Business cycle analysis with multivariate markov switching models.
Date: 2003-12-18
Institutions: Eurostat
Subjects: economic analysis, economic cycle, euro area

N
O

I
T
I

D
E

3
0
0
2

Business cycle 
analysis with 
multivariate Markov
switching models

E U R O P E A N
C O M M I S S I O N

 
Europe Direct is a service to help you find answers to your questions about the European Union

New freephone number:
00 800 6 7 8 9 10 11

A great deal of additional information on the European Union is available on the Internet.
It can be accessed through the Europa server (http://europa.eu.int).

Luxembourg: Office for Official Publications of the European Communities, 2003

ISBN 92-894-6830-0
ISSN 1725-4825

© European Communities, 2003

4TH EUROSTAT AND DG ECFIN  
COLLOQUIUM ON MODERN TOOLS FOR BUSINESS CYCLE ANALYSIS 

"GROWTH AND CYCLE IN THE EURO-ZONE" 

20 TO 22 OCTOBER 2003 

Luxembourg, European Parliament 
Hémicycle, Schuman building 

Business cycle analysis with  
multivariate markov switching models 

Jacques Anas, Centre d’Observation Economique, Paris, France 
Monica Billio, Dipartimento di Scienze Economiche, Università di Venezia, Italy  
Laurent Ferrara, Centre d’Observation Economique, Paris, France 
Marco Lo Duca, GRETA Associati, Venezia, Italy 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
BUSINESS CYCLE ANALYSIS  

WITH MULTIVARIATE MARKOV SWITCHING MODELS 

Jacques Anas*, Monica Billio**, Laurent Ferrara* and Marco Lo Duca*** 

*Centre d’Observation Economique, 
27 Avenue de Friedland, 75382 Paris Cedex 08, France 
janas@ccip.fr, lferrara@ccip.fr 

**Contact author 
Dipartimento di Scienze Economiche, Università di Venezia 
San Giobbe 873, 30121 Venezia, Italy 
billio@unive.it 

***GRETA Associati, 
S. Marco 3870, 30124 Venezia, Italy 
mloduca@greta.it 

29th January 2004 – Preliminary version 

Abstract 

The aim of  the  paper is   to describe the  cyclical phases of the economy    using  multivariate 
Markov  switching  models.  The  class  of  Markov  switching  models  can  be  extended  in  two 
main directions in a multivariate framework. In the first approach, the switching dynamics are 
introduced  by  way  of  one  common  latent  factor  (Diebold  and  Rudebusch,  1996).  In  the 
second approach, developed by Krolzig (1997), a VAR model with parameters depending on 
one common Markov chain is considered (MS VAR). We will extend the MS VAR approach 
allowing  for  the  presence  of  specific  Markov  chain  in  each  equation  of  the  VAR  (Multiple 
Markov Switching VAR models, MMS VAR).  
Dynamic factor models with regime switches, MS VAR and MMS VAR models allow for a 
multi-country or a multi-sector simultaneous analysis in the search of common phases which 
are  represented  by  the  states  of  the  switching  latent  factor.  Moreover,  in  the  MMS  VAR 
approach we explore the introduction of correlated Markov chains which allow us to evaluate 
the  relationships  among  phases  in  different  economies  or  sectors  and  introduce  causality 
relationships, which allow a more parsimonious representations.  
We apply the MMS model in order to study the relationship between cyclical phases of the 
industrial  production  in  the  U.S.  and  Euro  zone.  Moreover,  we  construct  a  MMS  model  in 
order to explore the cyclical relationship between the Euro zone industrial production and the 
industrial component of the European Sentiment Index (ESI). 

Subject Area N. 4: “Dating, detecting and forecasting turning points” 
Keywords:  Economic  cycles,  Multivariate  models,  Markov  switching  models,  Common 
latent factors, Causality, Euro-zone 
JEL Classification: C50, C32, E32 

 
 
 
 
 
 
 
 
 
1. Introduction 

Since  the  early  work  by  Burns  and  Mitchell  (1946),  many  attempts  have  been  made  to 

measure  and  forecast  business  cycles.  Recently,  approaches  based  on 

time  series 

econometrics,  have  emerged.  The  most  representative  works  are  the  Stock-Watson  model 

(Stock  and  Watson  1989,  1991,  1993)  which  focuses  on  comovements  among 

macroeconomic variables, and the univariate regime switching model developed by Hamilton 

(1989,  1990)  which  is  based  on  the  intuition  that  turning  points  and  changes  in  regime  are 

related.  Moreover,  Diebold  and  Rudebusch  (1996)  and  Kim  and  Nelson  (1998,  1999) 

synthesised  the  two  approaches  allowing  for  both  comovements  among  macroeconomic 

variables and switching regimes. 

The main advantage of Markov-switching processes, often advocated in the literature, is their 

ability to take into account the asymmetry of a time series and features such as non linearity 

and  the  persistence  of  extreme  observations:  these  features  are  crucial  in  business  cycle 

analysis. 

There  is  a  large  literature  that  uses  switching  regime  models  in  order  to  recognise  business 

cycle  phases.  The  starting  point  of  this  literature  is  that  there  is  a  relationship  between  the 

concepts of changes in cyclical phases and change in regime. Moreover, this relationship has 

been confirmed by empirical studies (Clements and Krolzig 1998 and 2000, Krolzig 1997a, 

1997b,  2000  and  2001,  Krolzig  and  Sensier  1999,  Krolzig,  Marcellino  and  Mizon  2000, 

Krolzig and Toro 2000 and  2001, Diebold and Rudebusch 1996 and 1999, Kim and Nelson 

1998  and  1999,  Chauvet  1998,  Kauffmann  2000,  Layton  1996,  Anas  and  Ferrara  2002, 

Harding and Pagan 2001a, Ferrara 2003). Therefore, contractions and expansions are modeled 

as switching regimes of the stochastic process generating the growth rate of some economic 

variables. 

In any case some words of caution have to be spent on the Markov switching approach to the 

business  cycle.  When  using  parametric  models,  such  as  MS-VAR  processes,  no  a  priori 

definition of business cycle is imposed: by means of the Markov switching model approach, 

different regimes are identified, indeed these regimes differ in terms of average growth rates 

and/or growth volatilities. In many cases the MS approach properly detects the classical cycle 

phases, but not necessarily this happens. The model only indicates some differentiation of the 

growth  rate  of  the  economy.  The  regime  of  low  growth  could  also  have  positive  mean:  the 

correct  detection  of  the  cycle  depends  on  the  features  of  the  analysed  series.  The  MS-VAR 

approach lets the data describe the features of the different phases of the economy. Actually, 

this approach simply represents the idea that economies are characterised by different phases. 

1 

Even if there is no perfect correspondence between cyclical phases and detected regimes this 

approach could be useful, especially in the case when detected regimes show an high level of 

persistence.  In  this  respect,  we  think  that  Markov-Switching  models  could  provide 

information useful to improve the perception of the current state of the economy. Moreover, 

useful information is provided by the transition matrix which describes how phases evolve: by 

modelling different economies or sectors with specific Markov chains (MMS VAR) we can 

get transition matrices providing useful information on how their phases are related. 

This  work  deals  with  the  multivariate  extensions  of  Markov-Switching  models  and  their 

reliability  in  Business  cycle  analysis.  We  extend  basic  MS  VAR  models  (Krolzig  1997) 

allowing  for  a  specific  Markov  chain  in  each  equation  of  the  VAR  (Multiple  Markov 

Switching VAR models, MMS VAR). Moreover in the MMS VAR approach we explore the 

introduction  of  correlated  Markov  chains  which  can  produce  useful  information  on  the 

relationship  between  phases  in  different  countries,  and  causality  relationships,  which  can 

allow more parsimonious representations. 

2. Multivariate extensions of Markov switching models: a review 

The class of Markov switching models can be extended in several directions in a multivariate 

framework. Concerning the description of business cycle phases and the detection of turning 

points, two main approaches have been proposed. The first one consists in jointly considering 

dynamic factor models and regime switching as proposed by Diebold and Rudebusch (1996). 

In this case, there are two levels of latent variables (the common factor and the Markov chain) 

and serious estimation difficulties arise. 

In the second approach, no common factors are used. The analysis of business cycle phases 

and the detection of turning points is provided by a multivariate version of the basic Markov 

switching AR model of Hamilton (1989). The class of Markov Switching VAR models (MS 

VAR) has been proposed by Krolzig (1997). 

2.1 Dynamic factor models 

Diebold and Rudebusch (1996) proposed to model simultaneously the two main stylised facts 

of the business cycle: (i) co-movements among economic variables through the cycle, and (ii) 

non linearity in the evolution of the business cycle. In fact, they tried to conciliate the recent 

works on two separate areas of the business cycle research, namely common factor extraction 

and non-linear modelling trough Markov-Switching models. 

2 

 
However,  the  introduction  of  a  latent  factor  in  the  switching  regime  model  makes  the 

estimation and the inference more difficult: this happens because the Hamilton’s filter is no 

longer useful. Then, the use of a dynamic factor model with regime switching requires more 

sophisticated  estimation  techniques.  From  a  methodological  point  of  view,  in  the  literature 

some solutions have been proposed.  

First  of  all,  the  use  of  approximations.  Following  Harrison  and  Stevens  (1976),  in  order  to 

avoid  that  the  number  of  states  increases  exponentially  at  each  iteration,  an  average  is 
performed, then the Hamilton’s filter can be adapted1.  

Secondly, simulation based approximations can be useful to obtain a filtering algorithm. In a 

classic  framework,  an  importance  sampling  technique  (see  Billio  and  Monfort,  1998)  or  a 
Markov Chain Monte Carlo approach can allow to obtain the maximum likelihood estimator2. 

