Turning point detection with bayesian panel Markov ...€¦ · Turning point detection 4 Abstract...

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

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

Main title

2015 edition

Turning point detection with bayesian panel

Markov-Switching VARMONICA BILLIO ROBERTO CASARIN HERMAN K VAN DIJK

GIAN LUIGI MAZZI FRANCESCO RAVAZZOLO 2016 edition


Markov-Switching VAR MONICA BILLIO ROBERTO CASARIN HERMAN K VAN DIJK

GIAN LUIGI MAZZI FRANCESCO RAVAZZOLO

2016 edition

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More information on the European Union is available on the Internet (httpeuropaeu) Luxembourg Publications Office of the European Union 2016 ISBN 978-92-79-61459-0 ISSN 2315-0807 doi 102785776839 Cat No KS-TC-16-016-EN-N copy European Union 2016 Reproduction is authorised provided the source is acknowledged For more information please consult httpeceuropaeueurostataboutour-partnerscopyright Copyright for the photograph of the cover copyShutterstock For reproduction or use of this photo permission must be sought directly from the copyright holder The information and views set out in this publication are those of the author(s) and do not necessarily reflect the official opinion of the European Union Neither the European Union institutions and bodies nor any person acting on their behalf may be held responsible for the use which may be made of the information contained therein

Table of contents

3 Turning point detection

Abstract 4

1 Introduction 5

2 A Panel Markov-switching VAR model 6

3 Bayesian Inference 8

31 Independent Priors 8

32 Hierarchical Prior 9

33 Gibbs sampler 10

34 Regime Probability Combination 12

4 Business Cycle Analysis 14

41 Data Description 14

42 Parameter Estimates 15

43 Turning Points 19

5 Conclusions 23

6 References 29

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


() Econometrics and Tinbergen Institutes Erasmus University Rotterdam)

() European Commission Eurostat

() Norges Bank

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 rdquointra-nationalrdquo business cycle (ie 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 US 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 ie 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

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 (eg the international common factors) the

strongly idiosyncratic factors which are idiosyncratic for all blocks and the weakly commonweakly

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 119910119894119905 isin ℝ119870 119894 = 1 hellip 119873 and 119905 = 1 hellip 119879 be a sequence of 119870-dimensional vectors of observations 119873 is the

number of units (countries) and 119879 the number of time observations We introduce a general specification of

the panel Markov-switching VAR (PMS-VAR) model



119962119946119957 = 119938119946(119956119946119957) + sum sum 119912119946119947119949119953119949=120783

119925119947=120783 (119956119946119957)119962119947119957minus119949 + 119915119946 (119956119946119957)119963119957 + 120634119946119957 (1)

119894 = 1 hellip 119873 with 휀119894119905~119977119870(120782 sum (119904119894119905)119894 ) and 119911119905 isin ℝ119866 a vector of variables common to all units

The 119904119894119905119905 are unit-specific and independent M-states Markov-chain processes with values in 1 hellip 119872 and

transition probability ℙ(119956119946119957 = 119896|119956119946119957minus1 = 119895) = 119901119894119896119895 119895 isin 1 hellip 119872 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 119912119946119947119949(119956119946119957) ne

0 for 119894 ne 119895 In order to define the parameter shifts more clearly and to simplify the exposition of the

inference procedure we introduce the indicator variable 120585119946119896119905 = 120575119896(119956119946119957) where

120633119948(119956119946119957) = 120783 119946119943 119956119946119957 = 119948120782 119848119853119841119838119851119856119842119852119838

for 119896 = 1 hellip 119872 119894 = 1 hellip 119873 and 119905 = 1 hellip 119879 and the vector of indicators 120643119946119905 = (1206431199461119905 hellip 120643119946119872119905)prime which collects

the information about the realizations of the 119894-th unit-specific Markov chain over the sample period The

indicators allow us to write the parameter shifts as

119886119946(119956119946119957) = sum 119886119872119896=1 119894119896

120643119946119896119905 119860119946119895119897(119956119946119957) = sum 119860119872119896=1 119894119895119897119896

120643119946119896119905

119863119946(119956119946119957) = sum 119863119872119896=1 119894119896

120643119946119896119905 Σ119946(119956119946119957) = sum Σ119872119896=1 119894119896

120643119946119896119905

In our applications we will assume the following restrictions hold 120124(휀119894119905휀119895119905prime ) = 119874119870times119870 with 119874119899times119898 the 119899 times 119898-

dimensional null matrix and there are no interdependencies among the same variable across units that is

119860119946119895119897 = 119860119946119895119897 120575119894(119895) + 119874119870times119870 (1 minus 120575119894(119895))

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 119886119946(119956119946119957) and volatilities Σ119946(119956119946119957) are driven by the regime-switching variables 119904119894119905119905 and

assume constant autoregressive coefficients 119860119946119897119896 = 119860119946119897 forall 119896 In the same spirit we assume that the

coefficients of the common variables do not change over time that is 119863119946119896 = 119863119946 forall 119896

Let 119894119905prime = (1 hellip 119962119894119905minus1

prime ⋯ 119962119894119905minus119901prime 119963119905

prime ) 119905 = 1 hellip 119879 be the sequence of (1 + 119870119901 + 119866)-dimensional column vectors

of regressors for the PMS-VAR model that includes the constant term the lagged dependent variables

and the set of common variables Moreover define the regressors 119882119894119905 = 119894119905prime ⨂119868119896 and coefficients 119860119946119896 =

(119886119946119896 1198601198941119896 ⋯ 119860119894119901119896 119863119894) matrices of dimension (119870(1 + 119870119901 + 119866) times 119870) and (119870 times 119870(1 + 119870119901 + 119866)) respectively

By using the allocation variables 120643119946119905 and the unit independence assumptions given above the PMS-VAR

model can be rewritten as

Bayesian Inference 3


119962119946119957 = 119912119946120783119934119946119957120643119946120783119957 + ⋯ + 119912119946119924119934119946119957120643119946119924119957 + 120634119946119957 120634119946119957~120029119922(120782 120622119946119957) (2)

or in a more compact form as 119910119894119905 = (120643119946119924119957⨂ 119882119894119905)119959119942119940(119861119946) + 휀119894119905 where 119861119946 = (119959119942119940(119860119946120783) 119959119942119940(119860119946120784) hellip

119959119942119940(119860119946119924)) 120622119946119957 = 120622(120643119946119924119957⨂ 119868119870) and 120622119946 = (1206221199461 hellip 120622119946119872) For reason of convenience we consider the partition

of the set of regressors 119894119905prime into 119872 + 1 subsets 1198940119905

prime and 119894119898119905prime 119898 = 1 hellip 119872 that are a 1198700 minusdimensional vector

of regressors with regime-invariant coefficients and 119872 vectors of 119870119898 regime-specific regressors with

regimedependent coefficients Under this assumption the previous model writes as

119962119946119957 = 119935119946120782120783120632119946120782 + 120643119946120783119957119935119946120783120783120632119946120783 + ⋯ + 120643119946119924119957119935119946119924120783120632119946119924 + 120634119946119957 (3)

where 1199351198940119905 = (1198940119905prime ⨂ 119868119870) and 119935119894119898119905 = (119894119898119905

prime ⨂ 119868119870)

3 Bayesian Inference

31 Independent Priors

We assume a conjugate priors for the coefficients and the variance of the panel MS-VAR For the

coefficients 120574119894120782 and 120574119894119898 we consider independent normals priors

120632119946120782 ~120029119922120782

(120632119946120782 120622119946120782) (4)

120632119946119924 ~120029119922119924

(120632119946119950 120622119946119950) 119950 = 120783 hellip 119924 (5)

119894 = 1 hellip 119873 We assume independence across units that is ℂ119900119907(120574119894120782 120574119895120782) = 120782 and ℂ119900119907(120574119894119898 120574119895119898) = 119874119870119898times119870119898

for 119894 ne 119895 For the inverse covariance matrix summinus1119894119898 we assume the Wishart priors

summinus120783119946119950 ~ 120038119922(119959119946119950120784 120624119946119950120784) 119950 = 120783 hellip 119924 (6)

with possibly regime-specific degrees of freedom 119959119894119950 and precision 120624119946119950 parameters We assume

ℂ119900119907(summinus1119894119898 summinus1

119894119898 ) = 1198741198701198982 times119870119898

2

When using Markov-switching processes one should deal with the identification issue associated to the

label switching problem See for example Celeux (1998) and Fruumlhwirth-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

Fruumlhwirth-Schnatter (2006) for a review) One of the most efficient approach is the permutation sampler

(see Fruumlhwirth-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 (eg recession and expansion) of the

business cycle In this work we follow this approach and include the constrains

1205741198941198951 lt 1205741198941198951 lt ⋯ lt 120574119894119895119872



for 119895 = 1 hellip 119870 that corresponds to a total ordering across the different regimes of the constant terms in

the equations of the system

For the rows 119953119946119895 119895 = 1 hellip 119872 of the transition probability matrix we assume the independent Dirichlet

distributions

119953119946119947 ~ 120019(119941119946120783 hellip 119941119946119924) (7)

with 119889119894119895 = 119889119894

32 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

120632119946120782 ~ 120029119922120782

(119929119946120782 120622119946120782) (8)

120782 ~ 120029119922119924120782

(120782 120622119946120782) (9)

120632119946119950 ~ 120029119922119950

(119950 120622119946119950) 119950 = 120783 hellip 119924 (10)

119950 ~ 120029119922119950

(119950 120622119946119950) 119950 = 120783 hellip 119924 (11)

119894 = 1 hellip 119873 where 119877119894 = 119868119896 We assume independence across units that is ℂ119900119907(120632119894119898 120632119895119898|120782) = 119874119870119898times119870119898 and

ℂ119900119907(120632119894119898 120632119895119898|119898) = 0 for 119894 ne 119895 For the inverse covariance matrix summinus1119894119898 we assume the Wishart priors

summinus120783119946119950 ~ 120038119922(119959119946119950120784 119946119950120784) 119950 = 120783 hellip 119924 (12)

119950minus120783~ 120038119922(119959119950120784 120624119950120784) 119950 = 120783 hellip 119924 (13)

119894 = 1 hellip 119873 that allow us to maintain the assumption of regime-specific degrees of freedom 119959119894119950 and

precision 120624119946119950 parameters We assume ℂ119900119907(summinus1119894119898 summinus1

119894119898 119950minus120783) = 119874119870119898

2 times1198701198982

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 120632119894119950 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 119953119946119895 119895 = 1 hellip 119872 of the transition probability matrix we

assume

119953119946119947 ~ 120019(119941119946120783 hellip 119941119946119924) (14)

with 119941119946120783 = 119889 that are conditionally independent and symmetric Dirichlet distributions We assume

119889 ~ ℬℯ(1212)



33 Gibbs sampler

We extend the Gibbs sampler of Krolzig (1997) and Fruumlhwirth-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 Fruumlhwirth-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

119901(119910|Ξ γ Σ) = (2120587)minus119879119870119873

2 prod |Σ119905|minus1

2 exp minus1

2sum 119854119905

prime Σ119905minus1119854119905

119879119905=1 119879

119905=1 (15)

where 119962prime = (11996211prime hellip 1199621198731

prime hellip 1199621119879prime hellip 119962119873119879

prime ) Ξ = (ξ11 hellip ξ1198731 hellip ξ1119879 hellip ξ119873119879) and

119854119905 = 119858119905 minus ((1 ξ1119905prime hellip ξ119873119905

prime ) ⨂ 119868119873119870) 119883119905120574 Under the independence assumption the likelihood factorises as

prod 119901(119962119894|Ξ119894 γ119894 Σ119894)119873119894=1 = prod (2120587)minus

119879119870

2119873119894=1 prod |Σ119905|minus

1

2 exp minus1

2sum 119854119894119905

prime Σ119894119905minus1119854119894119905

119879119905=1 119879

119905=1 (16)

where 119962119894prime = (1199621198941

prime hellip 119962119894119879prime ) Ξ = (ξ1198941 hellip ξ119894119879) 120632119894

prime = (1206321198941prime hellip 120632119894119872

prime ) 119854119894119905 = 119858119894119905 minus ((1 ξ119894119905prime )⨂119868119870)119883119894119905120574119894119905 and

119883119894119905 = (

1198831198940119905 1198831198941119905 0

⋮ ⋱1198831198940119905 0 119883119894119872119905

)

Let us introduce the auxiliary variables 1199621198940119905 = 119962119894119905 minus 120585119894111990511988311989411199051205741198941 + ⋯ + 1205851198941198721199051198831198941198721199051205741198941198721) and the notation 120632119894(minus119898) =

(1206321198941 hellip 120632119894119898minus1 120632119894119898+1 hellip 120632119894119872) and Σ119894(minus119898) = (Σ1198941 hellip Σ119894119898minus1 Σ119894119898+1 hellip Σ119894119872)

Then the full conditional distribution of the regime-independent parameter 1205741198940 is a normal with density

function

119891(1206321198940|119962119894 Ξ119894 γ119894 Σ119894) prop (17)

prop exp minus1

2sum sum (119962119946120782119957 minus 120632119946120782)primeΣ119894119905

minus1(119962119946120782119957 minus 120632119946120782)119879119905=1 minus

1

2(120632119946120782 minus 120632119946120782)119873

119894=1 Σ1198940minus1(120632119946120782 minus 120632119946120782)

prop exp minus1

21206321198940

prime (sum 119935119946120782119957prime 120506119957

minus120783119935119946120782119957 + 120506119957minus120783119879

119905=1 )120632119946120782 + 120632119946120782 (sum 119935119946120782119957prime 120506119946119957

minus120783119962119946120782119957 + 120506119946120782minus120783120632119946120782

119879119905=1 )

prop 1199771198700(120632

119946120782 120506119946120782)



where 120632119946120782

= 120506119946120782

minus120783(120506119946120782

minus120783120632119946120782 + sum 119935119946120782119957prime 120506119957

minus120783119935119946120782119957119879119905=1 ) and 120506119946120782

minus120783= (120506119946120782

minus120783120632119946120782 + sum 119935119946120782119957prime 120506119957

minus120783119935119946120782119957119879119905=1

The full conditional distributions of the regime-dependent parameters 120632119946119950 with 119898 = 1 hellip 119872 are normal with

density function

119891(120632119894119898|119962119894 Ξ119894 γ1198940 γ119894(minus119898) Σ) prop (18)

prop exp minus1

2sum 119854119894119905

prime Σ119905 119854119894119905 minus1

2(120632119946119950 minus 120632119946119950)prime119905isin119983119894119898

Σ119894119898minus1(120632119946119950 minus 120632119946119950)

prop exp minus1

21206321198941

prime (sum 119935119946119950119957prime 120506119957

minus120783119935119946119950119957 + 120506119946119950minus120783

119905isin119983119894119898)120632119946 + 120632119946

prime (sum 119935119946119950119957prime 120506119946119957

minus120783119962119946119950119957 + 120506119946119950minus120783120632119946119950119905isin119983119894119898

)

prop 119977119870119872(120632

119946119950 120506119946119950) (19)

where 120632119946119950

= 120506119946119950

minus120783(120506119946119950

minus120783120632119946119950 + sum 119935119946119950119957prime 120506119946119957

minus120783119935119946119950119957119905isin119983119894119898) and 120506119946119950

minus120783= (120506119946119950

minus120783 + sum 119935119946119950119957prime 120506119957

minus120783119935119946119950119957119905isin119983119894119898 where we defined

119983119894119898 = 119905 = 1 hellip 119879|120585119894119898119905 = 1 and 119962119946119950119957 = 119962119946119957 minus 119935119946120782119957120632119946120782

The full conditional distributions of the regime-dependent inverse variance-covariance matrix Σ119894119872 with

119898 = 1 hellip 119872 are Wishart distributions with density

119891(120622119894119898|119962119894 Ξ119894 γ1198940 γ119894) Σ119894(minus119898)) prop (20)

prop prod |Σ119894119905minus

1

2| exp minus1

2sum 119854119894119905

prime Σ119894119905minus1119854119894119905119905isin119983119894119898

|119879119905=1 Σ119894119898

minus1|120584119894119898+119870+1

2 exp minus1

2119905119903(120566119894119898Σ119894119898)

prop |Σ119894119898minus1|

120584119894119898+119879119894119898+119870+1

2 exp minus1

2119905119903 ((120566119894119898 + sum 119854119894119898119905119854119894119898119905

prime119905isin119983119894119898

)120506119946119950minus120783)

prop 119986119870(1205841198941198982 1205661198941198982) (21)

where 119983119894119898 = sum 120128(120585119894119898119905 = 1)119879119905=1 119958119946119950119957 = 119962119946119957 minus 119935119946120782119957120632119946120782 minus 119935119946119950119957120632119946119950 120584119894119898 = 120584119894 + 119879 and 120566119894119898 = 120566119894119898 + sum 119854119894119898119905119854119894119898119905

prime119905isin119983119894119898

The full conditional distribution of the 119896-th row of the transition matrix is

119891(119901119894119896|119962119894 Ξ119894 γ1198940 γ119894) prop prod 119901119894119896119895

119889119895119872119895=1 prod prod 119901

119894119896119895

120585119894119895119905120585119894119896119905119872119898=1

119879119905=1 (22)

prop 119967(1198891 + 1198731198941198961 hellip 119889119872 + 119873119894119896119872)

where

119873119894119896119872 = sum 120128(119904119894119905 = 119895)120128(119904119894119905minus1 = 119896)119879

119905=1

counts the number of transitions of the 119894-th chain from the 119896-th to the 119895-th state



The regime-switching process 119904119894119905119905=1

119879 is latent and has to be estimated We apply the forward filtering and

backward sampling technique described for example in Fruumlhwirth-Schnatter (2006)

34 Regime Probability Combination

Let ∆[01]119872 be the standard simplex and 120578119894119905 isin ∆[01]119872 119894 = 1 hellip 119873 and 119905 = 1 hellip 119879 be a sequence 119872 -dim

vectors of smoothing (or predictive) probabilities for the 119872 different regimes of the 119873 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 120636119905 isin ∆[01]119872 119905 = 1 hellip 119879 We define a general aggregation scheme as

a map 120601 ∶ ∆[01]119872119873 rarr ∆[01]119872

120636119905 = 120601 (1206361119905 hellip 120636119873119905) (23)

such that 120636119905 isin ∆[01]119872 that is 120636119905 can be interpreted as a probability

We consider here two alternative aggregation schemes

Equal weights

Let

119894119905 = arg 119898119886119909119896isin1hellip119872 1206361198941119905 hellip 120636119894119872119905

the MAP estimate of the unit-specific regime at time 119905 A simple aggregation method is

120636119896119905 =1

119873sum 120575119896

119873119894=1 (119894119905) (24)

119896 = 1 hellip 119872 where we assigned equal weights to the unit-specific regime probabilities When 119896 = 1 we get

a measure of the proportion of countries which are in a lsquostrong recessionrsquo regime

Unit-specific weights

Let 119894119905 as above then we define the second combination scheme

120636119896119905 = sum 120596119894119905120575119896119873119894=1 (119894119905) (25)



where in order to have a properly defined vector of probability we assume (1205961119905 hellip 120596119873119905)prime isin ∆[01]119873 The unit-

specific weight 120596119894119905 can be driven for example by the relative IPI growth rate or IPI size of the 119894-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

Business Cycle Analysis 4


4 Business Cycle Analysis

41 Data Description

As dependent variables in our PMS-VAR model we consider for 1199101198941119905 the Industrial Production Index (IPI)

and for 1199101198942119905 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

Begin dates of the series

Country IPI 3m-IR 10y-IR

Austria 1960M01 1989M06 1990M01

Belgium 1960M01 1960M01 1960M01

Finland 1960M01 1987M01 1988M01

France 1960M01 1970M01 1960M01

Germany 1960M01 1960M01 1960M01

Greece 1962M01 1997M06 2001M01

Ireland 1975M07 1984M01 1970M12

Italy 1960M01 1978M10 1991M03

Luxembourg 1960M01 1999M01 1993M10

Netherlands 1960M01 1986M01 1960M01

Portugal 1960M01 1992M01 1993M07

Spain 1965M01 1977M01 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



42 Parameter Estimates

The posterior distributions of the PMS-VAR model parameters are approximated through a kernel density

estimator applied to a sample of 1000 random draws from the posterior In order to generate 1000 iid

samples from the posterior we run the Gibbs sampler given in Section 3 for 110000 iterations discard

the first 10000 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 119899 = 110000 be the MCMC sample size and 1198991 = 10000 and 1198992 = 10000the sizes

of two non-overlapping sub-samples respectively For a parameter 120579of interest we let

1205791 =1

1198991sum 120579(119895)1198991

119895=1 1205792 =1

1198992sum 120579(119895)1198991

119895=119899+1minus1198992

be the MCMC sample means and 12 their variances estimated with the non-parametric estimator

1198942

119899119894

= Γ(0) + 2119899119894

119899119894 minus 1sum 119870(119895ℎ119894)Γ(119895)

ℎ119894

119895=1

Γ(119895) =1

119899119894

sum (120579(119896) minus 120579119894)(120579(119896minus119895)minus120579119894)prime119899119894

119896=119895+1

where we choose 119870(119909) to be the Parzen kernel (see Kim and Nelson (1999)) and ℎ1 = 100 and ℎ2 = 500

the bandwidths Then the following statistics

119862119863 =1minus2

radic121198991minus2

21198992

(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 = (ai1m ai2m)prime

(σi11m) and (σi22m) m = 1 hellip M and i = 1 hellip 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

region of the parameters γim = (ai1m ai2m)prime and sum = (120590119894119896119895119898)

119896lt119895119894119898 are given in Table 2-4

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 ai1m for the IPI

growth rate that do not differ to much across the regimes m = 1 hellip 3 (see coloured lines within each chart



in Figure 2) From Table 2 the average intercept values are -017 -027 and 02 for the first second and

third regime respectively

The rage of variation of the intercept parameters ai1m 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 -3635 -057 and

3365 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



Figure 2 Posterior distribution of the Markov-switching intercepts γim = (ai1m ai2m)prime i = 1 hellip N

m = 1 hellip 119872 for IPI growth rate (left column) and TS (right column)



Figure 3 Posterior distribution of the square root of the diagonal elements 120590119894119896119895119898 k j = 1 hellip 119870 with

k = j of the Markov-switching covariance matrices sum 119894 = 1 hellip 119873119894119898 and m = 1 hellip 119872 for IPI (left

column) and TS (right column)



Table 2 Posterior mean and credible intervals (in parenthesis) for the parameters γim = (ai1m ai2m)prime

and sum = (120590119894119895119896119898)119895lt119896119894119898 m = 1 (first regime) and i = 1 hellip 119873 which are driven by the Markov-switching

processes The estimates are obtained with 1000 draws that are the result of 110000 iterations of

the Gibbs sampler of a burn-in period of 10000 draws and a thinning procedure with a thinning

factor of 100 samples

Regime 1

Country i

AU 1 -20242 08453 1526 14309 00283

(-468-19) (-361564) (9251) (87237) (-413346)

BE 2 -01438 00127 5213 11209 -05786

(-39-01) (-5458) (452608) (89141) (-18264)

FI 3 -35669 01015 35118 13203 -0331

(-893-24) (-168213) (141564) (78211) (-414346)

FR 4 -02063 -00703 1601 10404 -00537

(-53-01) (-8966) (13197) (74143) (-7161)

GE 5 -01704 0046 29743 10345 -0253

(-41-01) (-781) (227364) (75142) (-12367)

GR 6 -41517 1013 15835 14095 -00286

(-724-124) (-177379) (92265) (85231) (-405412)

IR 7 -02211 00583 54996 1008 -00973

(-62-02) (-6478) (5608) (74133) (-1089)

IT 8 -36763 02934 17706 13322 -01696

(-633-11) (-207267) (114266) (8321) (-31424)

LU 9 -91082 0277 19724 12999 00593

(-1243-556) (-246299) (11336) (84202) (-556572)

NE 10 -46717 06634 15039 1463 00532

(-952-67) (-375521) (9251) (87237) (-445502)

PO 11 -47046 06718 17238 13152 -02942

(-814-142) (-189328) (102298) (83207) (-

433335)

SP 12 -05913 -00935 27753 11279 -02135

(-132-05) (-8765) (233328) (82152) (-128)

