Munich Personal RePEc Archive
Detection of the industrial business cycle using SETAR models
Ferrara, Laurent and Guégan, Dominique
September 2005
Online at https://mpra.ub.uni-muenchen.de/4389/
MPRA Paper No. 4389, posted 08 Aug 2007 UTC
using SETAR Models
Laurent Ferrara
and Dominique Guegan y
September13, 2005
Abstrat:
In this paper, we onsider a threshold time series model in order to take
into aount ertain stylized fats of the industrial business yle, suh as
asymmetriesinthephasesoftheyle. Ouraimistopointoutsomethresh-
olds under (over) whih a signal of turningpoint ould be given. First, we
introdue the various threshold models and we disuss both their statisti-
altheoretial and empirialproperties. Espeially,we review thelassial
tehniquesto estimatethe numberof regimes, thethreshold,the delayand
the parameters of the model. Then, we apply these models to the Euro-
zone industrial prodution index to detet, through a dynami simulation
approah,thedates ofpeaks and troughsinthebusinessyle.
Keywords:
Eonomiyle, turning pointdetetion, Thresholdmodel,Euro-zoneIPI.
JEL Classiation:
C22,C51,E32.
Centre d'Observation Eonomique de la CCIP and E.N.S. Cahan, MORA-IDHE,
CNRSUMR8533. Adress: COE,27AvenuedeFriedland,75382ParisCedex08. E-mail:
lferraraip.fr
y
E.N.S. Cahan, Departement d'Eonomie et Gestion, MORA-IDHE, CNRS UMR
8533, SeniorAademi Fellow del'Institut Europlae denane (IEF). Adress: 61Av-
enuedu President Wilson,94235,CahanCedex, Frane. E-mail: gueganeogest.ens-
ahan.fr
Reently,we witnessedthedevelopmentof newtoolsinbusinessyleanal-
ysis, mainly based on non-linear parametri modeling. Non-linear models
have thegreat advantage to beexibleenoughto take into aount ertain
stylized fats of the eonomi business yle, suh as asymmetries in the
phasesof the yle. In thisrespet, emphasis has been plaedon the lass
of non-linear dynami models that aommodate the possibility of regime
hanges. Espeially, Markov-Swithing models popularized by Hamilton
(1989) have been extensively used in business yle analysis in order to
desribe the eonomi utuations. Among the huge amount of empirial
studies,we an quote thepapers ofSihel(1994), Lahiriand Wang (1994),
Potter (1995), Chauvet and Piger (2003), Ferrara (2003), Clements and
Krolzig(2003) orAnasandFerrara(2004b) asregardstheUS eonomyand
thepapersofKrolzig (2001, 2004)orKrolzigand Toro(2001) ontheEuro-
zone eonomy. Generally, the output of these appliations is twofold. The
authorsaim eitherat datingtheturningpointsof theyle orat deteting
inreal-timetheurrent regime ofthe eonomy.
However, datingand detetingtheturningpointsof theylearequite dif-
ferent objetives. Datingisan ex post exeriseforwhihseveralparametri
and non-parametri methodologies are available. It turns out that simple
non-parametriproedures,suhasthefamousBryandBoshan(1971)pro-
edurestillusedbytheDatingCommitteeoftheNBER,aremoreonvenient
forthis kind of work (see Harding and Pagan, 2001, or Anas and Ferrara,
2004a, for a disussionon this issue). Real-time detetion refers mainlyto
short-term eonomi analysis, whih is not an easy task for pratitioners.
Indeed,several eonomi indiatorsarereleasedon aregularlymonthlyba-
sis,oreven onadailybasisasregardsthenanialsetor, addingvolatility
totheexistingvolatilityand thusleadingto anination oftheavailablein-
formationset. Moreover, thedataareoftenstronglyrevisedandthediverse
statistialmethods,suhasseasonaladjustmentorlteringtehniques,lead
to edge-eets.
BesidesthewellknownMarkov-Swithingapproah,otherparametrimod-
els have been proposed in the statistial literature to allow for dierent
regimes in business yle analysis. For instane, probit and logit models
have been used by Estrella and Mishkin (1998) to predit US reessions.
