Business Cycle Turning Points, a New Coincident Index, and Tests of Duration Dependence
Based on a Dynamic Factor Model with Regime Switching
Author(s): Chang-Jin Kim and Charles R. Nelson
Source: The Review of Economics and Statistics, Vol. 80, No. 2 (May, 1998), pp. 188-201
Published by: The MIT Press
Stable URL: http://www.jstor.org/stable/2646631
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BUSINESS CYCLE TURNING POINTS, A NEW COINCIDENT INDEX,
AND TESTS OF DURATION DEPENDENCE BASED ON A DYNAMIC
FACTOR MODEL WITH REGIME SWITCHING
Chang-JinKim and CharlesR. Nelson*
Abstract-The synthesisof thedynamicfactormodelof StockandWatson
(1989) and the regime-switchingmodel of Hamilton(1989) proposedby
Dieboldand Rudebusch(1996) potentiallyencompassesboth featuresof
the businesscycle identifiedby Burnsand Mitchell(1946): (1) comovementamongeconomicvariablesthroughthe cycle and(2) nonlinearityin
its evolution. However, maximum-likelihoodestimation has required
approximation.
Recentadvancesin multimoveGibbssamplingmethodolinferencein such non-Gaussian,
ogy open the way to approximation-free
nonlinearmodels. This paper estimates the model for U.S. data and
attemptsto addressthreequestions:Arebothfeaturesof thebusinesscycle
empiricallyrelevant?Mighttheimpliednew indexof coincidentindicators
be a usefulone in practice?Do the resultingestimatesof regimeswitches
show evidence of durationdependence?The answersto all threewould
appearto be yes.
I.
Introduction
JN THEIRpioneeringstudy, Burns and Mitchell (1946)
establishedtwo defining characteristicsof the business
cycle: (1) comovementamongeconomicvariablesthrough
the cycle and(2) nonlinearityin the evolutionof thebusiness
cycle, that is, regime switchingat the turningpoints of the
businesscycle. As notedby DieboldandRudebusch(1996),
these two aspectsof the businesscycle have generallybeen
consideredin isolation from one anotherin the literature.
Two of the most recent and influentialexamples include
Stock and Watson's (1989, 1991, 1993) linear dynamic
factor model, in which comovement among economic
variablesis capturedby a compositeindex, andHamilton's
(1989) regime-switchingmodel,featuringnonlinearityin an
individualeconomic variable.After an extensive surveyof
relatedliterature,andbasedon boththeoreticalandempirical foundations,Diebold and Rudebusch(1996) proposea
multivariatedynamicfactormodelwithregimeswitchingin
which the two key features of the business cycle are
encompassed.
The purposeof this paperis twofold. First, we develop
inferencein the DieboldandRudebusch
approximation-free
dynamicfactormodel with regime switching.As in Stock
and Watson(1991), the model can be cast in state-space
form, but nonlinearity(due to regime switching) in the
transitionequationimplies thatthe usual GaussianKalman
filtercannotbe applieddirectly.'Recentadvancesin Gibbs
samplingmethods,however,makeapproximation-free
inference2for non-Gaussian,nonlinearstate-spacemodels feasible. In particular,we take advantageof the multimove
Gibbssamplingmethodologyof Shephard(1994) andCarter
and Kohn (1994) and estimate the model in a Bayesian
frameworkin which the parameters,the regime that the
economy is in, and the dynamicfactoror compositeindex
areall treatedas unobservedrandomvariablesto be inferred
fromdataon individualindicators.Using this approach,we
reevaluatethe empiricalrelevanceof the two key featuresof
the businesscycle identifiedby Burnsand Mitchell(1946).
As by-products,we areableto calculatea new experimental
coincidentindex and the regimeprobabilitiesat each point
in time.
The second objectiveof this paperis to test for business
cycle durationdependence,whether the probabilityof a
transitionbetweenregimesdependson howlongtheeconomy
has been in a recession or boom. While most of existing
literatureon tests of business cycle durationdependence
deals with the issue within univariatecontexts, we offer
ways to deal with such issues within a multivariateand
Bayesiancontextby incorporatingnonzeroprobabilitiesof
durationdependencein the Diebold and Rudebuschdynamicfactormodelwithregimeswitching.
The structureof this paper is as follows. Section II
introducesthe model and discusses related econometric
issues. In section III the multimove Gibbs sampling of
Shephard(1994) and Carterand Kohn (1994) is appliedto
our model in order to obtain Bayesian estimates of the
unobservedregime-indicatorvariableand the unobserved
common growth component.The implied new composite
indexof coincidentindicatorsis presentedandthe empirical
relevance of regime switching and comovement among
economic variablesis evaluated.Section IV presentstests
for business cycle durationdependencethat exploit the
Gibbs samplingmethodology.The basic model is extended
to incorporatenonzeroprobabilitiesof durationdependence.
SectionV concludesthe paper.
approximatefilteringand smoothingalgorithmsfor nonlinearand nonGaussianstatemodelsare given in Harrisonand Stevens(1976), Gordon
Received for publicationJanuary2, 1997. Revision accepted for andSmith(1988),ShumwayandStoffer(1991),andKim(1993a,b, 1994).
Kim and Yoo (1995) and Chauvet(1995) apply Kim's (1993a, b, 1994)
publicationApril17, 1997.
* KoreaUniversityandUniversityof Washington,respectively.
approximatemaximum-likelihoodestimation method in estimating a
We thankthe editor,anonymousreferees,and seminarparticipantsat dynamicfactormodelwithregimeswitching.
2 A refereehas suggestedthat it is not entirelyaccurateto say thatthe
Universityof Pennsylvania,UCSD, Universityof Missouri,WayneState
Thekey approximaUniversity,and KoreanEconometricSociety meetingsfor helpful com- Gibbssamplingframeworkmakesno approximations.
tionin theGibbsframeworkis associatedwithdeclaringtheGibbschainto
ments,buttheresponsibilityfor errorsis entirelythe authors'.
I For a generalapproachto dealing with nonlinearand non-Gaussian have convergedto its steadystate.It is in the limit, as the length of the
errorvanishes.
state-space models, readersare referredto Kitagawa(1987). Various chaingoes to infinity,thatthe approximation
[ 188 ]
? 1998 by the Presidentand Fellows of HarvardCollege and the MassachusettsInstituteof Technology
BUSINESS CYCLES, A NEW COINCIDENTINDEX AND REGIME SWITCHING
II.
A Dynamic Factor Model of the Business Cycle
with Regime Switching
In the synthesisof the dynamicfactormodelof Stockand
Watson(1989, 1991, 1993) andthe regime-switchingmodel
of Hamilton(1989) proposedby Diebold and Rudebusch
(1996), the growthrateof each of four observedindividual
coincidentindicatorsdependson currentandlagged values
of anunobservedcommonfactor,whichis interpretedas the
compositeindexof coincidentindicators.The growthrateof
the indexis, in turn,dependentuponwhetherthe economyis
in therecessionstateor in the boom state.The modelmaybe
writtenas
AYit= Xi(L)ACt + Di + eit,
t = 1, 29.. . ,T
i = 1, 2, 3,4,
(1)
189
whereasV,,producesdeviationsfrom that long-rungrowth
rateaccordingto whetherthe economyis in a recessionor in
a boom.
This modelcan be cast into state-spaceform,butthereis
not a uniquestate-spacerepresentation.Dependingon our
purpose,we may choose a particularstate-spacerepresentation, andin this sectionwe focus on the followingone:
AY,= Htt + D
t
(6)
(7)
+ Ftt-1 + ut
= Mst + 5
where AYt = [AY1t AY2t AY3t AY4t]',D = [DI D2
D3 D4]', and the other tems are defined appropriately,
accordingto the specificationsof +~(L),*i(L), and Xi(L).
Assuming,for example,an AR(1) commoncomponentand
AR(1) individual components, and Xi(L) = Xi, i
whereAYitrepresentsthe firstdifferenceof the log of the ith we wouldhave
indicator, i = 1, ..., 4; Xi(L) is the polynomial in the lag
operator;ACtis the growthrateof the compositeindex;Di is
an intercept for the ith indicator; and eit is a process with
XI 1 0 0 O0
autoregressive(AR) representation,
Eit,
Eit
-iid N(O, 2).
(2)
st- 8) = vt,
vt
iid N(O, 1)
list= Po + plst,
pi > 0, St = log 1.
= 1]= pq
Pr [St =OSt-I
= 0] = q.
0
0
0
0
AC,
o
*II
0
0
0
elt
0
*21
0
O
O
0
P31 0
O
0
0
0
(4)
(
Transitionsbetweenregimesor statesof the economyare
thengovernedby the Markovprocess,
Pr [St = 1st-I
1
41
F= O
(St = 1), as follows:
0
1 0
X4 0 0 0
(3)
and vtand Eit are independentof one anotherfor all t and i,
while the varianceof vtis takento be unityfor identification
of the model.While 8 is constantover time, jst dependson
whetherthe economyis in a recession(St = 0) or in a boom
0
X3 0 0
Thus each indicatorAYit,i = 1, 2, 3, 4, consists of an
individualcomponent(Di + eit)anda linearcombinationof
currentand lagged values of the common factor or index
ACt. The index is assumed to be generatedby the AR
process,
4)(L)(ACt -
1
0
X2
Pi(L)eit=
,
e2t
e3,
,41
e4t
(1 -0%
,
-(1L)ps
0
0
Elt
Ut
1, 2, 3, 4,
=
E2t
,
Vt ~
E3t
Ms
0
0
,
t
=
0
0
0
0
This modeldiffersfromStockandWatson's(1991) linear
_E4t
the
factor
model
of
the
business
cycle by allowing
dynamic
meangrowthrateof the coincidentindex,givenby (pst + 8),
to switchbetweenthe two regimesof the businesscycle. We
Since the parametersD and8 overdeterminethe meansof
in
of
the
overdetermined
note that the means
variablesare
the processes, as alluded to above, the model is not
a
zero
on
the
this parameterization.
