona wayto estimate theparametersof theSETARmodelsand we spe ify some re ent literature whi h permits to implement qui kly the pro edure we usebelow.
Here,we assume rstthat themodel availableforourpurposeisa SETAR (2,p
1 ,p
2
) model des ribed by equation (1), with 1
= 2
= 1. As noted above, amajordiÆ ultyinapplyingTAR modelsis thespe i ationofthe thresholdvariable, whi hplaysa key rolein thenon-linearstru tureof the model. Sin e there is onlya nitenumber of hoi es for the parameters and d, the best hoi e an be done usingthe Akaike InformationCriterion (AIC), see Akaike (1974). This pro edure has beenproposedby Tong and Lim(1980)and isusedbyalargepartofthepra titioners dealingwiththis model.
Now, we assume that we observe a sequen e of data(Y 1
; ;Y n
) from the model (2). The equation (2) is a regression equation (albeit non-linear in parameters) and an appropriate estimation method is least squares (LS). Under the auxiliary assumption that the noise ("
t )
t
is a strong Gaussian whitenoise,theleastsquaresestimationisequivalent tothemaximum like-lihoodestimation. Sin etheregressionequation(2)isnon-linearand dis on-tinuous, theeasiest method to obtaintheLS estimates is to use sequential onditionalLS.Wewillusethisapproa hhere. Re allthat onditionalleast squaresleadto theminimization of:
n X Y t d ;t=1 (Y t 1;0 p 1 X i=1 1;i Y t i ) 2 + n X Y t d > ;t=1 (Y t 2;0 p 2 X i=1 2;i Y t i ) 2 ; (5) with respe t to 1 ; 2 ; ;d;p 1 ;p 2
. Generally, we rst assume that the au-toregressive ordersp
1 and p
2
areknown.
Re allthatChan(1993) proves,under geometri ergodi ityand someother regularity onditionsfor thepro ess (2), that the LS parameters estimates ofthispro esshavegoodproperties. Thethresholdparameteris onsistent, tends to the true value at rate n and, suitably normalized follows asymp-toti allya CompoundPoisson pro ess. Theother parameters of the model aren
1=2
onsistent andareasymptoti allyNormallydistributed. The lim-itationofthetheoryofChan(1993) on ernsthe onstru tionof on den e intervals for the threshold . Indeed, ifwe denote ^ the LS estimate for , Chan(1993) ndsthat(^ ) onverges indistributionto afun tionalofa CompoundPoisson pro ess and unfortunately, this representation depends upon a hostof nuisan e parameters, in ludingthemarginal distributionof (Y
t
extensionstothiswork anbefoundinHansen (2000),ClementsandSmith (2001) andGonzalo and Pitarakis(2002), forinstan e.
Inpra ti e, to determine theparameters ;d;p 1
;p 2
, we needto assume the existen eof amaximalpossibleorder P ofthetwo subregimesanda great-est possible delay D. The threshold parameter is estimated by using a grid-sear h pro edure. The grid points (
1 ;:::;
s
) are obtained using the quantiles of the sample under investigation. We use equally spa ed quan-tilesfromthe10(per ent)quantilesandendingatthe90(per ent)quantiles. Now,forea h xedpair(d;
i
),0<d<D,i=1;s,theappropriateTAR model isidenti ed. The AIC riterion is usedfor sele tionof the ordersp
1 and p
2
. Inthis ontext, itbe omes: AIC(p 1 ;p 2 ;d; )=ln( 1 n X ^ " 2 t )+2 p 1 +p 2 +2 n ; (6) where"^ t
denotestheresiduals whenwe usetheappropriatemodelforea h pair(d;
i
)from theLS approa h.