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 recessiongrowth

(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

43 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



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 s119894119905 = 1 moderate recession or moderate expansion s119894119905 = 2 and strong

expansion s119894119905 = 3) The smoothed


and sum = (120590119894119895119896119898)119895lt119896119894119898 m = 2 (second regime) and i = 1 hellip 119873 which are driven by the Markov-

switching processes The estimates are obtained with 1000 draws that are the result of 110000

iterations of the Gibbs sampler of a burn-in period of 10000 draws and a thinning procedure with a

thinning factor of 100 samples

Regime 2

Country i

AU 1 -35019 08714 15161 14219 -00547

(-679-104) (-312581) (92249) (87235) (-41433)

BE 2 -02779 00924 21663 05164 -00333

(-57-06) (-461) (193243) (3971) (-3425)

FI 3 00649 01184 22306 07454 -00026

(-677) (-4573) (197254) (55101) (-353)

FR 4 -02968 00977 12154 0578 -00166

(-6-05) (-2446) (111133) (4672) (-1511)

GE 5 -02357 00503 14882 05187 00129

(-46-05) (-338) (137162) (465) (-1415)

GR 6 0112 00451 22352 12036 -00777

(-7398) (-7485) (188263) (87157) (-8472)

IR 7 09287 13359 1617 14704 -00663

(-419617) (-299587) (91269) (88241) (-551491)

IT 8 -00771 00687 14874 07886 00347

(-6246) (-4859) (132168) (58103) (-2531)

LU 9 -0226 00602 32394 09625 00592

(-9546) (-6171) (29359) (68127) (-5667)

NE 10 01114 00366 25878 06758 -00144

(-352) (-3845) (24278) (589) (-2725)

PO 11 -26537 09577 1662 1431 -01303

(-789281) (-28486) (96271) (88233) (-47456)

SP 12 01082 00542 15429 06671 00041

(-3151) (-3447) (136173) (5285) (-221)

probabilities of the three regimes 119894119905 = ℙ(119894119905 = 1|1199101119879)for i = 1 hellip 119873 are given in Figures from 6 to 8 We

observe that the regimes are often highly persistent excluding few cases at the end of 80rsquos and beginning

of 90rsquos where few recessions were estimated with very short life see eg 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 70rsquos and beginning of 80rsquos 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 90rsquos The Irish



economy had experienced substantial changes from the 90rsquos 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 70rsquos the end of 80rsquos and the great financial crisis The latter index is

however highly volatile with some very short living false signals




and sum = (120590119894119895119896119898)119895lt119896119894119898 m = 3 (third regime) and i = 1 hellip 119873 which are driven by the Markov-switching




Regime 3

Country i

AU 1 03668 00344 18541 07307 -0027

(0484) (-451) (17202) (5297) (-2924)

BE 2 01099 00857 21417 05782 -00143

(0129) (-2542) (198232) (4672) (-1815)

FI 3 20781 00295 32925 14981 0273

(103588) (-137168) (1448) (105209) (-282328)

FR 4 02657 01885 17246 08334 -004

(01101) (-3983) (152) (65108) (-4236)

GE 5 0223 01074 19156 06455 004

(0164) (-3557) (17221) (528) (-1726)

GR 6 37886 08778 16864 15113 -00768

(132734) (-263509) (95283) (925) (-435401)

IR 7 73548 14607 14935 1496 -01523

(2641255) (-331623) (88247) (89245) (-427465)

IT 8 29927 12244 15941 15433 01307

(11668) (-228532) (93261) (9125) (-436466)

LU 9 54322 09447 16577 14504 -00798

(1771001) (-363571) (91299) (89239) (-53491)

NE 10 43871 09305 15606 14971 02215

(1384) (-386611) (88271) (89253) (-419455)

PO 11 08666 00554 25702 08614 -00181

(17137) (-4659) (234282) (64111) (-4237)

SP 12 30198 0508 15576 14046 01818

(109661) (-28942) (9426) (86237) (-318345)

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

Conclusions 5


Figure 4 Country-specific endogenous variables industrial production growth rate (IPI) and term

structure (TS) and Markov-switching (MS) processes 119904119894119905 i = 1 hellip 119873 and t = 1 hellip 119879

Conclusions 5


Figure 5 Smoothed probability (top) of being in the recession regime (regime 1) for the Markov-

switching processes 119904119894119905 i = 1 hellip 119873 and t = 1 hellip 119879 Proportion (middle) and weighted proportion

(bottom) of countries in a strong recession regime

Conclusions 5


Figure 6 First regime (recession) smoothed probabilities for the Markov-Switching processes 119904119894119905

i = 1 hellip 119873 and t = 1 hellip 119879

Conclusions 5


Figure 7 Second regime (moderate expansion) smoothed probabilities for the Markov-Switching

Processes 119904119894119905 i = 1 hellip 119873 and t = 1 hellip 119879

Conclusions 5


Figure 8 Third regime (strong expansion) smoothed probabilities for the Markov-switching

processes 119904119894119905 i = 1 hellip 119873 and t = 1 hellip 119879

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 76549ndash577

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 42133ndash167

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 1C47ndashC75

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 221ndash38

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

and estimation The Review of Economics and Statistics 82540ndash554

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

The Economic Journal 111C62ndashC85

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

Journal 1081009ndash1025

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

and Mixture Models Journal of the American Statistical Association 96(453)194ndash209

Fruumlhwirth-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 169ndash193 Oxford University Press Oxford

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

Econometrics 13ndash16

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

Review 38677ndash701

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

Papers ECO200822 European University Institute

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

business cycle Econometrica 57357ndash384

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

Economic Policy 1393ndash109

References 6


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

Economics 44289ndash313

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

86723ndash734

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 25309ndash344

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


Kim C J and Nelson C R (1999) Has the US economy become more stable A Bayesian approach

based on a Markov-switching model of the business cycle Review of Economics and Economic

Statistics 81608ndash616

Kim C J and Piger J (2000) Common stochastic trends common cycles and asymmetry in economic

fluctuations Working paper n 681 International Finance Division Federal Reserve Board

Semptember 2000

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

specific factors American Economic Review 931216ndash1239

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 113101ndash127

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 22129ndash162

Potter S M (1995) A Nonlinear Approach to US GNP Journal of Applied Econometrics 10109ndash125

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

Main title

2015 edition

Turning point detection with bayesian panel Markov-Switching VARMONICA 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 informationhttpeceuropaeueurostat

KS-TC-16-016-EN-N

ISBN 978-92-79-61459-0

Turning point detectionwith bayesian panel Markov-Switching VAR

Table of contents

Abstract

1 Introduction

2 A Panel Markov-switching VAR model



5 Conclusions

6 References


Markov-Switching VAR MONICA BILLIO ROBERTO CASARIN HERMAN K VAN DIJK

GIAN LUIGI MAZZI FRANCESCO RAVAZZOLO

2016 edition


Freephone number ()

00 800 6 7 8 9 10 11



Table of contents


Abstract 4

1 Introduction 5





33 Gibbs sampler 10






5 Conclusions 23

6 References 29

Abstract


Abstract



















() Norges Bank

Introduction 1






















































but not vice versa













































119962119946119957 = 119938119946(119956119946119957) + sum sum 119912119946119947119949119953119949=120783

119925119947=120783 (119956119946119957)119962119947119957minus119949 + 119915119946 (119956119946119957)119963119957 + 120634119946119957 (1)
















120633119948(119956119946119957) = 120783 119946119943 119956119946119957 = 119948120782 119848119853119841119838119851119856119842119852119838




119886119946(119956119946119957) = sum 119886119872119896=1 119894119896

120643119946119896119905 119860119946119895119897(119956119946119957) = sum 119860119872119896=1 119894119895119897119896

120643119946119896119905

119863119946(119956119946119957) = sum 119863119872119896=1 119894119896

120643119946119896119905 Σ119946(119956119946119957) = sum Σ119872119896=1 119894119896

120643119946119896119905



119860119946119895119897 = 119860119946119895119897 120575119894(119895) + 119874119870times119870 (1 minus 120575119894(119895))

















119962119946119957 = 119912119946120783119934119946119957120643119946120783119957 + ⋯ + 119912119946119924119934119946119957120643119946119924119957 + 120634119946119957 120634119946119957~120029119922(120782 120622119946119957) (2)







119962119946119957 = 119935119946120782120783120632119946120782 + 120643119946120783119957119935119946120783120783120632119946120783 + ⋯ + 120643119946119924119957119935119946119924120783120632119946119924 + 120634119946119957 (3)

where 1199351198940119905 = (1198940119905prime ⨂ 119868119870) and 119935119894119898119905 = (119894119898119905

prime ⨂ 119868119870)





120632119946120782 ~120029119922120782

(120632119946120782 120622119946120782) (4)

120632119946119924 ~120029119922119924

(120632119946119950 120622119946119950) 119950 = 120783 hellip 119924 (5)



summinus120783119946119950 ~ 120038119922(119959119946119950120784 120624119946119950120784) 119950 = 120783 hellip 119924 (6)



119894119898 ) = 1198741198701198982 times119870119898

2












1205741198941198951 lt 1205741198941198951 lt ⋯ lt 120574119894119895119872






distributions

119953119946119947 ~ 120019(119941119946120783 hellip 119941119946119924) (7)

with 119889119894119895 = 119889119894






120632119946120782 ~ 120029119922120782

(119929119946120782 120622119946120782) (8)

120782 ~ 120029119922119924120782

(120782 120622119946120782) (9)

120632119946119950 ~ 120029119922119950

(119950 120622119946119950) 119950 = 120783 hellip 119924 (10)

119950 ~ 120029119922119950

(119950 120622119946119950) 119950 = 120783 hellip 119924 (11)



summinus120783119946119950 ~ 120038119922(119959119946119950120784 119946119950120784) 119950 = 120783 hellip 119924 (12)

119950minus120783~ 120038119922(119959119950120784 120624119950120784) 119950 = 120783 hellip 119924 (13)



119894119898 119950minus120783) = 119874119870119898

2 times1198701198982







assume

119953119946119947 ~ 120019(119941119946120783 hellip 119941119946119924) (14)


119889 ~ ℬℯ(1212)



33 Gibbs sampler










119901(119910|Ξ γ Σ) = (2120587)minus119879119870119873


2 exp minus1

2sum 119854119905

prime Σ119905minus1119854119905

119879119905=1 119879

119905=1 (15)







119879119870

2119873119894=1 prod |Σ119905|minus

1

2 exp minus1

2sum 119854119894119905

prime Σ119894119905minus1119854119894119905

119879119905=1 119879

119905=1 (16)

where 119962119894prime = (1199621198941




119883119894119905 = (

1198831198940119905 1198831198941119905 0

⋮ ⋱1198831198940119905 0 119883119894119872119905

)




function

119891(1206321198940|119962119894 Ξ119894 γ119894 Σ119894) prop (17)

prop exp minus1



1

2(120632119946120782 minus 120632119946120782)119873

119894=1 Σ1198940minus1(120632119946120782 minus 120632119946120782)

prop exp minus1

21206321198940

prime (sum 119935119946120782119957prime 120506119957

minus120783119935119946120782119957 + 120506119957minus120783119879

119905=1 )120632119946120782 + 120632119946120782 (sum 119935119946120782119957prime 120506119946119957

minus120783119962119946120782119957 + 120506119946120782minus120783120632119946120782

119879119905=1 )

prop 1199771198700(120632

119946120782 120506119946120782)



where 120632119946120782

= 120506119946120782

minus120783(120506119946120782

minus120783120632119946120782 + sum 119935119946120782119957prime 120506119957

minus120783119935119946120782119957119879119905=1 ) and 120506119946120782

minus120783= (120506119946120782

minus120783120632119946120782 + sum 119935119946120782119957prime 120506119957

minus120783119935119946120782119957119879119905=1


density function

119891(120632119894119898|119962119894 Ξ119894 γ1198940 γ119894(minus119898) Σ) prop (18)

prop exp minus1

2sum 119854119894119905

prime Σ119905 119854119894119905 minus1

2(120632119946119950 minus 120632119946119950)prime119905isin119983119894119898

Σ119894119898minus1(120632119946119950 minus 120632119946119950)

prop exp minus1

21206321198941

prime (sum 119935119946119950119957prime 120506119957

minus120783119935119946119950119957 + 120506119946119950minus120783

119905isin119983119894119898)120632119946 + 120632119946

prime (sum 119935119946119950119957prime 120506119946119957

minus120783119962119946119950119957 + 120506119946119950minus120783120632119946119950119905isin119983119894119898

)

prop 119977119870119872(120632

119946119950 120506119946119950) (19)

where 120632119946119950

= 120506119946119950

minus120783(120506119946119950

minus120783120632119946119950 + sum 119935119946119950119957prime 120506119946119957

minus120783119935119946119950119957119905isin119983119894119898) and 120506119946119950

minus120783= (120506119946119950

minus120783 + sum 119935119946119950119957prime 120506119957


119983119894119898 = 119905 = 1 hellip 119879|120585119894119898119905 = 1 and 119962119946119950119957 = 119962119946119957 minus 119935119946120782119957120632119946120782



119891(120622119894119898|119962119894 Ξ119894 γ1198940 γ119894) Σ119894(minus119898)) prop (20)


1

2| exp minus1

2sum 119854119894119905


|119879119905=1 Σ119894119898

minus1|120584119894119898+119870+1

2 exp minus1

2119905119903(120566119894119898Σ119894119898)

prop |Σ119894119898minus1|

120584119894119898+119879119894119898+119870+1

2 exp minus1

2119905119903 ((120566119894119898 + sum 119854119894119898119905119854119894119898119905

prime119905isin119983119894119898

)120506119946119950minus120783)

prop 119986119870(1205841198941198982 1205661198941198982) (21)


prime119905isin119983119894119898


119891(119901119894119896|119962119894 Ξ119894 γ1198940 γ119894) prop prod 119901119894119896119895

119889119895119872119895=1 prod prod 119901

119894119896119895

120585119894119895119905120585119894119896119905119872119898=1

119879119905=1 (22)

prop 119967(1198891 + 1198731198941198961 hellip 119889119872 + 119873119894119896119872)

where

119873119894119896119872 = sum 120128(119904119894119905 = 119895)120128(119904119894119905minus1 = 119896)119879

119905=1














a map 120601 ∶ ∆[01]119872119873 rarr ∆[01]119872

120636119905 = 120601 (1206361119905 hellip 120636119873119905) (23)



Equal weights

Let



120636119896119905 =1

119873sum 120575119896

119873119894=1 (119894119905) (24)





120636119896119905 = sum 120596119894119905120575119896119873119894=1 (119894119905) (25)














41 Data Description
















December 2010



Austria 1960M01 1989M06 1990M01

Belgium 1960M01 1960M01 1960M01

Finland 1960M01 1987M01 1988M01

France 1960M01 1970M01 1960M01

Germany 1960M01 1960M01 1960M01

Greece 1962M01 1997M06 2001M01

Ireland 1975M07 1984M01 1970M12

Italy 1960M01 1978M10 1991M03

Luxembourg 1960M01 1999M01 1993M10


Portugal 1960M01 1992M01 1993M07

Spain 1965M01 1977M01 1980M01























1205791 =1

1198991sum 120579(119895)1198991

119895=1 1205792 =1

1198992sum 120579(119895)1198991

119895=119899+1minus1198992


1198942

119899119894

= Γ(0) + 2119899119894

119899119894 minus 1sum 119870(119895ℎ119894)Γ(119895)

ℎ119894

119895=1

Γ(119895) =1

119899119894


119896=119895+1



119862119863 =1minus2


21198992

(26)






































Regime 1

Country i

AU 1 -20242 08453 1526 14309 00283

(-468-19) (-361564) (9251) (87237) (-413346)

BE 2 -01438 00127 5213 11209 -05786

(-39-01) (-5458) (452608) (89141) (-18264)

FI 3 -35669 01015 35118 13203 -0331

(-893-24) (-168213) (141564) (78211) (-414346)

FR 4 -02063 -00703 1601 10404 -00537

(-53-01) (-8966) (13197) (74143) (-7161)

GE 5 -01704 0046 29743 10345 -0253

(-41-01) (-781) (227364) (75142) (-12367)

GR 6 -41517 1013 15835 14095 -00286

(-724-124) (-177379) (92265) (85231) (-405412)

IR 7 -02211 00583 54996 1008 -00973

(-62-02) (-6478) (5608) (74133) (-1089)

IT 8 -36763 02934 17706 13322 -01696

(-633-11) (-207267) (114266) (8321) (-31424)

LU 9 -91082 0277 19724 12999 00593

(-1243-556) (-246299) (11336) (84202) (-556572)

NE 10 -46717 06634 15039 1463 00532

(-952-67) (-375521) (9251) (87237) (-445502)

PO 11 -47046 06718 17238 13152 -02942

(-814-142) (-189328) (102298) (83207) (-

433335)

SP 12 -05913 -00935 27753 11279 -02135

(-132-05) (-8765) (233328) (82152) (-128)







43 Turning Points













Regime 2

Country i

AU 1 -35019 08714 15161 14219 -00547

(-679-104) (-312581) (92249) (87235) (-41433)

BE 2 -02779 00924 21663 05164 -00333

(-57-06) (-461) (193243) (3971) (-3425)

FI 3 00649 01184 22306 07454 -00026

(-677) (-4573) (197254) (55101) (-353)

FR 4 -02968 00977 12154 0578 -00166

(-6-05) (-2446) (111133) (4672) (-1511)

GE 5 -02357 00503 14882 05187 00129

(-46-05) (-338) (137162) (465) (-1415)

GR 6 0112 00451 22352 12036 -00777

(-7398) (-7485) (188263) (87157) (-8472)

IR 7 09287 13359 1617 14704 -00663

(-419617) (-299587) (91269) (88241) (-551491)

IT 8 -00771 00687 14874 07886 00347

(-6246) (-4859) (132168) (58103) (-2531)

LU 9 -0226 00602 32394 09625 00592

(-9546) (-6171) (29359) (68127) (-5667)

NE 10 01114 00366 25878 06758 -00144

(-352) (-3845) (24278) (589) (-2725)

PO 11 -26537 09577 1662 1431 -01303

(-789281) (-28486) (96271) (88233) (-47456)

SP 12 01082 00542 15429 06671 00041

(-3151) (-3447) (136173) (5285) (-221)












spread
















Regime 3

Country i

AU 1 03668 00344 18541 07307 -0027

(0484) (-451) (17202) (5297) (-2924)

BE 2 01099 00857 21417 05782 -00143

(0129) (-2542) (198232) (4672) (-1815)

FI 3 20781 00295 32925 14981 0273

(103588) (-137168) (1448) (105209) (-282328)

FR 4 02657 01885 17246 08334 -004

(01101) (-3983) (152) (65108) (-4236)

GE 5 0223 01074 19156 06455 004

(0164) (-3557) (17221) (528) (-1726)

GR 6 37886 08778 16864 15113 -00768

(132734) (-263509) (95283) (925) (-435401)

IR 7 73548 14607 14935 1496 -01523

(2641255) (-331623) (88247) (89245) (-427465)

IT 8 29927 12244 15941 15433 01307

(11668) (-228532) (93261) (9125) (-436466)

LU 9 54322 09447 16577 14504 -00798

(1771001) (-363571) (91299) (89239) (-53491)

NE 10 43871 09305 15606 14971 02215

(1384) (-386611) (88271) (89253) (-419455)

PO 11 08666 00554 25702 08614 -00181

(17137) (-4659) (234282) (64111) (-4237)

SP 12 30198 0508 15576 14046 01818

(109661) (-28942) (9426) (86237) (-318345)

Conclusions 5








Conclusions 5




Conclusions 5





Conclusions 5




Conclusions 5




Conclusions 5




References 6






University Press
































References 6





86723ndash734










Semptember 2000




Working paper 14292









Research London








Main title

2015 edition




KS-TC-16-016-EN-N

ISBN 978-92-79-61459-0


Table of contents

Abstract

1 Introduction




5 Conclusions

6 References


Freephone number ()

00 800 6 7 8 9 10 11



Table of contents


Abstract 4

1 Introduction 5





33 Gibbs sampler 10






5 Conclusions 23

6 References 29

Abstract


Abstract



















() Norges Bank

Introduction 1






















































but not vice versa













































119962119946119957 = 119938119946(119956119946119957) + sum sum 119912119946119947119949119953119949=120783

119925119947=120783 (119956119946119957)119962119947119957minus119949 + 119915119946 (119956119946119957)119963119957 + 120634119946119957 (1)
















120633119948(119956119946119957) = 120783 119946119943 119956119946119957 = 119948120782 119848119853119841119838119851119856119842119852119838




119886119946(119956119946119957) = sum 119886119872119896=1 119894119896

120643119946119896119905 119860119946119895119897(119956119946119957) = sum 119860119872119896=1 119894119895119897119896

120643119946119896119905

119863119946(119956119946119957) = sum 119863119872119896=1 119894119896

120643119946119896119905 Σ119946(119956119946119957) = sum Σ119872119896=1 119894119896

120643119946119896119905



119860119946119895119897 = 119860119946119895119897 120575119894(119895) + 119874119870times119870 (1 minus 120575119894(119895))

















119962119946119957 = 119912119946120783119934119946119957120643119946120783119957 + ⋯ + 119912119946119924119934119946119957120643119946119924119957 + 120634119946119957 120634119946119957~120029119922(120782 120622119946119957) (2)







119962119946119957 = 119935119946120782120783120632119946120782 + 120643119946120783119957119935119946120783120783120632119946120783 + ⋯ + 120643119946119924119957119935119946119924120783120632119946119924 + 120634119946119957 (3)

where 1199351198940119905 = (1198940119905prime ⨂ 119868119870) and 119935119894119898119905 = (119894119898119905

prime ⨂ 119868119870)





120632119946120782 ~120029119922120782

(120632119946120782 120622119946120782) (4)

120632119946119924 ~120029119922119924

(120632119946119950 120622119946119950) 119950 = 120783 hellip 119924 (5)



summinus120783119946119950 ~ 120038119922(119959119946119950120784 120624119946119950120784) 119950 = 120783 hellip 119924 (6)



119894119898 ) = 1198741198701198982 times119870119898

2












1205741198941198951 lt 1205741198941198951 lt ⋯ lt 120574119894119895119872






distributions

119953119946119947 ~ 120019(119941119946120783 hellip 119941119946119924) (7)

with 119889119894119895 = 119889119894






120632119946120782 ~ 120029119922120782

(119929119946120782 120622119946120782) (8)

120782 ~ 120029119922119924120782

(120782 120622119946120782) (9)

120632119946119950 ~ 120029119922119950

(119950 120622119946119950) 119950 = 120783 hellip 119924 (10)

119950 ~ 120029119922119950

(119950 120622119946119950) 119950 = 120783 hellip 119924 (11)



summinus120783119946119950 ~ 120038119922(119959119946119950120784 119946119950120784) 119950 = 120783 hellip 119924 (12)

119950minus120783~ 120038119922(119959119950120784 120624119950120784) 119950 = 120783 hellip 119924 (13)



119894119898 119950minus120783) = 119874119870119898

2 times1198701198982







assume

119953119946119947 ~ 120019(119941119946120783 hellip 119941119946119924) (14)


119889 ~ ℬℯ(1212)



33 Gibbs sampler










119901(119910|Ξ γ Σ) = (2120587)minus119879119870119873


2 exp minus1

2sum 119854119905

prime Σ119905minus1119854119905

119879119905=1 119879

119905=1 (15)







119879119870

2119873119894=1 prod |Σ119905|minus

1

2 exp minus1

2sum 119854119894119905

prime Σ119894119905minus1119854119894119905

119879119905=1 119879

119905=1 (16)

where 119962119894prime = (1199621198941




119883119894119905 = (

1198831198940119905 1198831198941119905 0

⋮ ⋱1198831198940119905 0 119883119894119872119905

)




function

119891(1206321198940|119962119894 Ξ119894 γ119894 Σ119894) prop (17)

prop exp minus1



1

2(120632119946120782 minus 120632119946120782)119873

119894=1 Σ1198940minus1(120632119946120782 minus 120632119946120782)

prop exp minus1

21206321198940

prime (sum 119935119946120782119957prime 120506119957

minus120783119935119946120782119957 + 120506119957minus120783119879

119905=1 )120632119946120782 + 120632119946120782 (sum 119935119946120782119957prime 120506119946119957

minus120783119962119946120782119957 + 120506119946120782minus120783120632119946120782