Imposing mean of
pst identified.However,when the dataare expressedas deviaprocess,8 determinesthe long-rungrowthrateof the index tions from means (Ayit = AYit- AY1), equations (1)-(3) or
THE REVIEW OF ECONOMICSAND STATISTICS
190
(6)-(7) can be replaced by
Ayi. = XA(L)Ac,+ eit
(8)
= Vt
(9)
4)(L)(Act -st)
where Act = ACt - 8, or in state-space form,
AY = H
( = M+
(10)
Fi* 1 + ut
(11)
where the first element of (* is given by Act instead of ACt.
Thus in computing the log likelihood based on the Kalman
filter, the terms contained in D and 8 can be concentrated out
of the likelihood function.
If the business cycle regime, St, were observable, then this
would be a linear Gaussian model and the procedures
described in Stock and Watson (1991) based on the likelihood function could be applied to: (1) estimate the parameters of the model based on the state-space representationin
equations (10) and (11); (2) recover Di and 8 from AY =
A Y4]Jbased on the steady-stateKalmangain;
[AY1 ...
and (3) calculate the composite coincident index Ct. However, since the regime is not directly observed, ratherit must
be inferred from the data, the usual Kalman filter cannot be
employed to carry out Stock and Watson's (1991) procedures. An intuitive explanation for this is as follows. Each
iteration of the Kalman filter produces a twofold increase in
the number of cases to consider, as explained in Harrison
and Stevens (1976), Gordon and Smith (1988), and Kim
(1993a, b, 1994). Since St takes on two possible values in
each time period, there would be 2T possible paths to
consider in evaluating the conditional log likelihood. For the
monthly data used in this paper, T is about 400, implying an
impractical computational burden. Thus to make Stock and
Watson's (1991) procedures operable, approximationsto the
Kalman filter are unavoidable.
Kim and Yoo (1995) and Chauvet (1995) employ the
approximate maximum-likelihood estimation method of
Kim (1993a, b, 1994) to estimate the nonlinear dynamic
factor model of the business cycle.3 Though the maximumlikelihood estimation based on approximations to the Kalman filter is straightforwardto implement, it is difficult to
judge the effects of the approximations on the parameter
estimates and on inferences pertaining to the unobserved
common component i* and the unobserved state St. The
effects of the approximationon the steady-state Kalman gain
are also unknown, and this uncertainty complicates the task
of decomposing A Yi into Di and 8, which is necessary to
extract the composite coincident index Ct = Act + Ct_1 + 8.
In addition, since inferences about St and i*, and thus Act,
are based on final parameterestimates of the model, effects
of parameteruncertainty are also unknown in this approach.
In contrast, the Gibbs sampling approach described in the
next section incorporates parameter uncertainty by casting
the model in a Bayesian framework, and no approximations
are used in estimating parameters or in making inferences
about S,, (*, and thus Act. In addition, it provides us with
more information than may be extracted from an approximate maximum-likelihood estimation of the model. For
example, posterior distributions of the parameters and other
variates contain important information, which we exploit in
sectionIV.
III. Bayesian Inference in the Dynamic Factor Model
with Regime Switching Using Gibbs Sampling-Fixed
Transition Probabilities
The objective is to infer from data on the four coincident
indicators: (1) the path of the growth rate in the composite
index Act, t = 1, 2, . . ., T, represented by ACT,(2) the path
of the Markov switching regime indicator St, t = 1, 2 ..., T,
represented by ST, and (3) all unknown parameters of the
model, represented by a K-dimensional vector 0. In the
Bayesian framework these are viewed as three vectors of
random variables, and inference amounts to summarizing
their joint distribution, conditional on historical data on the
four individual coincident indicator variables. For reasons
discussed above, this joint distribution is not readily obtained directly, but the fact that the distributionof any one of
these vectors, conditional on the other two and the data, can
be obtained opens the problem up to solution by Gibbs
sampling. In brief, Gibbs sampling4 is an iterative Monte
Carlo technique that generates a simulated sample from the
joint distribution of a set of random variables by generating
successive samples from their conditional distributions. In
the present case, Gibbs sampling proceeds by taking a
drawing from the conditional distribution of ACTgiven the
data STand 0; then a drawing from the conditional distribution of ST given the data, the prior drawing of AiT, and 0; and
then a drawing from the conditional distribution of 0 given
the data and prior drawings of AVT and ST. By successive
iteration, the procedure simulates (perhaps surprisingly) a
drawing from the joint distribution of the three vectors
consisting of the 2T + K variates of interest, given the data.
It is straightforwardthen to summarize the marginal distributions of any of these, given the data. Recently Carter and
Kohn (1994) and Shephard (1994) introduced an efficient
multimove Gibbs sampling approach for state-space models, with which we can generate all Act at once by taking
advantage of the time ordering of the state-space model.
They also show that their multimove Gibbs sampling
dominates Carlin et al.'s (1992) single-move Gibbs sampling, especially in terms of efficiency and the convergence
to the posterior distribution.
3 Kim andYoo (1995) andChauvet(1995) estimatemodifiedversionsof
the modelgiven in equations(8) and(9). Insteadof allowingthe meanof
the commoncomponentto be regimedependent,they allow an intercept 4 For a generalintroductionto Gibbs sampling,readersare referredto
termto be regimedependent.
GelfandandSmith(1990).
BUSINESS CYCLES, A NEW COINCIDENTINDEX AND REGIME SWITCHING
More specifically, the three steps involved in generating
the Gibbs sample in our model are as follows. First, conditional on ST = [SI S2 *
ST] and 0 (all unknown
parameters of the model), the model is a linear Gaussian
state-space model from which we can generate AiT =
[AC1 AC2 ...
ACT]' using the multimove Gibbs sampling technique. Second, conditional on AiT and 0, we can
then focus on equation (9) for the common component to
generate ST. This is possible because the common component is assumed to be distributedindependently of individual
components. The procedure provided by Albert and Chib
(1993) could be used, but as in the case of ACT, the
multimove Gibbs sampling can be implemented instead.
Third and finally, conditional on ACTand ST,we are left with
generating all unknown parameters of the model 0. As the
four equations in the model are independent of one another
except for the common component, we can treat them
separately. Given this structure, the problem of generating
the parameters of each equation essentially collapses to a
form of Bayes' regression with autocorrelated errors, as
proposed by Chib (1993). Further details of the procedures
used for generating AjT, ST, and 0 and, finally, calculation of
the new composite index of coincident indicators in levels
CG,t = 1, 2,. . . , T, are described in appendix A.
TABLE 1.-BAYESIAN
PRIOR AND POSTERIOR DISTRIBUTIONS
INFORMATIVE PRIORS ON FIXED TRANSITION PROBABILITIES
Prior
Posterior
Mean
SD
Mean
SD
MD
95% Bands
0.9
0.967
0.0
0.0
0.0
0.0
0.066
0.032
1
1
1
1
0.875
0.976
0.313
-0.002
-1.777
2.110
0.044
0.009
0.076
0.067
0.294
0.302
0.881
0.977
0.313
-0.002
-1.779
2.117
(0.775, 0.945)
(0.954, 0.991)
(0.165,0.471)
(-0.134,0.131)
(-2.346, -1.193)
(1.500,2.686)
0.568
0.333
0.038
0.104
0.565
0.336
0.568
-0.003
-0.030
0.238
0.036
0.066
0.064
0.033
0.568
-0.005
-0.030
0.237
(0.497, 0.641)
(-0.132, 0.125)
(-0.154, 0.096)
(0.175,0.309)
1
1
1
0.213
-0.302
-0.070
0.3 17
0.021
0.050
0.050
0.024
0.212
-0.302
-0.069
0.3 16
(0.173,0.254)
(-0.404, -0.204)
(-0.167, 0.031)
(0.273, 0.366)
0.0
0.0
0.0
1
1
1
0.442
-0.354
-0.154
0.660
0.034
0.053
0.052
0.051
0.441
-0.353
-0.154
0.658
(0.377,0.512)
(-0.458, -0.244)
(-0.258, -0.048)
(0.565,0.767)
0.0
0.0
0.0
0.0
0.0
0.0
1
1
1
1
1
1
0.120
0.006
0.023
0.026
-0.022
0.277
0.021
0.010
0.009
0.009
0.008
0.060
0.061
0.002
0.120
0.006
0.023
0.026
-0.021
0.280
0.021
(0.102,0.140)
(-0.011, 0.024)
(0.006, 0.040)
(0.012, 0.041)
(-0.141, 0.092)
(0.155, 0.390)
(0.017,0.025)
ACt
q
p
4)1
42
P0
Pi
8
Po + pi
AYit
X1
*11
412
92
-
0.0
0.0
0.0
1
1
1
_
(0.500,0.650)
(0.114,0.526)
AY2t
X2
421
422
U22
AY3t
X3
431
4P32
0.0
0.0
0.0
-
C32
AY4t
X40
The four monthly series for the United States used in the
estimation phase of this study are those used by the
Department of Commerce (DOC) to construct its composite
index of coincident indicators: industrial production (IP),
total personal income less transferpayments in 1987 dollars
(GMYXPQ), total manufacturing and trade sales in 1987
dollars (MTQ), and employees on nonagricultural payrolls
(LPNAG).5The time period is January 1960 throughJanuary
1995.