Finallythemodel with theparameters p 1 , p 2 , d and
that minimizethe AIC riterion anbe hosen. Sin efordi erentdtherearedi erentnumbers of valuesthat an be used forestimation, thefollowingadjustment should be done. With n d =max(d;P) itis: AIC(p 1 ;p 2 ;d ; )= min p 1 ;p 2 ;d; 1 n n d AIC(p 1 ;p 2 ;d; ): (7) Di erent algorithms have been proposed to improve the properties of the estimatesandthespeedofthealgorithmsandwe suggestthereaderto on-sultthem. ItispossibletouseaBayesianapproa hbasedonGibbssampler proposedbyTiaoand Tsay(1994), see also Potter (1999); graphi al pro e-dures lassifyingtheobservationswithoutknowingthethresholdvariableto estimatetheparameters,seeChen (1995); numeri alapproa hes, see Coak-ley, Fuertes and Perez (2003) or a Markov Chain Monte-Carlo approa h developed inparti ularbySo and Chen(2003).
Inthispaper, we onsider theTsaytest(1989) tojustify theuseof SETAR models. Hansen (1997) onsiders another approa h based on a likelihood ratio statisti and a Lagrange Multiplier test has been also proposed by Proietti (1998). To our knowledge, there is no test available to de ide be-tweenSETARmodelsandMarkov-Swit hing models.
4 Empiri al results
In this se tion, our aim is to apply a SETAR model to the Euro-zone In-dustrialProdu tionIndexinordertodete t thelowphasesoftheindustrial
doneintwo steps: rst we try to ndthebestSETARmodelfollowingthe method proposedinse tion 3 based on theAIC riterion. Se ondly we use themodelto dete t the periodsofexpansion and re ession. By omparing theresultstoreferen ere essiondates,we anassesstheabilityofthemodel to reprodu etheindustrialbusiness y lefeatures.
4.1 Data des ription
Theanalysisis arriedoutontheIPIseries onsideredinthepaperofAnas et al. (2003). This series is a proxy of the monthly aggregate Euro-zone IPIforthe12 ountries,beginninginJanuary1970 andendinginDe ember 2002. Thedata areworkingdayadjustedand seasonallyadjustedbyusing theTramo-SEATS model-basedmethodology, implementedintheDemetra software, whi hused aWiener-Kolmogorov lter(see forinstan eMaravall andPlanas, 1999). Moreover, theirregular partin ludingoutliershasbeen removedbyusingthesame methodology. Itisnoteworthythatthereisstill adebateamongstatisti iansabouttheimpa tof lteringmethodologieson thetimingof peaks and troughs inbusiness y le analysis(see forinstan e
1970
1973
1976
1979
1982
1985
1988
1991
1994
1997
2000
60
70
80
90
100
110
120
130
1970
1973
1976
1979
1982
1985
1988
1991
1994
1997
2000
-0.020
-0.015
-0.010
-0.005
0.000
0.005
0.010
0.015
Figure1: Euro12IPI(top)anditsmonthlygrowthrate(bottom),aswellasthereferen e industrialre essionperiods(shadedareas),fromJanuary1970 toDe ember2002.
-0.010
-0.005
0.000
0.005
0.010
0
20
40
60
80
100
120
140
Figure 2: Empiri al un onditional distribution of the IPI growth rate, from January 1970toDe ember2002.
Lahiriet al., 2004).
Theoriginalseries(X t
)ispresentedin gure1aswellasitsmonthlygrowth rate (Y
t
) de ned, for all t, by: Y t = log(X t ) log(X t 1 ). In gure 1, the shaded areas represent thereferen e industrialre ession dates. Several au-thorshaveproposedaturningpoint hronologyfortheEuro-zoneindustrial business y le, by usingdi erent statisti al te hniques and e onomi argu-ments. Forexample,werefer toAnas etal. (2003), who proposea lassi al NBER-basednon-parametri approa h, and to Artis et al. (2003), Krolzig (2004)orAnasandFerrara(2004b)whoapplyparametri Markov-Swit hing models. Generally, theindustrialre essiondatesaremoreorlesssimilar. In fa t, it turnsout that theEuro-zone experien ed ve industrial re essions: in 1974-75 and 1980-81 due to the rst and se ond oil sho ks, in 1981-82, in 1992-93, due to the Ameri an re ession and the Gulf war, and lastly in 2000-2001 be ause oftheglobale onomi slowdown aused itselfbytheUS re ession from Mar h 2001 to November 2001. It is noteworthy that, on-trarytoa ommonbeliefamonge onomists,theAsian risisin1997-98 has not aused anindustrialre ession inthewholeEuro-zone, butonlya slow-downoftheprodu tion. Finally,weretainasaben hmarkforourstudythe dates proposed byAnas et al. (2003) and summarizedin the rst olumn intable 4.