Thethresholdautoregressive(TAR)model,proposedrstbyTongandLim
(1980),hasbeenusedtodesribetheasymmetryobservedinthequartelyUS
realGNPbydierentauthors,suhasTiaoandTsay(1994), Potter(1995)
andProietti(1998) forinstane,andusingUSunemploymentmonthlydata
by Hansen (1997). In the TAR model, the transition variable may be ei-
theran exogenous variable,suhasa leadingindexforexample,ora linear
referredtoasaself-exitingthresholdautoregressive(SETAR)model. This
is the main dierene with the Markov-Swithing model whose generating
proessvariesaordingtothestatesofthelatentMarkovhain. Thesetwo
approahes are omplementary beause thenotion aptured under investi-
gation is notexatlythe same. Nevertheless, one of theinterest of SETAR
proesses lies on their preditability, see for instane De Goojier and De
Bruin(1999) and Clementsand Smith(2001). When dealing with SETAR
models, the transition is disrete, but smooth transition is also hosen to
studythebusinessylebysome authors. Then,we gettheso-alledSTAR
(smooth transition autoregressive) model, see for instane Terasvirta and
Anderson(1992) and vanDijk, Terasvirta and Franses (2002).
In this paper, we fous on the detetion of business yle turning points.
Ouraim is ratherto point outsome thresholdsunder (over)whiha signal
ofturning pointould be given. We adopt theSETARapproah beausea
thresholdmodelseems to be attrative interms of businessyle analysis.
Here,we proposea prospetive approah to detet the businessyleasan
alternative to othermore lassialparametri approahes.
Thispaperissplitintotwo parts. Inarststep(setiontwo),weintrodue
threshold models whih allow to apture states in a time series, then, in
setionthree,wespeifythemethodusedto estimatethedierentparame-
ters ofthethresholdmodels. Ina seond step,we applydierentthreshold
modelsto the Euro-zone industrialprodutionindex to detet thedates of
peaks and troughsinthebusinessyle (setionfour). By usingadynami
simulationapproah,wealsoprovideameasureofperformaneofourmodel
byomparisonto a benhmarkdatinghronology. Lastly,someonlusions
and furtherresearh diretions areproposed.
2 Desription of models whih apture states
In this setion, we speify some of the models whih permit to take into
aounttheexisteneofvariousstatesinthedata. Forsakeofsimpliity,we
desribethemodelsonlywithtworegimes,buttheyanbeeasilygeneralised
to more regimes. Fora reviewonerningthese kindsof proesses,werefer
to Tong (1990), Fransesand van Dijk(2000)and Guegan(2003).
2.1 Threshold proesses
The ovariane-stationary proess (Y
t
) follows a two-regimes threshold au-
toregressiveproess,denotedTAR(2,p
1 ,p
2
),ifitsatisesthefollowingequa-
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
proess, (Z
t
) the transition variable. Here, I(:) is the indiator funtion
suh that I(Z
t d
>) =1 ifZ
t d
> and zero otherwise. If, 8t, Z
t
=Y
t ,
theproessisreferredtoasself-exitingTARproess(SETAR).Foragiven
threshold and the position of the random variable Z
t d
with respet to
this threshold , the proess (Y
t
) follows here a partiular AR(p) model.
The model parameters are
j;i
, fori=0;:::;p
j
and j =1;2, the standard
varianeerrors
1 and
2
,thethreshold and thedelayd. Thismodelhas
beenintroduedrst 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 theunonditionalstationarydistributionoftheproess (Y
t ),
to get its analytial form is a non-trivial problem. However an impliit
solutionisalways availableif thestationary proess (Y
t
) an be onsidered
asan ergodiMarkovhainoverR n
. It isgiven foranyevent A,by:
(A)= Z
1
1
P(Ajx)(dx);
where denotes the limiting distribution of (Y
t
). For SETAR proesses
introduedin(1), dierentnumerialsolutionshave beenproposedto solve
thisproblem,seeJones (1978) andPemberton (1985). Inreality,weobtain
anapproximationofthisdistribution,omputingempiriallytheperentage
of points belonging to the rst regime or to the seond one. This method
gives an estimation of the unonditionalprobability (
1 or
2
) to be in a
givenregime.