Second-order autoregressive specifications are adopted
for the error processes of both the common component and
the four idiosyncratic components in equations (2) and (3).
Stock and Watson (1989, 1991) point out that the payroll
variable LPNAG may not be exactly coincident, lagging
slightly the unobserved common component. Following
them, three lags of Act are included in the LPNAG equation,
the fourth equation in equations (8). Readers are referred to
appendix B for the two alternative state-space representations of the model actually employed in this paper. Different
initial values of the parametersresulted in different speeds of
convergence for the Gibbs sampler. To be on the safe side,
the first 2000 draws in the Gibbs simulation process are
discarded, and then the next 8000 draws are saved and used
to calculate moments of the posterior distribution. This
procedure ensures that the results are not simply an artifact
of the initial values.6 The 95% posterior probability bands
191
X41
X42
X43
441
442
C2
Notes: (1) yU,Y2, y3, andY4,representIP, GMYXPQ,MTQ,and LPNAG,respectively.
(2) Priordistributionof O,2is improper.
(3) SD and MD refer to standarddeviation and median,respectively.
(4) 95% bandsrefers to 95% posteriorprobabilitybands.
are based on the 2.5th and the 97.5th percentiles of the 8000
simulated draws.
Table 1 presents the Bayesian prior and posterior distributions of the parameters. For each parameter, the prior
distribution is specified by mean and standard deviation
(SD), and the posterior by mean, standard deviation (SD),
median (MD), and the 95% posterior probability bands.
Rather tight prior distributions for p and q are designed to
incorporate what was known or believed about the average
duration of business cycle phases by the beginning of the
sample period. The average duration of recessions and
booms implied by the means of these priors are 10 months
and 33.3 months, respectively. As the model is estimated
with data expressed in deviation from means, under the
hypothesis of no regime switching we have Po = 0 and p0 +
, = 0. Standard errors and the 95% posterior probability
bands for these parameters seem to provide evidence in
S The abbreviations,
IP, GMYXPQ,
MTQ,andLPNAGareDRI variable favor of Burns and Mitchell's (1946) concept of the division
names.
of the business cycle into two separate phases.
6 Theinitialvaluesemployedarep = q = 0.9; I = +2 = 0; Xi = 0.5, i =
1, 2,3,4; X4j= Oj= 1, 2,3; ti-=
i = 1, 2,3,4,j = 1, 2;
F2
= 0.2, i= 1,
2, 3, 4. Gelman and Rubin (1 992) suggest that a single sequence of samples
may give a false impressionof convergence,no matterhow manydraws different initial values. The posterior distributions were robust with respect
are chosen. Thus following one of the referees, we also tried various to different initial values.
THE REVIEW OF ECONOMICSAND STATISTICS
192
FIGURE 2.-PROBABILITIES OF A RECESSION (UNIVARIATE MODEL;
INFORMATIVEPRIORS)
MODEL;
(MULTIVARIATE
FIGUREI.-PROBABILITIESOFA RECESSION
PRIORS)
INFORMATIVE
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2-
0.2
65
70
75
80
85
90
95
Figure 1 depicts the posterior probabilitythat the economy
was in the recession regime in each month as inferred from
the Gibbs sampling. The shaded areas represent the periods
of National Bureau of Economic Research (NBER) recessions (from peak to trough). The posterior probabilities are
in close agreement with the NBER reference cycle. To what
extent do these results reflect an ability of the model to
extract Burns and Mitchell's (1946) "comovement of economic variables" over the business cycle, or do these results
depend importantly on our use of tight priors for transition
probabilities?
In order to evaluate the marginal contribution of going
from a univariate analysis to a multivariate common factor
approach, we reestimated the posterior regime probabilities
from a univariate regime-switching model of the IP series,
one of the four series employed in this paper.The same tight
priors were given for the transition probabilities of the
univariate model, as in the multivariate case.7 Figure 2
shows the results. A univariate analysis results in significantly lower correlation between posterior regime probabilities and the NBER reference cycle.8
In orderto evaluate the marginalcontribution of the use of
tight priors for p and q, the multivariate model was
reestimated using noninformative priors on these parameters. Priors were chosen such that expected durations for
booms and recessions are both two months (i.e., expected
values of the priors on p and q were both specified as 0.5).
Except that the 95% posterior probability bands are somewhat larger than in the case of tight priors, results reportedin
table 2 are almost the same as in the case of tight priors. In
7In a univariate context, Filardo and Gordon (1993) take a similar
approach in evaluating the marginal contributions of the tight priors for
transition probabilities versus the use of leading indicator data in a model
of time-varying transition probabilities.
8 Filardo (1994) and Kim (1996) also report similar findings for
individual monthly data, based on maximum-likelihood estimation of the
univariate regime-switching model.
70
65
75
85
80
90
95
figure 3 posterior recession probabilities from the noninformative priors are depicted. One can hardly distinguish figure
3 from figure 1. Thus prior information about transition
probabilities does not play an important role in infering
PRIOR AND POSTERIOR DISTRIBUTIONS
TABLE 2.-BAYESIAN
NONINFORMATIVE PRIORS ON FIXED TRANSITION PROBABILITIES
Posterior
Prior
Mean
SD
Mean
SD
MD
95% Bands
PO
P1
0.5
0.5
0.0
0.0
0.0
0.0
0.289
0.289
1
1
1
1
0.808
0.946
0.341
0.001
-1.659
1.968
0.133
0.108
0.102
0.069
0.563
0.612
0.841
0.972
0.328
0.002
-1.779
2.097
(0.325, 0.938)
(0.568, 0.989)
(0.170, 0.575)
(-0.129,0.138)
(-2.435, -0.044)
(0.105, 2.740)
8
-
-
0.569
0.039
0.566
(0.501, 0.650)
-
-
0.308
0.192
0.316
(-0.029,0.529)
AC,
q
p
(+I
412
Po + pi
Aylt
XI
4'11
4'12
2
0.0
0.0
0.0
1
1
1
0.571
-0.012
-0.033
0.237
0.040
0.066
0.064
0.033
0.569
-0.011
-0.033
0.236
(0.498, 0.654)
(-0.139,0.121)
(-0.158, 0.094)
(0.175,0.305)
0.0
0.0
0.0
1
1
1
0.215 0.022
-0.303 0.052
-0.069 0.051
0.317 0.024
0.214
-0.302
-0.070
0.316
(0.174, 0.259)
(-0.405, -0.206)
(-0.168, 0.031)
(0.275.0.365)
0.445
-0.354
-0.153
0.659
(0.376, 0.522)
(-0.458, -0.249)
(-0.257, -0.049)
(0.567,0.768)
AV2
A2
'P21
1P22
U22
-
-
AV3t
A3
4'31
4132
a2
X40
X41
X42
A43
441
442
42
0.0
0.0
0.0
1
1
1
0.445
-0.354
-0.153
0.660
0.0
0.0
0.0
0.0
0.0
0.0
1
1
1
1
1
1
0.121 0.010
0.006 0.009
0.023 0.009
0.026 0.008
-0.020 0.058
0.282 0.060
0.021 0.002
-
0.037
0.053
0.053
0.052
0.121 (0.102,0.143)
0.006 (-0.012, 0.024)
0.024 (0.006, 0.040)
0.026 (0.011,0.042)
-0.019 (-0.136, 0.096)
0.283 (0.162, 0.394)
0.021 (0.017,0-025)
Notes: (1) Vl,, Y2,, y,, and y4, representIP, GMYXPQ,MTQ,and LPNAG,respectively.
(2) Priordistributionof o, is improper.
(3) SD and MD referto standarddeviation and median,respectively.
(4) 95% bandsrefersto 95% posteriorprobabilitybands.
BUSINESS CYCLES, A NEW COINCIDENTINDEX AND REGIME SWITCHING
FIGURE3.-PROBABILITIESOFA RECESSION
(MULTIVARIATE
MODEL;
NONINFORMATIVE
PRIORS)
1.0
...
.
......
FIGURE4.-NEW COINCIDENT
INDEXVERSUSDOC'SCOINCIDENT
INDEX
:.....
1v------.....
........~
.,,.
--
.. .. ...
'-:::
...
...
,
.... ,
...
.'.'.
.
,....
...
.....
......
.. .
:
75
80
85
90
.:..
:.
. *. :.:...:
.. ?
....:..
:
...
.:::.:::::..
:.:.:..::
....
.....
.............
.....
. ..
....
.. .
......
........
.. . ..::
:'.'.:
.
..
... ...
...
......
.. . . .
.......
........
0
DOC_CC
......
0.0
70
o
K
7
40~~ ~~-
.....
.........
..,....s
...
.
.. ,...,./..
..'.'
60-0
0.2
...
.::..:..:./
.:.:-'.::.:-,
.... :
"''.'.'.'.
40- %,....
...
-,-::.:-..
.... " ........
.:.....
.
.:
~~~~~~~.::
...
.... : .......'
:
.:,,':,.::,,....
0.4
...........
......
,.... ..,.
,....
.....
...
-.:-::
:-.:-..
.','
,,..,.
:
0:'.::.