To ensure stationarity, we are going to deal with the monthly industrial growthrate(Y
t
). Theun onditionalempiri aldistributionoftheIPIgrowth rate omputedbyusinganon-parametri kernelestimate(withthe Epane h-nikovkernel)ispresentedin gure2. Thereisa leareviden eofthreepeaks intheestimateddistribution. Thelowestpeakisduetothenegativegrowth
by periodsof low, but positive,growth rates, experien edforexample dur-ingtheeighties,whilethepeak orrespondingtothehighestvalueisrelated to periodsoffastgrowth. Itis noteworthythat, from1970 to 2002, periods of low growth rates seem to appear more frequently than periods of high growth rates. Moreover, this empiri al distribution is leary asymmetri (skewness equal to -0.9315) and with heavy tails (ex ess kurtosis equal to 2.4850). Consequently, the un onditional Gaussian assumption is strongly reje tedbyaJarque-Bera test.
4.2 Whole sample modelling
In this subse tion we t various SETAR models to the industrial growth rateseries(Y
t
),thatis,wemodelthegrowthoftheEuro-zoneindustry. We onsider rst atwo-regime model,thetransitionvariablebeingsu essively thelaggedseriesandthelagged di eren edseries. Then,we onsidera mul-tiple regime model by mixing the onditions on these previous series. For ea hmodel,we omparetheestimated regimeswiththereferen e re ession phasesinordertoassesstheabilityofthemodeltoreprodu ebusiness y le features.
4.2.1 Model 1
The rst SETAR model uses the lagged seriesY t d
as transition variable. Thus, we model the growth of the industrial produ tion a ording to the regimesofthelaggedgrowth. Thedelaydandthethreshold areestimated by usingthe methodology presented in theprevious se tion. However, the autoregressive lag p hasto be determineda priori. We pro eedbyusinga des endent stepwise approa hby onsidering rst p=12. Forallestimated models,itturnsoutthattheparameters orrespondingtoalaggreaterthan three are statisti ally notsigni ant bythe usualStudent test. Therefore, we imposethe hoi e p= 3 for all the models. The Tsay (1989) test with p = 3 reje ts the null of linearity for d = 1 and for 4 d 12, at the usualrisk =0:05, implyingthusthepresen e of two regimes. We getthe following estimates for and d : ^ = 0:0024 and
^
d = 1. We note from table 1 that, in the high regime, the persisten e is stronger, be ause the parameters orresponding to p = 2 and p = 3 in the low regime are not statisti allysigni ant and have beentherefore an elled, and thevarian e is smaller, whi h are expe ted results in business y le analysis. The full estimatedmodelisasfollows(estimatesand theirstandarderrors aregiven intable 1):
t t 1 [Yt 1> 0:0024℄ + (0:0025+1:3950Y t 1 0:8742Y t 2 +0:3318Y t 3 )I [Y t 1 > 0:0024℄ +" t : The empiri alun onditionalprobabilitiesof being inea h regime are
1 = 0:11 and
2
= 0:89, whi h is onsistent with the usualprobabilities of be-ing in re ession and expansion in business y le analysis. As regards the estimated re ession dates, we get them by assuming that the low regime mat hes with the re ession regime. The results are presented in gure 3 (topgraph) andtable 4along withthetwo other dating hronologies stem-mingfrom the models des ribed below. By omparison with the referen e dating hronology, we an observe that the results are basi ally identi al, ex eptthat we get asupplementaryofre ession in1977, lastingonlythree months. If we had to establish a dating hronology, thisperiod would not be retained as an industrialre essioninsofar asits duration is too shortin omparison with the minimum duration of a business y le phase, whi h generallyof sixmonths. However inthispaper, to avoid non-persistent sig-nals, we adopt the ensoring rule saying that a signal must stay at least three months to be re ognized asan estimated re ession phase. Thus, this supplementaryre essionin1977 isinterpretedasa falsesignal ofre ession. In the remaining of this paper, a re ession phase dete ted by the model but not present in the referen e hronology is interpretedas a false signal of re ession. Regarding thelast industrialre ession, themodelestimates a re essionperiod utintotwoparts. This an beinterpretedasafalse signal ofre overy. We alsonote thatthe otherestimated industrialre essions are shorter, espe ially the 1982 re ession but we get a rst signal of re ession inJanuary 1982 whi h was notpersistent. Otherwise,this model doesnot provideanyother false signalforindustrialre ession.