Otherwise, in a Bayesian framework a Markov chain Monte Carlo approach based on the data 
augmentation  principle  and  the  use  of  a  multi-move  sampler3  can  yield  samples  out  of  the 

joint posterior distribution of the unobservable common factor, the latent state variable and all 
model parameters, allows to obtain the parameter estimates and the latent factors dynamics4. . 

2.2 Markov switching VAR  

This  approach  does  not  necessitate  the  previous  identification  of  a  common  factor  and 

consists in directly considering the series in a multivariate model. Following Krolzig (1997), 

it  is  possible  to  consider  a  VAR  model  with  parameters  depending  on  a  common  Markov 

chain. This common latent variable is assumed to represent the phases of the common cycle 

and is used to detect turning points. 

The assumption of a single Markov chain is consistent with the existence of a common and 

coincident  cycle  in  the  analyzed  set  of  economies.  Using  a  single  Markov  chain  is 

undoubtedly a way of producing a description of the common cycle phases and aggregating 

turning  points:  in  some  way  the  model  finds  the  turning  point  that  best  fits  the  group  of 

turning  points  that  we  can  usually  observe  in  a  set  of  different  economies  when  there  is  a 

1 See Kim (1994), Kim and Nelson (1998 and 1999), Chauvet (1998) for the definition and application of the 

Kim’s approximated filter. 
2  See  the  Simulated  Likelihood  Ratio  Method  in  Billio,  Monfort  and  Robert  (1998),  or  the  Simulated  EM 

algorithm in Shephard (1993). 
3 See Carter and Kohn (1994), Frühwirth-Schnatter (1994), Shephard (1994), and de Jong and Shephard (1995). 
4 For some applications see Kim and Nelson (1999), Luginbuhl and de Vos (1999) and Kaufmann (2000). 

3 

                                                           
change  of  phase.  In  any  case,  using  this  class  of  models,  it  is  not  possible  to  evaluate  the 

relationship among phases in different sectors or economies. 

In  the  next  section,  we  deal  with  simple  MS  VAR  models  with  a  single  Markov  chain 

representing  the  common  cycle  (Artis,  Krolzig  and  Toro  2002)  and  models  with  multiple 

Markov chains (MMS VAR) which represent country or sector specific factors. Considering 

specific Markov chains and their relationships provide useful additional information that can 

be used in business cycle analysis. When dealing with multiple Markov chains, we consider 

different  specifications  of  the  transition  matrix,  involving  different  hypotheses  on  the 

relationship among the specific chains. 

3. Multiple Markov switching VAR (MMS VAR) models 

The  MS-VAR  with  a  single  Markov  chain  introduced  by  Krolzig  (1997)  is  the  simplest 

multivariate Markov switching model that can be used in business cycle analysis. To give an 

idea of this approach, we propose the following specification: 

(

y
t

−

(
µ

S

t

)
)

=

p
∑
φ
j
1
j
=

(

)

y
t

(

S

t

−

j

(
S
µ

−

t

−

j

)
)

+

ε
t

where 

ε
t

~

(
2
N σ
,0

(

S

t

))

,  yt  is  a  n-dimensional  vector  which  contains  the  endogenous 

variables  and  εt  contains  the  idiosyncratic  disturbances  of  each  endogenous  variable.  The 

common  latent  variable  St=1,..,M  follows  a  Markov  chain  with  constant  transition 

probabilities  and  is  assumed  to  represent  the  phases  of  the  common  cycle.  Usually,  in 

business  cycle  analysis,  only  the  mean  and  sometimes  the  covariance  matrix  is  supposed  to 

depend  on  the  common  latent  variable,  while  the  other  parameters  are  considered  to  be 

constant. 

Following  Krolzig  (1997),  it  is  also  possible  to  use  an  alternative  specification  where  the 

intercept  term  depends  on  the  Markov  chain  in  order  to  allow  a  different  dynamic  of  the 

variable yt: 

y

t

=

(
υ

S

)

+

t

p

∑

j

1
=

α
j

(

)
yS
t

t

−

j

+

ε
t

4 

 
 
 
 
 
 
 
The problems arising when dealing with this kind of model, concern the necessity to estimate 

a  large  set  of  parameters.  In  fact,  the  number  of  parameters  grows  exponentially  with  the 

number of regimes. In particular, we have to consider: 

- 

- 

the mean of each series in each regime: Mn parameters. 

the covariance matrix in each regime: M[n(n+1)/2]. 

-  M x (M-1) elements of the transition matrix of the Markov chain. 

For  example,  if  n  =  2  and  M  =  3  the  total  number  of  parameters  to  estimate  is  21.  This  is 

already a large number of parameters, but it is small in comparison with the models that are 

useful for practical applications. 

The assumption of a single Markov chain is consistent with the existence of a single common 

and coincident cycle in the analyzed set of economies. However, even if we can hypothesise a 

common business cycle, empirical analyses suggest that this cycle could not be synchronised. 

Using a single Markov chain is undoubtedly a way to aggregate turning points and produce a 

description of the common cycle phases: in some way, the model finds the turning point that 

best  fits  the  group  of  turning  points  that  we  can  usually  observe  in  a  set  of  different  series 

when there is a change of phase.  

To improve our comprehension of the connections among the phases of different economies 

or  sectors  and  then  to  produce  a  better  description  of  how  phases  evolve,  it  is  necessary  to 

consider  that  turning  points  are  not  always  coincident.  This  happens  when  different  series 

aredriven by different unobserved factors: for this reason, we are going  to consider models 

which explicitly use multiple Markov chains. 

3.1 Multiple Markov chains  
Even if a single business cycle in a group of economies (e.g. the European Monetary Union) 

exists,  there  could  be  no  synchronization  among  the  phases  of  the  different  economies  or 

among  those  of    different  sectors  of  a  single  economy.  In  order  to  take  into  account  this 

possibility, it is necessary to relax the hypothesis that a single Markov chain drives the shifts 

in different economies or sectors. In the general case, we can assume that the change in the 

regime  of  each  series  is  driven  by  a  specific  Markov  chain.  Then,  it  is  possible  to  provide 

indications on how to aggregate the turning points and to produce a single chronology. 

We then consider a VAR model in which each equation has the following form, with switches 
in the intercept term5 and in the variance:  

5 As for the previous model, we can consider a switching variance. 

5 

 
                                                           
y

it

=

(
υ

S

)

+

it

p

∑

j

1
=

α
j

y

it

−

j

+

ε
it

, 

where εit~N(0,σ2(St)) and i=1, … , n. 

The regime shift of each of the n endogenous variables is driven by a specific Markov chain 

with  M  regimes.  For  estimation  purposes,  the  n  specific  chains  are  combined  in  a  single 
Markov  chain  with  Mn  regimes.  The  regimes  of  the  joint  Markov  chain  are  all  the  possible 

combinations of the regimes of the specific chains: it is crucial to study the transition matrix 

of the joint process because it contains information on the relationship among the phases of 

the different series. Hence, the advantage of this approach is that the relationship among the 

phases in different countries is explicitly taken into account and described. 

Clearly,  the  number  of  parameters  that  have  to  be  estimated  rises  enormously  and  this 

produces serious difficulties in the estimation of the model. We have to estimate the following 

set of parameters: 

- 

- 

the mean and the variance of the series in each specific regime: 2Mn. 

the correlation between series in each regime: Mn(n-1)n/2. 

-  Mn(Mn-1) elements of the transition matrix of the common Markov chain. 

For example, if n = 2 and M = 3 and we impose no constraints on the transition matrix of the 

joint process, we will have a 9 x 9 transition matrix with 72 transition probabilities that have 

to be estimated. In this case the total number of parameters we have to estimate is 93. 

 The likelihood function could be characterised by many local maxima, and there are strong 

convergence problems (for details see Breunig and Pagan 2002 and Boldin 1996). Moreover, 

in business cycle analysis, short time series have often to be treated and the accuracy of the 

estimates could be even more compromised. 

By considering simple constraints based economic considerations or imposing a dependence 

structure based on a priori ideas, it is possible to reduce the number of parameters which are 

necessary to define the transition matrix. As we will see in section 4, most of the 72 transition 

probabilities are not different from zero and this will help us in estimating the model. It is also 

possible  to  formally  test  whether  the  constraints  are  consistent  with  the  data  and  thus 

understand whether we are working in the right direction. 

3.2 Modelling the transition matrix 
The aim is to describe the connections between the phases of cycles in different countries or 

in different sectors of the economy. Markov switching models with specific chains allow us to 

6 

 
describe these connections by making some assumptions on the transition matrix that drives 

the joint Markov process of all the specific factors.  

In the literature there are four possibilities (following Hamilton and Lin, 1996, and Susmel, 

1998): 

-  Common regimes (existence of a unique Markov chain); 

- 

Independent Markov chains; 

-  Related regimes; 

-  General specification. 

These allow the description of interesting phenomena but do not give the complete range of 

possibilities. In the following we propose also this strategies: 

-  To model the transition matrix; 

-  To consider dependent and correlated Markov chains (Billio, 2002). 

3.2.1 Independent and related Markov chains 
If we assume that there is no relationship among phases in different economies or sectors we 

can  use  independent  Markov  chains.  In  this  case  the  transition  matrix  of  the  joint  Markov 

process  is  given  by  the  Kronecker  product  of  the  transition  matrix  of  each  specific  chain. 

Suppose  for  example  that  we  have  two  Markov  chains,  st  and  zt.  (n=2)  with  three  regimes 

(M=3). Then we have the following result: 

s PPP
⊗=
z

where  P  is  the  9  x  9  matrix  of  the  joint  process  and  Pi  is  the  3  x  3  transition  matrix  of  the 

country  or  sector  specific  Markov  chain.  The  advantage  of  this  approach  is  the  limited 

number of parameters to estimate: in order to get the transition matrix of the joint process we 

only  have  to  estimate  the  transition  probabilities  of  the  Markov  chain  of  each  series. 

Considering the other parameters of the model, the total number of parameters to estimate is 

33. 

