Turning point detection with Bayesian panel Markov-switching VAR : 2016 edition.
Date: 2016-09-08
Institutions: Eurostat
Subjects: business activity, econometrics, economic analysis, economic fluctuation, economic forecasting, economic model, statistical method

Turning point detection 
with bayesian panel 
Markov-Switching VAR

MONICA BILLIO, ROBERTO CASARIN, HERMAN K. VAN DIJK, 
GIAN LUIGI MAZZI, FRANCESCO RAVAZZOLO

2016 edition

S T A T I S T I C A L 
W O R K I N G   P A P E R S

STATISTICAL WORKING PAPERSMain title 2015 editionTurning point detection  
with bayesian panel  
Markov-Switching VAR 
MONICA BILLIO, ROBERTO CASARIN, HERMAN K. VAN DIJK, 
 GIAN LUIGI MAZZI  FRANCESCO RAVAZZOLO 

2016 edition 

 
 
 
 
 
 
 
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ISBN: 978-92-79-61459-0 
ISSN: 2315-0807 
doi: 10.2785/776839 
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Table of contents 

Abstract ............................................................................................................................... 4 

1 

2 

3 

Introduction ................................................................................................................ 5 

A Panel Markov-switching VAR model ..................................................................... 6 

Bayesian Inference .................................................................................................... 8 

3.1 

Independent Priors ....................................................................................................................... 8 

3.2  Hierarchical Prior .......................................................................................................................... 9 

3.3  Gibbs sampler ............................................................................................................................. 10 

3.4  Regime Probability Combination ............................................................................................... 12 

4 

Business Cycle Analysis ......................................................................................... 14 

4.1  Data Description.......................................................................................................................... 14 

4.2 

4.3 

Parameter Estimates................................................................................................................... 15 

Turning Points ............................................................................................................................. 19 

5 

6 

Conclusions ............................................................................................................. 23 

References ............................................................................................................... 29 

 Turning point detection 

3 

 
 
 
 
 
 
 
Abstract 

Abstract 

This  paper  proposes  a  panel  Markov-Switching  (MS-)  VAR  model  suitable  for  a  multi-country  analysis  of 
the  business  cycle.  We  study  the  business  cycles  fluctuations  of  a  group  of  countries,  analyse  the 
transmission of shocks across cycles and predict the turning points of the country-specific cycles. We focus 
on  the  European  Union  (EU)  and  compare  the  results  obtained  by  analysing  the  EU  at  a  disaggregated 
level. We propose a forecast combination approach for aggregating the turning points of the EU countries 
in order to obtain a possibly better prediction of the turning points for the EU business cycle. A Bayesian 
approach has been applied to estimate the panel MS-VAR model and to forecast the turning points. 

Acknowledgements: Paper first presented at the Euro Area Business Cycle Network (EABCN) 
conference on Disaggregating the Business Cycle (Luxembourg) in October 2012. The paper has 
benefitted from the outcomes of the multi-annual PEEIs project financed by Eurostat.  

Authors: Monica Billio (*), Roberto Casarin (**), Herman K. van Dijk (***), Gian Luigi Mazzi (****) 
Francesco Ravazzolo (*****), 22 December 2011 

JEL classification code: C11, C15, C53, E37 

Keywords: Forecast Combination, Bayesian Model Averaging, panel VAR, Markov-Switching 

(*) University of Venice, GRETA Assoc. and School for Advanced Studies in Venice 
(**) University of Venice, GRETA Assoc. and School for Advanced Studies in Venice 
(***) Econometrics and Tinbergen Institutes, Erasmus University Rotterdam) 
(****) European Commission, Eurostat 
(*****) Norges Bank 

 Turning point detection 

4 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Introduction  1 

1  Introduction 

In  this  paper,  we  contribute  to  the  literature  on  the  analysis  of  the  business  cycle  of  large  panel  of 
countries.  The  analysis  of  the  world  business  cycle  has  been  proposed  by  Gregory  et  al.  (1997),  who 
consider  a  panel  of  trivariate  series  (output,  consumption  and  investment)  for  the  G7  countries  and 
estimate  dynamic  factor  model  featuring  a  common  (world)  cycle,  a  country  specific  component  and  a 
series specific (fully idiosyncratic) one. 

The specification of the model is based on an extension of the single index model of coincident indicators 
by Stock and Watson (1991). They conclude that both the world and the country specific factors captures a 
significant  amount  of  the  fluctuations.  Kose  et  al.  (2003)  reaches  similar conclusions,  using  a  larger  data 
set  on  60  countries  and  using  a  Bayesian  dynamic  factor  model.  They  conclude  that  real  output  growth 
depends on an international factor, a regional factor, plus an idiosyncratic one. The overall finding is again 
that the world factor explains a substantial fraction of economic fluctuations. In a recent paper, Kose et al. 
(2008)  find  however  that  the  relative  importance  of  the  common  factor has  been  declining  over time  and 
that  the  cycle  of  emerging  economies  has  become  decoupled  from  that  of  industrialized  countries.  Hess 
and Shin (1997, 1998) propose analysing the ”intra-national” business cycle (i.e. the co-movements within 
a country) in order to gain understanding of the transmission mechanism of shocks that enables to abstract 
from the trade frictions that affect international economics. They use disaggregated U.S. State level data on 
productivity growth for several industries and assess, by a descriptive decomposition technique, the role of 
the common intra-national cycle, that of the industry specific and the state-specific cycles.  

They conclude that the role of the state specific cycle is much reduced and sector specific shocks are more 
important  in  a  common  currency  area.  Lumsdaine  and  Prasad  (2003)  assess  the  relative  importance  of 
country specific versus common shocks, using industrial production growth for a set of 17 countries. They 
estimate the common component of international fluctuations by the aggregation with time-varying weights 
(derived  from  the  reciprocal  of  the  conditional  variance  of  the  series,  estimated  by  fitting  a  univariate 
GARCH  model),  which  aims  at  downweighting  the  idiosyncratic  variation,  of  the  industrial  production 
growth  rates.  In  the  present  paper  we  focus  on  the  business  cycle  of  the  European  Union  (EU)  and  the 
cycles  of  12  countries  of  the  EU.  First,  we  aim  to  measure  the  cycle  by  using  multivariate  series  and  to 
extract the turning points of the country-specific business cycles. Secondly, we investigate the similarities 
between the EU cycle at an aggregated level and the cycles of the 12 countries considered in our analysis. 

Another  aim  of  the  paper  is  to  verify  the  sources  business  cycle  co-movements,  i.e.  on  the  channels 
through  which  business  cycle  fluctuations  are  transmitted  across  countries  of  the  international  economic 
system. We will focus on the following sources of transmission: interest rates (financial sector) and the oil 
prices (world shocks). In this respect, the literature has focused on the determinants on two main sources: 
trade and financial integration. Theoretically, there is no consensus in the literature on the role of trade in 
the  international  transmission  of  shocks.  As  argued  by  Frankel  and  Rose  (1998),  on  the  one  hand  trade 
has a positive direct impact  on business cycle synchronisation, whilst on the other hand it could have an 
indirect negative effect through specialisation. Greater specialisation would lead to lower concordance, as 
countries  may  be  more  prone  to  sector-specific  and  idiosyncratic  (or  asymmetric)  shocks  (Bayoumi  and 
Eichengreen  (1993)).  As  a  consequence,  the  direction  of  the  link  between  trade  openness  and  business 
cycle  concordance  is  largely  regarded  as  an  empirical  issue.  Imbs  (2004)  estimates  a  simultaneous 
equations  system  to  explain  the  observed  cross-correlation  of,  say  output  growth,  using  explanatory 
variables that measure trade openness, financial integration and the degree of specialization. He concludes 
that trade has a strong effect on business cycle synchronization, but a sizable portion of this effect is found 
to actually work through intra-industry interlinkages. 

Financial integration  also  has  a  prevailing  direct  positive  effect  on synchronization.  Canova  and  Marrinan 
(1998) address a different question, as to whether the international business cycles originate from common 
shocks  or  from  a  common  propagation  mechanism.  Monfort  et  al.  (2003)  aim  at  disentangling  common 
shocks from spill-over effects. To this end, they estimate a Bayesian dynamic factor model for the G7 real 

 Turning point detection 

5 

 
 
 
 
 
A Panel Markov-switching VAR model  2 

output  growth,  featuring  a  global  common  factor  and  two  area  specific  (North-American  and  Continental 
European)  common  factors,  which,  being  modelled  as  a  VAR  process,  are  interdependent.  They  find 
empirical support for the presence of spill-over effects running from North-America to Continental Europe, 
but not vice versa. 