119879119905=1 )

prop 1199771198700(120632

119946120782 120506119946120782)



where 120632119946120782

= 120506119946120782

minus120783(120506119946120782

minus120783120632119946120782 + sum 119935119946120782119957prime 120506119957

minus120783119935119946120782119957119879119905=1 ) and 120506119946120782

minus120783= (120506119946120782

minus120783120632119946120782 + sum 119935119946120782119957prime 120506119957

minus120783119935119946120782119957119879119905=1


density function

119891(120632119894119898|119962119894 Ξ119894 γ1198940 γ119894(minus119898) Σ) prop (18)

prop exp minus1

2sum 119854119894119905

prime Σ119905 119854119894119905 minus1

2(120632119946119950 minus 120632119946119950)prime119905isin119983119894119898

Σ119894119898minus1(120632119946119950 minus 120632119946119950)

prop exp minus1

21206321198941

prime (sum 119935119946119950119957prime 120506119957

minus120783119935119946119950119957 + 120506119946119950minus120783

119905isin119983119894119898)120632119946 + 120632119946

prime (sum 119935119946119950119957prime 120506119946119957

minus120783119962119946119950119957 + 120506119946119950minus120783120632119946119950119905isin119983119894119898

)

prop 119977119870119872(120632

119946119950 120506119946119950) (19)

where 120632119946119950

= 120506119946119950

minus120783(120506119946119950

minus120783120632119946119950 + sum 119935119946119950119957prime 120506119946119957

minus120783119935119946119950119957119905isin119983119894119898) and 120506119946119950

minus120783= (120506119946119950

minus120783 + sum 119935119946119950119957prime 120506119957


119983119894119898 = 119905 = 1 hellip 119879|120585119894119898119905 = 1 and 119962119946119950119957 = 119962119946119957 minus 119935119946120782119957120632119946120782



119891(120622119894119898|119962119894 Ξ119894 γ1198940 γ119894) Σ119894(minus119898)) prop (20)


1

2| exp minus1

2sum 119854119894119905


|119879119905=1 Σ119894119898

minus1|120584119894119898+119870+1

2 exp minus1

2119905119903(120566119894119898Σ119894119898)

prop |Σ119894119898minus1|

120584119894119898+119879119894119898+119870+1

2 exp minus1

2119905119903 ((120566119894119898 + sum 119854119894119898119905119854119894119898119905

prime119905isin119983119894119898

)120506119946119950minus120783)

prop 119986119870(1205841198941198982 1205661198941198982) (21)


prime119905isin119983119894119898


119891(119901119894119896|119962119894 Ξ119894 γ1198940 γ119894) prop prod 119901119894119896119895

119889119895119872119895=1 prod prod 119901

119894119896119895

120585119894119895119905120585119894119896119905119872119898=1

119879119905=1 (22)

prop 119967(1198891 + 1198731198941198961 hellip 119889119872 + 119873119894119896119872)

where

119873119894119896119872 = sum 120128(119904119894119905 = 119895)120128(119904119894119905minus1 = 119896)119879

119905=1














a map 120601 ∶ ∆[01]119872119873 rarr ∆[01]119872

120636119905 = 120601 (1206361119905 hellip 120636119873119905) (23)



Equal weights

Let



120636119896119905 =1

119873sum 120575119896

119873119894=1 (119894119905) (24)





120636119896119905 = sum 120596119894119905120575119896119873119894=1 (119894119905) (25)














41 Data Description
















December 2010



Austria 1960M01 1989M06 1990M01

Belgium 1960M01 1960M01 1960M01

Finland 1960M01 1987M01 1988M01

France 1960M01 1970M01 1960M01

Germany 1960M01 1960M01 1960M01

Greece 1962M01 1997M06 2001M01

Ireland 1975M07 1984M01 1970M12

Italy 1960M01 1978M10 1991M03

Luxembourg 1960M01 1999M01 1993M10


Portugal 1960M01 1992M01 1993M07

Spain 1965M01 1977M01 1980M01























1205791 =1

1198991sum 120579(119895)1198991

119895=1 1205792 =1

1198992sum 120579(119895)1198991

119895=119899+1minus1198992


1198942

119899119894

= Γ(0) + 2119899119894

119899119894 minus 1sum 119870(119895ℎ119894)Γ(119895)

ℎ119894

119895=1

Γ(119895) =1

119899119894


119896=119895+1



119862119863 =1minus2


21198992

(26)






































Regime 1

Country i

AU 1 -20242 08453 1526 14309 00283

(-468-19) (-361564) (9251) (87237) (-413346)

BE 2 -01438 00127 5213 11209 -05786

(-39-01) (-5458) (452608) (89141) (-18264)

FI 3 -35669 01015 35118 13203 -0331

(-893-24) (-168213) (141564) (78211) (-414346)

FR 4 -02063 -00703 1601 10404 -00537

(-53-01) (-8966) (13197) (74143) (-7161)

GE 5 -01704 0046 29743 10345 -0253

(-41-01) (-781) (227364) (75142) (-12367)

GR 6 -41517 1013 15835 14095 -00286

(-724-124) (-177379) (92265) (85231) (-405412)

IR 7 -02211 00583 54996 1008 -00973

(-62-02) (-6478) (5608) (74133) (-1089)

IT 8 -36763 02934 17706 13322 -01696

(-633-11) (-207267) (114266) (8321) (-31424)

LU 9 -91082 0277 19724 12999 00593

(-1243-556) (-246299) (11336) (84202) (-556572)

NE 10 -46717 06634 15039 1463 00532

(-952-67) (-375521) (9251) (87237) (-445502)

PO 11 -47046 06718 17238 13152 -02942

(-814-142) (-189328) (102298) (83207) (-

433335)

SP 12 -05913 -00935 27753 11279 -02135

(-132-05) (-8765) (233328) (82152) (-128)







43 Turning Points













Regime 2

Country i

AU 1 -35019 08714 15161 14219 -00547

(-679-104) (-312581) (92249) (87235) (-41433)

BE 2 -02779 00924 21663 05164 -00333

(-57-06) (-461) (193243) (3971) (-3425)

FI 3 00649 01184 22306 07454 -00026

(-677) (-4573) (197254) (55101) (-353)

FR 4 -02968 00977 12154 0578 -00166

(-6-05) (-2446) (111133) (4672) (-1511)

GE 5 -02357 00503 14882 05187 00129

(-46-05) (-338) (137162) (465) (-1415)

GR 6 0112 00451 22352 12036 -00777

(-7398) (-7485) (188263) (87157) (-8472)

IR 7 09287 13359 1617 14704 -00663

(-419617) (-299587) (91269) (88241) (-551491)

IT 8 -00771 00687 14874 07886 00347

(-6246) (-4859) (132168) (58103) (-2531)

LU 9 -0226 00602 32394 09625 00592

(-9546) (-6171) (29359) (68127) (-5667)

NE 10 01114 00366 25878 06758 -00144

(-352) (-3845) (24278) (589) (-2725)

PO 11 -26537 09577 1662 1431 -01303

(-789281) (-28486) (96271) (88233) (-47456)

SP 12 01082 00542 15429 06671 00041

(-3151) (-3447) (136173) (5285) (-221)












spread
















Regime 3

Country i

AU 1 03668 00344 18541 07307 -0027

(0484) (-451) (17202) (5297) (-2924)

BE 2 01099 00857 21417 05782 -00143

(0129) (-2542) (198232) (4672) (-1815)

FI 3 20781 00295 32925 14981 0273

(103588) (-137168) (1448) (105209) (-282328)

FR 4 02657 01885 17246 08334 -004

(01101) (-3983) (152) (65108) (-4236)

GE 5 0223 01074 19156 06455 004

(0164) (-3557) (17221) (528) (-1726)

GR 6 37886 08778 16864 15113 -00768

(132734) (-263509) (95283) (925) (-435401)

IR 7 73548 14607 14935 1496 -01523

(2641255) (-331623) (88247) (89245) (-427465)

IT 8 29927 12244 15941 15433 01307

(11668) (-228532) (93261) (9125) (-436466)

LU 9 54322 09447 16577 14504 -00798

(1771001) (-363571) (91299) (89239) (-53491)

NE 10 43871 09305 15606 14971 02215

(1384) (-386611) (88271) (89253) (-419455)

PO 11 08666 00554 25702 08614 -00181

(17137) (-4659) (234282) (64111) (-4237)

SP 12 30198 0508 15576 14046 01818

(109661) (-28942) (9426) (86237) (-318345)

Conclusions 5








Conclusions 5




Conclusions 5





Conclusions 5




Conclusions 5




Conclusions 5




References 6






University Press
































References 6





86723ndash734










Semptember 2000




Working paper 14292









Research London








Main title

2015 edition




KS-TC-16-016-EN-N

ISBN 978-92-79-61459-0


Table of contents

Abstract

1 Introduction




5 Conclusions

6 References

Table of contents


Abstract 4

1 Introduction 5





33 Gibbs sampler 10






5 Conclusions 23

6 References 29

Abstract


Abstract



















() Norges Bank

Introduction 1






















































but not vice versa













































119962119946119957 = 119938119946(119956119946119957) + sum sum 119912119946119947119949119953119949=120783

119925119947=120783 (119956119946119957)119962119947119957minus119949 + 119915119946 (119956119946119957)119963119957 + 120634119946119957 (1)
















120633119948(119956119946119957) = 120783 119946119943 119956119946119957 = 119948120782 119848119853119841119838119851119856119842119852119838




119886119946(119956119946119957) = sum 119886119872119896=1 119894119896

120643119946119896119905 119860119946119895119897(119956119946119957) = sum 119860119872119896=1 119894119895119897119896

120643119946119896119905

119863119946(119956119946119957) = sum 119863119872119896=1 119894119896

120643119946119896119905 Σ119946(119956119946119957) = sum Σ119872119896=1 119894119896

120643119946119896119905



119860119946119895119897 = 119860119946119895119897 120575119894(119895) + 119874119870times119870 (1 minus 120575119894(119895))

















119962119946119957 = 119912119946120783119934119946119957120643119946120783119957 + ⋯ + 119912119946119924119934119946119957120643119946119924119957 + 120634119946119957 120634119946119957~120029119922(120782 120622119946119957) (2)







119962119946119957 = 119935119946120782120783120632119946120782 + 120643119946120783119957119935119946120783120783120632119946120783 + ⋯ + 120643119946119924119957119935119946119924120783120632119946119924 + 120634119946119957 (3)

where 1199351198940119905 = (1198940119905prime ⨂ 119868119870) and 119935119894119898119905 = (119894119898119905

prime ⨂ 119868119870)





120632119946120782 ~120029119922120782

(120632119946120782 120622119946120782) (4)

120632119946119924 ~120029119922119924

(120632119946119950 120622119946119950) 119950 = 120783 hellip 119924 (5)



summinus120783119946119950 ~ 120038119922(119959119946119950120784 120624119946119950120784) 119950 = 120783 hellip 119924 (6)



119894119898 ) = 1198741198701198982 times119870119898

2












1205741198941198951 lt 1205741198941198951 lt ⋯ lt 120574119894119895119872






distributions

119953119946119947 ~ 120019(119941119946120783 hellip 119941119946119924) (7)

with 119889119894119895 = 119889119894






120632119946120782 ~ 120029119922120782

(119929119946120782 120622119946120782) (8)

120782 ~ 120029119922119924120782

(120782 120622119946120782) (9)

120632119946119950 ~ 120029119922119950

(119950 120622119946119950) 119950 = 120783 hellip 119924 (10)

119950 ~ 120029119922119950

(119950 120622119946119950) 119950 = 120783 hellip 119924 (11)



summinus120783119946119950 ~ 120038119922(119959119946119950120784 119946119950120784) 119950 = 120783 hellip 119924 (12)

119950minus120783~ 120038119922(119959119950120784 120624119950120784) 119950 = 120783 hellip 119924 (13)



119894119898 119950minus120783) = 119874119870119898

2 times1198701198982







assume

119953119946119947 ~ 120019(119941119946120783 hellip 119941119946119924) (14)


119889 ~ ℬℯ(1212)



33 Gibbs sampler










119901(119910|Ξ γ Σ) = (2120587)minus119879119870119873


2 exp minus1

2sum 119854119905

prime Σ119905minus1119854119905

119879119905=1 119879

119905=1 (15)







119879119870

2119873119894=1 prod |Σ119905|minus

1

2 exp minus1

2sum 119854119894119905

prime Σ119894119905minus1119854119894119905

119879119905=1 119879

119905=1 (16)

where 119962119894prime = (1199621198941




119883119894119905 = (

1198831198940119905 1198831198941119905 0

⋮ ⋱1198831198940119905 0 119883119894119872119905

)




function

119891(1206321198940|119962119894 Ξ119894 γ119894 Σ119894) prop (17)

prop exp minus1



1

2(120632119946120782 minus 120632119946120782)119873

119894=1 Σ1198940minus1(120632119946120782 minus 120632119946120782)

prop exp minus1

21206321198940

prime (sum 119935119946120782119957prime 120506119957

minus120783119935119946120782119957 + 120506119957minus120783119879

119905=1 )120632119946120782 + 120632119946120782 (sum 119935119946120782119957prime 120506119946119957

minus120783119962119946120782119957 + 120506119946120782minus120783120632119946120782

119879119905=1 )

prop 1199771198700(120632

119946120782 120506119946120782)



where 120632119946120782

= 120506119946120782

minus120783(120506119946120782

minus120783120632119946120782 + sum 119935119946120782119957prime 120506119957

minus120783119935119946120782119957119879119905=1 ) and 120506119946120782

minus120783= (120506119946120782

minus120783120632119946120782 + sum 119935119946120782119957prime 120506119957

minus120783119935119946120782119957119879119905=1


density function

119891(120632119894119898|119962119894 Ξ119894 γ1198940 γ119894(minus119898) Σ) prop (18)

prop exp minus1

2sum 119854119894119905

prime Σ119905 119854119894119905 minus1

2(120632119946119950 minus 120632119946119950)prime119905isin119983119894119898

Σ119894119898minus1(120632119946119950 minus 120632119946119950)

prop exp minus1

21206321198941

prime (sum 119935119946119950119957prime 120506119957

minus120783119935119946119950119957 + 120506119946119950minus120783

119905isin119983119894119898)120632119946 + 120632119946

prime (sum 119935119946119950119957prime 120506119946119957

minus120783119962119946119950119957 + 120506119946119950minus120783120632119946119950119905isin119983119894119898

)

prop 119977119870119872(120632

119946119950 120506119946119950) (19)

where 120632119946119950

= 120506119946119950

minus120783(120506119946119950

minus120783120632119946119950 + sum 119935119946119950119957prime 120506119946119957

minus120783119935119946119950119957119905isin119983119894119898) and 120506119946119950

minus120783= (120506119946119950

minus120783 + sum 119935119946119950119957prime 120506119957


119983119894119898 = 119905 = 1 hellip 119879|120585119894119898119905 = 1 and 119962119946119950119957 = 119962119946119957 minus 119935119946120782119957120632119946120782



119891(120622119894119898|119962119894 Ξ119894 γ1198940 γ119894) Σ119894(minus119898)) prop (20)


1

2| exp minus1

2sum 119854119894119905


|119879119905=1 Σ119894119898

minus1|120584119894119898+119870+1

2 exp minus1

2119905119903(120566119894119898Σ119894119898)

prop |Σ119894119898minus1|

120584119894119898+119879119894119898+119870+1

2 exp minus1

2119905119903 ((120566119894119898 + sum 119854119894119898119905119854119894119898119905

prime119905isin119983119894119898

)120506119946119950minus120783)

prop 119986119870(1205841198941198982 1205661198941198982) (21)


prime119905isin119983119894119898


119891(119901119894119896|119962119894 Ξ119894 γ1198940 γ119894) prop prod 119901119894119896119895

119889119895119872119895=1 prod prod 119901

119894119896119895

120585119894119895119905120585119894119896119905119872119898=1

119879119905=1 (22)

prop 119967(1198891 + 1198731198941198961 hellip 119889119872 + 119873119894119896119872)

where

119873119894119896119872 = sum 120128(119904119894119905 = 119895)120128(119904119894119905minus1 = 119896)119879

119905=1














a map 120601 ∶ ∆[01]119872119873 rarr ∆[01]119872

120636119905 = 120601 (1206361119905 hellip 120636119873119905) (23)



Equal weights

Let



120636119896119905 =1

119873sum 120575119896

119873119894=1 (119894119905) (24)





120636119896119905 = sum 120596119894119905120575119896119873119894=1 (119894119905) (25)














41 Data Description
















December 2010



Austria 1960M01 1989M06 1990M01

Belgium 1960M01 1960M01 1960M01

Finland 1960M01 1987M01 1988M01

France 1960M01 1970M01 1960M01

Germany 1960M01 1960M01 1960M01

Greece 1962M01 1997M06 2001M01

Ireland 1975M07 1984M01 1970M12

Italy 1960M01 1978M10 1991M03

Luxembourg 1960M01 1999M01 1993M10


Portugal 1960M01 1992M01 1993M07

Spain 1965M01 1977M01 1980M01























1205791 =1

1198991sum 120579(119895)1198991

119895=1 1205792 =1

1198992sum 120579(119895)1198991

119895=119899+1minus1198992


1198942

119899119894

= Γ(0) + 2119899119894

119899119894 minus 1sum 119870(119895ℎ119894)Γ(119895)

ℎ119894

119895=1

Γ(119895) =1

119899119894


119896=119895+1



119862119863 =1minus2


21198992

(26)






































Regime 1

Country i

AU 1 -20242 08453 1526 14309 00283

(-468-19) (-361564) (9251) (87237) (-413346)

BE 2 -01438 00127 5213 11209 -05786

(-39-01) (-5458) (452608) (89141) (-18264)

FI 3 -35669 01015 35118 13203 -0331

(-893-24) (-168213) (141564) (78211) (-414346)

FR 4 -02063 -00703 1601 10404 -00537

(-53-01) (-8966) (13197) (74143) (-7161)

GE 5 -01704 0046 29743 10345 -0253

(-41-01) (-781) (227364) (75142) (-12367)

GR 6 -41517 1013 15835 14095 -00286

(-724-124) (-177379) (92265) (85231) (-405412)

IR 7 -02211 00583 54996 1008 -00973

(-62-02) (-6478) (5608) (74133) (-1089)

IT 8 -36763 02934 17706 13322 -01696

(-633-11) (-207267) (114266) (8321) (-31424)

LU 9 -91082 0277 19724 12999 00593

(-1243-556) (-246299) (11336) (84202) (-556572)

NE 10 -46717 06634 15039 1463 00532

(-952-67) (-375521) (9251) (87237) (-445502)

PO 11 -47046 06718 17238 13152 -02942

(-814-142) (-189328) (102298) (83207) (-

433335)

SP 12 -05913 -00935 27753 11279 -02135

(-132-05) (-8765) (233328) (82152) (-128)







43 Turning Points













Regime 2

Country i

AU 1 -35019 08714 15161 14219 -00547

(-679-104) (-312581) (92249) (87235) (-41433)

BE 2 -02779 00924 21663 05164 -00333

(-57-06) (-461) (193243) (3971) (-3425)

FI 3 00649 01184 22306 07454 -00026

(-677) (-4573) (197254) (55101) (-353)

FR 4 -02968 00977 12154 0578 -00166

(-6-05) (-2446) (111133) (4672) (-1511)

GE 5 -02357 00503 14882 05187 00129

(-46-05) (-338) (137162) (465) (-1415)

GR 6 0112 00451 22352 12036 -00777

(-7398) (-7485) (188263) (87157) (-8472)

IR 7 09287 13359 1617 14704 -00663

(-419617) (-299587) (91269) (88241) (-551491)

IT 8 -00771 00687 14874 07886 00347

(-6246) (-4859) (132168) (58103) (-2531)

LU 9 -0226 00602 32394 09625 00592

(-9546) (-6171) (29359) (68127) (-5667)

NE 10 01114 00366 25878 06758 -00144

(-352) (-3845) (24278) (589) (-2725)

PO 11 -26537 09577 1662 1431 -01303

(-789281) (-28486) (96271) (88233) (-47456)

SP 12 01082 00542 15429 06671 00041

(-3151) (-3447) (136173) (5285) (-221)












spread
















Regime 3

Country i

AU 1 03668 00344 18541 07307 -0027

(0484) (-451) (17202) (5297) (-2924)

BE 2 01099 00857 21417 05782 -00143

(0129) (-2542) (198232) (4672) (-1815)

FI 3 20781 00295 32925 14981 0273

(103588) (-137168) (1448) (105209) (-282328)

FR 4 02657 01885 17246 08334 -004

(01101) (-3983) (152) (65108) (-4236)

GE 5 0223 01074 19156 06455 004

(0164) (-3557) (17221) (528) (-1726)

GR 6 37886 08778 16864 15113 -00768

(132734) (-263509) (95283) (925) (-435401)

IR 7 73548 14607 14935 1496 -01523

(2641255) (-331623) (88247) (89245) (-427465)

IT 8 29927 12244 15941 15433 01307

(11668) (-228532) (93261) (9125) (-436466)

LU 9 54322 09447 16577 14504 -00798

(1771001) (-363571) (91299) (89239) (-53491)

NE 10 43871 09305 15606 14971 02215

(1384) (-386611) (88271) (89253) (-419455)

PO 11 08666 00554 25702 08614 -00181

(17137) (-4659) (234282) (64111) (-4237)

SP 12 30198 0508 15576 14046 01818

(109661) (-28942) (9426) (86237) (-318345)

Conclusions 5








Conclusions 5




Conclusions 5





Conclusions 5




Conclusions 5




Conclusions 5




References 6






University Press
































References 6





86723ndash734










Semptember 2000




Working paper 14292









Research London








Main title

2015 edition




KS-TC-16-016-EN-N

ISBN 978-92-79-61459-0


Table of contents

Abstract

1 Introduction




5 Conclusions

6 References

Abstract


Abstract



















() Norges Bank

Introduction 1






















































but not vice versa













































119962119946119957 = 119938119946(119956119946119957) + sum sum 119912119946119947119949119953119949=120783

119925119947=120783 (119956119946119957)119962119947119957minus119949 + 119915119946 (119956119946119957)119963119957 + 120634119946119957 (1)
















120633119948(119956119946119957) = 120783 119946119943 119956119946119957 = 119948120782 119848119853119841119838119851119856119842119852119838




119886119946(119956119946119957) = sum 119886119872119896=1 119894119896

120643119946119896119905 119860119946119895119897(119956119946119957) = sum 119860119872119896=1 119894119895119897119896

120643119946119896119905

119863119946(119956119946119957) = sum 119863119872119896=1 119894119896

120643119946119896119905 Σ119946(119956119946119957) = sum Σ119872119896=1 119894119896

120643119946119896119905



119860119946119895119897 = 119860119946119895119897 120575119894(119895) + 119874119870times119870 (1 minus 120575119894(119895))

















119962119946119957 = 119912119946120783119934119946119957120643119946120783119957 + ⋯ + 119912119946119924119934119946119957120643119946119924119957 + 120634119946119957 120634119946119957~120029119922(120782 120622119946119957) (2)







119962119946119957 = 119935119946120782120783120632119946120782 + 120643119946120783119957119935119946120783120783120632119946120783 + ⋯ + 120643119946119924119957119935119946119924120783120632119946119924 + 120634119946119957 (3)

where 1199351198940119905 = (1198940119905prime ⨂ 119868119870) and 119935119894119898119905 = (119894119898119905

prime ⨂ 119868119870)





120632119946120782 ~120029119922120782

(120632119946120782 120622119946120782) (4)

120632119946119924 ~120029119922119924

(120632119946119950 120622119946119950) 119950 = 120783 hellip 119924 (5)



summinus120783119946119950 ~ 120038119922(119959119946119950120784 120624119946119950120784) 119950 = 120783 hellip 119924 (6)