The  hypothesis  of  independent  cycles  could  be  not  consistent  with  the  cycle  in  Europe  and 

often it is not consistent when considering different sectors of an economy. This specification 

could in any case be useful if we need to formally test whether or not the specific cycles are 

related. For example could be useful before extracting a common factor from a group of series 

of  different  sectors.  In  fact,  it  is  possible  to  formally  test  the  null  of  related  Markov  chains 

against the alternative of independent Markov chains with a simple Likelihood Ratio test. In 

7 

 
 
this way it is possible to implement a selection strategy to choose the variables to include in 

the multivariate model: those showing no connections can be discarded. 

It must be underlined that the only difference between  this approach  and the  univariate MS 

model  is  that  a  regime  dependent  correlation  is  estimated.  We  can  get  the  same  transition 

matrix of the joint process considering only the transition matrices we get from the univariate 

analysis,  but  in  this  case  it  is  not  possible  to  formally  test  the  hypothesis  of  independence 

against the one of related chains. 

Concerning the related regime case, the forces that govern both cycles are the same, but are 

not  in  phase.  Two  sub-cases  can  be  considered:  the  first  cycle  shifts  before  so  zt=st-1  and 

causality  is  reversed,  namely  st=zt-1.  This  type  of  relation  strongly  simplify  the  transition 

matrix  since  the  number  of  parameter  is  simply  2  and  it  is  useful  to  describe  lead  or  lag 

relationships among the economies. 

3.2.2 Modelling the general transition matrix 
In  the  general  specification  no  a  priori  structure  on  the  transition  matrix  is  imposed.  In  the 

case of two series and specific Markov chains with three regimes each one, the joint process 

is characterised by the following 9 x 9 transition matrix: 

EE-EE 

EE-SE 

EE-RE 

EE-ES 

EE-SS 

EE-RS 

EE-ER 

EE-SR 

EE-RR 

SE-EE 

SE-SE 

SE-RE 

SE-ES 

SE-SS 

SE-RS 

SE-ER 

SE-SR 

SE-RR 

RE-EE 

RE-SE 

RE-RE  RE-ES 

RE-SS 

RE-RS 

RE-ER  RE-SR 

RE-RR 

ES-EE 

ES-SE 

ES-RE 

ES-ES 

ES-SS 

ES-RS 

ES-ER 

ES-SR 

ES-RR 

SS-EE 

SS-SE 

SS-RE 

SS-ES 

SS-SS 

SS-RS 

SS-ER 

SS-SR 

SS-RR 

RS-EE 

RS-SE 

RS-RE 

RS-ES 

RS-SS 

RS-RS 

RS-ER 

RS-SR 

RS-RR 

ER-EE 

ER-SE 

ER-RE 

ER-ES 

ER-SS 

ER-RS 

ER-ER 

ER-SR 

ER-RR 

SR-EE 

SR-SE 

SR-RE 

SR-ES 

SR-SS 

RR-EE  RR-SE 

RR-RE  RR-ES 

RR-SS 

SR-ER 

SR-RS 
SR-RR 
RR-RS  RR-ER  RR-SR   RR-RR 

SR-SR 

For  each  series,  the  regime  E  means  fast  expansion  (above  the  trend  growth  rate),  S  slow 

expansion (below the trend growth rate) and R recession. The elements of the matrix are the 

probabilities of a transition from a regime of the joint process to another. The regimes of the 

joint process are all the possible combinations of the three regimes (R, S ,E) of each series. 

In order to reduce the number of parameters that must be estimated it is possible to impose 

some constraints based on simple economic considerations. For example, it is reasonable to 

impose that it is not possible that an expansion is followed by a recession and vice versa. In 

this way, we impose that the expected relationship between turning points of different cycles 

(classical and growth) described by the ABCD approach (see Anas and Ferrara, 2002b, and 

8 

 
 
Anas, Billio, Ferrara and Lo Duca, 2003) is respected. Firstly, we will have the growth cycle 

peak then the business cycle peak and the latter will be followed by the business cycle trough 

and  finally  by  the  growth  cycle  trough.  If  we  proceed  in  this  way,  the  transition  matrix  is 

naturally reduced and only 40 elements have to be estimated: the total number of parameters 

then decreases to 61. The number of parameters to estimate is reduced, but it is still large and 

strong convergence problems remain.  

Even if there is the possibility to impose further constraints (for example: constant correlation, 

constant variance, other transition probabilities set to zero), it is difficult to have less than 40 

parameters.  Then,  from  a  practical  point  of  view,  we  are  limited  to  the  analysis  of  2  series 

with 3 regimes or 3 series with 2 regimes. For this reason this method could be used when 

dealing with two sectors of the economy of an area or with groups of countries. 

3.2.3 Markov switching VAR and Granger causality 
In  general,  in  a  VAR  context,  the  identification  of  causality  relations  allows  more 

parsimonious representations. In the literature there are very few attempts to address this issue 

in  Markov  switching  VAR  models:  one  exception  is  Warne  (2000),  who  analyses  how  the 

Granger causality concept allows improving the regime inference. 

In this context, we propose to describe causality relationships by working with Markov chains 

(see also Mosconi and Seri 2001 for binary data model). This is certainly useful in order to 

describe  the  relationships  between  leading  and  lagging  countries,  or  to  describe  the  relation 

between business surveys and macroeconomic variables. Using this approach it is possible to 

reduce  the  number  of  parameters  required  to  define  the  transition  matrix.  Moreover,  in  a 

multi-country/multi sector framework, this type of model allows us to explain the interactions 

among macro-areas. 

The independent and common cases present in the previous sections are not of interest if we 

would like to understand the causality between two cycles. Concerning the identification of 

the  transition  matrix,  with  3  states,  the  independent  case  requires  6  parameters,  while  the 

general  specification  requires  72  parameters.  Using  the  causality  approach  it  is  possible  to 

describe several correlated cases with a number of parameters comprising between 6 and 72.  

In  particular,  if  st  and  zt  are  two  Markov  chains  and  St  is  the  resulting  joint  process,  in  the 

general specification we can decompose the transition probabilities of St as follows: 

P(St | St-1)  = P(st ,zt | st-1 ,zt-1) 

= P(st | zt ,st-1 ,zt-1) P(zt | st-1 ,zt-1) 

9 

We can now define the Granger non causality for a Markov chain. Let st-1= st-1,st-2,...,s0. 

Strong one step ahead non causality (Granger non causality): st-1 does not strongly cause zt 

one step ahead, given zt-1 if: 

P(zt | st-1 ,zt-1)= P(zt | zt-1) 

This  means  that  we  can  assume  than  st-1  does  not  strongly  cause  zt  one  step  ahead  if  st-1 

contains no information on zt.  

Similarly, zt-1 does not strongly cause st one step ahead, given st-1 if: 

P(st | st-1 ,zt-1)= P(st | st-1) 

 ■ 

As one can see the non causality definition involves the marginal distributions conditional to 

the past: then, to study the causality it is necessary to consider the transition probabilities of 

the  Markov  process.  Let  we  see  how  we  can  construct  a  transition  matrix  starting  from  the 

non causality definition. 

Let  us  consider  specific  chains  (st  and  zt)  with  two  states:  the  independent  case  asks  for  4 

parameters  while  the  most  general  model  (see  the  above  general  specification)  representing 

P(St | St-1) involves 16 parameters to define the transition matrix. More precisely, since the 

sum  of  each  row  is  equal  to  one,  just  12  parameters  are  enough  to  describe  the  conditional 

distribution completely. 

Let  us  see  how  we  can  obtain  a  suitable  representation  with  a  number  of  parameters 

comprised between 4 and 12. 

The following vector represents the four possibilities of the joint Markov chain St: 

Xt=(1, st-1, zt-1, st-1 zt-1)’ 

=(1, st-1)’ ⊗ (1, zt-1)’ 

It is simply to verify that we can represent the joint probability of st ,zt as follows: 

P(st ,zt | st-1 ,zt-1)  = P(st | zt ,st-1 ,zt-1) P(zt | st-1 ,zt-1) 

exp
+

(
)
Y
α
1
t
(
Y
exp
α
1
t

=

1

)

1

X

exp
+

(
α
2
(
exp
α
2

)
t
)t
X

where  

Yt=(1, st)’ ⊗ (1, st-1)’ ⊗ (1, zt-1)’ 

10 

 
 
=(1, st-1, zt-1, st-1 zt-1, st, st zt-1, st st-1, st st-1 zt-1 )’ 

and Xt has already been defined. 

Now, Yt and Xt involve respectively 8 and 4 parameters, then we simply have an alternative 

parameterization of the transition matrix. Such parameterization is very useful since it allows 

us to simply impose the non causality restrictions by easily restricting the transition matrix to 

be described by a number of parameters comprised between 4 and 12. 

In particular, if st-1 does not strongly cause zt one step ahead, given zt-1, the Xt vector reduces to 

Xt=(1, zt-1)’ and the number of parameters reduces to 10. 

In this setting we can investigate if zt causes st in the Granger sense or the contrary without 

imposing common states, which is very useful for understanding which series can be useful in 

predicting the business cycle. Moreover, this type of decomposition allows us to describe all 

the previous specifications besides other even more interesting cases. 

4. Applications 

We  apply  the  MMS  model  using  different  specifications of the transition matrix in order to  

study    the  relationships  between  industrial  cyclical  phases  in  the  U.S.  and  Euro  zone. 

Moreover, we construct a MMS model in order to explore the cyclical relationships between 

the Euro zone industrial production and the industrial component of the European Sentiment 

Index (ESI). 