This  paper  also  contributes  to  the  literature  on  heterogeneity  in  cross-country  panel  data  models.  Panel 
datasets are appealing because they combine the information coming from the cross-section and the time-
series  dimension  of  the  data.  In  the  context  of  the  cross-country  panel  data  models,  the  more  recent 
approaches  have  focused  on  two  issues:  the  estimation  of  international  cycles  and  the  nature  of  the  co-
movements  using  relatively  large  dimensional  datasets  and  the  introduction  of  country  and  time 
heterogeneity in multi country vector autoregressive models. The first issue has been considered by Hallin 
and  Liska  (2008),  Pesaran  et  al.  (2004),  and  Dees  et  al.  (2007).  The  second  by  Canova  and  Ciccarelli 
(2006). Hallin and Liska (2008) extend the generalized dynamic factor model by Forni et al. (2000, 2001) to 
panel  of  time  series  with  block  structure, where the  blocks  are  represented  by  countries.  They  show  that 
the extension provides the means for the analysis of the interblock relationships, allowing the identification 
of strongly common factors, which are common to all the blocks (e.g. the international common factors), the 
strongly  idiosyncratic  factors,  which  are  idiosyncratic  for  all  blocks,  and  the  weakly  common/weakly 
idiosyncratic  factors,  that  are  common  to  at  least  one  block,  but  idiosyncratic  to  at  least  another.  Multi-
country VAR models provide a tool for examining the propagation of shocks across countries. Canova and 
Ciccarelli  (2006)  consider Bayesian  inference for multicountry  VAR models  with  time  varying  parameters, 
lagged interdependencies and country specific effects. They avoid the curse of dimensionality by a factor 
parameterization of the time varying VAR coefficients in terms of a number of random effects that are linear 
in the number of countries and series. The random coefficients are in turn driven by a common component, 
a  country  specific  component,  a  variable  specific  component  and  a  idiosyncratic  component.  The  factor 
loadings  assumed  to  evolve  according  to  a  stationary  vector  first  order  autoregression,  whereas  the 
idiosyncratic component is assumed to be serially uncorrelated. The disturbances driving the evolution of 
the  factors  are  also  allowed  to  be  heteroscedastic.  The  paper  proposes  a  Monte  Carlo  Markov  Chain 
sampling scheme to estimate the posterior distribution of the coefficients and to carry out impulse response 
analysis. Canova and Ciccarelli (2006) analyze the transmission of shocks in the G7 countries focusing on 
four  macroeconomic  variables:  real  growth,  inflation,  employment  growth  and  rent  inflation;  oil  prices  are 
considered as exogenous. In this paper, we build on Canova and Ciccarelli (2006) and extend their panel 
VAR  model  in  order  to  model  asymmetry  and  the  turning  points  in  the  business  cycles  of  different 
countries.  Our  paper  is  also  strictly  related  to  Kaufmann  (2010),  where  a  panel  of  univariate  Markov-
switching  (MS)  regression  models  is  considered.  The  early  contributions  in  the  business  cycle  literature 
consider  nonlinear  models  such  as  the  MS  models  (see  for  example  Goldfeld  and  Quandt  (1973)  and 
Hamilton  (1989))  and  the  threshold  autoregressive  models  (see  Tong  (1983)  and  Potter  (1995)),  both  of 
which are able to capture the asymmetry and the turning points in business cycle dynamics. In this paper 
we focus on the class of MS models. We take the models of Hamilton (1989) and Krolzig (2000) as points 
of departure and consider Markov-switching dynamics for the VAR coefficients and covariance matrices. 

The remainder of this paper is organized as follows. Section 2 presents the Bayesian panel MS-VAR model 
that  has  been  used  for  the  analysis.  Section  3  discusses  the  prior  choice  and  the  Bayesian  inference 
framework. Section 4 presents the empirical evidence on cross-country asymmetries in the business cycle 
and the comparison with the EU and US cycle. The same session presents the asymmetries in the shocks 
transmission mechanism. Finally, Section 5 concludes. 

2  A Panel Markov-switching VAR model 
Let 𝑦𝑖𝑡 ∈   ℝ𝐾, 𝑖 = 1, … , 𝑁 and , 𝑡 = 1, … , 𝑇 be a sequence of , 𝐾-dimensional vectors of observations. 𝑁 is the 
number of units (countries) and 𝑇 the number of time observations. We introduce a general specification of 
the panel Markov-switching VAR (PMS-VAR) model 

 Turning point detection 

6 

 
 
 
 
 
A Panel Markov-switching VAR model  2 

(𝒔𝒊𝒕)𝒚𝒋𝒕−𝒍 + 𝑫𝒊 (𝒔𝒊𝒕)𝒛𝒕 + 𝜺𝒊𝒕, 

(1) 

𝒚𝒊𝒕 =   𝒂𝒊(𝒔𝒊𝒕) +   ∑ ∑ 𝑨𝒊𝒋𝒍

𝑵
𝒋=𝟏

𝒑
𝒍=𝟏

𝑖 = 1, … , 𝑁, with 𝜀𝑖𝑡~𝒩𝐾(𝟎, ∑ (𝑠𝑖𝑡)

𝑖

) and 𝑧𝑡 ∈   ℝ𝐺 a vector of variables, common to all units. 

The {𝑠𝑖𝑡}𝑡 are unit-specific and independent M-states  Markov-chain processes with values in {1, … , 𝑀} and 
transition  probability ℙ(𝒔𝒊𝒕 = 𝑘|𝒔𝒊𝒕−1 = 𝑗) = 𝑝𝑖,𝑘𝑗, 𝑗 ∈   {1, … , 𝑀}.  We  assume  the  chains  are  stationary  and 
irreducible.  As  regards  to  the  choice  of  the  number  of  regimes,  we  notice  that  for  more  recent  data  one 
needs  an  adequate  business  cycle  model  with  more  than  two  regimes  (see  also  Clements  and  Krolzig 
(1998)) and a time-varying error variance. For example, Kim and Murray (2002) and Kim and Piger (2000) 
propose  a  three-regime  (recession,  high-growth,  and  normal-growth)  MS  model  while  Krolzig  (2000) 
suggests  the  use  of  a  model  with  regime-dependent  volatility  for  the  US  GDP.  In  our  paper  we  consider 
data  on  EU  industrial  production,  for  a  period  of  time  including  the  2009  recession  and  find  that  four 
regimes  (high-recession,  contraction,  normal-growth,  and  highgrowth)  are  necessary  to  capture  some 
important features of the US and EU cycle in the strong-recession phases. 

The  generality  of  the  propose  statistical  model  comes from  the  fact  that  the  coefficients  vary  both  across 
units  and  across  time.  Moreover  the  interdependencies  between  units  are  allowed  whenever  𝑨𝒊𝒋𝒍(𝒔𝒊𝒕)   ≠
0 for 𝑖  ≠ 𝑗 .  In  order  to  define  the  parameter  shifts  more  clearly  and  to  simplify  the  exposition  of  the 
inference procedure we introduce the indicator variable 𝜉𝒊𝑘𝑡 = 𝛿𝑘(𝒔𝒊𝒕), where 

𝜹𝒌(𝒔𝒊𝒕) = {

𝟏,       𝒊𝒇 𝒔𝒊𝒕 = 𝒌
𝟎     𝐨𝐭𝐡𝐞𝐫𝐰𝐢𝐬𝐞

for 𝑘 = 1, … , 𝑀, 𝑖 = 1, … , 𝑁, and 𝑡 = 1, … , 𝑇, and  the  vector  of  indicators 𝝃𝒊𝑡 = (𝝃𝒊1𝑡, … , 𝝃𝒊𝑀𝑡)′,  which  collects 
the  information  about  the  realizations  of  the 𝑖-th  unit-specific  Markov  chain  over  the  sample  period.  The 
indicators allow us to write the parameter shifts as 

𝑎𝒊(𝒔𝒊𝒕) = ∑

𝑎𝑀
𝑘=1 𝑖,𝑘 𝝃𝒊𝑘𝑡,   

𝐴𝒊𝑗𝑙(𝒔𝒊𝒕) = ∑

𝐴𝑀
𝑘=1

𝑖𝑗𝑙,𝑘

𝝃𝒊𝑘𝑡,

𝐷𝒊(𝒔𝒊𝒕) = ∑

𝐷𝑀
𝑘=1

𝑖,𝑘 𝝃𝒊𝑘𝑡,   

Σ𝒊(𝒔𝒊𝒕) = ∑

Σ𝑀
𝑘=1 𝑖,𝑘 𝝃𝒊𝑘𝑡.

′ ) = 𝑂𝐾×𝐾,  with 𝑂𝑛×𝑚 the 𝑛 × 𝑚-
In  our  applications  we  will  assume  the  following  restrictions  hold: 𝔼(𝜀𝑖𝑡𝜀𝑗𝑡
dimensional null matrix, and there are no interdependencies among the same variable across units, that is 
𝐴𝒊𝑗𝑙 = 𝐴𝒊𝑗𝑙 𝛿𝑖(𝑗) +   𝑂𝐾×𝐾 (1 − 𝛿𝑖(𝑗)). 

Clements and Krolzig (1998) found in an empirical study that most forecast errors are due to the constant 
terms in the prediction models. They suggest considering, for example, MS models with regime-dependent 
volatility.  In  this  paper,  we  follow  Krolzig  (2000)  and  Anas  et  al.  (2008)  and  assume  that  both  the  unit-
specific  intercepts, 𝑎𝒊(𝒔𝒊𝒕),  and  volatilities, Σ𝒊(𝒔𝒊𝒕),  are  driven  by  the  regime-switching  variables {𝑠𝑖𝑡}𝑡 and 
assume  constant  autoregressive  coefficients  𝐴𝒊𝑙,𝑘 = 𝐴𝒊𝑙 , ∀ 𝑘 .  In  the  same  spirit  we  assume  that  the 
coefficients of the common variables do not change over time, that is 𝐷𝒊,𝑘 = 𝐷𝒊 , ∀ 𝑘. 