119894119898 ) = 1198741198701198982 times119870119898

2












1205741198941198951 lt 1205741198941198951 lt ⋯ lt 120574119894119895119872






distributions

119953119946119947 ~ 120019(119941119946120783 hellip 119941119946119924) (7)

with 119889119894119895 = 119889119894






120632119946120782 ~ 120029119922120782

(119929119946120782 120622119946120782) (8)

120782 ~ 120029119922119924120782

(120782 120622119946120782) (9)

120632119946119950 ~ 120029119922119950

(119950 120622119946119950) 119950 = 120783 hellip 119924 (10)

119950 ~ 120029119922119950

(119950 120622119946119950) 119950 = 120783 hellip 119924 (11)



summinus120783119946119950 ~ 120038119922(119959119946119950120784 119946119950120784) 119950 = 120783 hellip 119924 (12)

119950minus120783~ 120038119922(119959119950120784 120624119950120784) 119950 = 120783 hellip 119924 (13)



119894119898 119950minus120783) = 119874119870119898

2 times1198701198982







assume

119953119946119947 ~ 120019(119941119946120783 hellip 119941119946119924) (14)


119889 ~ ℬℯ(1212)



33 Gibbs sampler










119901(119910|Ξ γ Σ) = (2120587)minus119879119870119873


2 exp minus1

2sum 119854119905

prime Σ119905minus1119854119905

119879119905=1 119879

119905=1 (15)







119879119870

2119873119894=1 prod |Σ119905|minus

1

2 exp minus1

2sum 119854119894119905

prime Σ119894119905minus1119854119894119905

119879119905=1 119879

119905=1 (16)

where 119962119894prime = (1199621198941




119883119894119905 = (

1198831198940119905 1198831198941119905 0

⋮ ⋱1198831198940119905 0 119883119894119872119905

)




function

119891(1206321198940|119962119894 Ξ119894 γ119894 Σ119894) prop (17)

prop exp minus1



1

2(120632119946120782 minus 120632119946120782)119873

119894=1 Σ1198940minus1(120632119946120782 minus 120632119946120782)

prop exp minus1

21206321198940

prime (sum 119935119946120782119957prime 120506119957

minus120783119935119946120782119957 + 120506119957minus120783119879

119905=1 )120632119946120782 + 120632119946120782 (sum 119935119946120782119957prime 120506119946119957

minus120783119962119946120782119957 + 120506119946120782minus120783120632119946120782

119879119905=1 )

prop 1199771198700(120632

119946120782 120506119946120782)



where 120632119946120782

= 120506119946120782

minus120783(120506119946120782

minus120783120632119946120782 + sum 119935119946120782119957prime 120506119957

minus120783119935119946120782119957119879119905=1 ) and 120506119946120782

minus120783= (120506119946120782

minus120783120632119946120782 + sum 119935119946120782119957prime 120506119957

minus120783119935119946120782119957119879119905=1


density function

119891(120632119894119898|119962119894 Ξ119894 γ1198940 γ119894(minus119898) Σ) prop (18)

prop exp minus1

2sum 119854119894119905

prime Σ119905 119854119894119905 minus1

2(120632119946119950 minus 120632119946119950)prime119905isin119983119894119898

Σ119894119898minus1(120632119946119950 minus 120632119946119950)

prop exp minus1

21206321198941

prime (sum 119935119946119950119957prime 120506119957

minus120783119935119946119950119957 + 120506119946119950minus120783

119905isin119983119894119898)120632119946 + 120632119946

prime (sum 119935119946119950119957prime 120506119946119957

minus120783119962119946119950119957 + 120506119946119950minus120783120632119946119950119905isin119983119894119898

)

prop 119977119870119872(120632

119946119950 120506119946119950) (19)

where 120632119946119950

= 120506119946119950

minus120783(120506119946119950

minus120783120632119946119950 + sum 119935119946119950119957prime 120506119946119957

minus120783119935119946119950119957119905isin119983119894119898) and 120506119946119950

minus120783= (120506119946119950

minus120783 + sum 119935119946119950119957prime 120506119957


119983119894119898 = 119905 = 1 hellip 119879|120585119894119898119905 = 1 and 119962119946119950119957 = 119962119946119957 minus 119935119946120782119957120632119946120782



119891(120622119894119898|119962119894 Ξ119894 γ1198940 γ119894) Σ119894(minus119898)) prop (20)


1

2| exp minus1

2sum 119854119894119905


|119879119905=1 Σ119894119898

minus1|120584119894119898+119870+1

2 exp minus1

2119905119903(120566119894119898Σ119894119898)

prop |Σ119894119898minus1|

120584119894119898+119879119894119898+119870+1

2 exp minus1

2119905119903 ((120566119894119898 + sum 119854119894119898119905119854119894119898119905

prime119905isin119983119894119898

)120506119946119950minus120783)

prop 119986119870(1205841198941198982 1205661198941198982) (21)


prime119905isin119983119894119898


119891(119901119894119896|119962119894 Ξ119894 γ1198940 γ119894) prop prod 119901119894119896119895

119889119895119872119895=1 prod prod 119901

119894119896119895

120585119894119895119905120585119894119896119905119872119898=1

119879119905=1 (22)

prop 119967(1198891 + 1198731198941198961 hellip 119889119872 + 119873119894119896119872)

where

119873119894119896119872 = sum 120128(119904119894119905 = 119895)120128(119904119894119905minus1 = 119896)119879

119905=1














a map 120601 ∶ ∆[01]119872119873 rarr ∆[01]119872

120636119905 = 120601 (1206361119905 hellip 120636119873119905) (23)



Equal weights

Let



120636119896119905 =1

119873sum 120575119896

119873119894=1 (119894119905) (24)





120636119896119905 = sum 120596119894119905120575119896119873119894=1 (119894119905) (25)














41 Data Description
















December 2010



Austria 1960M01 1989M06 1990M01

Belgium 1960M01 1960M01 1960M01

Finland 1960M01 1987M01 1988M01

France 1960M01 1970M01 1960M01

Germany 1960M01 1960M01 1960M01

Greece 1962M01 1997M06 2001M01

Ireland 1975M07 1984M01 1970M12

Italy 1960M01 1978M10 1991M03

Luxembourg 1960M01 1999M01 1993M10


Portugal 1960M01 1992M01 1993M07

Spain 1965M01 1977M01 1980M01























1205791 =1

1198991sum 120579(119895)1198991

119895=1 1205792 =1

1198992sum 120579(119895)1198991

119895=119899+1minus1198992


1198942

119899119894

= Γ(0) + 2119899119894

119899119894 minus 1sum 119870(119895ℎ119894)Γ(119895)

ℎ119894

119895=1

Γ(119895) =1

119899119894


119896=119895+1



119862119863 =1minus2


21198992

(26)






































Regime 1

Country i

AU 1 -20242 08453 1526 14309 00283

(-468-19) (-361564) (9251) (87237) (-413346)

BE 2 -01438 00127 5213 11209 -05786

(-39-01) (-5458) (452608) (89141) (-18264)

FI 3 -35669 01015 35118 13203 -0331

(-893-24) (-168213) (141564) (78211) (-414346)

FR 4 -02063 -00703 1601 10404 -00537

(-53-01) (-8966) (13197) (74143) (-7161)

GE 5 -01704 0046 29743 10345 -0253

(-41-01) (-781) (227364) (75142) (-12367)

GR 6 -41517 1013 15835 14095 -00286

(-724-124) (-177379) (92265) (85231) (-405412)

IR 7 -02211 00583 54996 1008 -00973

(-62-02) (-6478) (5608) (74133) (-1089)

IT 8 -36763 02934 17706 13322 -01696

(-633-11) (-207267) (114266) (8321) (-31424)

LU 9 -91082 0277 19724 12999 00593

(-1243-556) (-246299) (11336) (84202) (-556572)

NE 10 -46717 06634 15039 1463 00532

(-952-67) (-375521) (9251) (87237) (-445502)

PO 11 -47046 06718 17238 13152 -02942

(-814-142) (-189328) (102298) (83207) (-

433335)

SP 12 -05913 -00935 27753 11279 -02135

(-132-05) (-8765) (233328) (82152) (-128)







43 Turning Points













Regime 2

Country i

AU 1 -35019 08714 15161 14219 -00547

(-679-104) (-312581) (92249) (87235) (-41433)

BE 2 -02779 00924 21663 05164 -00333

(-57-06) (-461) (193243) (3971) (-3425)

FI 3 00649 01184 22306 07454 -00026

(-677) (-4573) (197254) (55101) (-353)

FR 4 -02968 00977 12154 0578 -00166

(-6-05) (-2446) (111133) (4672) (-1511)

GE 5 -02357 00503 14882 05187 00129

(-46-05) (-338) (137162) (465) (-1415)

GR 6 0112 00451 22352 12036 -00777

(-7398) (-7485) (188263) (87157) (-8472)

IR 7 09287 13359 1617 14704 -00663

(-419617) (-299587) (91269) (88241) (-551491)

IT 8 -00771 00687 14874 07886 00347

(-6246) (-4859) (132168) (58103) (-2531)

LU 9 -0226 00602 32394 09625 00592

(-9546) (-6171) (29359) (68127) (-5667)

NE 10 01114 00366 25878 06758 -00144

(-352) (-3845) (24278) (589) (-2725)

PO 11 -26537 09577 1662 1431 -01303

(-789281) (-28486) (96271) (88233) (-47456)

SP 12 01082 00542 15429 06671 00041

(-3151) (-3447) (136173) (5285) (-221)












spread
















Regime 3

Country i

AU 1 03668 00344 18541 07307 -0027

(0484) (-451) (17202) (5297) (-2924)

BE 2 01099 00857 21417 05782 -00143

(0129) (-2542) (198232) (4672) (-1815)

FI 3 20781 00295 32925 14981 0273

(103588) (-137168) (1448) (105209) (-282328)

FR 4 02657 01885 17246 08334 -004

(01101) (-3983) (152) (65108) (-4236)

GE 5 0223 01074 19156 06455 004

(0164) (-3557) (17221) (528) (-1726)

GR 6 37886 08778 16864 15113 -00768

(132734) (-263509) (95283) (925) (-435401)

IR 7 73548 14607 14935 1496 -01523

(2641255) (-331623) (88247) (89245) (-427465)

IT 8 29927 12244 15941 15433 01307

(11668) (-228532) (93261) (9125) (-436466)

LU 9 54322 09447 16577 14504 -00798

(1771001) (-363571) (91299) (89239) (-53491)

NE 10 43871 09305 15606 14971 02215

(1384) (-386611) (88271) (89253) (-419455)

PO 11 08666 00554 25702 08614 -00181

(17137) (-4659) (234282) (64111) (-4237)

SP 12 30198 0508 15576 14046 01818

(109661) (-28942) (9426) (86237) (-318345)

Conclusions 5








Conclusions 5




Conclusions 5





Conclusions 5




Conclusions 5




Conclusions 5




References 6






University Press
































References 6





86723ndash734










Semptember 2000




Working paper 14292









Research London








Main title

2015 edition




KS-TC-16-016-EN-N

ISBN 978-92-79-61459-0


Table of contents

Abstract

1 Introduction




5 Conclusions

6 References

Introduction 1






















































but not vice versa













































119962119946119957 = 119938119946(119956119946119957) + sum sum 119912119946119947119949119953119949=120783

119925119947=120783 (119956119946119957)119962119947119957minus119949 + 119915119946 (119956119946119957)119963119957 + 120634119946119957 (1)
















120633119948(119956119946119957) = 120783 119946119943 119956119946119957 = 119948120782 119848119853119841119838119851119856119842119852119838




119886119946(119956119946119957) = sum 119886119872119896=1 119894119896

120643119946119896119905 119860119946119895119897(119956119946119957) = sum 119860119872119896=1 119894119895119897119896

120643119946119896119905

119863119946(119956119946119957) = sum 119863119872119896=1 119894119896

120643119946119896119905 Σ119946(119956119946119957) = sum Σ119872119896=1 119894119896

120643119946119896119905



119860119946119895119897 = 119860119946119895119897 120575119894(119895) + 119874119870times119870 (1 minus 120575119894(119895))

















119962119946119957 = 119912119946120783119934119946119957120643119946120783119957 + ⋯ + 119912119946119924119934119946119957120643119946119924119957 + 120634119946119957 120634119946119957~120029119922(120782 120622119946119957) (2)







119962119946119957 = 119935119946120782120783120632119946120782 + 120643119946120783119957119935119946120783120783120632119946120783 + ⋯ + 120643119946119924119957119935119946119924120783120632119946119924 + 120634119946119957 (3)

where 1199351198940119905 = (1198940119905prime ⨂ 119868119870) and 119935119894119898119905 = (119894119898119905

prime ⨂ 119868119870)





120632119946120782 ~120029119922120782

(120632119946120782 120622119946120782) (4)

120632119946119924 ~120029119922119924

(120632119946119950 120622119946119950) 119950 = 120783 hellip 119924 (5)



summinus120783119946119950 ~ 120038119922(119959119946119950120784 120624119946119950120784) 119950 = 120783 hellip 119924 (6)



119894119898 ) = 1198741198701198982 times119870119898

2












1205741198941198951 lt 1205741198941198951 lt ⋯ lt 120574119894119895119872






distributions

119953119946119947 ~ 120019(119941119946120783 hellip 119941119946119924) (7)

with 119889119894119895 = 119889119894






120632119946120782 ~ 120029119922120782

(119929119946120782 120622119946120782) (8)

120782 ~ 120029119922119924120782

(120782 120622119946120782) (9)

120632119946119950 ~ 120029119922119950

(119950 120622119946119950) 119950 = 120783 hellip 119924 (10)

119950 ~ 120029119922119950

(119950 120622119946119950) 119950 = 120783 hellip 119924 (11)



summinus120783119946119950 ~ 120038119922(119959119946119950120784 119946119950120784) 119950 = 120783 hellip 119924 (12)

119950minus120783~ 120038119922(119959119950120784 120624119950120784) 119950 = 120783 hellip 119924 (13)



119894119898 119950minus120783) = 119874119870119898

2 times1198701198982







assume

119953119946119947 ~ 120019(119941119946120783 hellip 119941119946119924) (14)


119889 ~ ℬℯ(1212)



33 Gibbs sampler










119901(119910|Ξ γ Σ) = (2120587)minus119879119870119873


2 exp minus1

2sum 119854119905

prime Σ119905minus1119854119905

119879119905=1 119879

119905=1 (15)







119879119870

2119873119894=1 prod |Σ119905|minus

1

2 exp minus1

2sum 119854119894119905

prime Σ119894119905minus1119854119894119905

119879119905=1 119879

119905=1 (16)

where 119962119894prime = (1199621198941




119883119894119905 = (

1198831198940119905 1198831198941119905 0

⋮ ⋱1198831198940119905 0 119883119894119872119905

)




function

119891(1206321198940|119962119894 Ξ119894 γ119894 Σ119894) prop (17)

prop exp minus1



1

2(120632119946120782 minus 120632119946120782)119873

119894=1 Σ1198940minus1(120632119946120782 minus 120632119946120782)

prop exp minus1

21206321198940

prime (sum 119935119946120782119957prime 120506119957

minus120783119935119946120782119957 + 120506119957minus120783119879

119905=1 )120632119946120782 + 120632119946120782 (sum 119935119946120782119957prime 120506119946119957

minus120783119962119946120782119957 + 120506119946120782minus120783120632119946120782

119879119905=1 )

prop 1199771198700(120632

119946120782 120506119946120782)



where 120632119946120782

= 120506119946120782

minus120783(120506119946120782

minus120783120632119946120782 + sum 119935119946120782119957prime 120506119957

minus120783119935119946120782119957119879119905=1 ) and 120506119946120782

minus120783= (120506119946120782

minus120783120632119946120782 + sum 119935119946120782119957prime 120506119957

minus120783119935119946120782119957119879119905=1


density function

119891(120632119894119898|119962119894 Ξ119894 γ1198940 γ119894(minus119898) Σ) prop (18)

prop exp minus1

2sum 119854119894119905

prime Σ119905 119854119894119905 minus1

2(120632119946119950 minus 120632119946119950)prime119905isin119983119894119898

Σ119894119898minus1(120632119946119950 minus 120632119946119950)

prop exp minus1

21206321198941

prime (sum 119935119946119950119957prime 120506119957

minus120783119935119946119950119957 + 120506119946119950minus120783

119905isin119983119894119898)120632119946 + 120632119946

prime (sum 119935119946119950119957prime 120506119946119957

minus120783119962119946119950119957 + 120506119946119950minus120783120632119946119950119905isin119983119894119898

)

prop 119977119870119872(120632

119946119950 120506119946119950) (19)

where 120632119946119950

= 120506119946119950

minus120783(120506119946119950

minus120783120632119946119950 + sum 119935119946119950119957prime 120506119946119957

minus120783119935119946119950119957119905isin119983119894119898) and 120506119946119950

minus120783= (120506119946119950

minus120783 + sum 119935119946119950119957prime 120506119957


119983119894119898 = 119905 = 1 hellip 119879|120585119894119898119905 = 1 and 119962119946119950119957 = 119962119946119957 minus 119935119946120782119957120632119946120782



119891(120622119894119898|119962119894 Ξ119894 γ1198940 γ119894) Σ119894(minus119898)) prop (20)


1

2| exp minus1

2sum 119854119894119905


|119879119905=1 Σ119894119898

minus1|120584119894119898+119870+1

2 exp minus1

2119905119903(120566119894119898Σ119894119898)

prop |Σ119894119898minus1|

120584119894119898+119879119894119898+119870+1

2 exp minus1

2119905119903 ((120566119894119898 + sum 119854119894119898119905119854119894119898119905

prime119905isin119983119894119898

)120506119946119950minus120783)

prop 119986119870(1205841198941198982 1205661198941198982) (21)


prime119905isin119983119894119898


119891(119901119894119896|119962119894 Ξ119894 γ1198940 γ119894) prop prod 119901119894119896119895

119889119895119872119895=1 prod prod 119901

119894119896119895

120585119894119895119905120585119894119896119905119872119898=1

119879119905=1 (22)

prop 119967(1198891 + 1198731198941198961 hellip 119889119872 + 119873119894119896119872)

where

119873119894119896119872 = sum 120128(119904119894119905 = 119895)120128(119904119894119905minus1 = 119896)119879

119905=1














a map 120601 ∶ ∆[01]119872119873 rarr ∆[01]119872

120636119905 = 120601 (1206361119905 hellip 120636119873119905) (23)



Equal weights

Let



120636119896119905 =1

119873sum 120575119896

119873119894=1 (119894119905) (24)





120636119896119905 = sum 120596119894119905120575119896119873119894=1 (119894119905) (25)














41 Data Description
















December 2010



Austria 1960M01 1989M06 1990M01

Belgium 1960M01 1960M01 1960M01

Finland 1960M01 1987M01 1988M01

France 1960M01 1970M01 1960M01

Germany 1960M01 1960M01 1960M01

Greece 1962M01 1997M06 2001M01

Ireland 1975M07 1984M01 1970M12

Italy 1960M01 1978M10 1991M03

Luxembourg 1960M01 1999M01 1993M10


Portugal 1960M01 1992M01 1993M07

Spain 1965M01 1977M01 1980M01























1205791 =1

1198991sum 120579(119895)1198991

119895=1 1205792 =1

1198992sum 120579(119895)1198991

119895=119899+1minus1198992


1198942

119899119894

= Γ(0) + 2119899119894

119899119894 minus 1sum 119870(119895ℎ119894)Γ(119895)

ℎ119894

119895=1

Γ(119895) =1

119899119894


119896=119895+1



119862119863 =1minus2


21198992

(26)






































Regime 1

Country i

AU 1 -20242 08453 1526 14309 00283

(-468-19) (-361564) (9251) (87237) (-413346)

BE 2 -01438 00127 5213 11209 -05786

(-39-01) (-5458) (452608) (89141) (-18264)

FI 3 -35669 01015 35118 13203 -0331

(-893-24) (-168213) (141564) (78211) (-414346)

FR 4 -02063 -00703 1601 10404 -00537

(-53-01) (-8966) (13197) (74143) (-7161)

GE 5 -01704 0046 29743 10345 -0253

(-41-01) (-781) (227364) (75142) (-12367)

GR 6 -41517 1013 15835 14095 -00286

(-724-124) (-177379) (92265) (85231) (-405412)

IR 7 -02211 00583 54996 1008 -00973

(-62-02) (-6478) (5608) (74133) (-1089)

IT 8 -36763 02934 17706 13322 -01696

(-633-11) (-207267) (114266) (8321) (-31424)

LU 9 -91082 0277 19724 12999 00593

(-1243-556) (-246299) (11336) (84202) (-556572)

NE 10 -46717 06634 15039 1463 00532

(-952-67) (-375521) (9251) (87237) (-445502)

PO 11 -47046 06718 17238 13152 -02942

(-814-142) (-189328) (102298) (83207) (-

433335)

SP 12 -05913 -00935 27753 11279 -02135

(-132-05) (-8765) (233328) (82152) (-128)







43 Turning Points













Regime 2

Country i

AU 1 -35019 08714 15161 14219 -00547

(-679-104) (-312581) (92249) (87235) (-41433)

BE 2 -02779 00924 21663 05164 -00333

(-57-06) (-461) (193243) (3971) (-3425)

FI 3 00649 01184 22306 07454 -00026

(-677) (-4573) (197254) (55101) (-353)

FR 4 -02968 00977 12154 0578 -00166

(-6-05) (-2446) (111133) (4672) (-1511)

GE 5 -02357 00503 14882 05187 00129

(-46-05) (-338) (137162) (465) (-1415)

GR 6 0112 00451 22352 12036 -00777

(-7398) (-7485) (188263) (87157) (-8472)

IR 7 09287 13359 1617 14704 -00663

(-419617) (-299587) (91269) (88241) (-551491)

IT 8 -00771 00687 14874 07886 00347

(-6246) (-4859) (132168) (58103) (-2531)

LU 9 -0226 00602 32394 09625 00592

(-9546) (-6171) (29359) (68127) (-5667)

NE 10 01114 00366 25878 06758 -00144

(-352) (-3845) (24278) (589) (-2725)

PO 11 -26537 09577 1662 1431 -01303

(-789281) (-28486) (96271) (88233) (-47456)

SP 12 01082 00542 15429 06671 00041

(-3151) (-3447) (136173) (5285) (-221)












spread
















Regime 3

Country i

AU 1 03668 00344 18541 07307 -0027

(0484) (-451) (17202) (5297) (-2924)

BE 2 01099 00857 21417 05782 -00143

(0129) (-2542) (198232) (4672) (-1815)

FI 3 20781 00295 32925 14981 0273

(103588) (-137168) (1448) (105209) (-282328)

FR 4 02657 01885 17246 08334 -004

(01101) (-3983) (152) (65108) (-4236)

GE 5 0223 01074 19156 06455 004

(0164) (-3557) (17221) (528) (-1726)

GR 6 37886 08778 16864 15113 -00768

(132734) (-263509) (95283) (925) (-435401)

IR 7 73548 14607 14935 1496 -01523

(2641255) (-331623) (88247) (89245) (-427465)

IT 8 29927 12244 15941 15433 01307

(11668) (-228532) (93261) (9125) (-436466)

LU 9 54322 09447 16577 14504 -00798

(1771001) (-363571) (91299) (89239) (-53491)

NE 10 43871 09305 15606 14971 02215

(1384) (-386611) (88271) (89253) (-419455)

PO 11 08666 00554 25702 08614 -00181

(17137) (-4659) (234282) (64111) (-4237)

SP 12 30198 0508 15576 14046 01818

(109661) (-28942) (9426) (86237) (-318345)

Conclusions 5








Conclusions 5




Conclusions 5





Conclusions 5




Conclusions 5




Conclusions 5




References 6






University Press
































References 6





86723ndash734










Semptember 2000




Working paper 14292









Research London








Main title

2015 edition




KS-TC-16-016-EN-N

ISBN 978-92-79-61459-0


Table of contents

Abstract

1 Introduction




5 Conclusions

6 References






but not vice versa













































119962119946119957 = 119938119946(119956119946119957) + sum sum 119912119946119947119949119953119949=120783

119925119947=120783 (119956119946119957)119962119947119957minus119949 + 119915119946 (119956119946119957)119963119957 + 120634119946119957 (1)
