Lowregime Highregime [Y t 1 0:0024℄ [Y t 1 > 0:0024℄ ^ 0 -0.0049 0.0025 (0.0016) (0.0004) ^ 1 0.8103 1.3950 (0.0931) (0.0513) ^ 2 -0.8742 (0.0782) ^ 3 0.3318 (0.0513) ^ " 0.0021 0.0012
The se ond SETAR model uses the di eren ed lagged series as transition variable, that is we try to model the growth of the industrial produ tion a ording theregimes of its a eleration. We note thisseriesZ
t d , de ned su h as8t, Z t d = Y t 1 Y t d
. A tually, thisseries an be onsidered as a proxy of the a eleration of the IPI over d 1 months. It is interesting to investigate how the growth rate is related to the a eleration througha non-linear relationship. The Tsay test (1989) with p = 3 reje ts the null of linearity for 4 d 13, at the usual risk = 0:05, implying thus the presen e of two regimes. It turns out that the delay d orresponding to the minimum AIC is equal to d = 10. That is, the a eleration over nine months seems to be the mostsigni ant. The estimated model isgiven by thefollowingequation(estimatesandtheirstandarderrorsaregivenintable 2): Y t = ( 0:0051+0:8100Y t 1 )(1 I [Z t 10 > 0:0061℄ ) + (0:0024+1:7444Y t 1 1:3796Y t 2 +0:5567Y t 3 )I [Z t 10 > 0:0061℄ +" t : Hereagain,we observe thatthepersisten eisstronger inthehigherregime whilethevarian eissmallerandtheempiri alun onditionalprobabilitiesof being inea h regimeare exa tlyequalto thepreviousones. Theestimated industrial re ession dates, presented in gure 3 (middle graph) and table 4, slighty di er from the previous estimates. Indeed, we get another false signal of industrialre ession in 1998 due to the impa t of the Asian risis. Moreover, we notethatthe1977 re essionlastssevenmonths,butthe1982 re ession is only of two months. Therefore, by onsidering the ensoring ruleadoptedabove,thismodeldoesnotre ognizethisperiodasare ession. We also note that a non-persistent signal of re ession is given in Septem-ber 1995. Thus, by omparison with the referen e dating hronology, this model providestwo false signalsof re essionand misses the1982 re ession.
Lowregime Highregime [Zt 10 0:0061℄ [Zt 10 > 0:0061℄ ^ 0 -0.0051 0.0023 (0.0018) (0.0006) ^ 1 0.8100 1.7540 (0.0929) (0.0453) ^ 2 -1.3940 (0.0738) ^ 3 0.5630 (0.0454) ^ " 0.0023 0.0009
industrialre ession phases. This may be dueto the fa tthat the a elera-tion,although omputed over 9months, appearsto betoo volatile.