′

, 𝒛𝑡

′
, ⋯ , 𝒚𝑖𝑡−𝑝

′ = (1, … , 𝒚𝑖𝑡−1

′ ), 𝑡 = 1, … , 𝑇 be the sequence of (1 + 𝐾𝑝  + 𝐺)-dimensional column vectors 
Let 𝒘̅ 𝑖𝑡
of  regressors  for  the  PMS-VAR  model,  that  includes  the  constant  term,  the  lagged  dependent  variables, 
′ ⨂𝐼𝑘,  and  coefficients, 𝐴𝒊,𝑘 =
and  the  set  of  common  variables.  Moreover  define  the  regressors, 𝑊𝑖𝑡 = 𝑤̅𝑖𝑡
(𝑎𝒊,𝑘, 𝐴𝑖1,𝑘, ⋯ , 𝐴𝑖𝑝,𝑘, 𝐷𝑖),  matrices  of  dimension (𝐾(1 + 𝐾𝑝  + 𝐺) × 𝐾) and (𝐾 × 𝐾(1 + 𝐾𝑝  + 𝐺)) respectively. 
By using the allocation variables 𝝃𝒊𝑡 and the unit independence assumptions, given above, the PMS-VAR 
model can be rewritten as 

 Turning point detection 

7 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
𝒚𝒊𝒕 =   𝑨𝒊,𝟏𝑾𝒊𝒕𝝃𝒊𝟏𝒕 + ⋯ + 𝑨𝒊,𝑴𝑾𝒊𝒕𝝃𝒊𝑴𝒕   + 𝜺𝒊𝒕, 𝜺𝒊𝒕~𝓝𝑲(𝟎, 𝜮𝒊𝒕) 

(2) 

or  in  a  more  compact  form  as  𝑦𝑖𝑡 = (𝝃𝒊𝑴𝒕⨂ 𝑊𝑖𝑡)𝒗𝒆𝒄(𝐵𝒊) + 𝜀𝑖𝑡 ,  where  𝐵𝒊 = (𝒗𝒆𝒄(𝐴𝒊,𝟏), 𝒗𝒆𝒄(𝐴𝒊,𝟐), … ,

Bayesian Inference  3 

𝒗𝒆𝒄(𝐴𝒊,𝑴)),  𝜮𝒊𝒕 = 𝜮(𝝃𝒊𝑴𝒕⨂ 𝐼𝐾)  and 𝜮𝒊 = (𝜮𝒊1, … , 𝜮𝒊𝑀).  For  reason  of  convenience  we  consider the  partition 
of the set of regressors 𝑤̅𝑖𝑡
, 𝑚 = 1, … , 𝑀, that are a 𝐾0 −dimensional vector 
 and 𝑥̅𝑖𝑚𝑡
of  regressors  with  regime-invariant  coefficients  and  𝑀  vectors  of  𝐾𝑚  regime-specific  regressors  with 
regimedependent coefficients. Under this assumption the previous model writes as 

′
′  into 𝑀 + 1 subsets 𝑥̅𝑖0𝑡

′

𝒚𝒊𝒕 =   𝑿𝒊𝟎𝟏𝜸𝒊𝟎 + 𝝃𝒊𝟏𝒕𝑿𝒊𝟏𝟏𝜸𝒊𝟏 + ⋯ + 𝝃𝒊𝑴𝒕𝑿𝒊𝑴𝟏𝜸𝒊𝑴   + 𝜺𝒊𝒕, 

(3) 

where 𝑿𝑖0𝑡 = (𝑥̅𝑖0𝑡

′ ⨂ 𝐼𝐾)  and 𝑿𝑖𝑚𝑡 = (𝑥̅𝑖𝑚𝑡

′ ⨂ 𝐼𝐾). 

3  Bayesian Inference 

3.1  Independent Priors 

We  assume  a  conjugate  priors  for  the  coefficients  and  the  variance  of  the  panel  MS-VAR.  For  the 
coefficients 𝛾𝑖𝟎 and 𝛾𝑖𝑚 we consider independent normals priors 

𝜸𝒊𝟎 ~𝓝𝑲𝟎(𝜸𝒊𝟎, 𝜮𝒊𝟎) 

𝜸𝒊𝑴 ~𝓝𝑲𝑴(𝜸𝒊𝒎, 𝜮𝒊𝒎), 𝒎 = 𝟏, … , 𝑴 

(4) 

(5) 

𝑖 = 1, … , 𝑁.  We  assume  independence  across  units,  that  is: ℂ𝑜𝑣(𝛾𝑖𝟎, 𝛾𝑗𝟎) = 𝟎 and ℂ𝑜𝑣(𝛾𝑖𝑚, 𝛾𝑗𝑚) = 𝑂𝐾𝑚×𝐾𝑚, 
for 𝑖 ≠ 𝑗. For the inverse covariance matrix ∑−1

𝑖𝑚  we assume the Wishart priors 

∑−𝟏

𝒊𝒎  ~ 𝓦𝑲(𝒗𝒊𝒎/𝟐, 𝜰𝒊𝒎/𝟐), 𝒎 = 𝟏, … , 𝑴 

(6) 

with  possibly  regime-specific  degrees  of  freedom  𝒗𝑖𝒎  and  precision  𝜰𝒊𝒎  parameters.  We  assume 
ℂ𝑜𝑣(∑−1

2 . 

𝑖𝑚 , ∑−1

𝑖𝑚 ) = 𝑂𝐾𝑚

2 ×𝐾𝑚

When  using  Markov-switching  processes,  one  should  deal  with  the  identification  issue  associated  to  the 
label  switching  problem.  See  for example  Celeux  (1998)  and  Frühwirth-Schnatter (2001) for a  discussion 
on  the  effects  of  the  label  switching  and  the  unidentification  on  the  results  of  a  MCMC  based  Bayesian 
inference.  In  the  literature,  different  routes  have  been  proposed  for  dealing  with  the  label  switching  (see 
Frühwirth-Schnatter  (2006)  for  a  review).  One  of  the  most  efficient  approach  is  the  permutation  sampler 
(see  Frühwirth-Schnatter  (2001)),  which  can  be  applied  under  the  assumption  of  exchangeability  of  the 
posterior  distribution.  This  assumption  satisfied  when  assuming  symmetric  prior  on  the  transition 
probabilities of the switching process. As an alternative one could impose some identification constrains on 
the  parameters.  This  practice  is  largely  diffused  in  macroeconomics  and  is  related  to  the  natural 
interpretation  of  the  different  regimes  as  the  different  phases  (e.g.  recession  and  expansion)  of  the 
business cycle. In this work we follow this approach and include the constrains 

𝛾𝑖𝑗1 < 𝛾𝑖𝑗1 < ⋯ < 𝛾𝑖𝑗𝑀 

 Turning point detection 

8 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Bayesian Inference  3 

for 𝑗 = 1, … , 𝐾,  that  corresponds  to  a  total  ordering,  across  the  different  regimes,  of  the  constant  terms  in 
the equations of the system. 

For  the  rows  𝒑𝒊,𝑗 , 𝑗 = 1, … , 𝑀,  of  the  transition  probability  matrix  we  assume  the  independent  Dirichlet 
distributions 

 𝒑𝒊,𝒋 ~ 𝓓(𝒅𝒊,𝟏, … , 𝒅𝒊,𝑴) 

(7) 

with 𝑑𝑖,𝑗 =   𝑑𝑖. 

3.2  Hierarchical Prior 

As  an  alternative  to  the  independent  prior  assumption,  a  hierarchical  priors  could  be  used  as  in  Canova 
and  Ciccarelli  (2006).  This  prior  specification  strategy  allows  to  model  dependence  between  the  cross-
sectional units through common latent variables. We will not consider hierarchical priors in our applications 
and briefly describe here a possible specification for further extensions of our work 

𝜸𝒊𝟎 ~ 𝓝𝑲𝟎(𝑹𝒊𝜸̅𝟎, 𝜮𝒊𝟎),  

𝜸̅𝟎 ~ 𝓝𝑲𝑴𝟎(𝜸̅𝟎, 𝜮𝒊𝟎),  

𝜸𝒊𝒎 ~ 𝓝𝑲𝒎(𝜸̅𝒎, 𝜮𝒊𝒎), 𝒎 = 𝟏, … , 𝑴 

𝜸̅𝒎 ~ 𝓝𝑲𝒎(𝜸̅𝒎, 𝜮𝒊𝒎), 𝒎 = 𝟏, … , 𝑴 

(8) 

(9) 

(10) 

(11) 

𝑖 = 1, … , 𝑁, where  𝑅𝑖 =  𝐼𝑘. We assume independence across units, that is: ℂ𝑜𝑣(𝜸𝑖𝑚, 𝜸𝑗𝑚|𝜸̅𝟎) = 𝑂𝐾𝑚×𝐾𝑚 and 
ℂ𝑜𝑣(𝜸𝑖𝑚, 𝜸𝑗𝑚|𝜸̅𝑚) = 0, for 𝑖 ≠ 𝑗. For the inverse covariance matrix ∑−1

𝑖𝑚  we assume the Wishart priors 

∑−𝟏

𝒊𝒎  ~ 𝓦𝑲(𝒗𝒊𝒎/𝟐, 𝜰̅𝒊𝒎/𝟐), 𝒎 = 𝟏, … , 𝑴 

 𝜰̅𝒎

−𝟏~ 𝓦𝑲(𝒗𝒎/𝟐, 𝜰𝒎/𝟐), 𝒎 = 𝟏, … , 𝑴 

(12) 

(13) 

𝑖 = 1, … , 𝑁 ,  that  allow  us  to  maintain  the  assumption  of  regime-specific  degrees  of  freedom  𝒗𝑖𝒎  and 
precision 𝜰𝒊𝒎 parameters. We assume ℂ𝑜𝑣(∑−1

𝑖𝑚 ⃓ 𝛶̅𝒎

𝑖𝑚 , ∑−1

2 . 