On eah state, it is possible to propose more omplex stationary models
like ARMA(p,q) proesses, GARCH(p,q) proesses (see for instane Za-
koian,1994)orlongmemoryproesses(see Dufrenot, Gueganand Peiguin-
Feissolle,2005aand2005b). Notealsothattheregimesanbeharaterized
by hanges in the variane of the noise proess (see Pfann, Shotman and
Thernig, 1996).
Insteadof usingasharp transitionbetweenthetwostates, haraterisedby
theindiatorfuntionI(:),weanuseasmoothtransition. Thisisthebasi
idea ofthe smoothtransition autoregressive (STAR)proess. Inthat ase,
byusingthepreviousnotations,thetwo-regimesSTAR(2,p
1 ,p
2
)proess(Y
t )
followsthe reursive sheme:
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 funtion,forinstane thelogisti one:
G(Z
t d
;;)=
1
1+exp( (Z
t d ))
: (4)
Note that the transition funtion G is bounded between 0 and 1. The
parameteranbeinterpretedasthethresholdbetweenthetworegimesin
thesensethatthe logistifuntionhanges monotoniallyfrom 0to 1 with
respetto the value of thelagged exogeneous orendogenous variable Z
t d .
Theparameterdeterminesthesmoothnessofthehangeinthevalueofthe
logistifuntion,andthus,thesmoothnessofthetransitionofoneregimeto
theother. As beomesvery large,thelogistifuntion(4)approahesthe
indiator funtion I(Z
t d
> ). Consequently, the hange of G(Y
t d
;;)
from0to1beomesinstantaneousatZ
t d
=. ThenwendtheTARmodel
asapartiularaseofthisSTARmodel. When !0,thelogistifuntion
approahes a onstant (equal to 0.5) and when = 0, the STAR model
reduestoalinearARmodel. ThismodelhasbeendesribedbyTerasvirta
andAnderson(1992). OthergeneralisationsoftheSTARproesshavebeen
proposed, forinstane byreplaingthe logistifuntionbythe exponential
funtion or by using long memory dynamis in eah regime (see van Dijk,
Fransesand Paap,2002).
3 Estimation for SETAR models
Inthefollowing, weusetheSETARproessinordertomodeltheeonomi
business yle using the Euro-zone industrial prodution index. We now
desribetheestimationproedureweuse insetionfour.
TheTAR modelintroduedintheeighties'hasnotbeenwidelyusedinap-
pliationsuntilreently,primarilybeauseitwashardinpratietoidentify
thethresholdvariableand to estimatetheassoiatedvalues, and,seondly,
authors have proposed dierent ways to bypass this problem. We present
inthissetiona wayto estimate theparameters oftheSETARmodelsand
we speify some reent literature whih permits to implement quikly the
proedure we usebelow.
Here,we assume rstthat themodel availableforourpurposeisa SETAR
(2,p
1 ,p
2
) model desribed by equation (1), with
1
=
2
= 1. As noted
above, amajordiÆultyinapplyingTAR modelsis thespeiationofthe
thresholdvariable, whihplaysa key rolein thenon-linearstrutureof the
model. Sine there is onlya nitenumber of hoies for the parameters
and d, the best hoie an be done usingthe Akaike InformationCriterion
(AIC), see Akaike (1974). This proedure has beenproposedby Tong and
Lim(1980)and isusedbyalargepartofthepratitioners dealingwiththis
model.
Now, we assume that we observe a sequene 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 tothemaximumlike-
lihoodestimation. Sinetheregressionequation(2)isnon-linearanddison-
tinuous, theeasiest method to obtaintheLS estimates is to use sequential
onditionalLS.Wewillusethisapproahhere. Reallthatonditionalleast
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 respet to
1
;
2
;;d;p
1
;p
2
. Generally, we rst assume that the au-
toregressive ordersp
1 and p
2
areknown.
ReallthatChan(1993) proves,under geometriergodiityand someother
regularity onditionsfor theproess (2), that the LS parameters estimates
ofthisproesshavegoodproperties. Thethresholdparameterisonsistent,
tends to the true value at rate n and, suitably normalized follows asymp-
totiallya CompoundPoisson proess. Theother parameters of the model
aren 1=2
onsistent andareasymptotiallyNormallydistributed. The lim-
itationofthetheoryofChan(1993)onernstheonstrutionofondene
intervals for the threshold. Indeed, ifwe denote ^ the LS estimate for ,
Chan(1993) ndsthat(^ ) onverges indistributionto afuntionalofa
CompoundPoisson proess and unfortunately,this representation depends
upon a hostof nuisane parameters, inludingthemarginal distributionof
(Y
t
) and all theregression oeÆients. Hene, this theory does not yielda
extensionstothisworkan befoundinHansen (2000),ClementsandSmith
(2001) andGonzalo and Pitarakis(2002), forinstane.