4.1 Data set 

IPI Euro zone 

In  order  to  get  the  aggregate  Industrial  Production  index  at  the  Euro  zone  level,  a  back 

recalculation has been performed. The Industrial production indices at country level are taken 

from  the  GRETA  database.  They  represent  the  total  production  adjusted  by  working  days 

(WDA).  With  9  countries  starting  in  1970,  an  Euro9  aggregate  is  calculated  by  weighting 

adequately  the  9  indices.  Then,  a  Euro10  aggregate  is  calculated  from  1975  when  Ireland 

become available and a regression is performed to estimate a new Euro10 starting from 1970. 

Similarly,  a  Euro11  starting  in  1970  is  obtained  by  a  regression  of  Euro11  (available  since 

1977 by adding Greece) over Euro10. Finally, the Euro12 starting from 1970 is calculated by 

a regression of the available Euro12 in New-Chronos since 1985 over Euro11. The final data 

set covers the period from January 1970 to December 2002. 

11 

Data have then been pre-treated by using the TRAMO-SEATS method in Demetra6.  

IPI U.S.A. 

The  manufacturing  index  for  major  industry  groups  has  been  considered.  The  index  is 

published  monthly  by  the  Federal  Reserve  and  is  already  adjusted  for  seasonal  effects.  The 

data set covers the period from January 1919 to August 2003. 

ESI 

The  Economic  Sentiment  Index  (ESI)  is  released  monthly  and  is  a  combination  of  opinion 

surveys data computed by the European Commission since 1970. The index has been changed 

in  its  composition  and  weighting  scheme  in  1999  for  statistical  reasons  without  a  clear 

economic justification. Today, the ESI is composed of the confidence indices in the sectors of 

industry,  construction, 

retail  and  households. 

Its  economic 

interpretation 

is  not 

straightforward and the challenge is to evaluate how it relates to the European economic cycle 

(see  Batchelor  2002,  Doz  and  Lenglart  2001,  Vanhaelen,  Dresse  and  DeMulder  2000).  The 

data set covers the period from July 1985 to July 2003. 

4.2 A MMS model for the Euro zone and the U.S. industrial productions 

The MMS model introduced in section 3 aims  at describe the growth rate over three months 

of the industrial production in Europe and in the U.S.A. to detect cyclical phases and analyse 

the relationships between the two macro areas.  

Our aim is to simultaneously take into account fluctuations of the business and growth cycles 

using a single model (see for instance Anas and Ferrara, 2002b, and Anas, Billio, Ferrara and 

Lo Duca, 2003 for more detailed connections between the two cycles). According to this idea, 

for both Euro zone and the U.S.A. a specific Markov chain with three regimes is used. The 

three regimes have the following economic interpretation: 

−  Low  growth  regime,  Recession  (labeled  R):  the  regime  is  usually  characterized  by  a 

negative average growth rate and it is associated to the classical recessions. 

6 No trading day adjustment is used since data are already WDA. Sometimes, the airline model was imposed to 

avoid  a  non  parsimonious  model  or  to  avoid  too  many  outliers.  Similarly,  the  critical  limit  for  outliers  was 

sometimes  fixed  to  3.0  to  avoid  too  many  outliers.  Generally,  we  avoided  the  presence  of  level  shift  outliers 

except obviously in the case of the German series. The only outlier found in the Euro12 series was an additive 

outlier in June 1984. 

12 

 
 
 
                                                           
−  Intermediate growth regime, Slow Expansion (labeled S): we assume that the growth rate 

of  the  economy  is  below  its  trend  growth  rate  (low  phase  of  the  growth  cycle  without 

recession). 

−  High  growth  regime,  Fast  Expansion  (labeled  E):  we  assume  that  the  growth  rate  of  the 

economy is above its trend growth rate (high phase of the growth cycle). 

There is no theoretical reason which guarantees that it is possible to use this interpretation of 

the regimes, but in empirical applications the relationships between the three regimes and the 

phases  of  growth  and  business  cycle  is  quite  strong.  Moreover,  the  use  of  a  three  regime 

models  avoids  the  recession  regime  (R)  to  have  a  positive  mean.  Even  if,  as  mentioned, 

detected regimes and phases of the cycle could be not perfectly related, we think that using a 

three regime model with this interpretation improves our ability to understand the meaning of 

a regime. Furthermore, we noticed that if recessions are well dated with a two regime model, 

the introduction of a third regime has not a significant impact on the dates of the recessions: it 

usually splits in two regimes (fast growth and low growth) the single expansion regime found 

using  a  two  regime  model  (see  also  Krolzig,  2003  and  Artis,  Krolzig  and  Toro,  2002). 

However, it is noteworthy to underline that this interpretation of the three regimes implicitly 

assumes a constant long-term growth rate over the whole sample period. That is, the estimated 

growth cycle is the deviation from a linear trend, instead of being computed through a specific 

filter. Nonetheless, we can consider this specification as a local approximation. 

Once the parameters of the model have been estimated, in order to detect phases we simply 

assign  the  observation  at  time  t  to  the  regime  with  the  highest  smoothed  probability. 

Concerning turning points detection, a trough is the last observation of the low growth regime 

and a peak is the last observation of the high growth regime. 

In the case of our three regime model, we will proceed in the following way in order to date 

business and growth cycle phases: 

-  Concerning the business cycle, we obtain the smoothed probabilities of an expansion 

(labeled EX) by summing the smoothed probabilities of the second regime (S) and the 

third  regime  (E).  The  recession  regime  of  the  business  cycle  (labeled  RE)  is  the 

regime  R  of  the  starting  model.  In  this  way,  we  have  again  two  regimes  which  are 

supposed to represent the business cycle phases. 

-  Concerning the growth cycles, we obtain the smoothed probabilities of the under trend 

phase (labeled L) by summing the smoothed probabilities of the first regime (R) and 

the second regime (S). The high regime of the growth cycle (labeled H) is the regime 

13 

E of the starting model. In this way we have again two regimes which are supposed to 

represent the growth cycle phases. 

Starting  from  the  general  model  with  regime  dependent  mean  and  covariance  matrix  (no 

autoregression terms are considered) and 72 unrestricted conditional probabilities, we reduce 

the number of parameters by imposing some restrictions. In particular we set to zero all the 

transition probabilities that results very close to zero in the first estimation of the model. In 

the  final  specification  only  the  mean  and  the  variances  are  regime  dependent  while  the 

transition  matrix  is  reduced  to  only  17  unrestricted  elements  and  the  correlation  is  constant 

across  regimes  (see  tables  1  and  2  for  the  estimated  parameters  and  the  standard  errors).  In 

table 3 the test on the validity of the joint restrictions is reported. 

From table 1, it is possible to note that in the positive regimes (E and S) the mean growth rate 

over three months of the U.S. is greater than that of the Euro zone. In particular, in the regime 

E  the  average  growth  rate  for  the  U.S.  is  2%  while  for  the  Euro  zone  is  1.3%,  while  in  the 

regime S is 0.48% for the U.S. and 0.23% for the Euro Zone. In the negative regime (R) the 

growth  rate  for  the  U.S.  is  -1.44%  against  -1.3%  for  the  Euro  zone  which  means  that 

recessions  in  the  U.S.  are  on  average  more  severe.  Concerning  the  volatility  of  the  growth 

rate,  it  is  possible  to  note  that  in  Europe  the  volatility  is  greater  than  in  the  U.S.  The  most 

volatile regime is R, while the lowest is the intermediate regime S for both the areas. 

In figures 1 and 2 the detected low phases of the business and the growth cycle are reported 

for  both  the  areas.  In  particular,  concerning  figure  1  it  is  important  to  underline  the 

correspondence  between  regime  R  for  the  Euro  zone  and  for  the  U.S.  and  the  well  known 

episodes of recession. The first oil shock, the double dip at the beginning of the eighties, the 

recession in the U.S. at the beginning of the nineties, the Euro zone recession in 1992 and the 

possible 2001 recession are all well detected. Moreover, very low uncertainty is associated to 

the regime classification and by a visual inspection it is possible to note that the U.S. leads 

Europe in entering in recession. Concerning figure 2, it is possible to note that the slowdown 

periods emerging from the graph are strictly connected to the growth cycle low phases. For 

example, by comparing the reference chronology of the growth cycle published by the ECRI 
for the U.S.7 with the periods observed in the figure, it is possible to note that the slowdown 

of 1973-1975, the persistent low growth across the end of the seventies and at the beginning 

of  the  eighties,  the  slowdown  of  1984-1987  and  especially  the  slowdown  starting  in  April 

7 For the Euro zone there does not exist a reference chronology for both the growth and business cycles. 

14 

 
                                                           
2000 are well detected and turning points do not differ significantly. Moreover, it is possible 

to note that most of the times, the shifts to the low phase of the growth cycle in the U.S. lead 

those of the Euro zone. 

Both for the business and the growth cycles the low phases detected with the MMS approach 

are  not  erratic  but  display  a  very  high  persistence,  hence  regimes  should  be  considered 

informative by policy makers. 

The most interesting feature of the MMS model is that it allows us to estimate the transition 

matrix of the Markov chain determining the joint evolution of the phases in the Euro zone and 

in the U.S: this could produce useful information on the relationship between the cycle of the 

two areas. The final transition matrix is reported in table 4. It is noteworthy that we can accept 

the equality of zero of most of the transition probabilities. In many rows there are only two or 

three  elements  different  from  zero  and  this  means  that  there  is  little  uncertainty  about  the 

following regime. Let us consider for example the row SR, which reports the probabilities to 

go to another regime starting from the intermediate regime (S) for the Euro zone and from the 

recession regime (R) for the U.S. In this row only two probabilities are different form zero: 

that one of staying in SR and that one of getting to RR. We can thus conclude that the U.S. 

leads the Euro zone to a recession. Since the expected duration of SR is 2.98 months (see table 

5), we can conclude that when we are in the SR regime, Europe will move to the regime R in 3 

months, which is the average lead of the U.S. 