::::
on. :::
0.6-
.
......
......
....
~ ~~~~~.....
....
..
...
....
100-
0.8
65
193
.
8
.....
.
0
NEWCC
|
....
9
.
95
whether the economy is in the recession or the boom regime.
This, along with the substantially weaker results from the dent indexes implied by the two models with and without
univariate model, suggests that it is the ability to capture regime switching are plotted and compared. The two indexes
"comovement among economic variables" in a coincident are almost identical, except that since the 1970s, the Stock
index that accounts for the model's success in identifying the and Watson index seems to show higher growth than the new
experimental index from a dynamic factor model with
NBER turning points.
In figure 4 the composite coincident index implied by the
model is plotted together with the DOC's coincident index.9
TABLE 3.-BAYESIAN
PRIOR AND POSTERIOR DISTRIBUTIONS
Contemporaneous correlation between the first differences
STOCK AND WATSON (1991) MODEL; No REGIME SWITCHING
of the two series is 0.9825. Though the model-based index
Prior
Posterior
agrees closely with the DOC index, during recessions the
Mean SD
Mean
SD
MD
95% Bands
decline in the growth rate of the model-based index is much
sharperthan that of the DOC index. In the early portion of
AC,
1
0.0
0.533 0.069
0.534 (0.395, 0.669)
41
the sample, the growth rate of the model-based index during
0.0
1
0.029 0.062
0.029 (-0.091,0.153)
4)2
booms seems to be higher than that of the DOC index, but in
a
0.567 0.046
0.566 (0.483, 0.662)
the more recent portion of the sample, the pattern in the
Ay1t
1
0.0
XA
0.621 0.041
0.619 (0.544,0.705)
growth rates during booms seems to be reversed. After the
0.0
1
-0.032
0.065
-0.036
(-0.163, 0.095)
'pl
1982 recession, the model-based index seems to show
0.0
1
-0.054
0.065
-0.055
1112
(-0.176,0.074)
slower growth than the DOC coincident index.
o2
0.236 0.034
0.235 (0.173,0.303)
It might also be useful to compare the new coincident
AY2t
0.0
1
0.231 0.023
0.231 (0.188, 0.278)
X2
index with the one based on Stock and Watson's (1991)
1
0.0
-0.302
0.052
-0.303
1121
(-0.402, -0.202)
linear dynamic factor model. Table 3 presents the Bayesian
0.0
1
-0.065
-0.066
0.051
V122
(-0.163, 0.033)
o2
0.316 0.023
0.315 (0.274, 0.363)
prior and posterior distributions of the parameters for the
AY3t
Stock and Watson model without the feature of regime
0.0
1
0.479 0.038
0.477 (0.406,0.556)
X3
switching. The posterior distributions of parameters are
1
-0.358
0.053
-0.356
(-0.460, -0.253)
'P31 0.0
quite close to those from the model with regime switching.
-0.153
1
0.053
-0.153
'P32 0.0
(-0.256, -0.051)
2
0.668 0.052
0.666 (0.571,0.775)
One exception is that the sum of the AR coefficients
AY41
(1I + 42) for the common component is higher for the
1
0.133 0.010
0.133 (0.113, 0.153)
A40 0.0
1
Stock and Watson model. In figure 5 the composite coinci0.003 0.011
0.003 (-0.018, 0.024)
A41 0.0
-
-
-r-
9 The model-basedindex in figure4 is calculatedfirst by scaling AC,
from the model to have the same varianceas the growth in the DOC
coincidentindex.The scaledAC,is maintainedto have the samemeanas
the originalAC,.Thenthe new indexis adjustedsuchthatit is equalto the
value of the DOC index in January1970. The coincidentindex derived
with noninformativepriors was almost identicalto the one with tight
priors,andthusis not shownhere.
A42
A43
1141
'P42
o2
1
1
1
1
0.0
0.0
0.0
0.0
-
-
0.024
0.028
-0.003
0.293
0.021
0.010
0.008
0.060
0.060
0.002
0.024
0.028
-0.002
0.294
0.021
(0.004, 0.043)
(0.011,0.045)
(-0.120, 0.110)
(0.170,0.410)
(0.017,0.026)
Notes: (1) Yrl,Y21'y3, andy4,representIP, GMYXPQ,MTQ,and LPNAG,respectively.
(2) Priordistributionof ur?is improper.
(3) SD and MD referto standarddeviation and median,respectively.
(4) 95% bandsrefersto 95% posteriorprobabilitybands.
THE REVIEW OF ECONOMICSAND STATISTICS
194
tion of the businesscycle regimevariableSt:
INDEXVERSUSSTOCK
FIGURE5.-NEW COINCIDENT
INDEX
ANDWATSON'SCOINCIDENT
Pr [S, = 1] = Pr [S* - 0]
(12)
whereS* is a latentvariabledefinedas
* = Yo + ylSt-I
+ Y2,d2(l -St-I)NO,-
(13)
+ Y3,d3St- lNl,t I + ut
d2=0or1,
d3=Oor1
(14)
ut - iid N(O, 1)
40
65
70
75
80
85
90
(15)
where Nj,At1, j = 0, 1 are the durations up to time t - 1 of a
recessionorboom,respectively;andtheparametersY2,d2
and
which
will
be
discussed
in detail later,determinethe
Y3,d3,
of
business
duration
nature
cycle
dependence.In the above
95
probitspecification,the transitionprobabilitiesaregiven by:
NEW-CCI
-----SWCCI
Pr[S,= IISt-I==,Nl,t_I]
=Pr [S*'?O|S,1 = 1, Nl,t- ]
= Pr [u, -y - y - Y3,d3Nl,,t-]
regime switching. Overall, the Stock and Watson index
agrees more closely with the DOC index than our new
experimentalindex.
Pr [St = O|St-I = 0, No,1]
IV. Testsof BusinessCycleDurationDependence
Durationdependencein the business cycle and timevaryingtransitionprobabilitiesare relatedbut not identical
concepts.Durationdependenceasks whetherrecessionsor
booms "age," that is, are more likely to end as they last
longer. It is a special form of time-varyingtransition
probabilitiesin a regime-switchingmodel of the business
cycle. Diebold and Rudebusch(1990) and Diebold et al.
(1993) find evidence that postwarrecessionstend to have
positivedurationdependence,althoughboomsdo not. More
recently,Durlandand McCurdy(1994), using Hamilton's
(1989) univariateMarkov-switchingmodel of business
cycle, reach the same conclusion. Diebold et al. (1994),
Filardo(1994), and Filardoand Gordon(1993) deal with
anotherformof time-varyingtransitionprobabilities,specifying themas functionsof an exogenousvariablesuchas the
index of leading indicators.None of these papersspecifically deals with durationdependence in a multivariate
context. In this section we investigate the hypothesis of
durationdependenceby extendingthe dynamicfactormodel
in sectionII to incorporatenonzeroprobabilitiesof duration
dependencein the transitionprobabilities.
A. Extensionof the Model to Allowfor NonzeroProbabilities
of Business Cycle DurationDependence
To constructa formal test of business cycle duration
dependence,the transitionprobabilitiesin equation(5) are
replacedby the followingprobitspecificationfor the evolu-
(16)
Pr [S* < OjSt-1 = 0, No,t_]
= Pr [ut < Yo - Y2,d2NO,t-]I
(17)
-
When Y2,d2 = 0 and Y3,d3 = 0, we have fixed transition
probabilitiesor no business cycle durationdependence;
when Y2,d2> 0 and Y3,d3< 0, businesscycles are characterized by positive durationdependence.In orderto allow for
nonzeroprobabilitiesof positive durationdependence,we
adoptassumptionssimilarto thoseintroducedin Georgeand
McCulloch (1993) and Geweke (1994) in the context of
variable selection in regression. In particular,while we
assumeyo and yI each are nonzerowith priorprobability1,
we follow Geweke (1994) in adoptinga priorfor Y2,d2and
normalandpointmassat
Y3,d3thatis a mixtureof a truncated
0;
=
0,
if d2 = 0
N
Y2d211
?2)I[y2,d>O],
2
=
0,
if d3
Y3d31N(y3,
(?)I3d3<o'
=
Pr [Y3,d3 = 0]
=
0
if d3 = 1
Pr [d2= 0] = Pr [Y2,d2= ?]P2
Pr [d3= 0]
(18)
if d2
(19)
(20)
P3
(21)
BUSINESS CYCLES, A NEW COINCIDENTINDEX AND REGIME SWITCHING
TABLE4.-SENsrrIvrrIES OF POSTERIOR
PROBABILITIES
OF No DURATION
DEPENDENCE
TO DiFFERENTPRIORS
_.
.
0= !