Inpratie, to determine theparameters ;d;p
1
;p
2
,we needto assume the
existeneof amaximalpossibleorder P ofthetwo subregimesanda great-
est possible delay D. The threshold parameter is estimated by using a
grid-searh proedure. The grid points (
1
;:::;
s
) are obtained using the
quantiles of the sample under investigation. We use equally spaed quan-
tilesfromthe10(perent)quantilesandendingatthe90(perent)quantiles.
Now,foreahxedpair(d;
i
),0<d<D,i=1;s,theappropriateTAR
model isidentied. The AICriterion is usedfor seletionof the ordersp
1
and p
2
. Inthisontext, itbeomes:
AIC(p
1
;p
2
;d;)=ln(
1
n X
^
"
2
t )+2
p
1 +p
2 +2
n
; (6)
where"^
t
denotestheresiduals whenwe usetheappropriatemodelforeah
pair(d;
i
) fromtheLS approah.
Finallythemodelwith theparameters p
1 , p
2 , d
and
that minimizethe
AICriterionanbehosen. Sinefordierentdtherearedierentnumbers
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)
Dierent algorithms have been proposed to improve the properties of the
estimatesandthespeedofthealgorithmsandwe suggestthereaderto on-
sultthem. ItispossibletouseaBayesianapproahbasedonGibbssampler
proposedbyTiaoand Tsay(1994), see also Potter (1999); graphialproe-
dureslassifyingtheobservationswithoutknowingthethresholdvariableto
estimatetheparameters,seeChen (1995); numerialapproahes, seeCoak-
ley, Fuertes and Perez (2003) or a Markov Chain Monte-Carlo approah
developed inpartiularbySo and Chen(2003).
Inthispaper, weonsidertheTsaytest(1989) tojustify theuseof SETAR
models. Hansen (1997) onsiders another approah 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 deidebe-
tweenSETARmodelsandMarkov-Swithing models.
4 Empirial results
In this setion, our aim is to apply a SETAR model to the Euro-zone In-
dustrialProdutionIndexinordertodetet thelowphasesoftheindustrial
doneintwo steps: rst we try to ndthebestSETARmodelfollowingthe
method proposedinsetion 3 based on theAICriterion. Seondly we use
themodelto detet the periodsofexpansion and reession. By omparing
theresultstoreferenereessiondates,weanassesstheabilityofthemodel
to reproduetheindustrialbusinessylefeatures.
4.1 Data desription
TheanalysisisarriedoutontheIPIseriesonsideredinthepaperofAnas
et al. (2003). This series is a proxy of the monthly aggregate Euro-zone
IPIforthe12ountries,beginninginJanuary1970 andendinginDeember
2002. Thedata areworkingdayadjustedand seasonallyadjustedbyusing
theTramo-SEATS model-basedmethodology,implementedintheDemetra
software, whihused aWiener-Kolmogorovlter(see forinstaneMaravall
andPlanas, 1999). Moreover, theirregular partinludingoutliers hasbeen
removedbyusingthesame methodology. Itisnoteworthythatthereisstill
adebateamongstatistiiansabouttheimpatoflteringmethodologieson
thetimingof peaks and troughs inbusinessyle analysis(see forinstane
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),aswellasthereferene
industrialreessionperiods(shadedareas),fromJanuary1970 toDeember2002.
-0.010 -0.005 0.000 0.005 0.010 0
20 40 60 80 100 120 140
Figure 2: Empirial unonditional distribution of the IPI growth rate, from January
1970toDeember2002.
Lahiriet al., 2004).