4.2.3 Model 3
Lastly, the idea whi h appears to be natural is to ombine the two previ-ous SETAR models in a single model with two transition variables : the lagged growth rate and the a eleration. Therefore, the model possesses fourregimes and two thresholds
1 and
2
have to be estimated. The esti-mated model whi h minimizes theAIC is given by thefollowing equations (estimatesand theirstandarderrors aregiven intable 3) :
Regime1: ifY t 1 < 0:00148 and Z t 10 < 0:00076, then Y t = 0:0041+0:8273Y t 1 +" 1 t ; Regime2: ifY t 1 < 0:00148 and Z t 10 0:00076 Y t = 0:0017 0:0934Y t 1 +" 2 t ; Regime3: ifY t 1 0:00148 and Z t 10 < 0:00076 Y t =0:0010+0:6520Y t 1 +" 3 t ; Regime4: ifY t 1 0:00148 and Z t 10 0:00076 Y t =0:0036+1:3005Y t 1 0:7883Y t 2 +0:3321Y t 3 +" 4 t :
Regime1 Regime2 Regime3 Regime4 [Y t 1 < 0:0015℄ [Y t 1 < 0:0015℄ [Y t 1 0:0015℄ [Y t 1 0:0015℄ [Z t 10 < 0:0008℄ [Z t 10 0:0008℄ [Z t 10 < 0:0008℄ [Z t 10 0:0008℄ ^ 0 -0.0041 -0.0018 0.0010 0.0036 (0.0016) (0.0011) (0.0005) (0.0005) ^ 1 0.8273 -0.0934 0.6520 1.3005 (0.0784) (0.5282) (0.0749) (0.0660) ^ 2 -0.7883 (0.0974) ^ 3 0.3321 (0.0662) ^ " 0.0021 0.0028 0.0016 0.0012 Table3: Estimatesandstandarderrorsformodeldes ribedin4.2.3.
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Figure 3: Industrial re ession dates estimated by the 2-regime SETAR with lagged variableastransitionvariable(topgraph),bythe2-regimeSETARwithdi eren edlagged variableastransitionvariable(middlegraph)andbythe4-regimeSETAR(bottomgraph).
The two thresholds are estimated by using a double loop, but the delays of the model are xed a priori a ording the two previous estimated mod-els. Both estimated thresholds are negative but very lose to zero. The rst regime has an empiri al un onditional probability of 0.15 and should be onsideredat a rst sight asaperiodof re essionbe ausetheestimated re ession dates mat h the referen e re ession dates. However, the se ond regimeisalsomeaningful. Indeed,thisse ond regimepossessesan un ondi-tionalprobabilityof 0.02: only7 observationsover385 belongto thisstate. This is the reason why estimates and their standard errors in this regime shouldbetakenwith aution. Althoughthefrequen yofthisse ondregime is very low, this regime is persistent and appears in lusters. In fa t, this regime is very interesting be ause it orresponds to the end of a re ession
twi e: at the end of the 1974-75 re ession and at the end of the 1992-93 re ession. Thus,thesumofregime1andregime2 orrespondstothe indus-trialre ession phase. The thirdregime an be onsideredasa slowdownof theindustrialprodu tion,thatistheindustryisbelowitstrendgrowth rate without being in re ession. Lastly, when the series is in the high regime, we andedu e thattheindustrialgrowthrate isoverits trendgrowth rate. A tually, regime 3 and regime4 orrespond to thehigh phaseof the indus-trial business y le. It appears that onlythree regimes would be suÆ ient to des ribe the industrialbusiness y le. However, we de ide to keep four statesbe auseitgivesadeeperunderstandingoftheindustrialbusiness y- le features. As regards the dating results, the model provides almost the sameresultsthanthe rstmodel,thelastre essionperiodbeingnot utinto two parts (see gure 3, bottom graph, and table 4). However, this model presentssome non-persistent signalsofre ession.
After this whole sample analysis, we retain the third SETAR model with fourregimesforthedynami analysis,be auseitprovidesthemorea urate des riptionof theindustrialbusiness y le.