= 0-3;
TABLE 6.-SENSITIVITIES
195
OF POSTERIOR PROBABILITIES OF No DURATION
DEPENDENCE
TO DwiFERENT
PRIORS
(1)0= !1= 0.1;
2 =w3 = 0I.1
2 = 3 = 0-.05
Prior(P2 = P3)
Posterior
Pr [d2 = 0]: For Recessions
0.300
0.400
0.500
0.600
0.014
0.023
0.033
0.044
0.700
0.070
0.800
0.113
Prior(P2 = P3)
Posterior
Pr [d2 = 0]: For Recessions
0.300
0.400
0.500
0.600
0.032
0.036
0.020
0.049
0.700
0.089
0.800
0.153
Prior(P2 = P3)
Posterior
0.300
0.122
Pr [d3 = 0]: For Booms
0.400
0.500
0.600
0.196
0.276
0.355
0.700
0.460
0.800
0.595
Prior(P2 = P3)
Posterior
0.300
0.482
Pr [d3 = 0]: For Booms
0.400
0.500
0.600
0.549
0.673
0.802
0.700
0.807
0.800
0.876
Prior(P2 = P3)
Posterior
Pr [d2 = 0, d3 = 0]: For Joint Tests
0.300
0.500
0.400
0.600
0.001
0.004
0.008
0.015
0.700
0.028
0.800
0.054
Prior(P2 = p3)
Posterior
0.700
0.067
0.800
0.123
Pr [d2 = 0, d3 = 0]: For Joint Tests
where the subscriptI[ ] refers to an indicatorfunction
introducedto allow for positive business cycle duration
dependenceunderthe alternativehypothesis;andP2 andp3
are independentprior probabilitiesof no durationdependence for recessions and booms. Each of these prior
distributionsincludesthe possibilitythatthe durationvariables No,,-, andNl,,_1 are excludedfrom the model. Given
these priors, Bayesian tests of business cycle duration
dependenceare to read the posteriorprobabilitiesof no
durationdependencedirectlyfrom the proportionof posteriorsimulationsin whichd2 = 0 andd3 = 0.
B. Implementationof Gibbs Sampling
Replacingthe fixedtransitionprobabilitiesin equation(5)
by potentiallyduration-dependent
transitionprobabilitiesin
equations(16) and (17) requiresmodificationsof the basic
Gibbs samplerdescribedin appendixA. Specifically,the
procedure for generating St, t = 1, 2, . . , T, in appendix A.2
andthatfor generatingthe transitionprobabilitiesin appendix A.6 shouldbe modified.
Due to the time-varyingnatureof the transitionprobabilities, St, t = 1, 2, . . . , T, cannot be generated as a block as in
[S1
S*
...
0.300
0.008
0.400
0.017
0.500
0.021
0.600
0.036
ST*]'based on equation (1 3), by generating
ut from an appropriatetruncatedstandardnormaldistribution. Conditional on St = 1 and St-I = 1, for example, ut is
generatedfrom
Ut - N(O, 1)J[utl-(y0+y1
+yY3d3Nl
(22)
.t 1)]
wherethe subscriptI[ ] refersto an indicatorfunction.
Conditionalon ST = [SI
[S2
S*S3T],*
S2
...
ST] and, thus, on ST =
N1,-1 = [N1,1 N1,2
...
N1,T-1],
and N0,-I = [NO,I NO,2 ..
NO,T-1]', the variates in
equation (13) are independentof the data set and other
variatesin the system.This nice conditioningfeatureof the
Gibbs sampling allows us to focus on equation(13) for
generatingtransitionprobabilitiesand otherrelatedparameters for tests of businesscycle durationdependence.Thus
the transitionprobabilitiescan be generateddirectlyfrom
equations(16) and(17). AppendixC describesin detailhow
di, i = 2, 3, and y variables in equation (13) can be
generated.Generatingthe other variatesin the system is
exactlythe same as in the case of fixed transitionprobabilities, as describedin appendixA.
appendixA.2. Each St should be generatedone at a time C. EmpiricalResults
conditionalon Sj+t,j = 1, 2, . . ., T,andon othervariates.It
Of particularinterestin this sectionarethe proportionsof
is straightforwardto modify Albert and Chib's (1993)
procedure to achieve this goal. Once St, t = 1, 2, . . ., T, is posteriorsimulationsin which (1) d2 = 0 for tests of no
generated,we can count the durationof a recession or a durationdependencefor recessions;(2) d3 = 0 for a test of
boom (Nl,t- I or No, l1)up to month t - 1, t = 2, 3, . .., T. no durationdependencefor booms;and(3) d2 = d3= 0 for a
Also, given -yo, Y1, Y2,d2, Y3,d3, each generatedset ST can joint test of no business cycle durationdependence.We
be converted to a set of latent variables S * = adoptdifferentpriorprobabilities(P2 = P3 = 0.3, 0.4, 0.5,
0.6, 0.7, 0.8) of no durationdependenceand investigate
sensitivities of posteriorprobabilitiesto these priors.We
TABLE 5.-SENsITIvrrms
OF POSTERIOR PROBABILITIES OF No DURATION
and'Y3,1in equation
also adoptdifferentpriorsfor yo,'Yi,'Y2,1,
DEPENDENCE TO DIFFERENT PRIORS
(13) and investigate sensitivities of the results.10Prior
)22= I)3 = 0.1
!20= !1 = 0.3;
distributionsemployed for these parametes are yo Pr [d2 = 0]: For Recessions
=
Prior(P2 P3)
0.300
0.400
0.500
0.600
0.700
0.800
N(- 1.28,!O) and Yi
I N(3.24, w2) (theseareequivalentto
Posterior
0.029
0.042
0.067
0.081
0.114
0.187
assuming average durationsof recessions and booms to
Pr [d3 = 0]: For Booms
be 10 monthsand 33.3 monthsunderthe null hypothesisof
Prior(P2 = P3)
Posterior
0.300
0.405
0.400
0.533
Prior(P2 = P3)
Posterior
Pr [d2 = 0, d3 = 0]: For Joint Tests
0.300
0.400
0.600
0.500
0.010
0.018
0.040
0.056
0.500
0.637
0.600
0.734
0.700
0.767
0.800
0.874
0.700
0.080
0.800
0.162
10As mentionedin Georgeand McCulloch(1993), from a subjectivist
Bayesianstandpoint,one wouldwantto choose!t)j, j = 2, 3, largeenough
to give supportto values of -Yj,i,
j = 2, 3, thatare substantivelydifferent
from 0, but not so large that unrealisticvalues of Yyj,l,
j = 2, 3, are
supported.
THE REVIEW OF ECONOMICSAND STATISTICS
196
FIGURE 6.-SENSITIVITIES OF POSTERIOR PROBABILMES OF No DURATION
DEPENDENCE TO DIFFERENT PRIOR PROBABILITIES
w=
w=
0.3; W2
(1)3= 0.05
Posterior
Probabilities
PosteriorProbabilities
1.0
1.0-
0.8.
Pr[d3=0]
0.4-
0.6-
-
0.4-
___-~0.2-
Pr[d2=0]
~
^4~~~~~~~~~;~
0.5
0.4
0.6
0.7
Pr[d2=d3=0]
Probabilities
~~~~~~~Prior
0.8
(p2=p3)
no duration dependence, as in section III);
N(O,
Pr[d3=0]
__8
0.6-
0.0- t
0.3
FIGURE8.-SENSMVITIES OFPOSTERIOR
PROBABILITIES
OFNo DURATION
DEPENDENCE
TODIFFERENT
PRIORPROBABILITIES
WO= WI 0.1; W2 (03 = 0.1
(iJ2,O
and
Y3,1
Y2,1
- N(O, 2)J3, <o0 We try the
followingthreedifferentcombinationsof W:
(1) Case 1:
!o = w, = 0.3 and !2 = w3= 0.05.
(2) Case2: ?o= w =0.3 and j2 =
(3) Case3: !0 = w = 0.1 andW2=
W3 =
0.1
W3=-.1.
Throughoutthese experiments,we maintainthe priorsof
all the other variatesin the system to be the same as in
sectionIII(tables1 and2). Foreachof the above 18 different
sets of prior distributions,we run the Gibbs samplerfor
12,000 iterations.For most of the cases the Gibbs sampler
seems to convergein less than3000 iterations.To be on the
safe side, we discardthe first4000 andbase our inferences
on the last 8000 iterations.
Tables4 through6 summarizethe sensitivitiesof posterior probabilitiesof no durationdependenceto different
priors.These are also depictedin figures 6 through8. In
general,smallervalues of !j, the priorstandarddeviations
of y, the parametersresult in lower posteriorprobabilities
0.2-
Pr[d2=0]
0.0-a=S
0.3
0.4
0.5
0.6
0.7
Pr[d2=d3=0]
PriorProbabilities
0.8
(p2=p3)
for given priorprobabilities.Concerningtests of no duration
dependence for recessions, the results are robust with
respect to different priors: posterior probabilitiesrange
between0.014 and0.187, revealingstrongsampleinformation in favor of positive durationdependencebeyond that
containedin the priorprobabilities.Concerningtests of no
durationdependencefor booms, resultsfor case 1 tend to
reveal somewhat weak sample informationin favor of
positive durationdependence:posteriorprobabilitiesof no
durationdependencein case 1 range between 0.122 and
0.595. Resultsfromthe othertwo cases, however,revealno
such sampleinformationbeyondthatcontainedin the prior
probabilities.
V. Summaryand Conclusions
This paperestimatesa model that incorporatesthe two
key features of business cycles identifiedby Burns and
Mitchell (1946): comovementamong economic variables
and switching between distinct regimes of recession and
boom.A commonfactor,interpretedas a compositeindexof
coincidentindicatorsand estimatesof turningpoints from
one regime to the other were extracted from the four
OFNo DURATION
FIGURE7.-SENsrrIVMES OFPOSTERIOR
PROBABILITIES
monthlyDOCcoincidentindicators,utilizingthe multimove
TODIFFERENT
PRIORPROBABILITIES
DEPENDENCE
Gibbs samplingof Shephard(1994) and Carterand Kohn
wo = 01 = 0.3; 02 = 03 0.1
(1994) in a Bayesian framework.The empirical results
PosteriorProbabilities
demonstratethat both comovementamong indicatorsand
1.0
regime switching are importantfeatures of the business
cycle. Theresultingmodel-basedindexof coincidentindicaPr[d3=0]
0.8
torsprovidesa somewhatdifferentpictureof the strengthof
0.8-_recentbusinesscycles thandoes the DOC index. Estimates
0.6_
of turningpoints are sharperand agreemuch more closely
with the NBER dates than do estimatesfrom a univariate
0.4model.