The  analysis  of  the  remaining  elements  of  the  9x9  matrix  is  quite  tedious.  To  simplify,  we 

will analyse a reduced version of the matrix. Starting from the full transition matrix of the 9 

regime Markov chain, it is easy to get the 4 by 4 transition matrix referring to the business 

cycle. The same simplification can be performed if we want to analyse the growth cycle (see 

tables 6 and 7). 

Analysis of the business cycle: 

Table  6  gives  the  transition  matrix  for  the  business  cycle.  We  consider  the  following  two 

regimes for each chain: regime EX corresponds to the sum of regime E and regime S of the 

starting  model,  meaning  that  the  economy  is  expanding,  while  the  regime  RE  is  simply  the 

recession regime R. 

Let  us  start  considering  the  exit  of  EX-EX  regime:  this  regime  is  very  persistent  and  its 

expected duration is 30.5 months; it is striking that the only possible exit from EX-EX is EX-

RE,  which  means  that  the  U.S.  shifted  to  the  recession  regime.  Moreover  we  observe  that 

once we get EX-RE, there is a 84% probability to enter RE-RE, which indicates that both the 

15 

 
U.S. and the Euro zone are in the recession phase and only 16% probability to get back to EX-

EX. Since the expected duration of the EX-RE regime is 4 months, we can conclude that if the 

U.S. enters in the recession phase, there is an high probability that the Euro zone will follow 

in  4  months.  The  transition  matrix  is  clearly  indicating  a  lead  of  the  U.S.  in  entering 

recessions. 

Finally,  we  observe  that,  from  RE-RE  there  is  a  86%  probability  of  a  shift  to  the  EX-EX 

regime  which  means  synchronization  in  exiting  the  recession  and  14%  probability  to  get  to 

RE-EX which means that the U.S. switches in advance. It is striking that, once in RE-EX, the 

only possible exit is EX-EX which means that the U.S. drives the recovery. The average lead 

in this case is 8.5 months. 

We can conclude that, both in entering and exiting business cycle recessions, the U.S. leads 

the Euro zone or the shift are synchronised. 

Analysis of the growth cycle 

Table 7 gives the transition matrix for the growth cycle. Differently from the business cycle 

transition matrix, there is no clear evidence of a lead of the U.S. Starting from the above the 

trend phase for both areas (regime HH), there is the same probability to get HL and LH and no 

synchronised shift to the below trend phase (LL) is possible. From the regimes HL and LH it 

is possible to get LL but also to HH, so no clear lead is showed by any area in entering the low 

phase of the growth cycle. Consider now the LL regime: it may happen a synchronised exit to 

HH, but in the most of the cases, with a 89% probability, the exit is LH which means that the 

U.S. exit in advance. Once in LH, it is possible to move to SS again but the exit could also be 

HH:  we  can  thus  conclude  that  the  U.S.  lead  the  Euro  zone  in  exiting  the  low  phase  of  the 

growth cycle or the exit is synchronised. In any case, the Euro zone never leads the U.S. in 

exiting the low phase. 

The more complex dynamic described by the transition matrix of the growth cycle could be 

due to the fact that the low phase of the growth cycle is more connected to the area specific 

shocks  than  a  recession.  Recessions  in  fact  are  more  synchronised  and  in  most  of  the  cases 

seem to be the results of global shocks. 

In conclusion, the transition probabilities have to be studied with caution. Some of them may 

be influenced by the presence of specific cycles especially those concerning the growth cycle. 

On the contrary, when the U.S shift in a business cycle recession we expect the Euro zone to 

follow  within  a  few  months.  This  “catching  up”  effect  can  be  seen  if  we  consider  the 

evolution of the conditional mean of each area to the steady state mean (unconditional mean) 

16 

 
given a starting regime. The conditional mean at time t is the average of the means in different 

regimes  weighted  by  the  probability  of  the  corresponding  regime  at  time  t.  Starting  with  a 

specified  regime  (among  the  nine)  for  both  the  Euro  zone  and  the  U.S.  at  time  t=1  we 

multiply  i  times  the  9×9  transition  matrix  to  compute  the  conditional  probability  of  each 

regime  and  the  conditional  mean  at  time  t+i.  After  a  few  iterations  the  conditional 

probabilities and the conditional means converge to their steady state (unconditional) values. 

Table  8  shows  the  evolution  of  the  conditional  means  of  both  areas  starting  from  different 

regimes. It is possible to note that there is a strong inter-relationship between the two areas, in 

particular given a starting state for Europe, the evolution of the conditional mean of this area 

is quite different depending on which is the starting state for the U.S. Let us consider now the 

evolution of the conditional mean starting from regime ER and SR: it is evident the catching 

up  process.  If  we  start  from  ER,  the  U.S.  recover  to  the  steady  state  while  the  Euro  zone 

before  reaching  the  steady  state  experiences  a  period  of  low  growth.  The  same  happens 

starting from the SR regime, since it is evident that a deterioration of the economic condition 

in the Euro zone is expected. The opposite does not happen when the business condition in 

Europe is bad: starting from the RE and RS regimes, the U.S. are not strongly affected by the 

European conditions, in fact they reach directly their steady state growth without showing a 

catching up process. On the contrary, in the case of the regime RE, the Euro zone seems to 

exit from the bad condition very quickly. 

4.2 A causality analysis of the Euro zone and the U.S. industrial productions 

To  get  a  more  detailed  description  of  the  relationship  between  growth  cycles  phases  in  the 

U.S.  and  in  the  Euro  zone,  we  consider  the  industrial  production  indexes  without  the  trend. 

The trend has been extracted with the two stages Hodrick Prescott filter calibrated to maintain 

all  the  fluctuations  between  1.5  and  8  years  (see  Artis,  Marcellino  and  Proietti,  2002).  The 

series without the trend are reported in figure 3: peaks and troughs of the series correspond to 

peaks and troughs of the growth cycle. 

In  order  to  get  a  representation  of  the  growth  cycle  phases  in  terms  of  Markov  chains  we 

implement  an  MMS  VAR(1)  model.  This  produces  a  little  delay  in  detecting  turning  points 

but  in  any  case  the  phases  are  well  described.  Starting  from  the  general  model  with  regime 

dependent  mean  and  variances  and  12  unrestricted  conditional  probabilities,  we  reduce  the 

number  of  parameters  by  imposing  some  restrictions.  In  the  final  specification  only  the 

intercepts and the correlation are regime dependent, while the transition matrix is reduced to 

17 

 
only 6 unrestricted elements and the correlation is constant across regimes (see tables 9 and 

10 for the estimated parameters and the standard errors). In table 11 the test for the validity of 

the  joint  restrictions  is  reported.  Looking  at  table  9  it  is  possible  to  note  that  the  intercept 

terms associated to the low regime are negative for both series. 

Figure 4 reports the smoothed probability of being in the low phases of the growth cycle. If 

one compares figure 4 with figure 2 reporting the classification of the growth cycle derived 

from  the  MMS  model  with  9  regimes,  it  is  possible  to  note  that  phases  are  very  similar, 

moreover  in  figure  4  regimes  are  less  erratic:  as  mentioned,  this  is  because  the  9  regime 

model involves an approximation assuming a constant trend growth rate. 

Concerning  the  transition  matrix,  which  is  reported  in  table  12,  we  can  evidence  some 

important information. Let us start by considering the exit from the HH regime: there are two 

possible  exits,  LL  (18%  probability)  and  HL  (82%  probability).  If  we  move  to  LL  it  means 

that both areas switch to the low phase but if we go to HL it means that only the U.S. shift to 

the low regime. Moreover, we observe that once we get to FS we can only exit it by entering 

in LL. The transition matrix indicates that in most of the cases there is a lead of the U.S. in 

entering  the  below  trend  phase  or  at  least  the  shift  is  synchronized.  The  same  happens  in 

exiting the LL regime: we have a 21% probability to get HH and a 79% probability to get LH. 

In the latter case, the only possible exit is LL, hence the U.S. are leading again the Euro zone 

or at least the exits from the below trend phase are synchronised. 

Finally,  given  the  evidence  that  the  U.S.  lead  the  change  in  phase,  we  tested  the  null  that 

shifts  in  the  U.S.  do  not  cause  shifts  in  the  Euro  zone  (USA  NC  EU)  and  that  shifts  in  the 

Euro  zone  do  not  cause  shift  in  the  U.S.  (EU  NC  USA).  Results  of  the  test  are  reported  in 

table 13. As it was expected the null “USA NC EU” is rejected. However, also the null “EU 

NC USA” is rejected. 

4.3 A MMS model for the Euro zone industrial production and the industrial survey of 

the ESI 

In  figure  5,  the  standardised  industrial  component  of  the  ESI  in  levels  (SIND)  and  the 
standardised  Euro  zone  IPI  without  the  trend  component  (SIPI)  are  reported8.  Concerning 

peaks and troughs of the SIPI without the trend, they have a clear meaning since they indicate 

us that the growth rate is equal to the trend growth rate hence peaks and troughs of the SIPI 

8  The  trend  has  been  extracted  with  the  two  stages  Hodrick  Prescott  filter  calibrated  to  maintain  all  the 

fluctuations between 1.5 and 8 years (see Artis, Marcellino and Proietti, 2002). 

18 

 
                                                           
are the turning points of the growth cycle of the industrial production. By a visual inspection 

of figure 5, it is possible to note that starting from the beginning of the nineties until today, 

the turning points of the SIPI and those of the SIND are strictly connected. Moreover turning 

points of the survey seems to lead those of the growth cycle of the industrial production. 

To get a representation of the above mentioned cyclical phases in terms of Markov chains, we 

implement an MMS VAR(1) model. This produce a little delay in detecting turning points but 

in  any  case  the  phases  are  well  described.  Starting  from  the  general  model  with  regime 

dependent  means  and  covariance  matrix  and  12  unrestricted  conditional  probabilities,  we 

reduce the number of parameters by imposing some restrictions. In the final specification only 

the  intercepts  are  regime  dependent  while  the  transition  matrix  is  reduced  to  only  6 

unrestricted elements and the correlation is constant across regimes (see tables 14 and 15 for 

the estimated parameters and the standard errors). In table 16 the test for the validity of the 

joint restrictions is reported. 