120633119948(119956119946119957) = 120783 119946119943 119956119946119957 = 119948120782 119848119853119841119838119851119856119842119852119838




119886119946(119956119946119957) = sum 119886119872119896=1 119894119896

120643119946119896119905 119860119946119895119897(119956119946119957) = sum 119860119872119896=1 119894119895119897119896

120643119946119896119905

119863119946(119956119946119957) = sum 119863119872119896=1 119894119896

120643119946119896119905 Σ119946(119956119946119957) = sum Σ119872119896=1 119894119896

120643119946119896119905



119860119946119895119897 = 119860119946119895119897 120575119894(119895) + 119874119870times119870 (1 minus 120575119894(119895))

















119962119946119957 = 119912119946120783119934119946119957120643119946120783119957 + ⋯ + 119912119946119924119934119946119957120643119946119924119957 + 120634119946119957 120634119946119957~120029119922(120782 120622119946119957) (2)







119962119946119957 = 119935119946120782120783120632119946120782 + 120643119946120783119957119935119946120783120783120632119946120783 + ⋯ + 120643119946119924119957119935119946119924120783120632119946119924 + 120634119946119957 (3)

where 1199351198940119905 = (1198940119905prime ⨂ 119868119870) and 119935119894119898119905 = (119894119898119905

prime ⨂ 119868119870)





120632119946120782 ~120029119922120782

(120632119946120782 120622119946120782) (4)

120632119946119924 ~120029119922119924

(120632119946119950 120622119946119950) 119950 = 120783 hellip 119924 (5)



summinus120783119946119950 ~ 120038119922(119959119946119950120784 120624119946119950120784) 119950 = 120783 hellip 119924 (6)



119894119898 ) = 1198741198701198982 times119870119898

2












1205741198941198951 lt 1205741198941198951 lt ⋯ lt 120574119894119895119872






distributions

119953119946119947 ~ 120019(119941119946120783 hellip 119941119946119924) (7)

with 119889119894119895 = 119889119894






120632119946120782 ~ 120029119922120782

(119929119946120782 120622119946120782) (8)

120782 ~ 120029119922119924120782

(120782 120622119946120782) (9)

120632119946119950 ~ 120029119922119950

(119950 120622119946119950) 119950 = 120783 hellip 119924 (10)

119950 ~ 120029119922119950

(119950 120622119946119950) 119950 = 120783 hellip 119924 (11)



summinus120783119946119950 ~ 120038119922(119959119946119950120784 119946119950120784) 119950 = 120783 hellip 119924 (12)

119950minus120783~ 120038119922(119959119950120784 120624119950120784) 119950 = 120783 hellip 119924 (13)



119894119898 119950minus120783) = 119874119870119898

2 times1198701198982







assume

119953119946119947 ~ 120019(119941119946120783 hellip 119941119946119924) (14)


119889 ~ ℬℯ(1212)



33 Gibbs sampler










119901(119910|Ξ γ Σ) = (2120587)minus119879119870119873


2 exp minus1

2sum 119854119905

prime Σ119905minus1119854119905

119879119905=1 119879

119905=1 (15)







119879119870

2119873119894=1 prod |Σ119905|minus

1

2 exp minus1

2sum 119854119894119905

prime Σ119894119905minus1119854119894119905

119879119905=1 119879

119905=1 (16)

where 119962119894prime = (1199621198941




119883119894119905 = (

1198831198940119905 1198831198941119905 0

⋮ ⋱1198831198940119905 0 119883119894119872119905

)




function

119891(1206321198940|119962119894 Ξ119894 γ119894 Σ119894) prop (17)

prop exp minus1



1

2(120632119946120782 minus 120632119946120782)119873

119894=1 Σ1198940minus1(120632119946120782 minus 120632119946120782)

prop exp minus1

21206321198940

prime (sum 119935119946120782119957prime 120506119957

minus120783119935119946120782119957 + 120506119957minus120783119879

119905=1 )120632119946120782 + 120632119946120782 (sum 119935119946120782119957prime 120506119946119957

minus120783119962119946120782119957 + 120506119946120782minus120783120632119946120782

119879119905=1 )

prop 1199771198700(120632

119946120782 120506119946120782)



where 120632119946120782

= 120506119946120782

minus120783(120506119946120782

minus120783120632119946120782 + sum 119935119946120782119957prime 120506119957

minus120783119935119946120782119957119879119905=1 ) and 120506119946120782

minus120783= (120506119946120782

minus120783120632119946120782 + sum 119935119946120782119957prime 120506119957

minus120783119935119946120782119957119879119905=1


density function

119891(120632119894119898|119962119894 Ξ119894 γ1198940 γ119894(minus119898) Σ) prop (18)

prop exp minus1

2sum 119854119894119905

prime Σ119905 119854119894119905 minus1

2(120632119946119950 minus 120632119946119950)prime119905isin119983119894119898

Σ119894119898minus1(120632119946119950 minus 120632119946119950)

prop exp minus1

21206321198941

prime (sum 119935119946119950119957prime 120506119957

minus120783119935119946119950119957 + 120506119946119950minus120783

119905isin119983119894119898)120632119946 + 120632119946

prime (sum 119935119946119950119957prime 120506119946119957

minus120783119962119946119950119957 + 120506119946119950minus120783120632119946119950119905isin119983119894119898

)

prop 119977119870119872(120632

119946119950 120506119946119950) (19)

where 120632119946119950

= 120506119946119950

minus120783(120506119946119950

minus120783120632119946119950 + sum 119935119946119950119957prime 120506119946119957

minus120783119935119946119950119957119905isin119983119894119898) and 120506119946119950

minus120783= (120506119946119950

minus120783 + sum 119935119946119950119957prime 120506119957


119983119894119898 = 119905 = 1 hellip 119879|120585119894119898119905 = 1 and 119962119946119950119957 = 119962119946119957 minus 119935119946120782119957120632119946120782



119891(120622119894119898|119962119894 Ξ119894 γ1198940 γ119894) Σ119894(minus119898)) prop (20)


1

2| exp minus1

2sum 119854119894119905


|119879119905=1 Σ119894119898

minus1|120584119894119898+119870+1

2 exp minus1

2119905119903(120566119894119898Σ119894119898)

prop |Σ119894119898minus1|

120584119894119898+119879119894119898+119870+1

2 exp minus1

2119905119903 ((120566119894119898 + sum 119854119894119898119905119854119894119898119905

prime119905isin119983119894119898

)120506119946119950minus120783)

prop 119986119870(1205841198941198982 1205661198941198982) (21)


prime119905isin119983119894119898


119891(119901119894119896|119962119894 Ξ119894 γ1198940 γ119894) prop prod 119901119894119896119895

119889119895119872119895=1 prod prod 119901

119894119896119895

120585119894119895119905120585119894119896119905119872119898=1

119879119905=1 (22)

prop 119967(1198891 + 1198731198941198961 hellip 119889119872 + 119873119894119896119872)

where

119873119894119896119872 = sum 120128(119904119894119905 = 119895)120128(119904119894119905minus1 = 119896)119879

119905=1














a map 120601 ∶ ∆[01]119872119873 rarr ∆[01]119872

120636119905 = 120601 (1206361119905 hellip 120636119873119905) (23)



Equal weights

Let



120636119896119905 =1

119873sum 120575119896

119873119894=1 (119894119905) (24)





120636119896119905 = sum 120596119894119905120575119896119873119894=1 (119894119905) (25)














41 Data Description
















December 2010



Austria 1960M01 1989M06 1990M01

Belgium 1960M01 1960M01 1960M01

Finland 1960M01 1987M01 1988M01

France 1960M01 1970M01 1960M01

Germany 1960M01 1960M01 1960M01

Greece 1962M01 1997M06 2001M01

Ireland 1975M07 1984M01 1970M12

Italy 1960M01 1978M10 1991M03

Luxembourg 1960M01 1999M01 1993M10


Portugal 1960M01 1992M01 1993M07

Spain 1965M01 1977M01 1980M01























1205791 =1

1198991sum 120579(119895)1198991

119895=1 1205792 =1

1198992sum 120579(119895)1198991

119895=119899+1minus1198992


1198942

119899119894

= Γ(0) + 2119899119894

119899119894 minus 1sum 119870(119895ℎ119894)Γ(119895)

ℎ119894

119895=1

Γ(119895) =1

119899119894


119896=119895+1



119862119863 =1minus2


21198992

(26)






































Regime 1

Country i

AU 1 -20242 08453 1526 14309 00283

(-468-19) (-361564) (9251) (87237) (-413346)

BE 2 -01438 00127 5213 11209 -05786

(-39-01) (-5458) (452608) (89141) (-18264)

FI 3 -35669 01015 35118 13203 -0331

(-893-24) (-168213) (141564) (78211) (-414346)

FR 4 -02063 -00703 1601 10404 -00537

(-53-01) (-8966) (13197) (74143) (-7161)

GE 5 -01704 0046 29743 10345 -0253

(-41-01) (-781) (227364) (75142) (-12367)

GR 6 -41517 1013 15835 14095 -00286

(-724-124) (-177379) (92265) (85231) (-405412)

IR 7 -02211 00583 54996 1008 -00973

(-62-02) (-6478) (5608) (74133) (-1089)

IT 8 -36763 02934 17706 13322 -01696

(-633-11) (-207267) (114266) (8321) (-31424)

LU 9 -91082 0277 19724 12999 00593

(-1243-556) (-246299) (11336) (84202) (-556572)

NE 10 -46717 06634 15039 1463 00532

(-952-67) (-375521) (9251) (87237) (-445502)

PO 11 -47046 06718 17238 13152 -02942

(-814-142) (-189328) (102298) (83207) (-

433335)

SP 12 -05913 -00935 27753 11279 -02135

(-132-05) (-8765) (233328) (82152) (-128)







43 Turning Points













Regime 2

Country i

AU 1 -35019 08714 15161 14219 -00547

(-679-104) (-312581) (92249) (87235) (-41433)

BE 2 -02779 00924 21663 05164 -00333

(-57-06) (-461) (193243) (3971) (-3425)

FI 3 00649 01184 22306 07454 -00026

(-677) (-4573) (197254) (55101) (-353)

FR 4 -02968 00977 12154 0578 -00166

(-6-05) (-2446) (111133) (4672) (-1511)

GE 5 -02357 00503 14882 05187 00129

(-46-05) (-338) (137162) (465) (-1415)

GR 6 0112 00451 22352 12036 -00777

(-7398) (-7485) (188263) (87157) (-8472)

IR 7 09287 13359 1617 14704 -00663

(-419617) (-299587) (91269) (88241) (-551491)

IT 8 -00771 00687 14874 07886 00347

(-6246) (-4859) (132168) (58103) (-2531)

LU 9 -0226 00602 32394 09625 00592

(-9546) (-6171) (29359) (68127) (-5667)

NE 10 01114 00366 25878 06758 -00144

(-352) (-3845) (24278) (589) (-2725)

PO 11 -26537 09577 1662 1431 -01303

(-789281) (-28486) (96271) (88233) (-47456)

SP 12 01082 00542 15429 06671 00041

(-3151) (-3447) (136173) (5285) (-221)












spread
















Regime 3

Country i

AU 1 03668 00344 18541 07307 -0027

(0484) (-451) (17202) (5297) (-2924)

BE 2 01099 00857 21417 05782 -00143

(0129) (-2542) (198232) (4672) (-1815)

FI 3 20781 00295 32925 14981 0273

(103588) (-137168) (1448) (105209) (-282328)

FR 4 02657 01885 17246 08334 -004

(01101) (-3983) (152) (65108) (-4236)

GE 5 0223 01074 19156 06455 004

(0164) (-3557) (17221) (528) (-1726)

GR 6 37886 08778 16864 15113 -00768

(132734) (-263509) (95283) (925) (-435401)

IR 7 73548 14607 14935 1496 -01523

(2641255) (-331623) (88247) (89245) (-427465)

IT 8 29927 12244 15941 15433 01307

(11668) (-228532) (93261) (9125) (-436466)

LU 9 54322 09447 16577 14504 -00798

(1771001) (-363571) (91299) (89239) (-53491)

NE 10 43871 09305 15606 14971 02215

(1384) (-386611) (88271) (89253) (-419455)

PO 11 08666 00554 25702 08614 -00181

(17137) (-4659) (234282) (64111) (-4237)

SP 12 30198 0508 15576 14046 01818

(109661) (-28942) (9426) (86237) (-318345)

Conclusions 5








Conclusions 5




Conclusions 5





Conclusions 5




Conclusions 5




Conclusions 5




References 6






University Press
































References 6





86723ndash734










Semptember 2000




Working paper 14292









Research London








Main title

2015 edition




KS-TC-16-016-EN-N

ISBN 978-92-79-61459-0


Table of contents

Abstract

1 Introduction




5 Conclusions

6 References



119962119946119957 = 119938119946(119956119946119957) + sum sum 119912119946119947119949119953119949=120783

119925119947=120783 (119956119946119957)119962119947119957minus119949 + 119915119946 (119956119946119957)119963119957 + 120634119946119957 (1)
















120633119948(119956119946119957) = 120783 119946119943 119956119946119957 = 119948120782 119848119853119841119838119851119856119842119852119838




119886119946(119956119946119957) = sum 119886119872119896=1 119894119896

120643119946119896119905 119860119946119895119897(119956119946119957) = sum 119860119872119896=1 119894119895119897119896

120643119946119896119905

119863119946(119956119946119957) = sum 119863119872119896=1 119894119896

120643119946119896119905 Σ119946(119956119946119957) = sum Σ119872119896=1 119894119896

120643119946119896119905



119860119946119895119897 = 119860119946119895119897 120575119894(119895) + 119874119870times119870 (1 minus 120575119894(119895))

















119962119946119957 = 119912119946120783119934119946119957120643119946120783119957 + ⋯ + 119912119946119924119934119946119957120643119946119924119957 + 120634119946119957 120634119946119957~120029119922(120782 120622119946119957) (2)







119962119946119957 = 119935119946120782120783120632119946120782 + 120643119946120783119957119935119946120783120783120632119946120783 + ⋯ + 120643119946119924119957119935119946119924120783120632119946119924 + 120634119946119957 (3)

where 1199351198940119905 = (1198940119905prime ⨂ 119868119870) and 119935119894119898119905 = (119894119898119905

prime ⨂ 119868119870)





120632119946120782 ~120029119922120782

(120632119946120782 120622119946120782) (4)

120632119946119924 ~120029119922119924

(120632119946119950 120622119946119950) 119950 = 120783 hellip 119924 (5)



summinus120783119946119950 ~ 120038119922(119959119946119950120784 120624119946119950120784) 119950 = 120783 hellip 119924 (6)



119894119898 ) = 1198741198701198982 times119870119898

2












1205741198941198951 lt 1205741198941198951 lt ⋯ lt 120574119894119895119872






distributions

119953119946119947 ~ 120019(119941119946120783 hellip 119941119946119924) (7)

with 119889119894119895 = 119889119894






120632119946120782 ~ 120029119922120782

(119929119946120782 120622119946120782) (8)

120782 ~ 120029119922119924120782

(120782 120622119946120782) (9)

120632119946119950 ~ 120029119922119950

(119950 120622119946119950) 119950 = 120783 hellip 119924 (10)

119950 ~ 120029119922119950

(119950 120622119946119950) 119950 = 120783 hellip 119924 (11)



summinus120783119946119950 ~ 120038119922(119959119946119950120784 119946119950120784) 119950 = 120783 hellip 119924 (12)

119950minus120783~ 120038119922(119959119950120784 120624119950120784) 119950 = 120783 hellip 119924 (13)



119894119898 119950minus120783) = 119874119870119898

2 times1198701198982







assume

119953119946119947 ~ 120019(119941119946120783 hellip 119941119946119924) (14)


119889 ~ ℬℯ(1212)



33 Gibbs sampler










119901(119910|Ξ γ Σ) = (2120587)minus119879119870119873


2 exp minus1

2sum 119854119905

prime Σ119905minus1119854119905

119879119905=1 119879

119905=1 (15)







119879119870

2119873119894=1 prod |Σ119905|minus

1

2 exp minus1

2sum 119854119894119905

prime Σ119894119905minus1119854119894119905

119879119905=1 119879

119905=1 (16)

where 119962119894prime = (1199621198941




119883119894119905 = (

1198831198940119905 1198831198941119905 0

⋮ ⋱1198831198940119905 0 119883119894119872119905

)




function

119891(1206321198940|119962119894 Ξ119894 γ119894 Σ119894) prop (17)

prop exp minus1



1

2(120632119946120782 minus 120632119946120782)119873

119894=1 Σ1198940minus1(120632119946120782 minus 120632119946120782)

prop exp minus1

21206321198940

prime (sum 119935119946120782119957prime 120506119957

minus120783119935119946120782119957 + 120506119957minus120783119879

119905=1 )120632119946120782 + 120632119946120782 (sum 119935119946120782119957prime 120506119946119957

minus120783119962119946120782119957 + 120506119946120782minus120783120632119946120782

119879119905=1 )

prop 1199771198700(120632

119946120782 120506119946120782)



where 120632119946120782

= 120506119946120782

minus120783(120506119946120782

minus120783120632119946120782 + sum 119935119946120782119957prime 120506119957

minus120783119935119946120782119957119879119905=1 ) and 120506119946120782

minus120783= (120506119946120782

minus120783120632119946120782 + sum 119935119946120782119957prime 120506119957

minus120783119935119946120782119957119879119905=1


density function

119891(120632119894119898|119962119894 Ξ119894 γ1198940 γ119894(minus119898) Σ) prop (18)

prop exp minus1

2sum 119854119894119905

prime Σ119905 119854119894119905 minus1

2(120632119946119950 minus 120632119946119950)prime119905isin119983119894119898

Σ119894119898minus1(120632119946119950 minus 120632119946119950)

prop exp minus1

21206321198941

prime (sum 119935119946119950119957prime 120506119957

minus120783119935119946119950119957 + 120506119946119950minus120783

119905isin119983119894119898)120632119946 + 120632119946

prime (sum 119935119946119950119957prime 120506119946119957

minus120783119962119946119950119957 + 120506119946119950minus120783120632119946119950119905isin119983119894119898

)

prop 119977119870119872(120632

119946119950 120506119946119950) (19)

where 120632119946119950

= 120506119946119950

minus120783(120506119946119950

minus120783120632119946119950 + sum 119935119946119950119957prime 120506119946119957

minus120783119935119946119950119957119905isin119983119894119898) and 120506119946119950

minus120783= (120506119946119950

minus120783 + sum 119935119946119950119957prime 120506119957


119983119894119898 = 119905 = 1 hellip 119879|120585119894119898119905 = 1 and 119962119946119950119957 = 119962119946119957 minus 119935119946120782119957120632119946120782



119891(120622119894119898|119962119894 Ξ119894 γ1198940 γ119894) Σ119894(minus119898)) prop (20)


1

2| exp minus1

2sum 119854119894119905


|119879119905=1 Σ119894119898

minus1|120584119894119898+119870+1

2 exp minus1

2119905119903(120566119894119898Σ119894119898)

prop |Σ119894119898minus1|

120584119894119898+119879119894119898+119870+1

2 exp minus1

2119905119903 ((120566119894119898 + sum 119854119894119898119905119854119894119898119905

prime119905isin119983119894119898

)120506119946119950minus120783)

prop 119986119870(1205841198941198982 1205661198941198982) (21)


prime119905isin119983119894119898


119891(119901119894119896|119962119894 Ξ119894 γ1198940 γ119894) prop prod 119901119894119896119895

119889119895119872119895=1 prod prod 119901

119894119896119895

120585119894119895119905120585119894119896119905119872119898=1

119879119905=1 (22)

prop 119967(1198891 + 1198731198941198961 hellip 119889119872 + 119873119894119896119872)

where

119873119894119896119872 = sum 120128(119904119894119905 = 119895)120128(119904119894119905minus1 = 119896)119879

119905=1














a map 120601 ∶ ∆[01]119872119873 rarr ∆[01]119872

120636119905 = 120601 (1206361119905 hellip 120636119873119905) (23)



Equal weights

Let



120636119896119905 =1

119873sum 120575119896

119873119894=1 (119894119905) (24)





120636119896119905 = sum 120596119894119905120575119896119873119894=1 (119894119905) (25)














41 Data Description
















December 2010



Austria 1960M01 1989M06 1990M01

Belgium 1960M01 1960M01 1960M01

Finland 1960M01 1987M01 1988M01

France 1960M01 1970M01 1960M01

Germany 1960M01 1960M01 1960M01

Greece 1962M01 1997M06 2001M01

Ireland 1975M07 1984M01 1970M12

Italy 1960M01 1978M10 1991M03

Luxembourg 1960M01 1999M01 1993M10


Portugal 1960M01 1992M01 1993M07

Spain 1965M01 1977M01 1980M01























1205791 =1

1198991sum 120579(119895)1198991

119895=1 1205792 =1

1198992sum 120579(119895)1198991

119895=119899+1minus1198992


1198942

119899119894

= Γ(0) + 2119899119894

119899119894 minus 1sum 119870(119895ℎ119894)Γ(119895)

ℎ119894

119895=1

Γ(119895) =1

119899119894


119896=119895+1



119862119863 =1minus2


21198992

(26)






































Regime 1

Country i

AU 1 -20242 08453 1526 14309 00283

(-468-19) (-361564) (9251) (87237) (-413346)

BE 2 -01438 00127 5213 11209 -05786

(-39-01) (-5458) (452608) (89141) (-18264)

FI 3 -35669 01015 35118 13203 -0331

(-893-24) (-168213) (141564) (78211) (-414346)

FR 4 -02063 -00703 1601 10404 -00537

(-53-01) (-8966) (13197) (74143) (-7161)

GE 5 -01704 0046 29743 10345 -0253

(-41-01) (-781) (227364) (75142) (-12367)

GR 6 -41517 1013 15835 14095 -00286

(-724-124) (-177379) (92265) (85231) (-405412)

IR 7 -02211 00583 54996 1008 -00973

(-62-02) (-6478) (5608) (74133) (-1089)

IT 8 -36763 02934 17706 13322 -01696

(-633-11) (-207267) (114266) (8321) (-31424)

LU 9 -91082 0277 19724 12999 00593

(-1243-556) (-246299) (11336) (84202) (-556572)

NE 10 -46717 06634 15039 1463 00532

(-952-67) (-375521) (9251) (87237) (-445502)

PO 11 -47046 06718 17238 13152 -02942

(-814-142) (-189328) (102298) (83207) (-

433335)

SP 12 -05913 -00935 27753 11279 -02135

(-132-05) (-8765) (233328) (82152) (-128)







43 Turning Points













Regime 2

Country i

AU 1 -35019 08714 15161 14219 -00547

(-679-104) (-312581) (92249) (87235) (-41433)

BE 2 -02779 00924 21663 05164 -00333

(-57-06) (-461) (193243) (3971) (-3425)

FI 3 00649 01184 22306 07454 -00026

(-677) (-4573) (197254) (55101) (-353)

FR 4 -02968 00977 12154 0578 -00166

(-6-05) (-2446) (111133) (4672) (-1511)

GE 5 -02357 00503 14882 05187 00129

(-46-05) (-338) (137162) (465) (-1415)

GR 6 0112 00451 22352 12036 -00777

(-7398) (-7485) (188263) (87157) (-8472)

IR 7 09287 13359 1617 14704 -00663

(-419617) (-299587) (91269) (88241) (-551491)

IT 8 -00771 00687 14874 07886 00347

(-6246) (-4859) (132168) (58103) (-2531)

LU 9 -0226 00602 32394 09625 00592

(-9546) (-6171) (29359) (68127) (-5667)

NE 10 01114 00366 25878 06758 -00144

(-352) (-3845) (24278) (589) (-2725)

PO 11 -26537 09577 1662 1431 -01303

(-789281) (-28486) (96271) (88233) (-47456)

SP 12 01082 00542 15429 06671 00041

(-3151) (-3447) (136173) (5285) (-221)












spread
















Regime 3

Country i

AU 1 03668 00344 18541 07307 -0027

(0484) (-451) (17202) (5297) (-2924)

BE 2 01099 00857 21417 05782 -00143

(0129) (-2542) (198232) (4672) (-1815)