The approach in this paper provides us with more
Prld2=0]
0.2
information than may be extracted from maximum__----' .Pr[d2=d3=0]
likelihoodestimationof the model. In particular,a straight==-,
, PriorProbabilities forward extension of the base model leads naturallyto
0.0 0.3
(p2=p3)
0.7
0.8
0.5
0.6
0.4
Bayesian tests of business cycle durationdependence.By
BUSINESS CYCLES, A NEW COINCIDENTINDEX AND REGIME SWITCHING
takingadvantageof ideas set forthin Geweke(1994) in the
context of variableselection in regression,we extend the
base model to include nonzero probabilitiesof business
cycle durationdependence.
Whileexistingliterature(DieboldandRudebusch(1990),
Diebold et al. (1993), and Durlandand McCurdy(1994),
among others)reportevidence of positive durationdependence for recessionsand little evidencefor booms withina
univariatecontext,we confirmtheirresultswithina multivariate context.We find strongevidence of positive duration
dependencefor recessions,and the results are robustwith
respectto differentpriorsemployed.As for booms,however,
the results are not robust with respect to differentpriors
employed. Depending on the choice of priors, posterior
probabilitiesof no durationdependencetend to reveal a
certaindegree of sample informationin favor of positive
durationdependencefor booms. However,the evidence is
not strongenough.
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197
Gelman,A., andD. B. Rubin,"ASingleSequencefromtheGibbsSampler
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A. P. Dawid, and A. F. M. Smith, Bayesian Statistics (1992),
625-631.
George, EdwardI., and RobertE. McCulloch,"VariableSelection via
Gibbs Sampling," Journal of the American Statistical Association
88: 423, Theory and Methods (1993), 88 1-889.
Geweke, John, "VariableSelection and Model Comparisonin Regression," in J. 0. Berger,J. M. Bernardo,A. P. Dawid, andA. F. M.
Smith (eds.), Proceedings of the Fifth Valencia International
Meetings on Bayesian Statistics (1994).
Gordon,K., andA. F.M. Smith,"ModelingandMonitoringDiscontinuous
Changesin TimeSeries,"in J. C. Spall(ed.), BayesianAnalysisof
Time Series and Dynamic Linear Models (New York: Marcel
Dekker,1988),359-392.
Hamilton, James, "A New Approachto the Economic Analysis of
TimeSeriesandthe BusinessCycle,"Econometrica
Nonstationary
57: 2 (1989), 357-384.
Harrison,P. J., andC. F. Stevens, "BayesianForecasting,"Journalof the
Royal Statistical Society, ser. B, 38 (1976), 205-247.
Time Series Models with
Kim, Chang-Jin,"Unobserved-Component
Heteroskedasticity:
Changesin Regimeandthe
Markov-Switching
LinkbetweenInflationRatesandInflationUncertainty,"
Journalof
Business and Economic Statistics 11 (1993a), 341-349.
"Sourcesof MonetaryGrowthUncertaintyandEconomicActivModel with Heteroskedastic
ity: The Time-Varying-Parameter
Disturbances,"
thisREVIEW 75 (1993b),483-492.
Journalof
"DynamicLinearModels with Markov-Switching,"
Econometrics 60 (1994), 1-22.
"PredictingBusinessCycle Phaseswith Indexesof Leadingand
CoincidentIndicators:A MultivariateRegime-ShiftApproach,"
and Variance Shifts," Journal of Business and Economic Statistics
Journal of Economic Theory and Econometrics 2: 2 (1996), 1-27.
ing," Journal of the American Statistical Association 87: 418,
Theory and Methods (1992), 493-500.
82: 400, Theory and Method (Dec. 1987).
Kim,Myung-Jig,andJi-SungYoo, "New Indexof CoincidentIndicators:
11: 1 (1993), 1-15.
A MultivariateMarkovSwitchingFactorModelApproach,"JourBums,A. F., andW. C. Mitchell,"MeasuringBusinessCycles,"National
nal of Monetary Economics 36 (1995), 607-630.
Bureauof EconomicResearch,New York(1946).
State-SpaceModelingof NonstationCarlin,BradleyP., NicholasG. Polson, and David S. Stoffer,"A Monte Kitagawa,Genshiro,"Non-Gaussian
ary Time Series," Journal of the American Statistical Association
CarloApproachto Nonnormaland NonlinearState-SpaceModelShephard,Neil, "PartialNon-GaussianState Space," Biometrica 81
(1994), 115-131.
Carter,C. K., andP.Kohn,"OnGibbsSamplingfor StateSpaceModels," Shumway,
R. H., and D. S. Stoffer, "Dynamic Linear Models with
Biometrica81 (1994), 541-553.
Switching," Journal of the American Statistical Association 86
Chauvet,Marcelle,"An EconometricCharacterization
of BusinessCycle
(1991), 763-769.
Dynamicswith FactorStructureand Regime Switching,"Depart- Stock,JamesH., andMarkW. Watson,"New Indexesof Coincidentand
mentof Economics,Universityof Pennsylvania(1995).
LeadingEconomic Indicators,"in 0. Blanchardand S. Fischer
Chib,Siddhartha,"BayesRegressionwithAutocorrelated
Errors:A Gibbs
(eds.), NBER Macroeconomics, Annual (Cambridge, MA: MIT
Sampling Approach," Journal of Econometrics 58 (1993), 275Press,1989).
294.
"A ProbabilityModelof the CoincidentEconomicIndicators,"in
Diebold,FrancisX., J.-H.Lee, andG. C. Weinbach,"RegimeSwitching
K. Lahiri and G. H. Moore (eds.), Leading Economics Indicators:
with Time-VaryingTransitionProbabilities,"in C. Hargreaves
NewApproaches and Forecasting Records (Cambridge: Cambridge
(ed.), Nonstationary Time Series Analysis and Cointegration (OxUniversityPress,1991).
"A Procedurefor PredictingRecessionswith LeadingIndicators:
ford:OxfordUniversityPress,1994),283-302.
EconometricIssues and Recent Experience,"in J. H. Stock and
Diebold,FrancisX., andGlennRudebusch,"A Nonparametric
InvestigaM. W. Watson (eds.), Business Cycles, Indicators and Forecasting
tion of DurationDependencein the AmericanBusiness Cycle,"
(Chicago:Universityof ChicagoPressfor NBER, 1993),255-284.
Journal of Political Economy 98 (1990), 596-616.
"MeasuringBusinessCycles:A ModemPerspective,"thisREVIEW
78 (1996), 67-77.
Diebold, Francis X., Glenn Rudebusch,and D. E. Sichel, "Further
APPENDIX A
Evidence on Business-CycleDurationDependence,"in James
Stock and Mark Watson (eds.), Business Cycles, Indicators and
Forecasting (Chicago:Universityof Chicago Press for NBER,
1993),255-284.
Durland,J. Michael, and Thomas H. McCurdy,"Duration-Dependent
Transitionsin a MarkovModel of U.S. GNP Growth,"Journalof
Business and Economic Statistics 12: 3 (1994), 279-288.
Gibbs Samplingfor a Model with Fixed TransitionProbabilities
A.J Generating Act, t = 1, 2, .. ., T
As mentionedearlier,thereis morethanone way of castingthemodelin
equations(1)-(2) or (8)-(9) intostate-spaceform.Forthepresentpurpose
to theone in equations
we employanalternativestate-spacerepresentation
299-308.
Filardo,AndrewJ., andStephenF. Gordon,"BusinessCycle Durations," (10)-(11).1I If we multiplybothsides of equation(8) by *i(L), i = 1, 2, 3,
+ Ei,,i = 1, 2, 3, 4, where Ay,*,=
4, we have Ayi, = Xj(L)qij(L)Ac,
WorkingPaper,FederalResearchBank,KansasCity,MO (1993).
Gelfand,AlanE., andAdrianF. M. Smith, "Sampling-Based
Approaches
to CalculatingMarginal Densities," Journal of the American
I Thestate-spacerepresentation
in equations(10)-( 11)is usedlaterwhen
Statistical Association 85: 410, Theory and Methods (1990),
decomposingA YiintoDi and8 basedon the steady-stateKalmangain.
398-409.
Filardo,Andrew,"BusinessCycle PhasesandTheirTransitionalDynamics," Journal of Business and Economic Statistics 12 (1994),
THE REVIEW OF ECONOMICSAND STATISTICS
198
*i(L)Ayi,,resultingin the followingalternativestate-spacerepresentation: A.2 GeneratingSt, t = 1, 2,...,T
Ay*,= H*t, +
=tt
M* +
(10)'
Et
F*tt-I
OnceA/T = [Ac, ...
ACT] hasbeengenerated,
ST =
can be generatedbasedon the followingdistribution:
[S1
ST]'
...
(11)'
+ U*.