Figure 6 reports the smoothed probabilities of being in the low phase of the growth cycle. It is 

possible  to  note  that  there  is  a  clear  relationship  between  change  of  phase  detected  by  the 

model  and  the  turning  points  exhibited  by  the  series  in  figure  5.  Moreover,  a  lead  of  the 

survey is again visible.  

Analysing  the  transition  matrix  reported  in  table  17,  we  can  illustrate  some  important 

information. Let us start by considering the exit from the HH regime. The two possible exits 

are LL (60% probability) and HL (40% probability). If we go to LL it means that both areas 

switch  to  the  low  phase,  but  if  we  go  to  HL  it  means  that  only  the  survey  shift  to  the  low 

regime. Moreover, we observe that once we move to HL we can only exit by entering in LL. 

The transition matrix indicates that in some cases there is a lead of the survey in entering the 

below trend phase or at least the shifts are synchronized. The same happens in exiting the LL 

regime: we have a 43% probability to get HH and a 57% probability to get LH. In the latter 

case the only possible exit is LL, hence the survey leads again the industrial production or at 

least the two exits from the below trend phase are synchronised. 

Finally,  we  tested  the  null  that  shifts  in  the  survey  do  not  cause  shifts  in  the  Euro  zone  IPI 

growth cycle phase (IND NC IPI) and that shifts in the SIPI do not cause shift in the SIND. 

(IPI NC IND). Results of the test are reported in table 18. As it was expected the null “IND 

NC IPI” is rejected but also the null “IPI NC IND” is rejected. 

19 

 
 
5 Conclusions 

In  this  paper  we  extend  the  classical  MS  VAR  approach  to  the  business  cycle  proposed  by 

Krolzig (1997) introducing multiple Markov chain producing specific shifts in the parameters 

of each equation of the VAR (MMS VAR). Differently from the MS VAR approach, which 

allows  us  to  describe  the  common  cycle  and  the  evolution  of  its  phases,  the  MMS  VAR 

approach allows us to describe the relationship among cyclical phases of different countries or 

different sectors. Moreover we introduce a re-parameterization of the model which allows us 

to introduce causality relationships and test for them. 

The MMS VAR models and their different specifications are used to analyse the relationship 

between the industrial cycle in the Euro zone and in the U.S. and the relationship between the 

cyclical phases of the industrial production in the Euro zone and the phases exhibited by the 

industrial component of the ESI. 

Concerning  the  relationship  between  the  Euro  zone  and  the  U.S.  industrial  productions,  the 

estimated transition matrix of the MMS model shows that the U.S. most of the time leads the 

Euro  zone  in  both  the  business  and  the  growth  cycle.  Moreover,  we  conduct  a  study  of  the 

evolution of the conditional mean to its steady state value and we show that there is a strong 

inter-relationship  between  the  two  macro  areas.  In  particular,  the  evolution  of  the  business 

condition in the Euro zone is strongly affected by the current condition in the U.S. In order to 

evaluate the causality relationship between the growth cycles phases in the two macro areas, 

we estimate a MMS model for the Euro zone and the U.S. industrial productions without the 

trend components. The lead of the U.S. is graphically confirmed but the null of non causality 

is rejected in both directions.  

Finally, we consider a MMS model to study the relationship between the growth cycle phases 

of the industrial production in the Euro zone and the phases of the industrial component of the 

ESI  index.  The  estimated  transition  matrix  shows  that  change  in  phases  of  the  industrial 

survey and those of the industrial sector are at least synchronised and sometimes those of the 

survey are in advance (in particular in the last part of the sample). However, the null of non 

causality is again rejected in both directions. 

20 

References 

Anas J., Billio M., Ferrara L. and Lo Duca M. (2003), A turning point chronology for the Euro-zone classical 
and  growth  cycle,  to  be  presented  Colloquium  on  "Modern  Tools  for  Business  Cycle  Analysis", 
Luxembourg, October 2003. 

Anas,  J.  and  Ferrara,  L.  (2002a),  “Un  indicateur  d’entrée  et  sortie  de  récession  :application  aux  Etats-Unis”, 

Document de travail No. 58, Centre d’Observation Economique, Paris (www.coe.ccip.fr). 

Anas, J. and Ferrara, L. (2002b), “Detecting cyclical turning points: the ABCD approach and two probabilistic 
indicators ”, paper presented at the 26th Ciret Conference in Taiwan, October 2002 (www.coe.ccip.fr). 
Anas,  J.  and  Ferrara,  L.  (2002c),  “A  comparative  assessment  of  parametric  and  non-parametric  turning  points 
methods: the case of the Euro-zone economy”, paper presented at the Colloquium on “Modern Tools for 
Business Cycle Analysis”, Luxembourg, November 2002 (www.coe.ccip.fr). 

Artis  M.  J.,  Krolzig  H.  and  Toro  J.  (2002),  The  European  business  cycle,  Working  paper  ECO,  n°  99-24, 

European University Institute, Florence.  

Artis M. J., Marcellino M., Proietti T., (2002), Dating the Euro area Business Cycle, Working paper, European 

University Institute, Florence. 

Batchelor R. A., (2002) Confidence indexes and the probability of recession: a Markov switching model, Indian 

Economic Review, 36 (1) , 107-124. 

Billio M. (2002), Correlated Markov chains, manuscript GRETA, Venice. 

Billio M. and A. Monfort (1998), Switching state space models: likelihood, filtering and smoothing, Journal of 

Statistical Planning and Inference 68/1, 65-103. 

Billio  M.,  A.  Monfort  and  C.P.  Robert  (1998),  The  simulated  likelihood  ratio  (SLR)  method,  Document  de 

Travail du CREST 9828, Paris. 

Breunig R. and Pagan A. (2002), Specification Testing of Markov Switching Models, paper presented at the 13th 

(EC)² Conference “Model selection and evaluation”, Bologna, Italy, November 2002. 

Boldin  M.  D.  (1996),  A  check  on  the  robustness  of  Hamilton's  Markov  Switching  model  approach  to  the 
economic analysis of the business cycle, Studies in Nonlinear Dynamics and Econometrics 1, 35-46. 

Burns A. F. and Mitchell W. C. (1946), Measuring Business Cycles, NBER, Columbia University Press. 

Carter C. K. and P. Kohn (1994), On Gibbs sampling for state space models, Biometrika 84, 541–53. 

Chauvet  M.  (1998),  An  Econometric  Characterisation  of  Business  Cycle  Dynamics  with  Factor  Structure  and 

Regime Switching, International Economic Review 39, 969-996.  

Chauvet M. (1999), Stock Market Fluctuations and the Business Cycle, working paper  Department of Economics, 

University of California, Riverside (research.moore.sc.edu/workshop/papers/Chauvejr.rtf). 

Chen  S.W.  and  J.L.  Lin  (2000),  Identifying  turning  points  and  business  cycles  in  Taiwan:  A  multivariate 

dynamic Markov-switching factor model approach, Academia Economic Papers, 28, 289–321.  

Chin  D.  and  Geweke  J.  (2000),  Predicting  turning  points,  Working  Paper,  Federeal  Reserve  Bank  of 

Minneapolis. 

Clements  M.P.,  Krolzig  H.M.,  (2000),  Modelling  Business  Cycle  Features  using  Switching  Regime  Models, 

Discussion Paper, Institute of Economics and Statistics Oxford.  

Clements  M.P.,  Krolzig  H.M.,  (2003),  Business  cycle  asymmetries:  characterization  and  testing  based  on 
Markov-Switching autoregressions, Journal of Business and Economic Statistics,  21/1, 196-211. 

Davig T., (2002), Markov switching durations: an assessment of U.S. business cycles, Working Paper. 

Diebold  F.  X.  and  Rudebusch  G.  D.  (1996),  Measuring  business  cycles:  a  modern  perspective,  Review  of 

Economics and Statistics 78/1, 67-77. 

Doz  C.  and  Lenglart  F.,  (2001),  Dynamic  Factor  analysis:  estimation  and  test  with  an  application  to  the 
European business surveys, paper presented at the CEPR/ Banca d’Italia Conference in Rome, September 
2001. 

21 

 
Frühwirth-Schnatter S. (1994), Data augmentation and dynamic linear models. Journal of Time Series Analysis 

15, 183–202. 

Guegan D., (2003), Business and Financial Cycles: How to measure them and to measure their Interdependency? 
Note  de  recherché  IDHE-MORA  no.  07-2003,  paper  presented  at  the  Forecasting  Financial  Markets 
conference, Paris, June 2003. 

Guerrero  G.  (2002),  Markov-Switching  Models  of  Business  Cycle:  Can  the  Econometric  Model  detect  the 
Sciences  Economiques 

paper  EUREQua,  Maison 

growth 
(www.ecomod.net/conferences/ecomod2002/ papers/guerrero.pdf). 

  Working 

regime? 

des 

Hamilton J. D. (1989), A new approach to the economic analysis of non stationary time series and the business 

cycle, Econometrica 57/2, 357-384.  

Hamilton  J.D.  and  G.  Lin  (1996),  Stock  market  volatility  and  the  business  cycle,  Journal  of  Applied 

Econometrics 11, 573-593. 

Harrison P.J. and Stevens, C.F. (1976), Bayesian forecasting, Journal of the Royal Statistical Society B 38, 205-

247. 

Kaufmann S. (2000), Measuring business cycles with a dynamic Markov switching factor model: an assessment 

using Bayesian simulation methods, The Econometrics Journal 3/1, 39-65. 

Kholodilin  K.  (2001),  Latent  Leading  and  Coincident  Factors  Model  with  Markov-Switching  Dynamics, 

Economics Bulletin 3, 1-13. 