FI 3 20781 00295 32925 14981 0273

(103588) (-137168) (1448) (105209) (-282328)

FR 4 02657 01885 17246 08334 -004

(01101) (-3983) (152) (65108) (-4236)

GE 5 0223 01074 19156 06455 004

(0164) (-3557) (17221) (528) (-1726)

GR 6 37886 08778 16864 15113 -00768

(132734) (-263509) (95283) (925) (-435401)

IR 7 73548 14607 14935 1496 -01523

(2641255) (-331623) (88247) (89245) (-427465)

IT 8 29927 12244 15941 15433 01307

(11668) (-228532) (93261) (9125) (-436466)

LU 9 54322 09447 16577 14504 -00798

(1771001) (-363571) (91299) (89239) (-53491)

NE 10 43871 09305 15606 14971 02215

(1384) (-386611) (88271) (89253) (-419455)

PO 11 08666 00554 25702 08614 -00181

(17137) (-4659) (234282) (64111) (-4237)

SP 12 30198 0508 15576 14046 01818

(109661) (-28942) (9426) (86237) (-318345)

Conclusions 5








Conclusions 5




Conclusions 5





Conclusions 5




Conclusions 5




Conclusions 5




References 6






University Press
































References 6





86723ndash734










Semptember 2000




Working paper 14292









Research London








Main title

2015 edition




KS-TC-16-016-EN-N

ISBN 978-92-79-61459-0


Table of contents

Abstract

1 Introduction




5 Conclusions

6 References



119962119946119957 = 119912119946120783119934119946119957120643119946120783119957 + ⋯ + 119912119946119924119934119946119957120643119946119924119957 + 120634119946119957 120634119946119957~120029119922(120782 120622119946119957) (2)







119962119946119957 = 119935119946120782120783120632119946120782 + 120643119946120783119957119935119946120783120783120632119946120783 + ⋯ + 120643119946119924119957119935119946119924120783120632119946119924 + 120634119946119957 (3)

where 1199351198940119905 = (1198940119905prime ⨂ 119868119870) and 119935119894119898119905 = (119894119898119905

prime ⨂ 119868119870)





120632119946120782 ~120029119922120782

(120632119946120782 120622119946120782) (4)

120632119946119924 ~120029119922119924

(120632119946119950 120622119946119950) 119950 = 120783 hellip 119924 (5)



summinus120783119946119950 ~ 120038119922(119959119946119950120784 120624119946119950120784) 119950 = 120783 hellip 119924 (6)



119894119898 ) = 1198741198701198982 times119870119898

2












1205741198941198951 lt 1205741198941198951 lt ⋯ lt 120574119894119895119872






distributions

119953119946119947 ~ 120019(119941119946120783 hellip 119941119946119924) (7)

with 119889119894119895 = 119889119894






120632119946120782 ~ 120029119922120782

(119929119946120782 120622119946120782) (8)

120782 ~ 120029119922119924120782

(120782 120622119946120782) (9)

120632119946119950 ~ 120029119922119950

(119950 120622119946119950) 119950 = 120783 hellip 119924 (10)

119950 ~ 120029119922119950

(119950 120622119946119950) 119950 = 120783 hellip 119924 (11)



summinus120783119946119950 ~ 120038119922(119959119946119950120784 119946119950120784) 119950 = 120783 hellip 119924 (12)

119950minus120783~ 120038119922(119959119950120784 120624119950120784) 119950 = 120783 hellip 119924 (13)



119894119898 119950minus120783) = 119874119870119898

2 times1198701198982







assume

119953119946119947 ~ 120019(119941119946120783 hellip 119941119946119924) (14)


119889 ~ ℬℯ(1212)



33 Gibbs sampler










119901(119910|Ξ γ Σ) = (2120587)minus119879119870119873


2 exp minus1

2sum 119854119905

prime Σ119905minus1119854119905

119879119905=1 119879

119905=1 (15)







119879119870

2119873119894=1 prod |Σ119905|minus

1

2 exp minus1

2sum 119854119894119905

prime Σ119894119905minus1119854119894119905

119879119905=1 119879

119905=1 (16)

where 119962119894prime = (1199621198941




119883119894119905 = (

1198831198940119905 1198831198941119905 0

⋮ ⋱1198831198940119905 0 119883119894119872119905

)




function

119891(1206321198940|119962119894 Ξ119894 γ119894 Σ119894) prop (17)

prop exp minus1



1

2(120632119946120782 minus 120632119946120782)119873

119894=1 Σ1198940minus1(120632119946120782 minus 120632119946120782)

prop exp minus1

21206321198940

prime (sum 119935119946120782119957prime 120506119957

minus120783119935119946120782119957 + 120506119957minus120783119879

119905=1 )120632119946120782 + 120632119946120782 (sum 119935119946120782119957prime 120506119946119957

minus120783119962119946120782119957 + 120506119946120782minus120783120632119946120782

119879119905=1 )

prop 1199771198700(120632

119946120782 120506119946120782)



where 120632119946120782

= 120506119946120782

minus120783(120506119946120782

minus120783120632119946120782 + sum 119935119946120782119957prime 120506119957

minus120783119935119946120782119957119879119905=1 ) and 120506119946120782

minus120783= (120506119946120782

minus120783120632119946120782 + sum 119935119946120782119957prime 120506119957

minus120783119935119946120782119957119879119905=1


density function

119891(120632119894119898|119962119894 Ξ119894 γ1198940 γ119894(minus119898) Σ) prop (18)

prop exp minus1

2sum 119854119894119905

prime Σ119905 119854119894119905 minus1

2(120632119946119950 minus 120632119946119950)prime119905isin119983119894119898

Σ119894119898minus1(120632119946119950 minus 120632119946119950)

prop exp minus1

21206321198941

prime (sum 119935119946119950119957prime 120506119957

minus120783119935119946119950119957 + 120506119946119950minus120783

119905isin119983119894119898)120632119946 + 120632119946

prime (sum 119935119946119950119957prime 120506119946119957

minus120783119962119946119950119957 + 120506119946119950minus120783120632119946119950119905isin119983119894119898

)

prop 119977119870119872(120632

119946119950 120506119946119950) (19)

where 120632119946119950

= 120506119946119950

minus120783(120506119946119950

minus120783120632119946119950 + sum 119935119946119950119957prime 120506119946119957

minus120783119935119946119950119957119905isin119983119894119898) and 120506119946119950

minus120783= (120506119946119950

minus120783 + sum 119935119946119950119957prime 120506119957


119983119894119898 = 119905 = 1 hellip 119879|120585119894119898119905 = 1 and 119962119946119950119957 = 119962119946119957 minus 119935119946120782119957120632119946120782



119891(120622119894119898|119962119894 Ξ119894 γ1198940 γ119894) Σ119894(minus119898)) prop (20)


1

2| exp minus1

2sum 119854119894119905


|119879119905=1 Σ119894119898

minus1|120584119894119898+119870+1

2 exp minus1

2119905119903(120566119894119898Σ119894119898)

prop |Σ119894119898minus1|

120584119894119898+119879119894119898+119870+1

2 exp minus1

2119905119903 ((120566119894119898 + sum 119854119894119898119905119854119894119898119905

prime119905isin119983119894119898

)120506119946119950minus120783)

prop 119986119870(1205841198941198982 1205661198941198982) (21)


prime119905isin119983119894119898


119891(119901119894119896|119962119894 Ξ119894 γ1198940 γ119894) prop prod 119901119894119896119895

119889119895119872119895=1 prod prod 119901

119894119896119895

120585119894119895119905120585119894119896119905119872119898=1

119879119905=1 (22)

prop 119967(1198891 + 1198731198941198961 hellip 119889119872 + 119873119894119896119872)

where

119873119894119896119872 = sum 120128(119904119894119905 = 119895)120128(119904119894119905minus1 = 119896)119879

119905=1














a map 120601 ∶ ∆[01]119872119873 rarr ∆[01]119872

120636119905 = 120601 (1206361119905 hellip 120636119873119905) (23)



Equal weights

Let



120636119896119905 =1

119873sum 120575119896

119873119894=1 (119894119905) (24)





120636119896119905 = sum 120596119894119905120575119896119873119894=1 (119894119905) (25)














41 Data Description
















December 2010



Austria 1960M01 1989M06 1990M01

Belgium 1960M01 1960M01 1960M01

Finland 1960M01 1987M01 1988M01

France 1960M01 1970M01 1960M01

Germany 1960M01 1960M01 1960M01

Greece 1962M01 1997M06 2001M01

Ireland 1975M07 1984M01 1970M12

Italy 1960M01 1978M10 1991M03

Luxembourg 1960M01 1999M01 1993M10


Portugal 1960M01 1992M01 1993M07

Spain 1965M01 1977M01 1980M01























1205791 =1

1198991sum 120579(119895)1198991

119895=1 1205792 =1

1198992sum 120579(119895)1198991

119895=119899+1minus1198992


1198942

119899119894

= Γ(0) + 2119899119894

119899119894 minus 1sum 119870(119895ℎ119894)Γ(119895)

ℎ119894

119895=1

Γ(119895) =1

119899119894


119896=119895+1



119862119863 =1minus2


21198992

(26)






































Regime 1

Country i

AU 1 -20242 08453 1526 14309 00283

(-468-19) (-361564) (9251) (87237) (-413346)

BE 2 -01438 00127 5213 11209 -05786

(-39-01) (-5458) (452608) (89141) (-18264)

FI 3 -35669 01015 35118 13203 -0331

(-893-24) (-168213) (141564) (78211) (-414346)

FR 4 -02063 -00703 1601 10404 -00537

(-53-01) (-8966) (13197) (74143) (-7161)

GE 5 -01704 0046 29743 10345 -0253

(-41-01) (-781) (227364) (75142) (-12367)

GR 6 -41517 1013 15835 14095 -00286

(-724-124) (-177379) (92265) (85231) (-405412)

IR 7 -02211 00583 54996 1008 -00973

(-62-02) (-6478) (5608) (74133) (-1089)

IT 8 -36763 02934 17706 13322 -01696

(-633-11) (-207267) (114266) (8321) (-31424)

LU 9 -91082 0277 19724 12999 00593

(-1243-556) (-246299) (11336) (84202) (-556572)

NE 10 -46717 06634 15039 1463 00532

(-952-67) (-375521) (9251) (87237) (-445502)

PO 11 -47046 06718 17238 13152 -02942

(-814-142) (-189328) (102298) (83207) (-

433335)

SP 12 -05913 -00935 27753 11279 -02135

(-132-05) (-8765) (233328) (82152) (-128)







43 Turning Points













Regime 2

Country i

AU 1 -35019 08714 15161 14219 -00547

(-679-104) (-312581) (92249) (87235) (-41433)

BE 2 -02779 00924 21663 05164 -00333

(-57-06) (-461) (193243) (3971) (-3425)

FI 3 00649 01184 22306 07454 -00026

(-677) (-4573) (197254) (55101) (-353)

FR 4 -02968 00977 12154 0578 -00166

(-6-05) (-2446) (111133) (4672) (-1511)

GE 5 -02357 00503 14882 05187 00129

(-46-05) (-338) (137162) (465) (-1415)

GR 6 0112 00451 22352 12036 -00777

(-7398) (-7485) (188263) (87157) (-8472)

IR 7 09287 13359 1617 14704 -00663

(-419617) (-299587) (91269) (88241) (-551491)

IT 8 -00771 00687 14874 07886 00347

(-6246) (-4859) (132168) (58103) (-2531)

LU 9 -0226 00602 32394 09625 00592

(-9546) (-6171) (29359) (68127) (-5667)

NE 10 01114 00366 25878 06758 -00144

(-352) (-3845) (24278) (589) (-2725)

PO 11 -26537 09577 1662 1431 -01303

(-789281) (-28486) (96271) (88233) (-47456)

SP 12 01082 00542 15429 06671 00041

(-3151) (-3447) (136173) (5285) (-221)












spread
















Regime 3

Country i

AU 1 03668 00344 18541 07307 -0027

(0484) (-451) (17202) (5297) (-2924)

BE 2 01099 00857 21417 05782 -00143

(0129) (-2542) (198232) (4672) (-1815)

FI 3 20781 00295 32925 14981 0273

(103588) (-137168) (1448) (105209) (-282328)

FR 4 02657 01885 17246 08334 -004

(01101) (-3983) (152) (65108) (-4236)

GE 5 0223 01074 19156 06455 004

(0164) (-3557) (17221) (528) (-1726)

GR 6 37886 08778 16864 15113 -00768

(132734) (-263509) (95283) (925) (-435401)

IR 7 73548 14607 14935 1496 -01523

(2641255) (-331623) (88247) (89245) (-427465)

IT 8 29927 12244 15941 15433 01307

(11668) (-228532) (93261) (9125) (-436466)

LU 9 54322 09447 16577 14504 -00798

(1771001) (-363571) (91299) (89239) (-53491)

NE 10 43871 09305 15606 14971 02215

(1384) (-386611) (88271) (89253) (-419455)

PO 11 08666 00554 25702 08614 -00181

(17137) (-4659) (234282) (64111) (-4237)

SP 12 30198 0508 15576 14046 01818

(109661) (-28942) (9426) (86237) (-318345)

Conclusions 5








Conclusions 5




Conclusions 5





Conclusions 5




Conclusions 5




Conclusions 5




References 6






University Press
































References 6





86723ndash734










Semptember 2000




Working paper 14292









Research London








Main title

2015 edition




KS-TC-16-016-EN-N

ISBN 978-92-79-61459-0


Table of contents

Abstract

1 Introduction




5 Conclusions

6 References






distributions

119953119946119947 ~ 120019(119941119946120783 hellip 119941119946119924) (7)

with 119889119894119895 = 119889119894






120632119946120782 ~ 120029119922120782

(119929119946120782 120622119946120782) (8)

120782 ~ 120029119922119924120782

(120782 120622119946120782) (9)

120632119946119950 ~ 120029119922119950

(119950 120622119946119950) 119950 = 120783 hellip 119924 (10)

119950 ~ 120029119922119950

(119950 120622119946119950) 119950 = 120783 hellip 119924 (11)



summinus120783119946119950 ~ 120038119922(119959119946119950120784 119946119950120784) 119950 = 120783 hellip 119924 (12)

119950minus120783~ 120038119922(119959119950120784 120624119950120784) 119950 = 120783 hellip 119924 (13)



119894119898 119950minus120783) = 119874119870119898

2 times1198701198982







assume

119953119946119947 ~ 120019(119941119946120783 hellip 119941119946119924) (14)


119889 ~ ℬℯ(1212)



33 Gibbs sampler










119901(119910|Ξ γ Σ) = (2120587)minus119879119870119873


2 exp minus1

2sum 119854119905

prime Σ119905minus1119854119905

119879119905=1 119879

119905=1 (15)







119879119870

2119873119894=1 prod |Σ119905|minus

1

2 exp minus1

2sum 119854119894119905

prime Σ119894119905minus1119854119894119905

119879119905=1 119879

119905=1 (16)

where 119962119894prime = (1199621198941




119883119894119905 = (

1198831198940119905 1198831198941119905 0

⋮ ⋱1198831198940119905 0 119883119894119872119905

)




function

119891(1206321198940|119962119894 Ξ119894 γ119894 Σ119894) prop (17)

prop exp minus1



1

2(120632119946120782 minus 120632119946120782)119873

119894=1 Σ1198940minus1(120632119946120782 minus 120632119946120782)

prop exp minus1

21206321198940

prime (sum 119935119946120782119957prime 120506119957

minus120783119935119946120782119957 + 120506119957minus120783119879

119905=1 )120632119946120782 + 120632119946120782 (sum 119935119946120782119957prime 120506119946119957

minus120783119962119946120782119957 + 120506119946120782minus120783120632119946120782

119879119905=1 )

prop 1199771198700(120632

119946120782 120506119946120782)



where 120632119946120782

= 120506119946120782

minus120783(120506119946120782

minus120783120632119946120782 + sum 119935119946120782119957prime 120506119957

minus120783119935119946120782119957119879119905=1 ) and 120506119946120782

minus120783= (120506119946120782

minus120783120632119946120782 + sum 119935119946120782119957prime 120506119957

minus120783119935119946120782119957119879119905=1


density function

119891(120632119894119898|119962119894 Ξ119894 γ1198940 γ119894(minus119898) Σ) prop (18)

prop exp minus1

2sum 119854119894119905

prime Σ119905 119854119894119905 minus1

2(120632119946119950 minus 120632119946119950)prime119905isin119983119894119898

Σ119894119898minus1(120632119946119950 minus 120632119946119950)

prop exp minus1

21206321198941

prime (sum 119935119946119950119957prime 120506119957

minus120783119935119946119950119957 + 120506119946119950minus120783

119905isin119983119894119898)120632119946 + 120632119946

prime (sum 119935119946119950119957prime 120506119946119957

minus120783119962119946119950119957 + 120506119946119950minus120783120632119946119950119905isin119983119894119898

)

prop 119977119870119872(120632

119946119950 120506119946119950) (19)

where 120632119946119950

= 120506119946119950

minus120783(120506119946119950

minus120783120632119946119950 + sum 119935119946119950119957prime 120506119946119957

minus120783119935119946119950119957119905isin119983119894119898) and 120506119946119950

minus120783= (120506119946119950

minus120783 + sum 119935119946119950119957prime 120506119957


119983119894119898 = 119905 = 1 hellip 119879|120585119894119898119905 = 1 and 119962119946119950119957 = 119962119946119957 minus 119935119946120782119957120632119946120782



119891(120622119894119898|119962119894 Ξ119894 γ1198940 γ119894) Σ119894(minus119898)) prop (20)


1

2| exp minus1

2sum 119854119894119905


|119879119905=1 Σ119894119898

minus1|120584119894119898+119870+1

2 exp minus1

2119905119903(120566119894119898Σ119894119898)

prop |Σ119894119898minus1|

120584119894119898+119879119894119898+119870+1

2 exp minus1

2119905119903 ((120566119894119898 + sum 119854119894119898119905119854119894119898119905

prime119905isin119983119894119898

)120506119946119950minus120783)

prop 119986119870(1205841198941198982 1205661198941198982) (21)


prime119905isin119983119894119898


119891(119901119894119896|119962119894 Ξ119894 γ1198940 γ119894) prop prod 119901119894119896119895

119889119895119872119895=1 prod prod 119901

119894119896119895

120585119894119895119905120585119894119896119905119872119898=1

119879119905=1 (22)

prop 119967(1198891 + 1198731198941198961 hellip 119889119872 + 119873119894119896119872)

where

119873119894119896119872 = sum 120128(119904119894119905 = 119895)120128(119904119894119905minus1 = 119896)119879

119905=1














a map 120601 ∶ ∆[01]119872119873 rarr ∆[01]119872

120636119905 = 120601 (1206361119905 hellip 120636119873119905) (23)



Equal weights

Let



120636119896119905 =1

119873sum 120575119896

119873119894=1 (119894119905) (24)





120636119896119905 = sum 120596119894119905120575119896119873119894=1 (119894119905) (25)














41 Data Description
















December 2010



Austria 1960M01 1989M06 1990M01

Belgium 1960M01 1960M01 1960M01

Finland 1960M01 1987M01 1988M01

France 1960M01 1970M01 1960M01

Germany 1960M01 1960M01 1960M01

Greece 1962M01 1997M06 2001M01

Ireland 1975M07 1984M01 1970M12

Italy 1960M01 1978M10 1991M03

Luxembourg 1960M01 1999M01 1993M10


Portugal 1960M01 1992M01 1993M07

Spain 1965M01 1977M01 1980M01























1205791 =1

1198991sum 120579(119895)1198991

119895=1 1205792 =1

1198992sum 120579(119895)1198991

119895=119899+1minus1198992


1198942

119899119894

= Γ(0) + 2119899119894

119899119894 minus 1sum 119870(119895ℎ119894)Γ(119895)

ℎ119894

119895=1

Γ(119895) =1

119899119894


119896=119895+1



119862119863 =1minus2


21198992

(26)






































Regime 1

Country i

AU 1 -20242 08453 1526 14309 00283

(-468-19) (-361564) (9251) (87237) (-413346)

BE 2 -01438 00127 5213 11209 -05786

(-39-01) (-5458) (452608) (89141) (-18264)

FI 3 -35669 01015 35118 13203 -0331

(-893-24) (-168213) (141564) (78211) (-414346)

FR 4 -02063 -00703 1601 10404 -00537

(-53-01) (-8966) (13197) (74143) (-7161)

GE 5 -01704 0046 29743 10345 -0253

(-41-01) (-781) (227364) (75142) (-12367)

GR 6 -41517 1013 15835 14095 -00286

(-724-124) (-177379) (92265) (85231) (-405412)

IR 7 -02211 00583 54996 1008 -00973

(-62-02) (-6478) (5608) (74133) (-1089)

IT 8 -36763 02934 17706 13322 -01696

(-633-11) (-207267) (114266) (8321) (-31424)

LU 9 -91082 0277 19724 12999 00593

(-1243-556) (-246299) (11336) (84202) (-556572)

NE 10 -46717 06634 15039 1463 00532

(-952-67) (-375521) (9251) (87237) (-445502)

PO 11 -47046 06718 17238 13152 -02942

(-814-142) (-189328) (102298) (83207) (-

433335)

SP 12 -05913 -00935 27753 11279 -02135

(-132-05) (-8765) (233328) (82152) (-128)







43 Turning Points













Regime 2

Country i

AU 1 -35019 08714 15161 14219 -00547

(-679-104) (-312581) (92249) (87235) (-41433)

BE 2 -02779 00924 21663 05164 -00333

(-57-06) (-461) (193243) (3971) (-3425)

FI 3 00649 01184 22306 07454 -00026

(-677) (-4573) (197254) (55101) (-353)

FR 4 -02968 00977 12154 0578 -00166

(-6-05) (-2446) (111133) (4672) (-1511)

GE 5 -02357 00503 14882 05187 00129

(-46-05) (-338) (137162) (465) (-1415)

GR 6 0112 00451 22352 12036 -00777

(-7398) (-7485) (188263) (87157) (-8472)

IR 7 09287 13359 1617 14704 -00663

(-419617) (-299587) (91269) (88241) (-551491)

IT 8 -00771 00687 14874 07886 00347

(-6246) (-4859) (132168) (58103) (-2531)

LU 9 -0226 00602 32394 09625 00592

(-9546) (-6171) (29359) (68127) (-5667)

NE 10 01114 00366 25878 06758 -00144

(-352) (-3845) (24278) (589) (-2725)

PO 11 -26537 09577 1662 1431 -01303

(-789281) (-28486) (96271) (88233) (-47456)

SP 12 01082 00542 15429 06671 00041

(-3151) (-3447) (136173) (5285) (-221)












spread
















Regime 3

Country i

AU 1 03668 00344 18541 07307 -0027

(0484) (-451) (17202) (5297) (-2924)

BE 2 01099 00857 21417 05782 -00143

(0129) (-2542) (198232) (4672) (-1815)

FI 3 20781 00295 32925 14981 0273

(103588) (-137168) (1448) (105209) (-282328)

FR 4 02657 01885 17246 08334 -004

(01101) (-3983) (152) (65108) (-4236)

GE 5 0223 01074 19156 06455 004

(0164) (-3557) (17221) (528) (-1726)

GR 6 37886 08778 16864 15113 -00768

(132734) (-263509) (95283) (925) (-435401)

IR 7 73548 14607 14935 1496 -01523

(2641255) (-331623) (88247) (89245) (-427465)

IT 8 29927 12244 15941 15433 01307

(11668) (-228532) (93261) (9125) (-436466)

LU 9 54322 09447 16577 14504 -00798

(1771001) (-363571) (91299) (89239) (-53491)

NE 10 43871 09305 15606 14971 02215

(1384) (-386611) (88271) (89253) (-419455)

PO 11 08666 00554 25702 08614 -00181

(17137) (-4659) (234282) (64111) (-4237)

SP 12 30198 0508 15576 14046 01818

(109661) (-28942) (9426) (86237) (-318345)