Assuming AR(1) common and idiosyncraticcomponents and for
P (STI AYT, ACT) = P (STI AY T, ACT)
Xi(L) = Xi, for example, we have
T- I
f
p (SIAy*,Ac s +,)
(A.6)
T- I
AYit
X2 - 2X21
FAc, 1
= P(STI ACT)
'E2(
I P (StI AC,, S.+1).
t= ,
AY3*
X3
-X331
AY4
X4
-X4p4UJ4
[AcT
1
LAC-I
[4(L)ps[1
0
[AC,t Ij
'f)1
E3t
1Act
+ [1
I
This impliesthatwe may focus only on equation(9) to generateST,as
the distributionof ST is independentof Ay*, given AjT. To generateS, we
can adoptthe followingsteps.
Step 1: Run Hamilton's(1989) basic filter for equation(9) to get
p(S,|A/t) andp(SfJAcj,-) for t = 1, 2, . . ., T and save them. The last
iterationof the filter provides us with P(STI AMT), from which ST is
generated.
[v1-
0 AAct-2
Step 2:
Now consider the following joint distribution of T =
and given
[A1 t2 63 ** T]', given AyT*= [Ay* Ay* ...Ay*l'
thepriordistributionof to:
To generate S, conditional on Ai, and S,+, t = T - 1, T - 2,
, 1, we employthe followingresult:
p (St|A,tS,+X)
T- 1
P (ZTI4YT*) = P *1 AYT*)II P (tt IAS.t*tt+ I
=
p(S,+,
IJS,)p(S,tfAc,)
IS
p(S,+
)
tA,
(A.7)
(A.1)
*=1
If we pay attentionto the time orderingof the state-spacemodel as in
CarterandKohn(1994), generationof ZTfromP(ZTI 5*) canbe donein the
following sequence.First,we can generatetT fromP(*T1 Ay*). Then, for
whereP(S,+I St) is the transitionprobability,and the othertermson the
right-handside havebeensavedfromstep 1.
= T-1,
T-2, . . ., 1, we can generate tt from p(CtIAy*, ,t+l). As the A.3 GeneratingA1,i = 1, 2, 3, 412
state-space model in equations (10)'-(11)' is linear and Gaussian,
conditionalon S,, t = 1, 2, . . ., T,we can takeadvantageof the Gaussian
Given ACT, ST, and all other parametersof the model, Xi can be
Kalmanfilterto obtainP(tT| Ay*) andp(t, IAy*, tt+l).
generatedbasedon equation(8) multipliedby 4ii(L),i = 1, 2, 3, 4,
Noticealso thatall otherelementsof t, exceptfor the firstelement,Ac,,
are associatedwith elements of t-I by the identity matrix.The joint
= XAc* + Eit, i = 1, 2, 3, 4
(A.8)
i*t
densityin equation(A.1) can thereforebe rewrittenalternativelyin terms
of the first element of t,, Ac,,
wherey,*,=
andAc* =
DefiningA5YandAC*to be the
Jj(L)Ayj, and right-hand-side
1i(L)Ac,.
vectors of left-hand-side
variables,respectively,in
P (ZTIAYT*) =P (AiTIAYT*)
equation (A.8), Xi can be generated from the following posterior
T- I
(A.2)
distributions:
=
P(ACTIiY T)
7 Pp(AC,tI
AS,
ACt+I).
t=1
Keeping these relationshipsin mind, we can proceed to generate
Ac,, t = 1, 2, . . , T, as follows.
*)- (A-'ai + u7AC'I~
Xi - N((A-'i + ,C-2AeC*
Cr-2Ae*,Ay)
ACflA7a
i
i =1, 2, 3, 4
(Ai7 + a2AC*IAC*)-I1),
Step1: Runthe Kalmanfilteralgorithmto calculate(ttI= E(ttIAy)
andV,t,= var(t|I y*,*)fort = 1, 2, . . ., Tandsavethem.Thelastiteration wherethepriordistributions
for Xi,i = 1, 2, 3, 4, aregivenby
of the Kalmanfilterprovidesus withtTI TandVTIT, andthe (1, 1) elements
of TIT andVTIT canbe usedto generateACT.
Step 2:
For t = T - 1, T-2,
. . ., 1, given t,I,and V,t,, if we treat the
Xi -N(ai, Ai),
i = 1, 2, 3,4.
(
(A.9)
(A.1I0)
generatedAc,t+as an additionalobservationto the system,the distribution
p(, IAy*, Ac,t+) is easilyderivedby applyingtheupdatingequationsof the
Kalman filter. As Ac,+,, the first element of t,+ I, is given by
Act+I = 4(L)pst+ + F * (1)t, + p*+I(1)
A.4 Generatingqiand oi , i = 1, 2, 3, 4
(A.3)
is the firstelementof u*
whereF*(1) is the firstrow of F* and
u,*+I(I)
updatingequationsarederivedas
tt It,,t+ I=
t It + Vt ItF * (1)r,IR,
Vtt,,c+, = V,t, - V,t,F*(1)'F*(l)V,t1,/R,
(A.4)
GivenAi, andA,,equation(8) canbe rewrittenas
Ayi
-
XAc, = ei,,
(A.l1)
By multiplyingbothsides of equation(A. 11)by Ii(L), we get
(A.5)
where q, = Avc,+1- (L),ps - F*(1)4,1, and R, = F*(1)V,itF*(1)' +
and
var [u,+e(1)J.Thenthe (1, i -elementsof
can be used
R=F,
t,F,
to generateAc,, fort + T- 1, T-2, 2..
,c1.
i = 1, 2, 3, 4.
j(L)Zt =
Eit,
i = 1, 2, 3,4
12Forexpositionalpurposewe assumethatX1(L)=
(A.12)
X,,i = 1, 2, 3, 4.
BUSINESS CYCLES, A NEW COINCIDENTINDEX AND REGIME SWITCHING
or assuming that i4j(L) = 1 Z
4JilZ-I + 42Z4-I +
i
* -
L-
distribution:
[p* p,] can be generatedfromthe followingmultivariate
LnI
+ 4Jin,Zt-. + Eit,
(A.21)
(
where Z, = Ayit - XAc,. Defining Z to be the vector of the Z,, and X to be
the matrixof the right-hand-sidevariables,the posteriordistributionof
I
is given by
i= H'ii
*. * 4*i,,]
N((J1T'i+
&2k1k)
(|
i
f-7 + cr,fcfo-l
,i - N(ot*, A*)1[v1>o].
Ti,fi,
Values of p can thereforebe simulatedfrom the above truncated
multivariatenormaldistribution.
i = 1, 2,3, 4
i= 1, 2, 3, 4.
(A.22)
(A.14)
wherethepriordistributionfor 4,jhas the multivariate
normalform
j- N
+
wherethe subscriptI[ ] refersto the indicatorfunctionon [PI> 0], and
wherethepriordistributionfor ,i has the multivariatenormalform
+
Or2rt2),
+ Q*'G*)
j1-~N((A*-l + Q*'Q*)l(A*-la*
(A.13)
1, 2, 3, 4
199
Generating Transition Probabilities p and q
A.6
(A.15)
Using the above posteriordistributionof Iji, values of Jjj can be
generated,given cr,i = 1, 2, 3, 4. Then,given the generatedvaluesof 4,i,
the followingposteriordistributions
of cr4canbe employedto generatecr2:
We follow AlbertandChib(1993) in derivingthe conditionaldistribution of p andq. GivenSTandthe initialstate,let the transitionsfromstate
S,_I = i to S, = j be denotedby nij.Thenthelikelihoodfunctionis givenby
L(q, p)
= qnoo(
-q)fqolpnll
(1
-
p)lo.
(A.23)
We thereforetake the independentbeta family of distributionsas a
conjugatepriorfor eachof thetransitionprobabilities
(A.16)
ir(q, p) x q'0(1 - q)uolp"ll (I
i= 1,2,3,4
whereIG denotesthe inversegammadistribution,the priordistributionis
given by IG (v,/2,f/2), and the hyperparameters
vi andf, are known. v
reflectsthe strengthof thepriorof or.
(A.24)
Combiningthe likelihoodfunctionandthe prior,we get the conditional
distributionof [q,p] as a productof the following independentbeta
fromwhichwe can generatep andq:
distributions,
[q
A.S
-p)UlO*
IST] - ,(uoo + noo, uol + no, )
(A.25)
Generating ,uo, pul,and 4
(A.26)
[P1ST]
,(uII + nll, ulo + n10).
Given the generatedvalues of ACT and ST,the last step of the Gibbs
in equation(9). A convenient
samplingalgorithmis to generateparameters
conditioningfeatureof the Gibbs samplingapproachis that it makes
A. 7 Calculation of the Composite Coincident Index
equation(9) linear,conditionalon St.
We first generate + = [14 ...
and
,,]', given po
j'l. Rewriting
The mean of each coincidentvariable(AwYi)consists of the portion
equation(9) for thecase of anAR (n) process,we have
due to the idisyncraticcomponent[Di in equation(1)] and the portion
dueto thecommoncomponent[8 in equation(2)]. Wenow discusshow we
= 41(/IC,1t-I(Act mightdecomposeA Yi,i = 1, 2, 3, 4, into its two components.
s')
PSI
As mentionedearlier,conditionalon ST,the state-spacemodelsgiven
(A.17)
.,+ Vt.
by equations(10)-(11) and (10)'-(11)' are linear and Gaussian.The
4)n (AC,-n
J-n)
is thereforereadilyavailable,as shown in
methodof decomposingA_Yi
LettingG denotethe vectorof the left-hand-sidevariablesand Q the Stock andWatson(1991). At each runof Gibbssampling,conditionalon
all the parametersof the modelandST as generatedfromequations(10)'
matrixof the right-hand-side
variables,the multivariatenormaldistribu- and(11)',
we applythe Kalmanfilterrecursionto equations(10) and(11).
tionfromwhich+ is generatedis givenby
Denotingby K* the steady-stateKalmangain fromthe filter,the meanof
each series AYiis decomposedinto the two componentsin the following
(A.18) way:'3
+ - N((A-' + Q'Q)-'(A-'ot + Q 'G), (A-l + Q'Q)-')
wherethepriordistributionfor (Phas the multivariatenormalform
(P- N(cx,A).