Theoriginalseries(X
t
)ispresentedingure1aswellasitsmonthlygrowth
rate (Y
t
) dened, for all t, by: Y
t
= log(X
t
) log(X
t 1
). In gure 1, the
shaded areas represent thereferene industrialreession dates. Several au-
thorshaveproposedaturningpointhronologyfortheEuro-zoneindustrial
businessyle, by usingdierent statistial tehniques and eonomi argu-
ments. Forexample,werefer to Anasetal. (2003), who proposealassial
NBER-basednon-parametri approah, and to Artis et al. (2003), Krolzig
(2004)orAnasandFerrara(2004b)whoapplyparametriMarkov-Swithing
models. Generally,theindustrialreessiondatesaremoreorlesssimilar. In
fat, it turnsout that theEuro-zone experiened ve industrial reessions:
in 1974-75 and 1980-81 due to the rst and seond oil shoks, in 1981-82,
in 1992-93, due to the Amerian reession and the Gulf war, and lastly in
2000-2001 beause oftheglobaleonomi slowdownaused itselfbytheUS
reession from Marh 2001 to November 2001. It is noteworthy that, on-
trarytoa ommonbeliefamongeonomists,theAsianrisisin1997-98 has
notaused anindustrialreession inthewholeEuro-zone,butonlya slow-
downoftheprodution. Finally,weretainasabenhmarkforourstudythe
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
). TheunonditionalempirialdistributionoftheIPIgrowth
rateomputedbyusinganon-parametrikernelestimate(withtheEpaneh-
nikovkernel)ispresentedingure2. Thereisalearevideneofthreepeaks
intheestimateddistribution. Thelowestpeakisduetothenegativegrowth
by periodsof low, but positive,growth rates, experienedforexample dur-
ingtheeighties,whilethepeakorrespondingtothehighestvalueisrelated
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 empirial distribution is leary asymmetri
(skewness equal to -0.9315) and with heavy tails (exess kurtosis equal to
2.4850). Consequently, the unonditional Gaussian assumption is strongly
rejetedbyaJarque-Bera test.
4.2 Whole sample modelling
In this subsetion we t various SETAR models to the industrial growth
rateseries(Y
t
),thatis,wemodelthegrowthoftheEuro-zoneindustry. We
onsiderrst atwo-regime model,thetransitionvariablebeingsuessively
thelaggedseriesandthelagged dierenedseries. Then,weonsideramul-
tiple regime model by mixing the onditions on these previous series. For
eahmodel,we omparetheestimated regimeswiththereferene reession
phasesinordertoassesstheabilityofthemodeltoreproduebusinessyle
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 prodution aording to the
regimesofthelaggedgrowth. Thedelaydandthethresholdareestimated
by usingthe methodology presented in theprevious setion. However, the
autoregressive lag p hasto be determineda priori. We proeedbyusinga
desendent stepwise approahbyonsideringrst p=12. Forallestimated
models,itturnsoutthattheparametersorrespondingtoalaggreaterthan
three are statistially notsigniant bythe usualStudent test. Therefore,
we imposethe hoie p= 3 for all the models. The Tsay (1989) test with
p = 3 rejets the null of linearity for d = 1 and for 4 d 12, at the
usualrisk=0:05, implyingthusthepresene 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 persistene is stronger, beause the
parameters orresponding to p = 2 and p = 3 in the low regime are not
statistiallysigniant and have beentherefore anelled, and thevariane
is smaller, whih are expeted results in business yle 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 empirialunonditionalprobabilitiesof being ineah regime are
1
=
0:11 and
2
= 0:89, whih is onsistent with the usualprobabilities of be-
ing in reession and expansion in business yle analysis. As regards the
estimated reession dates, we get them by assuming that the low regime
mathes with the reession regime. The results are presented in gure 3
(topgraph) andtable 4along withthetwo other datinghronologies stem-
mingfrom the models desribed below. By omparison with the referene
dating hronology, we an observe that the results are basially idential,
exeptthat we get asupplementaryofreession in1977, lastingonlythree
months. If we had to establish a dating hronology, thisperiod would not
be retained as an industrialreessioninsofar asits duration is too shortin
omparison with the minimum duration of a business yle phase, whih
generallyof sixmonths. However inthispaper, to avoid non-persistentsig-
nals, we adopt the ensoring rule saying that a signal must stay at least
three months to be reognized asan estimated reession phase. Thus, this
supplementaryreessionin1977 isinterpretedasa falsesignal ofreession.