−𝟏) = 𝑂𝐾𝑚

2 ×𝐾𝑚

Modeling  dependence  between  the  chains  is  a  difficult  issues  to  deal  with.  The  hierarchical  prior 
specification  allow  us  to  introduce  dependence  between  the  unit-specific  Markov-chains.  In  a  hierarchical 
prior setting there are many ways to introduce dependence. 

With  the  above  given  specification  of  the  coefficients 𝜸𝑖𝒎 it  is  possible  to  have  dependence  between  the 
different  regimes.  Another  way  to  introduce  dependence  is  through  a  hierarchical  prior  for  the  transition 
matrices.  In  particular  for  the  i-th  unit,  the  rows  𝒑𝒊,𝑗, 𝑗 = 1, … , 𝑀,  of  the  transition  probability  matrix  we 
assume 

 𝒑𝒊,𝒋. ~ 𝓓(𝒅𝒊,𝟏, … , 𝒅𝒊,𝑴) 

(14) 

with  𝒅𝒊,𝟏 = 𝑑 ,  that  are  conditionally  independent  and  symmetric  Dirichlet  distributions.  We  assume 
𝑑 ~ ℬℯ(1/2,1/2). 

 Turning point detection 

9 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Bayesian Inference  3 

3.3  Gibbs sampler 

We  extend  the  Gibbs  sampler  of  Krolzig  (1997)  and  Frühwirth-Schnatter  (2006)  to  our  PMS-VAR  model 
with the informative priors given in the previous sections. Under both the independent and hierarchical prior 
settings  the  full  conditional  posterior  distributions  of  the  equation-specific  blocks  of  parameters  are 
independent.  Thus  the  Gibbs  sampler  can  be  iterated  over  different  blocks  of  parameters  avoiding  the 
computational  difficulties  associated  with  the  inversions  of  large  covariance  matrices.  We  give  the  full 
conditional distributions of the parameters in Eq. 2. We apply a further blocking step. We follow the Markov-
switching  regression  framework  in  Frühwirth-Schnatter  (2006)  and  separate  the  unit-specific  parameters 
into two different blocks: the regime-independent parameters and the regime-specific parameters. 

The likelihood function associated to the PMS-VAR model is 

𝑝(𝑦|Ξ, γ, Σ) = (2𝜋)−

𝑇𝐾𝑁

2 ∏ |Σ𝑡|−
𝑇
𝑡=1

1
2 exp {−

1

2

𝑇
∑ 𝐮𝑡
𝑡=1

′ Σ𝑡

−1𝐮𝑡

}

(15) 

′
, … , 𝒚1𝑇

, … , 𝒚𝑁𝑇

′ ), Ξ = (ξ11, … , ξ𝑁1, … , ξ1𝑇, … , ξ𝑁𝑇) and 

′ ) ⨂ 𝐼𝑁𝐾) 𝑋𝑡𝛾. Under the independence assumption, the likelihood factorises as 

′
where 𝒚′ = (𝒚11
′
𝐮𝑡 = 𝐲𝑡 − ((1, ξ1𝑡

, … , 𝒚𝑁1
, … , ξ𝑁𝑡

′

∏ 𝑝(𝒚𝑖|Ξ𝑖, γ𝑖, Σ𝑖)

𝑁
𝑖=1

= ∏ (2𝜋)−
𝑁
𝑖=1

𝑇𝐾
2

∏ |Σ𝑡|−

𝑇
𝑡=1

1
2 exp {−

1

2

𝑇
∑ 𝐮𝑖𝑡
𝑡=1

′ Σ𝑖𝑡

−1𝐮𝑖𝑡

}

(16) 

where 𝒚𝑖

′ = (𝒚𝑖1

′ , … , 𝒚𝑖𝑇

′ ), Ξ = (ξ𝑖1, … ξ𝑖𝑇), 𝜸𝑖

′ = (𝜸𝑖1

′ , … , 𝜸𝑖𝑀

′ ), 𝐮𝑖𝑡 = 𝐲𝑖𝑡 − ((1, ξ𝑖𝑡

′ )⨂𝐼𝐾)𝑋𝑖𝑡𝛾𝑖𝑡 and 

𝑋𝑖𝑡 = (

𝑋𝑖0𝑡 𝑋𝑖1𝑡
⋱
0

⋮
𝑋𝑖0𝑡

0

) 

𝑋𝑖𝑀𝑡

Let us introduce the auxiliary variables 𝒚𝑖0𝑡 = 𝒚𝑖𝑡 − 𝜉𝑖1𝑡𝑋𝑖1𝑡𝛾𝑖1 + ⋯ + 𝜉𝑖𝑀𝑡𝑋𝑖𝑀𝑡𝛾𝑖𝑀1) and the notation 𝜸𝑖(−𝑚) =
(𝜸𝑖1, … , 𝜸𝑖𝑚−1, 𝜸𝑖𝑚+1 … , 𝜸𝑖𝑀) and Σ𝑖(−𝑚) = (Σ𝑖1, … , Σ𝑖𝑚−1, Σ𝑖𝑚+1 … , Σ𝑖𝑀). 

Then  the  full  conditional  distribution  of  the  regime-independent  parameter 𝛾𝑖0  is  a  normal  with  density 
function 

𝑓(𝜸𝑖0|𝒚𝑖, Ξ𝑖, γ𝑖, Σ𝑖) ∝ 

(17) 

∝ exp {−

∝ exp {−

1

2

1

2

∑ ∑ (𝒚𝒊𝟎𝒕 − 𝜸𝒊𝟎)′Σ𝑖𝑡

𝑇
𝑡=1

𝑁
𝑖=1

−1(𝒚𝒊𝟎𝒕 − 𝜸𝒊𝟎)

−

1

2

(𝜸𝒊𝟎 − 𝜸𝒊𝟎)

−1(𝜸𝒊𝟎 − 𝜸𝒊𝟎)} 
Σ𝑖0

′ (∑ 𝑿𝒊𝟎𝒕
𝑇
𝜸𝑖0
𝑡=1

′ 𝚺𝒕

−𝟏𝑿𝒊𝟎𝒕 + 𝚺𝒕

−𝟏

)𝜸𝒊𝟎 + 𝜸𝒊𝟎 (∑ 𝑿𝒊𝟎𝒕

𝑇
𝑡=1

′ 𝚺𝒊𝒕

−𝟏𝒚𝒊𝟎𝒕 + 𝚺𝒊𝟎

−𝟏𝜸𝒊𝟎

)} 

∝ 𝒩𝐾0(𝜸𝒊𝟎, 𝚺𝒊𝟎) 

 Turning point detection 

10 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
−𝟏
where 𝜸𝒊𝟎 = 𝚺𝒊𝟎

(𝚺𝒊𝟎

−𝟏𝜸𝒊𝟎 + ∑ 𝑿𝒊𝟎𝒕

𝑇
𝑡=1

′ 𝚺𝒕

−𝟏𝑿𝒊𝟎𝒕

−𝟏
) and 𝚺𝒊𝟎

= (𝚺𝒊𝟎

−𝟏𝜸𝒊𝟎 + ∑ 𝑿𝒊𝟎𝒕

𝑇
𝑡=1

′ 𝚺𝒕

−𝟏𝑿𝒊𝟎𝒕

Bayesian Inference  3 

. 

The full conditional distributions of the regime-dependent parameters 𝜸𝒊𝒎, with 𝑚 = 1, … , 𝑀 are normal with 
density function 

𝑓(𝜸𝑖𝑚|𝒚𝑖, Ξ𝑖, γ𝑖0, γ𝑖(−𝑚), Σ) ∝ 

∝ exp {−

∝ exp {−

1

2

1

2

∑

𝑡∈𝒯𝑖𝑚

′ Σ𝑡 𝐮𝑖𝑡 −
𝐮𝑖𝑡

1

2

(𝜸𝒊𝒎 − 𝜸𝒊𝒎)′

−1(𝜸𝒊𝒎 − 𝜸𝒊𝒎)} 
Σ𝑖𝑚

′ (∑

𝜸𝑖1

𝑡∈𝒯𝑖𝑚

𝑿𝒊𝒎𝒕

′ 𝚺𝒕

−𝟏
−𝟏𝑿𝒊𝒎𝒕 + 𝚺𝒊𝒎

)𝜸𝒊 + 𝜸𝒊

′ (∑

𝑡∈𝒯𝑖𝑚

𝑿𝒊𝒎𝒕

′ 𝚺𝒊𝒕

−𝟏𝒚𝒊𝒎𝒕 + 𝚺𝒊𝒎

−𝟏𝜸𝒊𝒎

(18) 

(19) 

)} 

∝ 𝒩𝐾𝑀(𝜸𝒊𝒎, 𝚺𝒊𝒎) 

−𝟏
where  𝜸𝒊𝒎 = 𝚺𝒊𝒎
𝒯𝑖𝑚 = {𝑡 = 1, … , 𝑇|𝜉𝑖𝑚𝑡 = 1} and 𝒚𝒊𝒎𝒕 = 𝒚𝒊𝒕 − 𝑿𝒊𝟎𝒕𝜸𝒊𝟎. 