Kim C.J. (1994), Dynamic linear models with Markov-switching, Journal of Econometrics 60, 1-22. 

Kim C. J., Morley J. and Piger J. (2002), A Markov-Switching Model of Business Cycle Dynamics with a Post-

Recession “Bounce-Back” Effect, wc.wustl.edu/workingpapers/Kim_Morley_Piger.pdf. 

Kim C.J. and Nelson C. R. (1998), Business cycles turning points, a new coincident index and tests of duration 
dependence based on a dynamic factor model with regime switching, Review of Economics and Statistics 
80, 188-201. 

Kim, C.J. and C. R. Nelson (1999), State-Space Models with Regime Switching. Cambridge, MIT Press. 

Kim C.J., Piger J., Startz R., (2003), Estimation of Markov regime switching regression models with endogenous 

switching, working paper 2003-015A, Federal Reserve Bank of St.Luis. 

Krolzig H.M. (1997), Markov-switching vector autoregressions. Modelling, statistical inference and applications 

to business cycle analysis, Springer, Berlin. 

Krolzig  H.M.  and  Toro  J.  (2001),  Classical  and  Modern  Business  Cycle  Measurement:  The  European  Case. 
Working  paper  Institute  of  Economics  and  Statistics  Oxford  (www.econ.ox.ac.uk/research/hendry/ 
paper/HMKJTeurosty.pdf). 

Krolzig H.M. (2001), Markov-Switching procedures for dating the Euro-zone business cycle, Quarterly Journal 

of Economic Research, 3, 339-351. 

Krolzig,  H.-M.  (2003),  Constructing  turning  point  chronologies  with  Markov-switching  vector  autoregressive 
models:  the  euro-zone  business  cycle,  in:  Eurostat  (ed.),  Proceedings  on  “Modern  Tools  for  Business 
Cycle Analysis”, Monography in Official Statistics.  

Layton A.P. and Smith D. (2000), A further note to the three phases of the US business cycle, Journal of Applied 

Economics, 2000, 32, 1133-1143. 

Layton A.P. and Katsuura M. (2001a), A new business cycle turning point signalling system using the Markov 

Switching model, Applied Economics 33, 59-70. 

Layton  A.P.  and  M.  Katsuura  (2001b),  Comparison  of  Regime  Switching,  Probit  and  Logit  Models  in  Dating 

and Forecasting US Business Cycles, International Journal of Forecasting 17, 403-17. 

Luginbuhl  R.  and  de  Vos  A.  (1999),  Bayesian  Analysis  of  an  Unobserved-Component  Time  Series  Model  of 
GDP with Markov-Switching and Time-Varying Growths, Journal of Business and Economic Statistics 
17/4, 456-65. 

Mosconi 

R., 

Seri 

R. 

(2001), 

Non-Causality 

in 

Bivariate 

Binary 

Time 

Series, 

http://www.crest.fr/pageperso/lfa/seri/SeriFr.HTM. 

Shephard  N.  (1993),  Fitting  non-linear  time  series  models,  with  applications  to  stochastic  variance  models, 

Journal of Applied Econometrics 8, 135-152. 

22 

Shephard N. (1994), Partial non-Gaussian state space, Biometrika 81, 115–31. 

Smith P.A. and Summers P.M, (2002), How well do Markov switching models describe actual business cycles? 

The case of synchronisation, Working Paper University of Melbourne.. 

Susmel  R.  (1998),  Switching  volatilities  in  international  equity  markets,  manuscript  Department  of  Finance, 

University of Houston. 

Stock J.H. and Watson M. W. (1989), New indexes of coincident and leading indicators, in: Macroeconomics 

Annual, vol. 4, MIT Press. 

Vanhaelen,  J.-J.,  Dresse,  L.  and  DeMulder,  J.  (2000),  The  Belgian  industrial  confidence  indicator:  leading 
indicator of economic activity in the Euro area, Economic Surveys and Data Analysis, Proceedings of the 
25th CIRET Conference in Paris, October 2000, OECD Publications, Paris.  

Warne A. (2000), Causality and Regime Inference in a Markov Switching VAR, Sveriges Riksbank. 

23 

 
Table 1:  Estimated parameters and standard errors of the MMS model for the Euro zone and the U.S. Industrial 
Productions. The growth rate over three months has been considered. 

Parameter Estimate 

Standard 
error 

µE,EU 
µS,EU 
µR,EU 
µE,USA 
µS,USA 
µR,USA 
σE,EU 
σS,EU 
σR,EU 
σE,USA 
σS,USA 
σR,USA 
ρ 

0.0133 
0.0023 
-0.0130 
0.0200 
0.0048 
-0.0144 
0.0053 
0.0032 
0.0088 
0.0090 
0.0063 
0.0236 
0.2441 

0.0004 
0.0003 
0.0011 
0.0009 
0.0009 
0.0029 
0.0003 
0.0002 
0.0007 
0.0005 
0.0009 
0.0019 
0.0929 

Table 2: Estimated transition probabilities. 

Transition 
probability

Estimate 

Standard 
error 

EE to SE 
SE to SE 
SS to SE 
SE to RE 
RE to RE 
EE to ES 
ES to ES 
SE to SS 
SS to SS 
RS to RS 
RR to RS 
ES to ER 
ER to ER 
SS to SR 
ER to SR 
SR to RR 
RR to RR 

0.0540 
0.6018 
0.1685 
0.0205 
0.7822 
0.0471 
0.8813 
0.1977 
0.7503 
0.9199 
0.0209 
0.0824 
0.6401 
0.0543 
0.2457 
0.3351 
0.8563 

0.0237 
0.0982 
0.0566 
0.0204 
0.1953 
0.0208 
0.0514 
0.0461 
0.0639 
0.0771 
0.0208 
0.0455 
0.1828 
0.0317 
0.1464 
0.1169 
0.0507 

Table 3: Joint test for the restrictions imposed to the general model. 

L_unr 
3042.62 

L_res 
3016.66 

χ2(63) 
51.92 

P-value 
0.84 

24 

 
 
 
 
 
 
Table 4: Final specification of the transition matrix. 

EE 
SE 
RE 
ES 
SS 
RS 
ER 
SR 
RR 

EE 
0.90 
0.18 
0.00 
0.00 
0.03 
0.00 
0.00 
0.00 
0.00 

SE 
0.05 
0.60 
0.22 
0.04 
0.17 
0.00 
0.00 
0.00 
0.00 

RE 
0.00 
0.02 
0.78 
0.00 
0.00 
0.00 
0.00 
0.00 
0.00 

ES 
0.05 
0.00 
0.00 
0.88 
0.00 
0.00 
0.11 
0.00 
0.00 

SS 
0.00 
0.20 
0.00 
0.00 
0.75 
0.08 
0.00 
0.00 
0.12 

RS 
0.00 
0.00 
0.00 
0.00 
0.00 
0.92 
0.00 
0.00 
0.02 

ER 
0.00 
0.00 
0.00 
0.08 
0.00 
0.00 
0.64 
0.00 
0.00 

SR 
0.00 
0.00 
0.00 
0.00 
0.05 
0.00 
0.25 
0.66 
0.00 

RR 
0.00 
0.00 
0.00 
0.00 
0.00 
0.00 
0.00 
0.34 
0.86 

Table 5: Unconditional probabilities and expected duration of each regime. 

Unconditional 
Probabilities 
0.29 
0.13 
0.01 
0.15 
0.18 
0.03 
0.03 
0.05 
0.12 

Expected 
Duration
9.89 
2.51 
4.59 
8.43 
4.00 
12.49 
2.78 
2.98 
6.96 

EE 
SE 
RE 
ES 
SS 
RS 
ER 
SR 
RR 

Table 6: Transition matrix for the business cycle phases; unconditional probabilities and expected duration of 
each regime. 

EX-EX 
RE-EX 
EX-RE 
RE-RE 

EX-EX 
0.97 
0.12 
0.04 
0.12 

RE-EX 
0.00 
0.88 
0.00 
0.02 

EX-RE 
0.03 
0.00 
0.75 
0.00 

RE-RE  Unc Prob Exp Dur 
0.74 
0.05 
0.09 
0.12 

30.47 
8.43 
4.00 
7.14 

0.00 
0.00 
0.21 
0.86 

Table  7:  Transition  matrix  for  the  growth  cycle  phases;  unconditional  probabilities  and  expected  duration  of 
each regime. 

HH 
LH 
HL 
LL 

HH 
0.90 
0.16 
0.00 
0.01 

LH 
0.05 
0.65 
0.03 
0.08 

HL 
0.05 
0.00 
0.92 
0.00 

LL 
0.00 
0.18 
0.05 
0.91 

Unc Prob Exp Dur 

0.29 
0.15 
0.18 
0.39 

9.89 
2.90 
13.28 
11.17 

25 

 
 
 
 
 
 
 
 
 
 
 
 
Figure 1: Smoothed probabilities of a business cycle recession in the Euro zone (EU) and in the U.S. (USA). 

EU

USA

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

4
0

-
0
7
9
1

0
1

-
1
7
9
1

4
0

-
3
7
9
1

0
1

-
4
7
9
1

4
0

-
6
7
9
1

0
1

-
7
7
9
1

4
0

-
9
7
9
1

0
1

-
0
8
9
1

4
0

-
2
8
9
1

0
1

-
3
8
9
1

4
0

-
5
8
9
1

0
1

-
6
8
9
1

4
0

-
8
8
9
1

0
1

-
9
8
9
1

4
0

-
1
9
9
1

0
1

-
2
9
9
1

4
0

-
4
9
9
1

0
1

-
5
9
9
1

4
0

-
7
9
9
1

0
1

-
8
9
9
1

4
0

-
0
0
0
2

0
1

-
1
0
0
2

Figure 2: Smoothed probabilities of the low phase of the growth cycle in the Euro zone (EU) and in the U.S. 
(USA). 