4.3 Dynami analysis
To be useful for short-term e onomi analysis, an e onomi indi ator re-quiresatleasttwoqualities: itmustbereliableandmustprovideareadable signal as soon as possible. Thus, there is a well known trade-o between advan e and reliabilityfor the e onomi indi ators. By usingthe previous
Referen e Model1 Model2 Model3 Peak m41974 m61974 m61974 m61974 Trough m51975 m51975 m31975 m61975 Peak - m31977 m121976 m31977 Trough - m61977 m71977 m71977 Peak m21980 m41980 m31980 m31980 Trough m11981 m101980 m101980 m111980 Peak m101981 m51982 m61982 m61982 Trough m121982 m121982 m81982 m121982 Peak m11992 m41992 m71992 m41992 Trough m51993 m51993 m11993 m61993 Peak - - m71998 -Trough - - m111998 -Peak m122000 m22001 m12001 m22001 Trough m122001 m122001 m102001 m122001
Table 4: Referen eand estimateddating hronologies stemmingfromthe 3 onsidered SETARmodels.
signal for the turning points of the industrial business y le in a dynami analysis.
In thispart, we onsider theprevious IPI series from January 1970 to De- ember1999,andweaddprogressivelyamonthlydatauntilDe ember2002. Forea hstep,were-estimatethemodelandwe lassifytheobservationsinto oneofthefourregimes. Thus,byusingthe on lusionsofthewhole-sample analysis,iftheobservations fall into regime 1or regime2, we an on lude thattheindustryisinare essionphase. We areawarethatatruereal-time analysisshouldbedonebyusinghistori allyreleaseddata(see forinstan e Chauvet andPiger,2003)inorderto take therevisionsand theedge-e e ts of the statisti al treatments of the raw data into a ount. However, su h seriesarevery diÆ ultto ndine onomi databases.
The dynami ally estimated re ession period is presented in gure 4. We observe thisperiodmat heswiththe2001 re essionperiodestimatedinthe whole-sampleanalysis. Thisfa t points outthe stabilityof the model. In-deed,wedete t a peak inthebusiness y le inFebruary2001 and atrough inDe ember2001. However, itmustbenotedthatafalsesignalofa hange in regime is emitted in August 2001 but lasts only one month. Knowing thatasignalmustbepersistenttobereliable,wehaveto proposeanadho real-time de ision rule. Thus, it is advo ated to wait at least two months before sendingasignal ofa hange inregime. We also notethattheexitof there essionisveryfast, be ausetheobservationsgodire tlyfromregime1 inDe ember2001 to regime 4 inJanuary 2002. Moreover, we observe that theDe ember2002 observation fallsinto regime3.
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Figure 4: Euro12 IPI and the dinami ally estimated re ession period (shaded area), fromJanuary2000toDe ember2002.
This paper is an exploratory analysis of the ability of SETAR models to reprodu e the business y le stylised fa ts. The results are promising. It appears that these non-linear models allow to identify the turning points of the Euro-zone industrial business y le and an thus be useful for real-time dete tion. However, a true real-time analysis should be extended by usinghistori allyreleaseddata, asusedinthere ent paperofChauvetand Piger(2003) as regardsthe US GDP and employment. This truereal-time analysis would also allow to he k the robustness of the model over time. Espe ially, the stabilityof the estimated thresholds ould be interestingto investigate. Iftheunstabilityise e tive,aninnovativetime-varyingSETAR ouldbeintrodu ed. Unfortunately, su hdataarenotsystemati allystored in data bases and are therefore very diÆ ult to get, espe ially as regards the Euro-zone. As another exampleof appli ation, onsumer and business surveys seem to be good andidatesfor real-timeanalysis through SETAR modelsbe ausetheyaretimelyreleasedandarenotgenerallyrevised. Last, it ould be interesting to ompare the dete tions made by these threshold models with analogous dete tions made by other models, su h as Markov-Swit hingorlogitmodels,basedonthesamedataset,andto omparetheir goodnessof tthrougha simulation study.
A knowledments
Theauthorswouldlike to thanktheeditor andtwo anonymousreferees for their helpful suggestions as well as the organizers and parti ipants of the fourthColloquiumon Modern Tools for Business Cy leAnalysis in Luxem-bourg, O tober2003.
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