−𝟏𝜸𝒊𝒎 + ∑

−𝟏𝑿𝒊𝒎𝒕

′ 𝚺𝒊𝒕

(𝚺𝒊𝒎

𝑿𝒊𝒎𝒕

𝑡∈𝒯𝑖𝑚

−𝟏
)  and  𝚺𝒊𝒎

= (𝚺𝒊𝒎

−𝟏 + ∑

𝑡∈𝒯𝑖𝑚

𝑿𝒊𝒎𝒕

′ 𝚺𝒕

−𝟏𝑿𝒊𝒎𝒕

,  where  we  defined 

The  full  conditional  distributions  of  the  regime-dependent  inverse  variance-covariance  matrix Σ𝑖𝑀 ,  with 
𝑚 = 1, … , 𝑀  are Wishart distributions with density 

𝑓(𝜮𝑖𝑚|𝒚𝑖, Ξ𝑖, γ𝑖0, γ𝑖), Σ𝑖(−𝑚)) ∝ 

𝑇
∝ ∏ |Σ𝑖𝑡
𝑡=1

−

1
2| exp {−

1

2

∑

𝑡∈𝒯𝑖𝑚

′ Σ𝑖𝑡
𝐮𝑖𝑡

−1𝐮𝑖𝑡

} |

−1|
Σ𝑖𝑚

𝜈𝑖𝑚+𝐾+1
2

exp {−

1

2

𝑡𝑟(𝛶𝑖𝑚Σ𝑖𝑚)} 

−1|
∝ |Σ𝑖𝑚

𝜈𝑖𝑚+𝑇𝑖𝑚+𝐾+1
2

exp {−

1

2

𝑡𝑟  ((𝛶𝑖𝑚 + ∑

𝑡∈𝒯𝑖𝑚

𝐮𝑖𝑚𝑡𝐮𝑖𝑚𝑡

′

)𝚺𝒊𝒎

−𝟏)} 

∝ 𝒲𝐾(𝜈𝑖𝑚/2, 𝛶𝑖𝑚/2) 

(20) 

(21) 

𝑇
where  𝒯𝑖𝑚 = ∑ 𝕀(𝜉𝑖𝑚𝑡 = 1)
𝑡=1
The full conditional distribution of the 𝑘-th row of the transition matrix is 

,  𝒖𝒊𝒎𝒕 = 𝒚𝒊𝒕 − 𝑿𝒊𝟎𝒕𝜸𝒊𝟎 − 𝑿𝒊𝒎𝒕𝜸𝒊𝒎 ,  𝜈𝑖𝑚 = 𝜈𝑖 + 𝑇  and  𝛶𝑖𝑚 = 𝛶𝑖𝑚 + ∑

𝐮𝑖𝑚𝑡𝐮𝑖𝑚𝑡

′

. 

𝑡∈𝒯𝑖𝑚

𝑓(𝑝𝑖,𝑘.|𝒚𝑖, Ξ𝑖, γ𝑖0, γ𝑖)   ∝ 

𝑑𝑗
𝑀
∏ 𝑝𝑖,𝑘𝑗
𝑗=1

𝑇
∏ ∏
𝑡=1

𝑀
𝑚=1

𝜉𝑖𝑗𝑡𝜉𝑖𝑘𝑡
𝑝𝑖,𝑘𝑗

(22) 

∝ 

𝒟(𝑑1 + 𝑁𝑖,𝑘1, … , 𝑑𝑀 + 𝑁𝑖,𝑘𝑀) 

where 

𝑇

𝑁𝑖,𝑘𝑀 = ∑ 𝕀(𝑠𝑖𝑡 = 𝑗)𝕀(𝑠𝑖,𝑡−1 = 𝑘)

𝑡=1

counts the number of transitions of the 𝑖-th chain from the 𝑘-th to the 𝑗-th state. 

 Turning point detection 

11 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Bayesian Inference  3 

The regime-switching process {𝑠𝑖,𝑡}
backward sampling technique described for example in Frühwirth-Schnatter (2006). 

 is latent and has to be estimated. We apply the forward filtering and 

𝑡=1

𝑇

3.4  Regime Probability Combination 

Let  ∆[0,1]𝑀   be  the  standard  simplex  and  𝜂𝑖,𝑡 ∈ ∆[0,1]𝑀, 𝑖 = 1, … , 𝑁   and  𝑡 = 1, … , 𝑇   be  a  sequence  𝑀 -dim 
vectors of smoothing (or predictive) probabilities for the 𝑀 different regimes of the 𝑁 unit-specific Markov-
chains  used  in  the  PMS-VAR  model.  These  probabilities  reveal  information  on  the  dynamics  of  the 
endogenous variables both at the unit-specific and aggregated levels. We propose a method to summarize 
the information contents of the different units. We combine the smoothing (or predictive) probabilities and 
get a new probability vector sequence, 𝜼𝑡 ∈ ∆[0,1]𝑀, 𝑡 = 1, … , 𝑇. We define a general aggregation scheme as 
a map 𝜙 ∶ ∆[0,1]𝑀

𝑁 →   ∆[0,1]𝑀 

𝜼𝑡 = 𝜙 (𝜼1,𝑡, … , 𝜼𝑁,𝑡) 

(23) 

such that 𝜼𝑡 ∈ ∆[0,1]𝑀, that is 𝜼𝑡  can be interpreted as a probability. 

We consider here two alternative aggregation schemes: 

  Equal weights 

Let 

𝑠̂𝑖,𝑡 = arg 𝑚𝑎𝑥𝑘∈{1,…,𝑀} {𝜼𝑖1,𝑡, … , 𝜼𝑖𝑀,𝑡}

the MAP estimate of the unit-specific regime at time 𝑡. A simple aggregation method is 

𝜼𝑘𝑡 =

1

𝑁

𝑁
∑ 𝛿𝑘
𝑖=1

(𝑠̂𝑖,𝑡) 

(24) 

𝑘 = 1, … , 𝑀, where we assigned equal weights to the unit-specific regime probabilities. When 𝑘 = 1 we get 
a measure of the proportion of countries which are in a ‘strong recession’ regime. 

  Unit-specific weights 

Let 𝑠̂𝑖,𝑡 as above, then we define the second combination scheme 

𝑁
𝜼𝑘𝑡 = ∑ 𝜔𝑖𝑡𝛿𝑘
𝑖=1

(𝑠̂𝑖,𝑡) 

(25) 

 Turning point detection 

12 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Bayesian Inference  3 

where, in order to have a properly defined vector of probability, we assume (𝜔1𝑡, … , 𝜔𝑁𝑡)′ ∈ ∆[0,1]𝑁. The unit-
specific weight 𝜔𝑖𝑡 can be driven, for example, by the relative IPI growth rate or IPI size of the 𝑖-th unit in 
the sample with respect to the other units. 

Figure 1: Top: log-change in percent (top chart) of the EU area Industrial Production Indexes (IPI). 
Middle: term spread (TS), that is the difference between 3-month and 10-year interest rates. All 
variables are at a monthly frequency for the period: January 1960 to December 2010. Black lines: 
average value of the variable across countries. Gray lines: maximum and minimum values across 
countries. Bottom: square of the IPI log-change series. 

 Turning point detection 

13 

 
 
 
 
 
 
 
 
 
 
Business Cycle Analysis  4 

4  Business Cycle Analysis 

4.1  Data Description 

As  dependent  variables  in  our  PMS-VAR  model  we  consider  for 𝑦𝑖1,𝑡 the  Industrial  Production  Index  (IPI) 
and for 𝑦𝑖2,𝑡  the short term (3 months) and long term (10 years) interest rate differentials, for the EU area. 
All  data  are  from  the  Eurostat  and  OECD  databases  and  are  sampled  at  a  monthly  frequency,  from 
January 1960 to December 2010. 

As our aim is to analyse the individual contribution of the EU countries to the fluctuations of the EU area 
business  cycle,  we  do  not  consider  the  variables  at  the  Euro  zone  level,  but  at  a  country  level.  More 
specifically we consider IPI and interest rates for 12 countries: Austria, Belgium, Finland, France, Germany, 
Greece,  Ireland,  Italy,  Luxembourg,  Netherlands,  Portugal  and  Spain.  Data  for  the  EU  countries  are 
seasonally  adjusted  and  working  day  adjusted.  The  data  are available  with  different sample  sizes  for the 
EU countries  (see  Table  1).  The  problem  of  sample  with  different sizes  has  been  handled  in  a  Bayesian 
setting,  through  a  suitable  specification  of  the  prior  distribution  (see  Section  3).  Moreover,  since  Phillips-
Perron  and  Dickey-Fuller  stationarity  tests  point  out  the  non-stationarity  of  the  IPI,  we  considered  in  our 
analysis the log-changes of the IPI index. 

Table 1: Begin date for the series of the Industrial Production Index (IPI) and of the 3-months (3m-
IR) and 10-years (10y-IR) interest rates in 12 countries of the EU. The end date for all of the series is 
December, 2010. 

Country 

Austria 

Belgium 

Finland 

France 

Germany 

Greece 

Ireland 

Italy 

Luxembourg 

Netherlands 

Portugal 

Spain 

Begin dates of the series 

IPI 

1960M01 

1960M01 

1960M01 

1960M01 

1960M01 

1962M01 

1975M07 

1960M01 

1960M01 

1960M01 

1960M01 

1965M01 

3m-IR 

1989M06 

1960M01 

1987M01 

1970M01 

1960M01 

1997M06 

1984M01 

1978M10 

1999M01 

1986M01 

1992M01 

1977M01 

10y-IR 

1990M01 

1960M01 

1988M01 

1960M01 

1960M01 

2001M01 

1970M12 

1991M03 

1993M10 

1960M01 

1993M07 

1980M01 

Another aim of the analysis is to capture the shock transmission mechanism from the financial sector to the 
real one. We consider, as a source of financial shocks, the spread between long and short interest rates. 
For the EU countries, interest rate data are available with different sample sizes (see Table 1). As a source 
of global shocks for the EU area we consider log-changes in the oil West Texas Index (WTI) of spot prices, 
that is available from the Bloomberg database, from January 1961. 