Conclusions 5








Conclusions 5




Conclusions 5





Conclusions 5




Conclusions 5




Conclusions 5




References 6






University Press
































References 6





86723ndash734










Semptember 2000




Working paper 14292









Research London








Main title

2015 edition




KS-TC-16-016-EN-N

ISBN 978-92-79-61459-0


Table of contents

Abstract

1 Introduction




5 Conclusions

6 References



33 Gibbs sampler










119901(119910|Ξ γ Σ) = (2120587)minus119879119870119873


2 exp minus1

2sum 119854119905

prime Σ119905minus1119854119905

119879119905=1 119879

119905=1 (15)







119879119870

2119873119894=1 prod |Σ119905|minus

1

2 exp minus1

2sum 119854119894119905

prime Σ119894119905minus1119854119894119905

119879119905=1 119879

119905=1 (16)

where 119962119894prime = (1199621198941




119883119894119905 = (

1198831198940119905 1198831198941119905 0

⋮ ⋱1198831198940119905 0 119883119894119872119905

)




function

119891(1206321198940|119962119894 Ξ119894 γ119894 Σ119894) prop (17)

prop exp minus1



1

2(120632119946120782 minus 120632119946120782)119873

119894=1 Σ1198940minus1(120632119946120782 minus 120632119946120782)

prop exp minus1

21206321198940

prime (sum 119935119946120782119957prime 120506119957

minus120783119935119946120782119957 + 120506119957minus120783119879

119905=1 )120632119946120782 + 120632119946120782 (sum 119935119946120782119957prime 120506119946119957

minus120783119962119946120782119957 + 120506119946120782minus120783120632119946120782

119879119905=1 )

prop 1199771198700(120632

119946120782 120506119946120782)



where 120632119946120782

= 120506119946120782

minus120783(120506119946120782

minus120783120632119946120782 + sum 119935119946120782119957prime 120506119957

minus120783119935119946120782119957119879119905=1 ) and 120506119946120782

minus120783= (120506119946120782

minus120783120632119946120782 + sum 119935119946120782119957prime 120506119957

minus120783119935119946120782119957119879119905=1


density function

119891(120632119894119898|119962119894 Ξ119894 γ1198940 γ119894(minus119898) Σ) prop (18)

prop exp minus1

2sum 119854119894119905

prime Σ119905 119854119894119905 minus1

2(120632119946119950 minus 120632119946119950)prime119905isin119983119894119898

Σ119894119898minus1(120632119946119950 minus 120632119946119950)

prop exp minus1

21206321198941

prime (sum 119935119946119950119957prime 120506119957

minus120783119935119946119950119957 + 120506119946119950minus120783

119905isin119983119894119898)120632119946 + 120632119946

prime (sum 119935119946119950119957prime 120506119946119957

minus120783119962119946119950119957 + 120506119946119950minus120783120632119946119950119905isin119983119894119898

)

prop 119977119870119872(120632

119946119950 120506119946119950) (19)

where 120632119946119950

= 120506119946119950

minus120783(120506119946119950

minus120783120632119946119950 + sum 119935119946119950119957prime 120506119946119957

minus120783119935119946119950119957119905isin119983119894119898) and 120506119946119950

minus120783= (120506119946119950

minus120783 + sum 119935119946119950119957prime 120506119957


119983119894119898 = 119905 = 1 hellip 119879|120585119894119898119905 = 1 and 119962119946119950119957 = 119962119946119957 minus 119935119946120782119957120632119946120782



119891(120622119894119898|119962119894 Ξ119894 γ1198940 γ119894) Σ119894(minus119898)) prop (20)


1

2| exp minus1

2sum 119854119894119905


|119879119905=1 Σ119894119898

minus1|120584119894119898+119870+1

2 exp minus1

2119905119903(120566119894119898Σ119894119898)

prop |Σ119894119898minus1|

120584119894119898+119879119894119898+119870+1

2 exp minus1

2119905119903 ((120566119894119898 + sum 119854119894119898119905119854119894119898119905

prime119905isin119983119894119898

)120506119946119950minus120783)

prop 119986119870(1205841198941198982 1205661198941198982) (21)


prime119905isin119983119894119898


119891(119901119894119896|119962119894 Ξ119894 γ1198940 γ119894) prop prod 119901119894119896119895

119889119895119872119895=1 prod prod 119901

119894119896119895

120585119894119895119905120585119894119896119905119872119898=1

119879119905=1 (22)

prop 119967(1198891 + 1198731198941198961 hellip 119889119872 + 119873119894119896119872)

where

119873119894119896119872 = sum 120128(119904119894119905 = 119895)120128(119904119894119905minus1 = 119896)119879

119905=1














a map 120601 ∶ ∆[01]119872119873 rarr ∆[01]119872

120636119905 = 120601 (1206361119905 hellip 120636119873119905) (23)



Equal weights

Let



120636119896119905 =1

119873sum 120575119896

119873119894=1 (119894119905) (24)





120636119896119905 = sum 120596119894119905120575119896119873119894=1 (119894119905) (25)














41 Data Description
















December 2010



Austria 1960M01 1989M06 1990M01

Belgium 1960M01 1960M01 1960M01

Finland 1960M01 1987M01 1988M01

France 1960M01 1970M01 1960M01

Germany 1960M01 1960M01 1960M01

Greece 1962M01 1997M06 2001M01

Ireland 1975M07 1984M01 1970M12

Italy 1960M01 1978M10 1991M03

Luxembourg 1960M01 1999M01 1993M10


Portugal 1960M01 1992M01 1993M07

Spain 1965M01 1977M01 1980M01























1205791 =1

1198991sum 120579(119895)1198991

119895=1 1205792 =1

1198992sum 120579(119895)1198991

119895=119899+1minus1198992


1198942

119899119894

= Γ(0) + 2119899119894

119899119894 minus 1sum 119870(119895ℎ119894)Γ(119895)

ℎ119894

119895=1

Γ(119895) =1

119899119894


119896=119895+1



119862119863 =1minus2


21198992

(26)






































Regime 1

Country i

AU 1 -20242 08453 1526 14309 00283

(-468-19) (-361564) (9251) (87237) (-413346)

BE 2 -01438 00127 5213 11209 -05786

(-39-01) (-5458) (452608) (89141) (-18264)

FI 3 -35669 01015 35118 13203 -0331

(-893-24) (-168213) (141564) (78211) (-414346)

FR 4 -02063 -00703 1601 10404 -00537

(-53-01) (-8966) (13197) (74143) (-7161)

GE 5 -01704 0046 29743 10345 -0253

(-41-01) (-781) (227364) (75142) (-12367)

GR 6 -41517 1013 15835 14095 -00286

(-724-124) (-177379) (92265) (85231) (-405412)

IR 7 -02211 00583 54996 1008 -00973

(-62-02) (-6478) (5608) (74133) (-1089)

IT 8 -36763 02934 17706 13322 -01696

(-633-11) (-207267) (114266) (8321) (-31424)

LU 9 -91082 0277 19724 12999 00593

(-1243-556) (-246299) (11336) (84202) (-556572)

NE 10 -46717 06634 15039 1463 00532

(-952-67) (-375521) (9251) (87237) (-445502)

PO 11 -47046 06718 17238 13152 -02942

(-814-142) (-189328) (102298) (83207) (-

433335)

SP 12 -05913 -00935 27753 11279 -02135

(-132-05) (-8765) (233328) (82152) (-128)







43 Turning Points













Regime 2

Country i

AU 1 -35019 08714 15161 14219 -00547

(-679-104) (-312581) (92249) (87235) (-41433)

BE 2 -02779 00924 21663 05164 -00333

(-57-06) (-461) (193243) (3971) (-3425)

FI 3 00649 01184 22306 07454 -00026

(-677) (-4573) (197254) (55101) (-353)

FR 4 -02968 00977 12154 0578 -00166

(-6-05) (-2446) (111133) (4672) (-1511)

GE 5 -02357 00503 14882 05187 00129

(-46-05) (-338) (137162) (465) (-1415)

GR 6 0112 00451 22352 12036 -00777

(-7398) (-7485) (188263) (87157) (-8472)

IR 7 09287 13359 1617 14704 -00663

(-419617) (-299587) (91269) (88241) (-551491)

IT 8 -00771 00687 14874 07886 00347

(-6246) (-4859) (132168) (58103) (-2531)

LU 9 -0226 00602 32394 09625 00592

(-9546) (-6171) (29359) (68127) (-5667)

NE 10 01114 00366 25878 06758 -00144

(-352) (-3845) (24278) (589) (-2725)

PO 11 -26537 09577 1662 1431 -01303

(-789281) (-28486) (96271) (88233) (-47456)

SP 12 01082 00542 15429 06671 00041

(-3151) (-3447) (136173) (5285) (-221)












spread
















Regime 3

Country i

AU 1 03668 00344 18541 07307 -0027

(0484) (-451) (17202) (5297) (-2924)

BE 2 01099 00857 21417 05782 -00143

(0129) (-2542) (198232) (4672) (-1815)

FI 3 20781 00295 32925 14981 0273

(103588) (-137168) (1448) (105209) (-282328)

FR 4 02657 01885 17246 08334 -004

(01101) (-3983) (152) (65108) (-4236)

GE 5 0223 01074 19156 06455 004

(0164) (-3557) (17221) (528) (-1726)

GR 6 37886 08778 16864 15113 -00768

(132734) (-263509) (95283) (925) (-435401)

IR 7 73548 14607 14935 1496 -01523

(2641255) (-331623) (88247) (89245) (-427465)

IT 8 29927 12244 15941 15433 01307

(11668) (-228532) (93261) (9125) (-436466)

LU 9 54322 09447 16577 14504 -00798

(1771001) (-363571) (91299) (89239) (-53491)

NE 10 43871 09305 15606 14971 02215

(1384) (-386611) (88271) (89253) (-419455)

PO 11 08666 00554 25702 08614 -00181

(17137) (-4659) (234282) (64111) (-4237)

SP 12 30198 0508 15576 14046 01818

(109661) (-28942) (9426) (86237) (-318345)

Conclusions 5








Conclusions 5




Conclusions 5





Conclusions 5




Conclusions 5




Conclusions 5




References 6






University Press
































References 6





86723ndash734










Semptember 2000




Working paper 14292









Research London








Main title

2015 edition




KS-TC-16-016-EN-N

ISBN 978-92-79-61459-0


Table of contents

Abstract

1 Introduction




5 Conclusions

6 References



where 120632119946120782

= 120506119946120782

minus120783(120506119946120782

minus120783120632119946120782 + sum 119935119946120782119957prime 120506119957

minus120783119935119946120782119957119879119905=1 ) and 120506119946120782

minus120783= (120506119946120782

minus120783120632119946120782 + sum 119935119946120782119957prime 120506119957

minus120783119935119946120782119957119879119905=1


density function

119891(120632119894119898|119962119894 Ξ119894 γ1198940 γ119894(minus119898) Σ) prop (18)

prop exp minus1

2sum 119854119894119905

prime Σ119905 119854119894119905 minus1

2(120632119946119950 minus 120632119946119950)prime119905isin119983119894119898

Σ119894119898minus1(120632119946119950 minus 120632119946119950)

prop exp minus1

21206321198941

prime (sum 119935119946119950119957prime 120506119957

minus120783119935119946119950119957 + 120506119946119950minus120783

119905isin119983119894119898)120632119946 + 120632119946

prime (sum 119935119946119950119957prime 120506119946119957

minus120783119962119946119950119957 + 120506119946119950minus120783120632119946119950119905isin119983119894119898

)

prop 119977119870119872(120632

119946119950 120506119946119950) (19)

where 120632119946119950

= 120506119946119950

minus120783(120506119946119950

minus120783120632119946119950 + sum 119935119946119950119957prime 120506119946119957

minus120783119935119946119950119957119905isin119983119894119898) and 120506119946119950

minus120783= (120506119946119950

minus120783 + sum 119935119946119950119957prime 120506119957


119983119894119898 = 119905 = 1 hellip 119879|120585119894119898119905 = 1 and 119962119946119950119957 = 119962119946119957 minus 119935119946120782119957120632119946120782



119891(120622119894119898|119962119894 Ξ119894 γ1198940 γ119894) Σ119894(minus119898)) prop (20)


1

2| exp minus1

2sum 119854119894119905


|119879119905=1 Σ119894119898

minus1|120584119894119898+119870+1

2 exp minus1

2119905119903(120566119894119898Σ119894119898)

prop |Σ119894119898minus1|

120584119894119898+119879119894119898+119870+1

2 exp minus1

2119905119903 ((120566119894119898 + sum 119854119894119898119905119854119894119898119905

prime119905isin119983119894119898

)120506119946119950minus120783)

prop 119986119870(1205841198941198982 1205661198941198982) (21)


prime119905isin119983119894119898


119891(119901119894119896|119962119894 Ξ119894 γ1198940 γ119894) prop prod 119901119894119896119895

119889119895119872119895=1 prod prod 119901

119894119896119895

120585119894119895119905120585119894119896119905119872119898=1

119879119905=1 (22)

prop 119967(1198891 + 1198731198941198961 hellip 119889119872 + 119873119894119896119872)

where

119873119894119896119872 = sum 120128(119904119894119905 = 119895)120128(119904119894119905minus1 = 119896)119879

119905=1














a map 120601 ∶ ∆[01]119872119873 rarr ∆[01]119872

120636119905 = 120601 (1206361119905 hellip 120636119873119905) (23)



Equal weights

Let



120636119896119905 =1

119873sum 120575119896

119873119894=1 (119894119905) (24)





120636119896119905 = sum 120596119894119905120575119896119873119894=1 (119894119905) (25)














41 Data Description
















December 2010



Austria 1960M01 1989M06 1990M01

Belgium 1960M01 1960M01 1960M01

Finland 1960M01 1987M01 1988M01

France 1960M01 1970M01 1960M01

Germany 1960M01 1960M01 1960M01

Greece 1962M01 1997M06 2001M01

Ireland 1975M07 1984M01 1970M12

Italy 1960M01 1978M10 1991M03

Luxembourg 1960M01 1999M01 1993M10


Portugal 1960M01 1992M01 1993M07

Spain 1965M01 1977M01 1980M01























1205791 =1

1198991sum 120579(119895)1198991

119895=1 1205792 =1

1198992sum 120579(119895)1198991

119895=119899+1minus1198992


1198942

119899119894

= Γ(0) + 2119899119894

119899119894 minus 1sum 119870(119895ℎ119894)Γ(119895)

ℎ119894

119895=1

Γ(119895) =1

119899119894


119896=119895+1



119862119863 =1minus2


21198992

(26)






































Regime 1

Country i

AU 1 -20242 08453 1526 14309 00283

(-468-19) (-361564) (9251) (87237) (-413346)

BE 2 -01438 00127 5213 11209 -05786

(-39-01) (-5458) (452608) (89141) (-18264)

FI 3 -35669 01015 35118 13203 -0331

(-893-24) (-168213) (141564) (78211) (-414346)

FR 4 -02063 -00703 1601 10404 -00537

(-53-01) (-8966) (13197) (74143) (-7161)

GE 5 -01704 0046 29743 10345 -0253

(-41-01) (-781) (227364) (75142) (-12367)

GR 6 -41517 1013 15835 14095 -00286

(-724-124) (-177379) (92265) (85231) (-405412)

IR 7 -02211 00583 54996 1008 -00973

(-62-02) (-6478) (5608) (74133) (-1089)

IT 8 -36763 02934 17706 13322 -01696

(-633-11) (-207267) (114266) (8321) (-31424)

LU 9 -91082 0277 19724 12999 00593

(-1243-556) (-246299) (11336) (84202) (-556572)

NE 10 -46717 06634 15039 1463 00532

(-952-67) (-375521) (9251) (87237) (-445502)

PO 11 -47046 06718 17238 13152 -02942

(-814-142) (-189328) (102298) (83207) (-

433335)

SP 12 -05913 -00935 27753 11279 -02135

(-132-05) (-8765) (233328) (82152) (-128)







43 Turning Points













Regime 2

Country i

AU 1 -35019 08714 15161 14219 -00547

(-679-104) (-312581) (92249) (87235) (-41433)

BE 2 -02779 00924 21663 05164 -00333

(-57-06) (-461) (193243) (3971) (-3425)

FI 3 00649 01184 22306 07454 -00026

(-677) (-4573) (197254) (55101) (-353)

FR 4 -02968 00977 12154 0578 -00166

(-6-05) (-2446) (111133) (4672) (-1511)

GE 5 -02357 00503 14882 05187 00129

(-46-05) (-338) (137162) (465) (-1415)

GR 6 0112 00451 22352 12036 -00777

(-7398) (-7485) (188263) (87157) (-8472)

IR 7 09287 13359 1617 14704 -00663

(-419617) (-299587) (91269) (88241) (-551491)

IT 8 -00771 00687 14874 07886 00347

(-6246) (-4859) (132168) (58103) (-2531)

LU 9 -0226 00602 32394 09625 00592

(-9546) (-6171) (29359) (68127) (-5667)

NE 10 01114 00366 25878 06758 -00144

(-352) (-3845) (24278) (589) (-2725)

PO 11 -26537 09577 1662 1431 -01303

(-789281) (-28486) (96271) (88233) (-47456)

SP 12 01082 00542 15429 06671 00041

(-3151) (-3447) (136173) (5285) (-221)












spread
















Regime 3

Country i

AU 1 03668 00344 18541 07307 -0027

(0484) (-451) (17202) (5297) (-2924)

BE 2 01099 00857 21417 05782 -00143

(0129) (-2542) (198232) (4672) (-1815)

FI 3 20781 00295 32925 14981 0273

(103588) (-137168) (1448) (105209) (-282328)

FR 4 02657 01885 17246 08334 -004

(01101) (-3983) (152) (65108) (-4236)

GE 5 0223 01074 19156 06455 004

(0164) (-3557) (17221) (528) (-1726)

GR 6 37886 08778 16864 15113 -00768

(132734) (-263509) (95283) (925) (-435401)

IR 7 73548 14607 14935 1496 -01523

(2641255) (-331623) (88247) (89245) (-427465)

IT 8 29927 12244 15941 15433 01307

(11668) (-228532) (93261) (9125) (-436466)

LU 9 54322 09447 16577 14504 -00798

(1771001) (-363571) (91299) (89239) (-53491)

NE 10 43871 09305 15606 14971 02215

(1384) (-386611) (88271) (89253) (-419455)

PO 11 08666 00554 25702 08614 -00181

(17137) (-4659) (234282) (64111) (-4237)

SP 12 30198 0508 15576 14046 01818

(109661) (-28942) (9426) (86237) (-318345)

Conclusions 5








Conclusions 5




Conclusions 5





Conclusions 5




Conclusions 5




Conclusions 5




References 6






University Press
































References 6





86723ndash734










Semptember 2000




Working paper 14292









Research London








Main title

2015 edition




KS-TC-16-016-EN-N

ISBN 978-92-79-61459-0


Table of contents

Abstract

1 Introduction




5 Conclusions

6 References













a map 120601 ∶ ∆[01]119872119873 rarr ∆[01]119872

120636119905 = 120601 (1206361119905 hellip 120636119873119905) (23)



Equal weights

Let



120636119896119905 =1

119873sum 120575119896

119873119894=1 (119894119905) (24)





120636119896119905 = sum 120596119894119905120575119896119873119894=1 (119894119905) (25)














41 Data Description
















December 2010



Austria 1960M01 1989M06 1990M01

Belgium 1960M01 1960M01 1960M01

Finland 1960M01 1987M01 1988M01

France 1960M01 1970M01 1960M01

Germany 1960M01 1960M01 1960M01

Greece 1962M01 1997M06 2001M01

Ireland 1975M07 1984M01 1970M12

Italy 1960M01 1978M10 1991M03

Luxembourg 1960M01 1999M01 1993M10


Portugal 1960M01 1992M01 1993M07

Spain 1965M01 1977M01 1980M01























1205791 =1

1198991sum 120579(119895)1198991

119895=1 1205792 =1

1198992sum 120579(119895)1198991

119895=119899+1minus1198992


1198942

119899119894

= Γ(0) + 2119899119894

119899119894 minus 1sum 119870(119895ℎ119894)Γ(119895)

ℎ119894

119895=1

Γ(119895) =1

119899119894


119896=119895+1



119862119863 =1minus2


21198992

(26)






































Regime 1

Country i

AU 1 -20242 08453 1526 14309 00283

(-468-19) (-361564) (9251) (87237) (-413346)

BE 2 -01438 00127 5213 11209 -05786

(-39-01) (-5458) (452608) (89141) (-18264)

FI 3 -35669 01015 35118 13203 -0331

(-893-24) (-168213) (141564) (78211) (-414346)

FR 4 -02063 -00703 1601 10404 -00537

(-53-01) (-8966) (13197) (74143) (-7161)

GE 5 -01704 0046 29743 10345 -0253

(-41-01) (-781) (227364) (75142) (-12367)

GR 6 -41517 1013 15835 14095 -00286

(-724-124) (-177379) (92265) (85231) (-405412)

IR 7 -02211 00583 54996 1008 -00973

(-62-02) (-6478) (5608) (74133) (-1089)

IT 8 -36763 02934 17706 13322 -01696

(-633-11) (-207267) (114266) (8321) (-31424)

LU 9 -91082 0277 19724 12999 00593

(-1243-556) (-246299) (11336) (84202) (-556572)

NE 10 -46717 06634 15039 1463 00532

(-952-67) (-375521) (9251) (87237) (-445502)

PO 11 -47046 06718 17238 13152 -02942

(-814-142) (-189328) (102298) (83207) (-

433335)

SP 12 -05913 -00935 27753 11279 -02135

(-132-05) (-8765) (233328) (82152) (-128)







43 Turning Points













Regime 2

Country i

AU 1 -35019 08714 15161 14219 -00547

(-679-104) (-312581) (92249) (87235) (-41433)

BE 2 -02779 00924 21663 05164 -00333

(-57-06) (-461) (193243) (3971) (-3425)

FI 3 00649 01184 22306 07454 -00026

(-677) (-4573) (197254) (55101) (-353)

FR 4 -02968 00977 12154 0578 -00166

(-6-05) (-2446) (111133) (4672) (-1511)

GE 5 -02357 00503 14882 05187 00129

(-46-05) (-338) (137162) (465) (-1415)

GR 6 0112 00451 22352 12036 -00777

(-7398) (-7485) (188263) (87157) (-8472)

IR 7 09287 13359 1617 14704 -00663

(-419617) (-299587) (91269) (88241) (-551491)

IT 8 -00771 00687 14874 07886 00347

(-6246) (-4859) (132168) (58103) (-2531)

LU 9 -0226 00602 32394 09625 00592

(-9546) (-6171) (29359) (68127) (-5667)

NE 10 01114 00366 25878 06758 -00144

(-352) (-3845) (24278) (589) (-2725)

PO 11 -26537 09577 1662 1431 -01303

(-789281) (-28486) (96271) (88233) (-47456)

SP 12 01082 00542 15429 06671 00041

(-3151) (-3447) (136173) (5285) (-221)












spread
















Regime 3

Country i

AU 1 03668 00344 18541 07307 -0027

(0484) (-451) (17202) (5297) (-2924)

BE 2 01099 00857 21417 05782 -00143

(0129) (-2542) (198232) (4672) (-1815)

FI 3 20781 00295 32925 14981 0273

(103588) (-137168) (1448) (105209) (-282328)

FR 4 02657 01885 17246 08334 -004

(01101) (-3983) (152) (65108) (-4236)

GE 5 0223 01074 19156 06455 004

(0164) (-3557) (17221) (528) (-1726)

GR 6 37886 08778 16864 15113 -00768

(132734) (-263509) (95283) (925) (-435401)

IR 7 73548 14607 14935 1496 -01523

(2641255) (-331623) (88247) (89245) (-427465)

IT 8 29927 12244 15941 15433 01307

(11668) (-228532) (93261) (9125) (-436466)

LU 9 54322 09447 16577 14504 -00798

(1771001) (-363571) (91299) (89239) (-53491)

NE 10 43871 09305 15606 14971 02215

(1384) (-386611) (88271) (89253) (-419455)

PO 11 08666 00554 25702 08614 -00181

(17137) (-4659) (234282) (64111) (-4237)

SP 12 30198 0508 15576 14046 01818

(109661) (-28942) (9426) (86237) (-318345)