In the remaining of this paper, a reession phase deteted by the model
but not present in the referene hronology is interpretedas a false signal
of reession. Regarding thelast industrialreession,themodelestimates a
reessionperiodutintotwoparts. Thisan beinterpretedasafalsesignal
ofreovery. We alsonote thatthe otherestimated industrialreessions are
shorter, espeially the 1982 reession but we get a rst signal of reession
inJanuary 1982 whih was notpersistent. Otherwise,this model doesnot
provideanyother falsesignalforindustrialreession.
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
Table1: Estimatesandstandarderrorsformodeldesribedin4.2.1.
The seond SETAR model uses the dierened lagged series as transition
variable, that is we try to model the growth of the industrial prodution
aording theregimes of its aeleration. We note thisseriesZ
t d
, dened
suh as8t, Z
t d
= Y
t 1 Y
t d
. Atually, thisseries an be onsidered as
a proxy of the aeleration of the IPI over d 1 months. It is interesting
to investigate how the growth rate is related to the aeleration througha
non-linear relationship. The Tsay test (1989) with p = 3 rejets the null
of linearity for 4 d 13, at the usual risk = 0:05, implying thus the
presene of two regimes. It turns out that the delay d orresponding to
the minimum AIC is equal to d = 10. That is, the aeleration over nine
months seems to be the mostsigniant. 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 thatthepersisteneisstronger inthehigherregime
whilethevarianeissmallerandtheempirialunonditionalprobabilitiesof
being ineah regimeare exatlyequalto thepreviousones. Theestimated
industrial reession dates, presented in gure 3 (middle graph) and table
4, slighty dier from the previous estimates. Indeed, we get another false
signal of industrialreession in 1998 due to the impat of the Asianrisis.
Moreover, we notethatthe1977 reessionlastssevenmonths,butthe1982
reession is only of two months. Therefore, by onsidering the ensoring
ruleadoptedabove,thismodeldoesnotreognizethisperiodasareession.
We also note that a non-persistent signal of reession is given in Septem-
ber 1995. Thus, by omparison with the referene dating hronology, this
model providestwo falsesignalsof reessionand misses the1982 reession.
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
Table2: Estimatesandstandarderrorsformodeldesribedin4.2.2.
industrialreession phases. This may be dueto the fatthat the aelera-
tion,althoughomputed over 9months, appearsto betoo volatile.
4.2.3 Model 3
Lastly, the idea whih 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 aeleration. Therefore, the model possesses
fourregimes and two thresholds
1 and
2
have to be estimated. The esti-
mated model whih 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: Estimatesandstandarderrorsformodeldesribedin4.2.3.
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 60
70 80 90 100 110 120 130
1970 1973 1976 1979 1982 1985 1988 1991 1994 1997 2000 60
70 80 90 100 110 120 130
Figure 3: Industrial reession dates estimated by the 2-regime SETAR with lagged
variableastransitionvariable(topgraph),bythe2-regimeSETARwithdierenedlagged
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 aording the two previous estimated mod-
els. Both estimated thresholds are negative but very lose to zero. The
rst regime has an empirial unonditionalprobability of 0.15 and should
be onsideredat arst sight asaperiodof reessionbeausetheestimated
reession dates math the referene reession dates. However, the seond
regimeisalsomeaningful. Indeed,thisseond regimepossessesanunondi-
tionalprobabilityof 0.02: only7 observationsover385 belongto thisstate.
This is the reason why estimates and their standard errors in this regime
shouldbetakenwithaution. Althoughthefrequenyofthisseondregime
is very low, this regime is persistent and appears in lusters. In fat, this
regime is very interesting beause it orresponds to the end of a reession
twie: at the end of the 1974-75 reession and at the end of the 1992-93
reession. Thus,thesumofregime1andregime2orrespondstotheindus-
trialreession phase. The thirdregime an be onsideredasa slowdownof
theindustrialprodution,thatistheindustryisbelowitstrendgrowth rate
without being in reession. Lastly, when the series is in the high regime,
we andedue thattheindustrialgrowthrate isoverits trendgrowth rate.