We apply the proposed PMS-VAR model to IPI grow rate and term spread series (upper and mid charts in 
Figure 1). The presence of time-varying volatility and volatility clustering (bottom chart in Figure 1) suggests 
that the model should account for different regimes in the volatility level. 

 Turning point detection 

14 

 
 
 
 
 
 
Business Cycle Analysis  4 

4.2  Parameter Estimates 

The posterior distributions of the PMS-VAR model parameters are approximated through a kernel density 
estimator applied to a sample of 1,000 random draws from the posterior. In order to generate 1,000 i.i.d. 
samples from  the  posterior, we  run  the  Gibbs  sampler, given  in  Section  3, for 110,000  iterations,  discard 
the first 10,000 draws to avoid dependence from the initial condition, and finally apply a thinning procedure 
with  a  factor  of  100  samples,  to  reduce  the  dependence  between  consecutive  Markov-chain  draws.  As 
regards  to  the  number  of  iterations,  we  should  say  that  the  choice  of  the  initial  sample  size  and  the 
convergence detection of the Gibbs sampler remain open issues (see Robert and Casella (1999)). In our 
application  we  choose  the  sample  size  on  the  basis  of  both  a  graphical  inspection  of  the  MCMC 
progressive  averages  and  the  application  of  the  convergence  diagnostic  (CD)  statistics  proposed  in 
Geweke (1992). We let 𝑛 = 110,000 be the MCMC sample size and 𝑛1 = 10,000 and 𝑛2 = 10,000the sizes 
of two non-overlapping sub-samples, respectively. For a parameter 𝜃of interest, we let 

𝜃̂1 =

1

𝑛1

𝑛1
∑ 𝜃(𝑗)
𝑗=1

, 𝜃̂2 =

1

𝑛2

∑

𝑛1
𝑗=𝑛+1−𝑛2

𝜃(𝑗)

, 

be the MCMC sample means and 𝜎̂1

2 their variances estimated with the non-parametric estimator 

2

𝜎̂𝑖
𝑛𝑖

= Γ̂(0) +  

2𝑛𝑖
𝑛𝑖 − 1

∑ 𝐾(𝑗/ℎ𝑖)Γ̂(𝑗)

ℎ𝑖

𝑗=1

Γ̂(𝑗) =

1
𝑛𝑖

∑ (𝜃(𝑘) − 𝜃̂𝑖)(𝜃(𝑘−𝑗)−𝜃̂𝑖)′

𝑛𝑖

𝑘=𝑗+1

where  we  choose 𝐾(𝑥) to  be  the  Parzen  kernel  (see  Kim  and  Nelson  (1999))  and ℎ1 = 100 and ℎ2 = 500 
the bandwidths. Then the following statistics 

𝐶𝐷 =

𝜃̂1−𝜃̂2
2/𝑛1−𝜎̂2

√𝜎̂1

2/𝑛2

(26) 

converges  in  distribution  to  a  standard  normal  (see  Geweke  (1992)),  under  the  null  hypothesis  that  the 
MCMC chain has converged. 

Figures  2  and  3  show  the  approximated  posterior  distributions  of  the  parameters  γim = (ai1,m, ai2,m)′ , 
(σi11,m) and (σi22,m), m = 1, … , M and i = 1, … , N,  that  represent  the  value  of  the  unit-  and  variable-specific 
time-varying  intercepts  and  volatilities  of  the  PMS-VAR  model.  The  posterior  mean  and  the  credibility 
 and ∑ = (𝜎𝑖𝑘𝑗,𝑚)
region of the parameters γim = (ai1,m, ai2,m)

, are given in Table 2-4. 

𝑖𝑚

′

𝑘<𝑗

As  regards  to  the  intercept  posterior  ((see  first  column  of  Figure  2)),  there  are  at  least  two  groups  of 
countries.  The  first  one  is  Belgium,  France  and  Germany,  with  intercept  parameters,  ai1,m,  for  the  IPI 
growth rate, that do not differ to much across the regimes, m = 1, … ,3 (see coloured lines within each chart 

 Turning point detection 

15 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Business Cycle Analysis  4 

in Figure 2). From Table 2 the average intercept values are -0.17, -0.27 and 0.2 for the first, second and 
third regime, respectively. 

The  rage  of  variation  of  the  intercept  parameters, ai1,m ,  of  the  remaining  group  of  countries,  that  are 
Austria,  Finland,  Greece,  Ireland,  Italy,  Luxembourg,  Netherland,  Portugal  and  Spain,  differ  substantially 
across  the  regimes,  in  terms  of  location  and  shape.  The  average  intercept  values  are  -3.635,  -0.57  and 
3.365 in the first, second and third regime, respectively. 

Within the second group, Austria, Portugal and Spain have similar intercept posteriors, in terms of location 
and dispersion, across the first (strong recession) and the second regime (moderate growth or recession). 
The posterior distribution of the unit- and variable-specific  

 Turning point detection 

16 

 
 
 
 
 
 
 
Figure 2: Posterior distribution of the Markov-switching intercepts, γim = (ai1,m, ai2,m)

′

, i = 1, … , N,  

m = 1, … , 𝑀 for IPI growth rate (left column) and TS (right column). 

Business Cycle Analysis  4 

 Turning point detection 

17 

 
 
 
 
 
 
 
 
Figure 3: Posterior distribution of the square root of the diagonal elements, 𝜎𝑖𝑘𝑗,𝑚, k, j = 1, … , 𝐾, with 
k = j,  of  the  Markov-switching  covariance  matrices, ∑ , 𝑖 = 1, … , 𝑁
,  and m = 1, … , 𝑀 ,,  for  IPI  (left 
column) and TS (right column). 

𝑖𝑚

Business Cycle Analysis  4 

 Turning point detection 

18 

 
 
 
 
 
 
 
 
Business Cycle Analysis  4 

𝑖𝑚

′
Table 2: Posterior mean and credible intervals (in parenthesis) for the parameters, γim = (ai1,m, ai2,m)
and ∑ = (𝜎𝑖𝑗𝑘,𝑚)
, m = 1 (first  regime)  and i = 1, … , 𝑁,  which  are  driven  by  the  Markov-switching 
processes. The estimates are obtained with 1,000 draws, that are the result of 110,000 iterations of 
the  Gibbs  sampler,  of  a  burn-in  period  of  10,000  draws  and  a  thinning  procedure  with  a  thinning 
factor of 100 samples. 

𝑗<𝑘

Country 

AU 

BE 

FI 

FR 

GE 

GR 

IR 

IT 

LU 

NE 

PO 

i 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

Regime 1 

-2.0242 

0.8453 

1.526 

1.4309 

0.0283 

(-4.68,-.19) 

(-3.61,5.64) 

(.9,2.51) 

(.87,2.37) 

(-4.13,3.46) 

-0.1438 

0.0127 

5.213 

1.1209 

-0.5786 

(-.39,-.01) 

(-.54,.58) 

(4.52,6.08) 

(.89,1.41) 

(-1.82,.64) 

-3.5669 

0.1015 

3.5118 

1.3203 

-0.331 

(-8.93,-.24) 

(-1.68,2.13) 

(1.41,5.64) 

(.78,2.11) 

(-4.14,3.46) 

-0.2063 

-0.0703 

1.601 

1.0404 

-0.0537 

(-.53,-.01) 

(-.89,.66) 

(1.3,1.97) 

(.74,1.43) 

(-.71,.61) 

-0.1704 

0.046 

2.9743 

1.0345 

-0.253 

(-.41,-.01) 

(-.7,.81) 

(2.27,3.64) 

(.75,1.42) 

(-1.23,.67) 

-4.1517 

1.013 

1.5835 

1.4095 

-0.0286 

(-7.24,-1.24) 

(-1.77,3.79) 

(.92,2.65) 

(.85,2.31) 

(-4.05,4.12) 

-0.2211 

0.0583 

5.4996 

1.008 

-0.0973 

(-.62,-.02) 

(-.64,.78) 

(5.,6.08) 

(.74,1.33) 

(-1.08,.9) 

-3.6763 

0.2934 

1.7706 

1.3322 

-0.1696 

(-6.33,-1.1) 

(-2.07,2.67) 

(1.14,2.66) 

(.83,2.1) 

(-3.14,2.4) 

-9.1082 

0.277 

1.9724 

1.2999 

0.0593 

(-12.43,-5.56) 

(-2.46,2.99) 

(1.1,3.36) 

(.84,2.02) 

(-5.56,5.72) 

-4.6717 

0.6634 

1.5039 

1.463 

0.0532 

(-9.52,-.67) 

(-3.75,5.21) 

(.9,2.51) 

(.87,2.37) 

(-4.45,5.02) 

-4.7046 

0.6718 

1.7238 

1.3152 

-0.2942 

(-8.14,-1.42) 

(-1.89,3.28) 

(1.02,2.98) 

(.83,2.07) 

(-
4.33,3.35) 

SP 

12 

-0.5913 

-0.0935 

2.7753 

1.1279 

-0.2135 

(-1.32,-.05) 

(-.87,.65) 

(2.33,3.28) 

(.82,1.52) 

(-1.2,.8) 

volatilities (see first column of Fig. 3 in the different regimes (different line within the same chart) are quite 
different across regimes. Belgium, Finland, Germany, Ireland and Spain exhibit a high volatility (red lines) 
associated with the first regime (recession) with respect to the volatility of the moderate recession/growth 
(green  line)  and  expansion  regimes.  The  posterior  distribution  of  the  volatilities  of  the  first  and  second 
regime  are  quite  similar,  for  Austria  and  Portugal,  while  for  Belgium  and  Ireland,  the  volatilities  in  the 
second and third regime are similar. For Italy all of the three regimes exhibit similar volatility features. 