=
where E' = [1 0 0 ...
0], H is the selection matrix in the
measurementequation(10), F is the transitionmatrixin equation(11),
k is the dimensionof the F matrix,and AY = [AY, AY2 AY3 AY4]
andan initialvalueof C0,we areableto
Once8 is identified,forgiven ACT,
generateournew compositecoincidentindex'4 C, = Ac, + C,t_ + 8, t = 1,
(A.20)
2,.. . , T.
(Ac, - 4P1Ac,t - * ** - (lAC,f_?)
PO*+ P, (S, -
where p * = po(1
*P1S,I
- ()nSt,,)
(A.27)
(A.19)
Given(P,equation(9) can againbe rewrittenas
=
E' [Kk- (Ik - K*H)F]-'K*AY
+ Vt
-p).
Letting G* be the vector of left-hand-1pside variables and Q* the matrix of right-hand-sidevariables, ,i =
13
14
For details, refer to Stock and Watson (1991, pp. 70-71).
The index is unitless and identified up to an arbitrarychoice of CO.
THE REVIEW OF ECONOMICSAND STATISTICS
200
APPENDIX B
B.2 State-Space Representation 2
State-Space Representationsof the EmpiricalModel
equation:
(1) Measurement
B.1 State-Space Representation 1
~
AC,
~
Ay2*
0
X1
10000000
000
AY2r _ A2
X3
Ay3n
00
0
0
O O 1
0
0 0 0 0 1
0
0
0
0
0
0
I3
O
1X40 A41 x42 X43 0
Ay4t
e0A
0
1
0
0
0
0/
0
40
e2
0
0
0
A
O O Oee1 ,_1
0
- X2422
- 2X21
X2
2
Act
AY~
- 1Pi1 -Xi1iP2 0
kItl
equation:
(1) Measurement
4
43
Act2 +2r
0 e2,t-I
ACt-3
e3,
ACt-4
e3s,t- I
-Ct-5
E3t
LE4tJ
e4t
+ X41,
X*1=-X4041
*wherey, = Yit- ' lYi,t-I- 'i,2Yii-2, i = 1, 2, 3, 4;
+ X42,X*43= -X4142 - X42141+ X43,X44 =
42 = -X40*42 - X41I]41
-4242- X43141,andX*45
=-X43142
e4,t-
(2) Transitionequation:
(2) Transitionequation:
Ac,
Ac-
(P(L)ps.
0
Ac
Ac
Ls
PLp,
h
)
ACt-2
0
Act-I
0
1
0
0
Act3
0
AC-2
0
0
1
0 0 0 0
elt
0
ACt-3
0
00
ACt4
0
0
0
0
1 0
0
0
0
0
0
0
0
0
el,,-,
e2
1
0
00Ac5
?
0
0
0
1 0 0 0
1
0
e2,t+ I
0?vtC
Ac?1
e3t
O 0
e3Ot
I
0
ACt02
e4t
0
e4,t-I
0
ACt-3
14)0
00
0
10 0 0
0
01
00
0
00
010
0 0
0
0
00
0
0
0
?
?
00
Act-,
vt
0
0
0
ACt-2
0
?
0
0
ACt-3
0
0
00
ACt-4
0
00
el,t-2
0
00
0
0
?
00
1
000
000
0
0 0 0 0 0
04
0
012 02 0 0 44
2p, 2(
where~~~~~~~~~~~~~~~~~~~~~~~~~~hr
)spls =)s_
4)()L
4)2PSI-2',(
0
0
0
00 00 00 00
00
00
0 0 0 0
00
00
0
ACtO4
0
0
+
X
1
0 0 1
41314132
0
'1
~~31
0
0
0
AC-
0
AC6
Ih1
eto
ee
[C i"Y,'Y,'d,add
PlaiNho
IX~
0
2=0add
itrs
nti
pedxaetepseirpoaiiiso
e2,t-2
dependence.
forobbltiest
of
Du
dration
Gibbsth
Saplseing
e3,O,
that
of
e3,t-0,IthatGis,bth
Saposeior
porobblTiest noudrationDependence.h h
e2,t-
E2t
Anti
94
cappocn
bznrthd.iszapendi
iS an strihtfoSrw
,.ard appliatio of Gewekeula
BUSINESS CYCLES, A NEW COINCIDENTINDEX AND REGIME SWITCHING
201
C.l Generatingyo,vy,y2,1Jand y3,j
Forexample,d2canbe generatedfromtheconditionaldistribution
First,conditionalon Y2,d2and'Y3,d3.we generatey* = [^Yo YI]'.Define
Y1and Xl as the matricesof the left-hand-sideand the right-hand-side
S T*,ST,N0,-1,N1,-1)
f(d2 IY*, Y3,d3.
=
variables in a regression, S* -Y2d2(2 - S )No,- -y3,d3SlNl,t-l
T. Given the priordistribution-y*
yo + y1St-I + ut, t = 2, 3.,
N(-y*,W*2), the posteriordistributionof y* is given by the following
multivariatenormaldistribution:
.y*
Ny*
= (X*-2
wheredistribution(C.5) is Bernoulliwithprobability
(C.6)
Pr [d2 = 1Yl*, 3d3, ST ST.N,-l. N,,-,Ii IP2=
P2 + (1 -P2)-
*)(C.
1)
_
+
andj* = XwY).
Second, conditional on j* = [yo 'Y]' 'Y3,d3. and on d2 =
where
(C.5)
+ XX1)f
1,
we generate'Y2,1.DefineY2andX2as the matricesof the left-hand-sideand
- Y3,d3St-1
the right-hand-side
variablesin a regressionS* - yo --S
aO
HereP2 is the priorprobability,Pr [d2= 0], of no durationdependencefor
recessionsand
a, = f(-Y2,d2I,y*, Y3,d3,S9T.,
ST, NoT1, Nl,-.i,
d2 =
(C.7)
0)J[Y21,>0
Nl,t- = y2,1(I - StO)NO-,I + u,, t = 2, 3, ... , T. Assuming the prior is
given by a truncatednormal distribution,,Y2,1 - N(y2, )I) [Y2>O] the
(C.8)
d2 = 0).
ao =f(Y2,d21Y*, Y3,d3.ST. ST. No,-1.NR,_1,
posteriordistributionof -Y2,1is denotedby the followingtruncatednormal
The termal/ao in the right-handside of equation(C.6) is the conditional
distribution:
Bayes factorin favorof d2 = 1 (Y2,d2> 0) versusd2 = 0 (-Y2,d2= 0). This
'Y2,1 - N(^y,())1w10
(C.2) conditionalBayes factorcan be shownas15
where the subscript I[ *] refers to an indicator function, -W2
+
(-2
+ X2X2',)-, and 2 = (J2(2z2X2Y2)
Finally,conditionalon y* = [-Yo YI"', Y2,d2'andon d3 = 1, we generate
'Y3,1- Define Y3 and X3 as the matrices of the left-hand-side and
ao
=-2 exp/
\2-i
2
2wj
/
cI(
\2
Il
I-
L
(C.9)
Thusbasedon
where(1( *) is the c.d.f of the standardnormaldistribution.
a comparisonof a drawingfromthe uniformdistributionon [O,1] withthe
probabilitycalculatedfromequation(C.6),d2is generatedas 0 or 1.
prioris given by a truncatednormaldistribution,Y3,1 - N(y3, !)J(1)y3<0I
In a similarway, d3 can be generatedusing the following posterior
the posteriordistributionof -Y3,1is denotedby the following truncated
probability:
normaldistribution:
the right-hand-side variables in a regression, S* - yo - ySt-I IS_NoN- I + u, t = 2, 3 . . ., T. Assumingthe
Y2d3(I- S_I)Not-I = Y3,
'Y3,1
- N(^y, -S )1e,
p,'<0j
(C.3)
,I
(C.10)
=
b,
where the subscript I[ *] refers to an indicator function, W3
3(X3 2'Y3+ X3Y3).
(S!2 + X3X3)-, and 3 =
p3 + (1-P3)-
=
definedas in equations(C.7) and(C.8);
whereboandb1areappropriately
p3 is the prior probability,Pr [d3= 0], of no durationdependencefor
booms; and the conditionalBayes factorin favor of d3 = 1 (Y3,d3< 0)
versusd3 = 0 (Y3,d3= 0) is givenby
C.2 Generatingd2and d3
As in GeorgeandMcCulloch(1993) andGeweke(1994), d2andd3are
generatedcomponentwiseby samplingconsecutivelyfromthe conditional
distribution
f(dIdjj, y ST*ST,No,-_,N1,1 ),
Pr [d3= 1JY*, Y2,d2.
ST.ST. No,-I,N
i = 2, 3.
Wefollow the approachin Geweke(1994) in generatingd2andd3.
bo
2AW3 2)(t)[
(
(C.4)
15Fora proof,referto Geweke(1994).
(C.11)