EU

USA

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

4
0

0
1

4
0

0
1

4
0

0
1

4
0

0
1

4
0

0
1

4
0

0
1

4
0

0
1

4
0

0
1

4
0

0
1

4
0

0
1

4
0

0
1

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

0
7
9
1

1
7
9
1

3
7
9
1

4
7
9
1

6
7
9
1

7
7
9
1

9
7
9
1

0
8
9
1

2
8
9
1

3
8
9
1

5
8
9
1

6
8
9
1

8
8
9
1

9
8
9
1

1
9
9
1

2
9
9
1

4
9
9
1

5
9
9
1

7
9
9
1

8
9
9
1

0
0
0
2

1
0
0
2

26 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Table 8: Impulse response analysis. Evolution of the conditional mean in the Euro zone (EU) and in the U.S.A. (USA) given a starting regime. 

EU 

USA

STARTING FROM EE

EU 

USA

STARTING FROM ES

EU 

USA

STARTING FROM ER

0.025
0.020
0.015
0.010
0.005
0.000
-0.005
-0.010
-0.015

0.025

0.015

0.005

-0.005

-0.015

0.025

0.015

0.005

-0.005

-0.015

1

5

9 13 17 21 25 29 33 37 41 45 49

1

5

9 13 17 21 25 29 33 37 41 45 49

1

5

9 13 17 21 25 29 33 37 41 45 49

EU 

USA

STARTING FROM SE

EU 

USA

STARTING FROM SS

EU 

USA

STARTING FROM SR

0.025

0.015

0.005

-0.005

-0.015

0.025

0.015

0.005

-0.005

-0.015

0.025

0.015

0.005

-0.005

-0.015

1 10 19 28 37 46 55 64 73 82 91 100

1

5

9 13 17 21 25 29 33 37 41 45 49

1

5

9 13 17 21 25 29 33 37 41 45 49

EU 

USA

STARTING FROM RE

EU 

USA

STARTING FROM RS

EU 

USA

STARTING FROM RR

0.025

0.015

0.005

-0.005

-0.015

0.025

0.015

0.005

-0.005

-0.015

0.025

0.015

0.005

-0.005

-0.015

1

5

9 13 17 21 25 29 33 37 41 45 49

1

5

9 13 17 21 25 29 33 37 41 45 49

1

5

9 13 17 21 25 29 33 37 41 45 49

27 

 
 
 
 
 
 
 
 
 
 
 
 
Figure 3: Standardised Industrial Production Indexes without the trend component for Euro zone (SDEU) and the U.S. 
(SDUSA). 

SDUSA

SDEU

3.0

2.0

1.0

0.0

-1.0

-2.0

-3.0

-4.0

2
0

-
0
7
9
1

8
0

-
1
7
9
1

2
0

-
3
7
9
1

8
0

-
4
7
9
1

2
0

-
6
7
9
1

8
0

-
7
7
9
1

2
0

-
9
7
9
1

8
0

-
0
8
9
1

2
0

-
2
8
9
1

8
0

-
3
8
9
1

2
0

-
5
8
9
1

8
0

-
6
8
9
1

2
0

-
8
8
9
1

8
0

-
9
8
9
1

2
0

-
1
9
9
1

8
0

-
2
9
9
1

2
0

-
4
9
9
1

8
0

-
5
9
9
1

2
0

-
7
9
9
1

8
0

-
8
9
9
1

2
0

-
0
0
0
2

8
0

-
1
0
0
2

Table 9: Estimated parameters and standard errors of the MMS model with 4 regimes for the Euro zone and the U.S. 
standardised Industrial Production Indexes without the trend component. 

Parameter Estimate 

υF,EU 
υS,EU 
υF,USA 
υS,USA 
σEU 
σUSA 
ФEU 
ФUSA 
ρE,E 
ρS,E 
ρE,S 
ρS,S 

0.0745 
-0.0807 
0.0625 
-0.0778 
0.0085 
0.0185 
0.9969 
0.9844 
0.9203 
0.7338 
0.9196 
0.9124 

Standard 
error 
0.0064 
0.0068 
0.0064 
0.0064 
0.0061 
0.0027 
0.0046 
0.0045 
0.0172 
0.0415 
0.0147 
0.0190 

Table 10: Estimated transition probabilities and standard errors. 

Transition 
probability 
LH to LH 
LL to LH 
HH to HL 
HH to LL 
HL to LL 
LL to LL 

Estimate 

0.9282 
0.1104 
0.0861 
0.0233 
0.0861 
0.8636 

Standard 
error 
0.0248 
0.0355 
0.0296 
0.0164 
0.0277 
0.0397 

28

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Table 11: Test for the validity of the joint restrictions imposed to the general model. 

L_unr 
1224.84 

L_res 
1224.84 

χ2(6) 
0.00 

p-value 
0.00 

Table 12: Final specification of the transition matrix and expected duration of each regime. 

HH 
LH 
HL 
LL 

HH 
0.89 
0.07 
0.00 
0.03 

LH 
0.00 
0.93 
0.00 
0.11 

HL 
0.09 
0.00 
0.91 
0.00 

LL 
0.02 
0.00 
0.09 
0.86 

Exp Dur 
9.14 
13.92 
11.62 
7.33 

Table 13: Non causality tests. 

Null 
USA NC EU 
EU NC USA 

L unr 
1227.04 
1227.04 

L null 
1155.24 
1173.18 

χ2(2) 
143.60 
107.72 

p-value 
0.00 
0.00 

Figure  4:  Smoothed  probabilities  of  the  low  phase  of  the  growth  cycle  in  the  Euro  zone  (SDEU)  and  in  the  U.S. 
(SDUSA). 

EU

USA

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

2
0

8
0

2
0

8
0

2
0

8
0

2
0

8
0

2
0

8
0

2
0

8
0

2
0

8
0

2
0

8
0

2
0

8
0

2
0

8
0

2
0

8
0

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

0
7
9
1

1
7
9
1

3
7
9
1

4
7
9
1

6
7
9
1

7
7
9
1

9
7
9
1

0
8
9
1

2
8
9
1

3
8
9
1

5
8
9
1

6
8
9
1

8
8
9
1

9
8
9
1

1
9
9
1

2
9
9
1

4
9
9
1

5
9
9
1

7
9
9
1

8
9
9
1

0
0
0
2

1
0
0
2

29

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Figure  5:  Standardised  Industrial  Production  Index  without  the  trend  component  for  the  Euro  zone  (SIPI)  and 
standardised level of the industrial survey (SIND). 

SIND

SIPI

3.0

2.0

1.0

0.0

-1.0

-2.0

-3.0

-4.0

2
0

-

5
8
9
1

2
1
-

5
8
9
1

0
1

8
0

6
0

4
0

2
0

-

-

-

-

-

6
8
9
1

7
8
9
1

8
8
9
1

9
8
9
1

0
9
9
1

2
1
-

0
9
9
1

0
1

8
0

6
0

4
0

2
0

-

-

-

-

-

1
9
9
1

2
9
9
1

3
9
9
1

4
9
9
1

5
9
9
1

2
1
-

5
9
9
1

0
1

8
0

6
0

4
0

2
0

-

-

-

-

-

6
9
9
1

7
9
9
1

8
9
9
1

9
9
9
1

0
0
0
2

2
1
-

0
0
0
2

0
1

8
0

-

-

1
0
0
2

2
0
0
2

Table  14:  Estimated  parameters  and  standard  errors  of  the  MMS  model  with  4  regimes  of  Euro  zone  standardised 
Industrial Production Indexes without the trend component and standardised industrial survey. 

Parameter Estimate 

Standard 
error 

υF,IPI 
υS,IPI 
υF,IND 
υS,IND 
σIPI 
σIND 
ФIPI 
ФIND 
ρ 

0.0819 
-0.1574 
0.0436 
-0.1336 
0.0147 
0.0429 
0.9883 
0.9556 
0.5761 

0.0091 
0.0125 
0.0163 
0.0214 
0.0051 
0.0061 
0.0081 
0.0112 
0.0600 

Table 15: Estimated transition probabilities and standard errors. 

Transition 
probability

Estimate 

Standard 
error 

LH to LH 
LL to LH 
HH to HL
HH to LL 
HL to LL 
LL to LL 

0.5735 
0.0411 
0.0215 
0.0245 
0.0912 
0.9287 

0.2394 
0.0418 
0.0162 
0.0165 
0.0633 
0.0312 

30

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Table 16: Test for the validity of the joint restrictions imposed to the general model. 

L_unr 
514.17 

L_res 
512.66 

χ2(9) 
3.02 

p-value 
0.96 

Table 17: Final specification of the transition matrix and expected duration of each regime. 

HH 
LH 
HL 
LL 

HH 
0.95 
0.43 
0.00 
0.03 

LH 
0.00 
0.57 
0.00 
0.04 

HL 
0.02 
0.00 
0.91 
0.00 

LL 
0.03 
0.00 
0.09 
0.93 

Exp Dur 
21.73 
2.34 
10.97 
14.03 

Table 18: Non causality tests. 

TEST 
IND NC IPI 
IPI NC IND 

F unr 
512.82 
512.69 

F null 
508.96 
508.42 

χ2(2) 
7.72 
8.53 

p-value 
0.02 
0.01 

Figure  6:  Smoothed  probabilities  of  the  low  phase  of  the  growth  cycle  in  the  Euro  zone  (PSIPI)  and  smoothed 
probabilities of the low phase of the standardised industrial survey (PSIND). 

PSIPI

PSIND

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

1985 -12
1990 -12
2000 -12
1995 -12
1988 - 06
1985 - 02
1993 - 06
1987 - 08
1990 - 02
1999 - 04
1995 - 02
1998 - 06
1992 - 08
1997 - 08
2000 - 02
2001 - 10
2002 - 08
1986 - 10
1989 - 04
1991 - 10
1994 - 04
1996 - 10

31