4.3  Turning Points 

The PMS-VAR model allows us to study the business cycles fluctuations of each country in the panel, to 
analyse  the  transmission  of  shocks  across  cycles  and  predict  the  turning  points  of  the  country-specific 

 Turning point detection 

19 

 
 
 
 
 
 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
 
cycles. The red lines in Figure 4 present the country-specific cycles in terms of a 3-regime Markov-chain. 
The regimes are: strong recession, s𝑖𝑡 = 1, moderate recession or moderate expansion, s𝑖𝑡 = 2,, and strong 
expansion, s𝑖𝑡 = 3,). The smoothed 

Business Cycle Analysis  4 

𝑖𝑚

′
Table 3: Posterior mean and credible intervals (in parenthesis) for the parameters, γim = (ai1,m, ai2,m)
and  ∑ = (𝜎𝑖𝑗𝑘,𝑚)
,  m = 2  (second  regime)  and  i = 1, … , 𝑁 ,  which  are  driven  by  the  Markov-
switching  processes.  The  estimates  are  obtained  with  1,000  draws,  that  are  the  result  of  110,000 
iterations of the Gibbs sampler, of a burn-in period of 10,000 draws and a thinning procedure with a 
thinning factor of 100 samples. 

𝑗<𝑘

Country 

AU 

BE 

FI 

FR 

GE 

GR 

IR 

IT 

LU 

NE 

PO 

SP 

i 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

Regime 2 

-3.5019 

0.8714 

1.5161 

1.4219 

-0.0547 

(-6.79,-1.04) 

(-3.12,5.81) 

(.92,2.49) 

(.87,2.35) 

(-4.14,3.3) 

-0.2779 

0.0924 

2.1663 

0.5164 

-0.0333 

(-.57,-.06) 

(-.4,.61) 

(1.93,2.43) 

(.39,.71) 

(-.34,.25) 

0.0649 

0.1184 

2.2306 

0.7454 

-0.0026 

(-.6,.77) 

(-.45,.73) 

(1.97,2.54) 

(.55,1.01) 

(-.35,.3) 

-0.2968 

0.0977 

1.2154 

0.578 

-0.0166 

(-.6,-.05) 

(-.24,.46) 

(1.11,1.33) 

(.46,.72) 

(-.15,.11) 

-0.2357 

0.0503 

1.4882 

0.5187 

0.0129 

(-.46,-.05) 

(-.3,.38) 

(1.37,1.62) 

(.4,.65) 

(-.14,.15) 

0.112 

0.0451 

2.2352 

1.2036 

-0.0777 

(-.73,.98) 

(-.74,.85) 

(1.88,2.63) 

(.87,1.57) 

(-.84,.72) 

0.9287 

1.3359 

1.617 

1.4704 

-0.0663 

(-4.19,6.17) 

(-2.99,5.87) 

(.91,2.69) 

(.88,2.41) 

(-5.51,4.91) 

-0.0771 

0.0687 

1.4874 

0.7886 

0.0347 

(-.62,.46) 

(-.48,.59) 

(1.32,1.68) 

(.58,1.03) 

(-.25,.31) 

-0.226 

0.0602 

3.2394 

0.9625 

0.0592 

(-.95,.46) 

(-.61,.71) 

(2.9,3.59) 

(.68,1.27) 

(-.56,.67) 

0.1114 

0.0366 

2.5878 

0.6758 

-0.0144 

(-.3,.52) 

(-.38,.45) 

(2.4,2.78) 

(.5,.89) 

(-.27,.25) 

-2.6537 

0.9577 

1.662 

1.431 

-0.1303 

(-7.89,2.81) 

(-2.8,4.86) 

(.96,2.71) 

(.88,2.33) 

(-4.7,4.56) 

0.1082 

0.0542 

1.5429 

0.6671 

0.0041 

(-.31,.51) 

(-.34,.47) 

(1.36,1.73) 

(.52,.85) 

(-.2,.21) 

probabilities of the three regimes, 𝑝̂𝑖,𝑡 = ℙ(𝑠̃𝑖,𝑡 = 1|𝑦1:𝑇),for i = 1, … , 𝑁 are given in Figures from 6 to 8. We 
observe that the regimes are often highly persistent, excluding few cases at the end of 80’s and beginning 
of 90’s where few recessions were estimated with very short life, see e.g. Finland and Ireland. On average, 
regime  2  is  the  most  probable  as  we  could  anticipate  since  its  definition  can  fit  both  light  recession  and 
expansion  periods.  The  70’s  and  beginning  of  80’s  are  the  most  volatile,  with  several  periods  of  strong 
recessions, but also strong expansion. The great moderation and the great financial crisis in 2008-2009 are 
also evident. The exception is Ireland which is estimated to be in regime 1 from the end of 90’s. The Irish 

 Turning point detection 

20 

 
 
 
 
 
 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
 
Business Cycle Analysis  4 

economy  had  experienced  substantial  changes from  the  90’s  switching from  farms  and  light  industries  to 
services.  Our  model  suggests  that  the  Irish  economy  was  underperforming  conditional  to  the  low  term 
spread. 

In order to have a measure of the contagion of the recession within the EU area we apply the combination 
methods  given  in  Equations  24  and  25.  Both  measures  in  Figure 5  indicate  that  the  great financial  crisis 
was the period with longer and stronger recession period. However, the equal weight averages shows that 
not  all  countries  were  in  the  deepest  point  at  the  same  time,  calling  for  an  analysis  which  allow  for  the 
possibility of leading and lagging countries. 

The weighted average is close to 1 in several periods and it is 1 for several consecutive months during the 
first  and  second  oil  shock  in  the  70’s,  the  end  of  80’s  and  the  great  financial  crisis.  The  latter  index  is, 
however, highly volatile with some very short living false signals. 

 Turning point detection 

21 

 
 
 
 
 
 
 
𝑖𝑚

′
Table 4: Posterior mean and credible intervals (in parenthesis) for the parameters, γim = (ai1,m, ai2,m)
and ∑ = (𝜎𝑖𝑗𝑘,𝑚)
, m = 3 (third regime) and i = 1, … , 𝑁, which are driven by the Markov-switching 
processes. The estimates are obtained with 1,000 draws, that are the result of 110,000 iterations of 
the Gibbs sampler, of a burn-in period of 10,000 draws and a thinning procedure with a thinning 
factor of 100 samples. 

𝑗<𝑘

Business Cycle Analysis  4 

Country 

AU 

BE 

FI 

FR 

GE 

GR 

IR 

IT 

LU 

NE 

PO 

SP 

i 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

Regime 3 

0.3668 

0.0344 

1.8541 

0.7307 

-0.027 

(.04,.84) 

(-.4,.51) 

(1.7,2.02) 

(.52,.97) 

(-.29,.24) 

0.1099 

0.0857 

2.1417 

0.5782 

-0.0143 

(.01,.29) 

(-.25,.42) 

(1.98,2.32) 

(.46,.72) 

(-.18,.15) 

2.0781 

0.0295 

3.2925 

1.4981 

0.273 

(1.03,5.88) 

(-1.37,1.68) 

(1.4,4.8) 

(1.05,2.09) 

(-2.82,3.28) 

0.2657 

0.1885 

1.7246 

0.8334 

-0.04 

(.01,1.01) 

(-.39,.83) 

(1.5,2.) 

(.65,1.08) 

(-.42,.36) 

0.223 

0.1074 

1.9156 

0.6455 

0.04 

(.01,.64) 

(-.35,.57) 

(1.7,2.21) 

(.52,.8) 

(-.17,.26) 

3.7886 

0.8778 

1.6864 

1.5113 

-0.0768 

(1.32,7.34) 

(-2.63,5.09) 

(.95,2.83) 

(.9,2.5) 

(-4.35,4.01) 

7.3548 

1.4607 

1.4935 

1.496 

-0.1523 

(2.64,12.55) 

(-3.31,6.23) 

(.88,2.47) 

(.89,2.45) 

(-4.27,4.65) 

2.9927 

1.2244 

1.5941 

1.5433 

0.1307 

(1.1,6.68) 

(-2.28,5.32) 

(.93,2.61) 

(.91,2.5) 

(-4.36,4.66) 

5.4322 

0.9447 

1.6577 

1.4504 

-0.0798 

(1.77,10.01) 

(-3.63,5.71) 

(.91,2.99) 

(.89,2.39) 

(-5.3,4.91) 

4.3871 

0.9305 

1.5606 

1.4971 

0.2215 

(1.3,8.4) 

(-3.86,6.11) 

(.88,2.71) 

(.89,2.53) 

(-4.19,4.55) 

0.8666 

0.0554 

2.5702 

0.8614 

-0.0181 

(.17,1.37) 

(-.46,.59) 

(2.34,2.82) 

(.64,1.11) 

(-.42,.37) 

3.0198 

0.508 

1.5576 

1.4046 

0.1818 

(1.09,6.61) 

(-2.89,4.2) 

(.94,2.6) 

(.86,2.37) 

(-3.18,3.45) 

 Turning point detection 

22 

 
 
 
 
 
 
 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
 
 
 
 
Conclusions  5 

5  Conclusions 

We  propose  a  new  Bayesian  panel  VAR  model  with  unit-specific  Markov-switching  latent  factors.  We 
discuss  the  choice  of  the  prior  with  particular  attention  to  the  case  that  some  variable  are  missing.  We 
apply  the  resulting  panel  MS-VAR  model  and  the  simulation  based  Bayesian  inference  procedure  to  the 
analysis of the contributions of the EU countries to the fluctuations of the EU business cycle. We extract the 
turning  points  of  the  unit-specific  business  cycle  and  propose  an  aggregation  technique  for  the 
reconstruction of the EU turning points. 