Atually,regime 3 and regime4 orrespond to thehigh phaseof theindus-
trial businessyle. It appears that onlythree regimes would be suÆient
to desribe the industrialbusiness yle. However, we deide to keep four
statesbeauseitgivesadeeperunderstandingoftheindustrialbusinessy-
le features. As regards the dating results, the model provides almost the
sameresultsthantherstmodel,thelastreessionperiodbeingnotutinto
two parts (see gure 3, bottom graph, and table 4). However, this model
presentssome non-persistent signalsofreession.
After this whole sample analysis, we retain the third SETAR model with
fourregimesforthedynamianalysis,beauseitprovidesthemoreaurate
desriptionof theindustrialbusinessyle.
4.3 Dynami analysis
To be useful for short-term eonomi analysis, an eonomi indiator re-
quiresatleasttwoqualities: itmustbereliableandmustprovideareadable
signal as soon as possible. Thus, there is a well known trade-o between
advane and reliabilityfor the eonomi indiators. By usingthe previous
Referene 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: Refereneand estimateddatinghronologies stemmingfromthe 3onsidered
SETARmodels.
signal for the turning points of the industrial business yle in a dynami
analysis.
In thispart, we onsidertheprevious IPI series from January 1970 to De-
ember1999,andweaddprogressivelyamonthlydatauntilDeember2002.
Foreahstep,were-estimatethemodelandwelassifytheobservationsinto
oneofthefourregimes. Thus,byusingtheonlusionsofthewhole-sample
analysis,iftheobservations fallinto regime 1or regime2, we an onlude
thattheindustryisinareessionphase. We areawarethatatruereal-time
analysisshouldbedonebyusinghistoriallyreleaseddata(see forinstane
ChauvetandPiger,2003)inorderto take therevisionsand theedge-eets
of the statistial treatments of the raw data into aount. However, suh
seriesarevery diÆultto ndineonomi databases.
The dynamially estimated reession period is presented in gure 4. We
observe thisperiodmatheswiththe2001 reessionperiodestimatedinthe
whole-sampleanalysis. Thisfat points outthe stabilityof the model. In-
deed,wedetet a peak inthebusinessyle inFebruary2001 and atrough
inDeember2001. However, itmustbenotedthatafalsesignalofahange
in regime is emitted in August 2001 but lasts only one month. Knowing
thatasignalmustbepersistenttobereliable,wehaveto proposeanadho
real-time deision rule. Thus, it is advoated to wait at least two months
before sendingasignal ofa hange inregime. We also notethattheexitof
thereessionisveryfast, beausetheobservationsgodiretlyfromregime1
inDeember2001 to regime 4 inJanuary 2002. Moreover, we observe that
theDeember2002 observation fallsinto regime3.
2000 2001 2002 2003
115 116 117 118 119 120
Figure 4: Euro12 IPI and the dinamially estimated reession period (shaded area),
fromJanuary2000toDeember2002.
This paper is an exploratory analysis of the ability of SETAR models to
reprodue the business yle stylised fats. The results are promising. It
appears that these non-linear models allow to identify the turning points
of the Euro-zone industrial business yle and an thus be useful for real-
time detetion. However, a true real-time analysis should be extended by
usinghistoriallyreleaseddata, asusedinthereent paperofChauvetand
Piger(2003) as regardsthe US GDP and employment. This truereal-time
analysis would also allow to hek the robustness of the model over time.
Espeially,the stabilityof the estimated thresholdsould be interestingto
investigate. Iftheunstabilityiseetive,aninnovativetime-varyingSETAR
ouldbeintrodued. Unfortunately,suhdataarenotsystematiallystored
in data bases and are therefore very diÆult to get, espeially as regards
the Euro-zone. As another exampleof appliation,onsumer and business
surveys seem to be good andidatesfor real-timeanalysis through SETAR
modelsbeausetheyaretimelyreleasedandarenotgenerallyrevised. Last,
it ould be interesting to ompare the detetions made by these threshold
models with analogous detetions made by other models, suh as Markov-
Swithingorlogitmodels,basedonthesamedataset,andtoomparetheir
goodnessof tthrougha simulation study.
Aknowledments
Theauthorswouldlike to thanktheeditor andtwo anonymousreferees for
their helpful suggestions as well as the organizers and partiipants of the
fourthColloquiumon Modern Tools for Business CyleAnalysis inLuxem-
bourg, Otober2003.
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