Conclusions 5








Conclusions 5




Conclusions 5





Conclusions 5




Conclusions 5




Conclusions 5




References 6






University Press
































References 6





86723ndash734










Semptember 2000




Working paper 14292









Research London








Main title

2015 edition




KS-TC-16-016-EN-N

ISBN 978-92-79-61459-0


Table of contents

Abstract

1 Introduction




5 Conclusions

6 References














41 Data Description
















December 2010



Austria 1960M01 1989M06 1990M01

Belgium 1960M01 1960M01 1960M01

Finland 1960M01 1987M01 1988M01

France 1960M01 1970M01 1960M01

Germany 1960M01 1960M01 1960M01

Greece 1962M01 1997M06 2001M01

Ireland 1975M07 1984M01 1970M12

Italy 1960M01 1978M10 1991M03

Luxembourg 1960M01 1999M01 1993M10


Portugal 1960M01 1992M01 1993M07

Spain 1965M01 1977M01 1980M01























1205791 =1

1198991sum 120579(119895)1198991

119895=1 1205792 =1

1198992sum 120579(119895)1198991

119895=119899+1minus1198992


1198942

119899119894

= Γ(0) + 2119899119894

119899119894 minus 1sum 119870(119895ℎ119894)Γ(119895)

ℎ119894

119895=1

Γ(119895) =1

119899119894


119896=119895+1



119862119863 =1minus2


21198992

(26)






































Regime 1

Country i

AU 1 -20242 08453 1526 14309 00283

(-468-19) (-361564) (9251) (87237) (-413346)

BE 2 -01438 00127 5213 11209 -05786

(-39-01) (-5458) (452608) (89141) (-18264)

FI 3 -35669 01015 35118 13203 -0331

(-893-24) (-168213) (141564) (78211) (-414346)

FR 4 -02063 -00703 1601 10404 -00537

(-53-01) (-8966) (13197) (74143) (-7161)

GE 5 -01704 0046 29743 10345 -0253

(-41-01) (-781) (227364) (75142) (-12367)

GR 6 -41517 1013 15835 14095 -00286

(-724-124) (-177379) (92265) (85231) (-405412)

IR 7 -02211 00583 54996 1008 -00973

(-62-02) (-6478) (5608) (74133) (-1089)

IT 8 -36763 02934 17706 13322 -01696

(-633-11) (-207267) (114266) (8321) (-31424)

LU 9 -91082 0277 19724 12999 00593

(-1243-556) (-246299) (11336) (84202) (-556572)

NE 10 -46717 06634 15039 1463 00532

(-952-67) (-375521) (9251) (87237) (-445502)

PO 11 -47046 06718 17238 13152 -02942

(-814-142) (-189328) (102298) (83207) (-

433335)

SP 12 -05913 -00935 27753 11279 -02135

(-132-05) (-8765) (233328) (82152) (-128)







43 Turning Points













Regime 2

Country i

AU 1 -35019 08714 15161 14219 -00547

(-679-104) (-312581) (92249) (87235) (-41433)

BE 2 -02779 00924 21663 05164 -00333

(-57-06) (-461) (193243) (3971) (-3425)

FI 3 00649 01184 22306 07454 -00026

(-677) (-4573) (197254) (55101) (-353)

FR 4 -02968 00977 12154 0578 -00166

(-6-05) (-2446) (111133) (4672) (-1511)

GE 5 -02357 00503 14882 05187 00129

(-46-05) (-338) (137162) (465) (-1415)

GR 6 0112 00451 22352 12036 -00777

(-7398) (-7485) (188263) (87157) (-8472)

IR 7 09287 13359 1617 14704 -00663

(-419617) (-299587) (91269) (88241) (-551491)

IT 8 -00771 00687 14874 07886 00347

(-6246) (-4859) (132168) (58103) (-2531)

LU 9 -0226 00602 32394 09625 00592

(-9546) (-6171) (29359) (68127) (-5667)

NE 10 01114 00366 25878 06758 -00144

(-352) (-3845) (24278) (589) (-2725)

PO 11 -26537 09577 1662 1431 -01303

(-789281) (-28486) (96271) (88233) (-47456)

SP 12 01082 00542 15429 06671 00041

(-3151) (-3447) (136173) (5285) (-221)












spread
















Regime 3

Country i

AU 1 03668 00344 18541 07307 -0027

(0484) (-451) (17202) (5297) (-2924)

BE 2 01099 00857 21417 05782 -00143

(0129) (-2542) (198232) (4672) (-1815)

FI 3 20781 00295 32925 14981 0273

(103588) (-137168) (1448) (105209) (-282328)

FR 4 02657 01885 17246 08334 -004

(01101) (-3983) (152) (65108) (-4236)

GE 5 0223 01074 19156 06455 004

(0164) (-3557) (17221) (528) (-1726)

GR 6 37886 08778 16864 15113 -00768

(132734) (-263509) (95283) (925) (-435401)

IR 7 73548 14607 14935 1496 -01523

(2641255) (-331623) (88247) (89245) (-427465)

IT 8 29927 12244 15941 15433 01307

(11668) (-228532) (93261) (9125) (-436466)

LU 9 54322 09447 16577 14504 -00798

(1771001) (-363571) (91299) (89239) (-53491)

NE 10 43871 09305 15606 14971 02215

(1384) (-386611) (88271) (89253) (-419455)

PO 11 08666 00554 25702 08614 -00181

(17137) (-4659) (234282) (64111) (-4237)

SP 12 30198 0508 15576 14046 01818

(109661) (-28942) (9426) (86237) (-318345)

Conclusions 5








Conclusions 5




Conclusions 5





Conclusions 5




Conclusions 5




Conclusions 5




References 6






University Press
































References 6





86723ndash734










Semptember 2000




Working paper 14292









Research London








Main title

2015 edition




KS-TC-16-016-EN-N

ISBN 978-92-79-61459-0


Table of contents

Abstract

1 Introduction




5 Conclusions

6 References















1205791 =1

1198991sum 120579(119895)1198991

119895=1 1205792 =1

1198992sum 120579(119895)1198991

119895=119899+1minus1198992


1198942

119899119894

= Γ(0) + 2119899119894

119899119894 minus 1sum 119870(119895ℎ119894)Γ(119895)

ℎ119894

119895=1

Γ(119895) =1

119899119894


119896=119895+1



119862119863 =1minus2


21198992

(26)






































Regime 1

Country i

AU 1 -20242 08453 1526 14309 00283

(-468-19) (-361564) (9251) (87237) (-413346)

BE 2 -01438 00127 5213 11209 -05786

(-39-01) (-5458) (452608) (89141) (-18264)

FI 3 -35669 01015 35118 13203 -0331

(-893-24) (-168213) (141564) (78211) (-414346)

FR 4 -02063 -00703 1601 10404 -00537

(-53-01) (-8966) (13197) (74143) (-7161)

GE 5 -01704 0046 29743 10345 -0253

(-41-01) (-781) (227364) (75142) (-12367)

GR 6 -41517 1013 15835 14095 -00286

(-724-124) (-177379) (92265) (85231) (-405412)

IR 7 -02211 00583 54996 1008 -00973

(-62-02) (-6478) (5608) (74133) (-1089)

IT 8 -36763 02934 17706 13322 -01696

(-633-11) (-207267) (114266) (8321) (-31424)

LU 9 -91082 0277 19724 12999 00593

(-1243-556) (-246299) (11336) (84202) (-556572)

NE 10 -46717 06634 15039 1463 00532

(-952-67) (-375521) (9251) (87237) (-445502)

PO 11 -47046 06718 17238 13152 -02942

(-814-142) (-189328) (102298) (83207) (-

433335)

SP 12 -05913 -00935 27753 11279 -02135

(-132-05) (-8765) (233328) (82152) (-128)







43 Turning Points













Regime 2

Country i

AU 1 -35019 08714 15161 14219 -00547

(-679-104) (-312581) (92249) (87235) (-41433)

BE 2 -02779 00924 21663 05164 -00333

(-57-06) (-461) (193243) (3971) (-3425)

FI 3 00649 01184 22306 07454 -00026

(-677) (-4573) (197254) (55101) (-353)

FR 4 -02968 00977 12154 0578 -00166

(-6-05) (-2446) (111133) (4672) (-1511)

GE 5 -02357 00503 14882 05187 00129

(-46-05) (-338) (137162) (465) (-1415)

GR 6 0112 00451 22352 12036 -00777

(-7398) (-7485) (188263) (87157) (-8472)

IR 7 09287 13359 1617 14704 -00663

(-419617) (-299587) (91269) (88241) (-551491)

IT 8 -00771 00687 14874 07886 00347

(-6246) (-4859) (132168) (58103) (-2531)

LU 9 -0226 00602 32394 09625 00592

(-9546) (-6171) (29359) (68127) (-5667)

NE 10 01114 00366 25878 06758 -00144

(-352) (-3845) (24278) (589) (-2725)

PO 11 -26537 09577 1662 1431 -01303

(-789281) (-28486) (96271) (88233) (-47456)

SP 12 01082 00542 15429 06671 00041

(-3151) (-3447) (136173) (5285) (-221)












spread
















Regime 3

Country i

AU 1 03668 00344 18541 07307 -0027

(0484) (-451) (17202) (5297) (-2924)

BE 2 01099 00857 21417 05782 -00143

(0129) (-2542) (198232) (4672) (-1815)

FI 3 20781 00295 32925 14981 0273

(103588) (-137168) (1448) (105209) (-282328)

FR 4 02657 01885 17246 08334 -004

(01101) (-3983) (152) (65108) (-4236)

GE 5 0223 01074 19156 06455 004

(0164) (-3557) (17221) (528) (-1726)

GR 6 37886 08778 16864 15113 -00768

(132734) (-263509) (95283) (925) (-435401)

IR 7 73548 14607 14935 1496 -01523

(2641255) (-331623) (88247) (89245) (-427465)

IT 8 29927 12244 15941 15433 01307

(11668) (-228532) (93261) (9125) (-436466)

LU 9 54322 09447 16577 14504 -00798

(1771001) (-363571) (91299) (89239) (-53491)

NE 10 43871 09305 15606 14971 02215

(1384) (-386611) (88271) (89253) (-419455)

PO 11 08666 00554 25702 08614 -00181

(17137) (-4659) (234282) (64111) (-4237)

SP 12 30198 0508 15576 14046 01818

(109661) (-28942) (9426) (86237) (-318345)

Conclusions 5








Conclusions 5




Conclusions 5





Conclusions 5




Conclusions 5




Conclusions 5




References 6






University Press
































References 6





86723ndash734










Semptember 2000




Working paper 14292









Research London








Main title

2015 edition




KS-TC-16-016-EN-N

ISBN 978-92-79-61459-0


Table of contents

Abstract

1 Introduction




5 Conclusions

6 References




























Regime 1

Country i

AU 1 -20242 08453 1526 14309 00283

(-468-19) (-361564) (9251) (87237) (-413346)

BE 2 -01438 00127 5213 11209 -05786

(-39-01) (-5458) (452608) (89141) (-18264)

FI 3 -35669 01015 35118 13203 -0331

(-893-24) (-168213) (141564) (78211) (-414346)

FR 4 -02063 -00703 1601 10404 -00537

(-53-01) (-8966) (13197) (74143) (-7161)

GE 5 -01704 0046 29743 10345 -0253

(-41-01) (-781) (227364) (75142) (-12367)

GR 6 -41517 1013 15835 14095 -00286

(-724-124) (-177379) (92265) (85231) (-405412)

IR 7 -02211 00583 54996 1008 -00973

(-62-02) (-6478) (5608) (74133) (-1089)

IT 8 -36763 02934 17706 13322 -01696

(-633-11) (-207267) (114266) (8321) (-31424)

LU 9 -91082 0277 19724 12999 00593

(-1243-556) (-246299) (11336) (84202) (-556572)

NE 10 -46717 06634 15039 1463 00532

(-952-67) (-375521) (9251) (87237) (-445502)

PO 11 -47046 06718 17238 13152 -02942

(-814-142) (-189328) (102298) (83207) (-

433335)

SP 12 -05913 -00935 27753 11279 -02135

(-132-05) (-8765) (233328) (82152) (-128)







43 Turning Points













Regime 2

Country i

AU 1 -35019 08714 15161 14219 -00547

(-679-104) (-312581) (92249) (87235) (-41433)

BE 2 -02779 00924 21663 05164 -00333

(-57-06) (-461) (193243) (3971) (-3425)

FI 3 00649 01184 22306 07454 -00026

(-677) (-4573) (197254) (55101) (-353)

FR 4 -02968 00977 12154 0578 -00166

(-6-05) (-2446) (111133) (4672) (-1511)

GE 5 -02357 00503 14882 05187 00129

(-46-05) (-338) (137162) (465) (-1415)

GR 6 0112 00451 22352 12036 -00777

(-7398) (-7485) (188263) (87157) (-8472)

IR 7 09287 13359 1617 14704 -00663

(-419617) (-299587) (91269) (88241) (-551491)

IT 8 -00771 00687 14874 07886 00347

(-6246) (-4859) (132168) (58103) (-2531)

LU 9 -0226 00602 32394 09625 00592

(-9546) (-6171) (29359) (68127) (-5667)

NE 10 01114 00366 25878 06758 -00144

(-352) (-3845) (24278) (589) (-2725)

PO 11 -26537 09577 1662 1431 -01303

(-789281) (-28486) (96271) (88233) (-47456)

SP 12 01082 00542 15429 06671 00041

(-3151) (-3447) (136173) (5285) (-221)












spread
















Regime 3

Country i

AU 1 03668 00344 18541 07307 -0027

(0484) (-451) (17202) (5297) (-2924)

BE 2 01099 00857 21417 05782 -00143

(0129) (-2542) (198232) (4672) (-1815)

FI 3 20781 00295 32925 14981 0273

(103588) (-137168) (1448) (105209) (-282328)

FR 4 02657 01885 17246 08334 -004

(01101) (-3983) (152) (65108) (-4236)

GE 5 0223 01074 19156 06455 004

(0164) (-3557) (17221) (528) (-1726)

GR 6 37886 08778 16864 15113 -00768

(132734) (-263509) (95283) (925) (-435401)

IR 7 73548 14607 14935 1496 -01523

(2641255) (-331623) (88247) (89245) (-427465)

IT 8 29927 12244 15941 15433 01307

(11668) (-228532) (93261) (9125) (-436466)

LU 9 54322 09447 16577 14504 -00798

(1771001) (-363571) (91299) (89239) (-53491)

NE 10 43871 09305 15606 14971 02215

(1384) (-386611) (88271) (89253) (-419455)

PO 11 08666 00554 25702 08614 -00181

(17137) (-4659) (234282) (64111) (-4237)

SP 12 30198 0508 15576 14046 01818

(109661) (-28942) (9426) (86237) (-318345)

Conclusions 5








Conclusions 5




Conclusions 5





Conclusions 5




Conclusions 5




Conclusions 5




References 6






University Press
































References 6





86723ndash734










Semptember 2000




Working paper 14292









Research London








Main title

2015 edition




KS-TC-16-016-EN-N

ISBN 978-92-79-61459-0


Table of contents

Abstract

1 Introduction




5 Conclusions

6 References











Regime 2

Country i

AU 1 -35019 08714 15161 14219 -00547

(-679-104) (-312581) (92249) (87235) (-41433)

BE 2 -02779 00924 21663 05164 -00333

(-57-06) (-461) (193243) (3971) (-3425)

FI 3 00649 01184 22306 07454 -00026

(-677) (-4573) (197254) (55101) (-353)

FR 4 -02968 00977 12154 0578 -00166

(-6-05) (-2446) (111133) (4672) (-1511)

GE 5 -02357 00503 14882 05187 00129

(-46-05) (-338) (137162) (465) (-1415)

GR 6 0112 00451 22352 12036 -00777

(-7398) (-7485) (188263) (87157) (-8472)

IR 7 09287 13359 1617 14704 -00663

(-419617) (-299587) (91269) (88241) (-551491)

IT 8 -00771 00687 14874 07886 00347

(-6246) (-4859) (132168) (58103) (-2531)

LU 9 -0226 00602 32394 09625 00592

(-9546) (-6171) (29359) (68127) (-5667)

NE 10 01114 00366 25878 06758 -00144

(-352) (-3845) (24278) (589) (-2725)

PO 11 -26537 09577 1662 1431 -01303

(-789281) (-28486) (96271) (88233) (-47456)

SP 12 01082 00542 15429 06671 00041

(-3151) (-3447) (136173) (5285) (-221)












spread
















Regime 3

Country i

AU 1 03668 00344 18541 07307 -0027

(0484) (-451) (17202) (5297) (-2924)

BE 2 01099 00857 21417 05782 -00143

(0129) (-2542) (198232) (4672) (-1815)

FI 3 20781 00295 32925 14981 0273

(103588) (-137168) (1448) (105209) (-282328)

FR 4 02657 01885 17246 08334 -004

(01101) (-3983) (152) (65108) (-4236)

GE 5 0223 01074 19156 06455 004

(0164) (-3557) (17221) (528) (-1726)

GR 6 37886 08778 16864 15113 -00768

(132734) (-263509) (95283) (925) (-435401)

IR 7 73548 14607 14935 1496 -01523

(2641255) (-331623) (88247) (89245) (-427465)

IT 8 29927 12244 15941 15433 01307

(11668) (-228532) (93261) (9125) (-436466)

LU 9 54322 09447 16577 14504 -00798

(1771001) (-363571) (91299) (89239) (-53491)

NE 10 43871 09305 15606 14971 02215

(1384) (-386611) (88271) (89253) (-419455)

PO 11 08666 00554 25702 08614 -00181

(17137) (-4659) (234282) (64111) (-4237)

SP 12 30198 0508 15576 14046 01818

(109661) (-28942) (9426) (86237) (-318345)

Conclusions 5








Conclusions 5




Conclusions 5





Conclusions 5




Conclusions 5




Conclusions 5




References 6






University Press
































References 6





86723ndash734










Semptember 2000




Working paper 14292









Research London








Main title

2015 edition




KS-TC-16-016-EN-N

ISBN 978-92-79-61459-0


Table of contents

Abstract

1 Introduction




5 Conclusions

6 References





spread
















Regime 3

Country i

AU 1 03668 00344 18541 07307 -0027

(0484) (-451) (17202) (5297) (-2924)

BE 2 01099 00857 21417 05782 -00143

(0129) (-2542) (198232) (4672) (-1815)

FI 3 20781 00295 32925 14981 0273

(103588) (-137168) (1448) (105209) (-282328)

FR 4 02657 01885 17246 08334 -004

(01101) (-3983) (152) (65108) (-4236)

GE 5 0223 01074 19156 06455 004

(0164) (-3557) (17221) (528) (-1726)

GR 6 37886 08778 16864 15113 -00768

(132734) (-263509) (95283) (925) (-435401)

IR 7 73548 14607 14935 1496 -01523

(2641255) (-331623) (88247) (89245) (-427465)

IT 8 29927 12244 15941 15433 01307

(11668) (-228532) (93261) (9125) (-436466)

LU 9 54322 09447 16577 14504 -00798

(1771001) (-363571) (91299) (89239) (-53491)

NE 10 43871 09305 15606 14971 02215

(1384) (-386611) (88271) (89253) (-419455)

PO 11 08666 00554 25702 08614 -00181

(17137) (-4659) (234282) (64111) (-4237)

SP 12 30198 0508 15576 14046 01818

(109661) (-28942) (9426) (86237) (-318345)

Conclusions 5








Conclusions 5




Conclusions 5





Conclusions 5




Conclusions 5




Conclusions 5




References 6






University Press
































References 6





86723ndash734










Semptember 2000




Working paper 14292









Research London








Main title

2015 edition




KS-TC-16-016-EN-N

ISBN 978-92-79-61459-0


Table of contents

Abstract

1 Introduction




5 Conclusions

6 References








Regime 3

Country i

AU 1 03668 00344 18541 07307 -0027

(0484) (-451) (17202) (5297) (-2924)

BE 2 01099 00857 21417 05782 -00143

(0129) (-2542) (198232) (4672) (-1815)

FI 3 20781 00295 32925 14981 0273

(103588) (-137168) (1448) (105209) (-282328)

FR 4 02657 01885 17246 08334 -004

(01101) (-3983) (152) (65108) (-4236)

GE 5 0223 01074 19156 06455 004

(0164) (-3557) (17221) (528) (-1726)

GR 6 37886 08778 16864 15113 -00768

(132734) (-263509) (95283) (925) (-435401)

IR 7 73548 14607 14935 1496 -01523

(2641255) (-331623) (88247) (89245) (-427465)

IT 8 29927 12244 15941 15433 01307

(11668) (-228532) (93261) (9125) (-436466)

LU 9 54322 09447 16577 14504 -00798

(1771001) (-363571) (91299) (89239) (-53491)

NE 10 43871 09305 15606 14971 02215

(1384) (-386611) (88271) (89253) (-419455)

PO 11 08666 00554 25702 08614 -00181

(17137) (-4659) (234282) (64111) (-4237)

SP 12 30198 0508 15576 14046 01818

(109661) (-28942) (9426) (86237) (-318345)

Conclusions 5








Conclusions 5




Conclusions 5





Conclusions 5




Conclusions 5




Conclusions 5




References 6






University Press
































References 6





86723ndash734










Semptember 2000




Working paper 14292









Research London








Main title

2015 edition




KS-TC-16-016-EN-N

ISBN 978-92-79-61459-0


Table of contents

Abstract

1 Introduction




5 Conclusions

6 References

Conclusions 5








Conclusions 5




Conclusions 5





Conclusions 5




Conclusions 5




Conclusions 5




References 6






University Press
































References 6





86723ndash734










Semptember 2000




Working paper 14292









Research London








Main title

2015 edition




KS-TC-16-016-EN-N

ISBN 978-92-79-61459-0


Table of contents

Abstract

1 Introduction




5 Conclusions

6 References

Conclusions 5




Conclusions 5





Conclusions 5




Conclusions 5




Conclusions 5




References 6






University Press
































References 6





86723ndash734










Semptember 2000




Working paper 14292









Research London








Main title

2015 edition




KS-TC-16-016-EN-N

ISBN 978-92-79-61459-0


Table of contents

Abstract

1 Introduction




5 Conclusions

6 References

Conclusions 5





Conclusions 5




Conclusions 5




Conclusions 5




References 6






University Press
































References 6





86723ndash734










Semptember 2000




Working paper 14292









Research London








Main title

2015 edition




KS-TC-16-016-EN-N

ISBN 978-92-79-61459-0


Table of contents

Abstract

1 Introduction




5 Conclusions

6 References

Conclusions 5




Conclusions 5




Conclusions 5




References 6






University Press
































References 6





86723ndash734










Semptember 2000




Working paper 14292









Research London








Main title

2015 edition




KS-TC-16-016-EN-N

ISBN 978-92-79-61459-0


Table of contents

Abstract

1 Introduction




5 Conclusions

6 References

Conclusions 5




Conclusions 5




References 6






University Press
































References 6





86723ndash734










Semptember 2000




Working paper 14292









Research London








Main title

2015 edition




KS-TC-16-016-EN-N

ISBN 978-92-79-61459-0


Table of contents

Abstract

1 Introduction




5 Conclusions

6 References

Conclusions 5




References 6






University Press
































References 6





86723ndash734










Semptember 2000




Working paper 14292









Research London








Main title

2015 edition




KS-TC-16-016-EN-N

ISBN 978-92-79-61459-0


Table of contents

Abstract

1 Introduction




5 Conclusions

6 References

References 6






University Press
































References 6





86723ndash734










Semptember 2000




Working paper 14292









Research London








Main title

2015 edition




KS-TC-16-016-EN-N

ISBN 978-92-79-61459-0


Table of contents

Abstract

1 Introduction




5 Conclusions

6 References

References 6





86723ndash734










Semptember 2000




Working paper 14292









Research London








Main title

2015 edition




KS-TC-16-016-EN-N

ISBN 978-92-79-61459-0


Table of contents

Abstract

1 Introduction




5 Conclusions

6 References

Main title

2015 edition




KS-TC-16-016-EN-N

ISBN 978-92-79-61459-0


Table of contents

Abstract

1 Introduction




5 Conclusions

6 References

Turning point detection with bayesian panel Markov ...€¦ · Turning point detection 4 Abstract...

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