 Turning point detection 

23 

 
 
 
 
 
 
 
 
Figure  4:  Country-specific  endogenous  variables:  industrial  production  growth  rate  (IPI)  and  term 
structure (TS); and Markov-switching (MS) processes 𝑠𝑖,𝑡, i = 1, … , 𝑁 and t = 1, … , 𝑇. 

Conclusions  5 

 Turning point detection 

24 

 
 
 
 
 
 
 
 
 
Figure  5:  Smoothed  probability  (top)  of  being  in  the  recession  regime  (regime  1)  for  the  Markov-
switching  processes  𝑠𝑖,𝑡 ,  i = 1, … , 𝑁  and  t = 1, … , 𝑇 .  Proportion  (middle)  and  weighted  proportion 
(bottom) of countries in a "strong recession" regime. 

Conclusions  5 

 Turning point detection 

25 

 
 
 
 
 
 
Figure 6: First regime (recession) smoothed probabilities for the Markov-Switching processes 𝑠𝑖,𝑡, 
i = 1, … , 𝑁 and t = 1, … , 𝑇. 

Conclusions  5 

 Turning point detection 

26 

 
 
 
 
 
 
 
 
Figure 7: Second regime (moderate expansion) smoothed probabilities for the Markov-Switching 
Processes 𝑠𝑖,𝑡, i = 1, … , 𝑁 and t = 1, … , 𝑇. 

Conclusions  5 

 Turning point detection 

27 

 
 
 
 
 
 
 
 
Figure 8: Third regime (strong expansion) smoothed probabilities for the Markov-switching 
processes 𝑠𝑖,𝑡, i = 1, … , 𝑁 and t = 1, … , 𝑇. 

Conclusions  5 

 Turning point detection 

28 

 
 
 
 
 
 
 
 
References  6 

6  References 

Anas, J., Billio, M., Ferrara, L., and Mazzi, G. L. (2008). A System for Dating and Detecting Turning Points 

in the Euro Area. The Manchester School, 76:549–577. 

Bayoumi, T. and Eichengreen, B. (1993). Shocking Aspects of European Monetary Unification. In Giavazzi, 
F.  and  Torres,  F.,  editors,  The  Transition  to  Economic  and  Monetary  Union  in  Europe.  Cambridge 
University Press. 

Canova, F. and Ciccarelli, M. (2006).  Estimating  Multi-Country VAR Models. ECB working paper No 603. 

European Central Bank, Frankfurt. 

Canova,  F.  and  Marrinan,  J.  (1998).  Sources  and  propagation  of  international  cycles:  common  shocks  or 

transmission? Journal of International Economics, 42:133–167. 

Celeux, G. (1998). Bayesian Inference for Mixture: The Label Switching Problem. Preprint INRIA 

Clements, M. P. and Krolzig, H. M. (1998). A comparison of the forecast performances of Markov-switching 

and threshold autoregressive models of US GNP. Econometrics Journal, 1:C47–C75. 

Dees, S., Di Mauro, F., Pesaran, M., and Smith, L. (2007). Exploring the international linkages of the Euro 

area: a global VAR analysis. Journal of Applied Econometrics, 22:1–38. 

Forni, M., Hallin, M., Lippi, M., and Reichlin, L. (2000). The generalized dynamic factor model: identification 

and estimation. The Review of Economics and Statistics, 82:540–554. 

Forni, M., Hallin, M., Lippi, M., and Reichlin, L. (2001). Coincident and leading indicators for the euro area. 

The Economic Journal, 111:C62–C85. 

Frankel, J. and Rose, A. (1998). The Endogeneity of the Optimum Currency Area Criterion. The Economic 

Journal, 108:1009–1025. 

Frühwirth-Schnatter, S. (2001). Markov Chain Monte Carlo Estimation of Classical and Dynamic Switching 

and Mixture Models. Journal of the American Statistical Association, 96(453):194–209. 

Frühwirth-Schnatter, S. (2006). Mixture and Markov-swithing Models. Springer, New York. 

Geweke, J. (1992). Evaluating the accuracy of sampling-based approaches to the calculation of posterior 
moments.  In  Bernardo,  J.  M.,  Berger,  J.  O.,  Dawid,  A.  P.,  and  Smith,  A.  F.  M.,  editors,  Bayesian 
Statistics 4, pages 169–193. Oxford University Press, Oxford. 

Goldfeld,  S.  M.  and  Quandt,  R.  E.  (1973).  A  Markov  Model  for  Switching  Regression.  Journal  of 

Econometrics, 1:3–16. 

Gregory, A., Head, A., and Raynauld, J. (1997). Measuring world business cycles. International Economic 

Review, 38:677–701. 

Hallin, M. and Liska, R. (2008). Dynamic Factors in the Presence of Block Structure. Economics Working 

Papers ECO2008/22, European University Institute. 

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

business cycle. Econometrica, 57:357–384. 

Hess,  G.  D.  and  Shin,  K.  (1997).  International  and  Intranational  Business  Cycles.  Oxford  Review  of 

Economic Policy, 13:93–109. 

 Turning point detection 

29 

 
 
 
 
 
References  6 

Hess, G. D. and Shin, K. (1998). Intranational business cycles in the United States. Journal of International 

Economics, 44:289–313. 

Imbs,  J.  (2004).  Trade,  finance,  specialization  and  synchronization.  Review  of  Economics  and  Statistics, 

86:723–734. 

Kaufmann, S. (2010). Dating and forecasting turning points by bayesian clustering with dynamic structure: 

A suggestion with an application to austrian data. Journal of Applied Econometrics, 25:309–344. 

Kim,  C.  J.  and  Murray,  C.  J.  (2002).  Permanent  and  Transitory  Components  of  Recessions.  Empirical 

Economics, 27:163–183. 

Kim, C. J. and Nelson, C. R. (1999). Has the  U.S. economy become  more stable?  A Bayesian approach 
based  on  a  Markov-switching  model  of  the  business  cycle.  Review  of  Economics  and  Economic 
Statistics, 81:608–616. 

Kim, C. J. and Piger, J. (2000). Common stochastic trends, common cycles, and asymmetry in economic 
International  Finance  Division,  Federal  Reserve  Board, 

fluctuations.  Working  paper,  n.  681, 
Semptember 2000. 

Kose,  M.,  Otrok,  C.,  andWhiteman,  C.  (2003).  International  business  cycles:  world,  region,  and  country-

specific factors. American Economic Review, 93:1216–1239. 

Kose, M., Otrok, C., and Whiteman, C. (2008). Global business cycles: convergence or decoupling? NBER 

Working paper 14292. 

Krolzig,  H.-M.  (1997).  Markov  Switching  Vector  Autoregressions.  Modelling,  Statistical  Inference  and 

Application to Business Cycle Analysis. Springer, Berlin. 

Krolzig,  H.-M.  (2000).  Predicting  Markov-Switching  Vector  Autoregressive  Processes.  Nuffield  College 

Economics Working Papers, 2000-WP31. 

Lumsdaine,  R.  and  Prasad,  E.  (2003).  Identifying  the  common  component  of  international  economic 

fluctuations: a new approach. Economic Journal, 113:101–127. 

Monfort, A., Renne, J., R., R., and Vitale, G. (2003). Is economic activity in the G7 synchronized? common 
shocks  versus  spillover  effects.  CEPR  Discussion  Paper  No.  4119,  Centre  for  Economic  Policy 
Research, London. 

Pesaran, M., Schuermann, T., and Weiner, S. (2004). Modelling regional interdependencies using a global 
error correcting macroeconometric model. Journal of Business and Economic Statistics, 22:129–162. 

Potter, S. M. (1995). A Nonlinear Approach to U.S. GNP. Journal of Applied Econometrics, 10:109–125. 

Robert, C. P. and Casella, G. (1999). Monte Carlo Statistical Methods. Springer Verlag, New York. 

Stock, J. H. and Watson, M. W. (1991). A probability model of the coincident economic indicators. In Lahiri 

K., M. G., editor, Leading Economic Indicators. Cambridge University Press, New York. 

Tong, H. (1983). Threshold Models in Non-Linear Time-Series Models. Springer-Verlag, New York. 

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Turning point detection  
with bayesian panel  
Markov-Switching VAR
MONICA BILLIO, ROBERTO CASARIN, HERMAN K. VAN DIJK, 
GIAN LUIGI MAZZI, FRANCESCO RAVAZZOLO 

This paper proposes a panel Markov-Switching (MS-) VAR model suitable 
for a multi-country analysis of the business cycle. We study the business 
cycles fluctuations of a group of countries, analyse the transmission 
of shocks across cycles and predict the turning points of the country-
specific cycles. 

For more information
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