DIRECTION GÉNÉRALE DES ÉTUDES ET DES RELATIONS INTERNATIONALES

National natural rates of interest and the single monetary policy in the Euro Area

Sébastien Fries1, Jean-Stéphane Mésonnier2, Sarah Mouabbi3 & Jean-Paul Renne4

Working Paper 611

December 2016

ABSTRACT

Using a semi-structural approach, we jointly estimate time-varying national natural real rates of interest for the largest four economies of the Euro area over 1999-2016 and discuss the associated challenges for the single monetary policy. We find evidence of an increased dispersion of real equilibrium rates across major Euro area economies during the Euro area sovereign debt crisis. This dispersion translated into significantly diverging national real interest rate gaps, a synthetic metrics of the monetary policy stance, between core and Southern countries, notably Spain. Real interest rate gaps have nonetheless converged towards zero in all four economies as of 2014, suggesting that it took the acceleration of unconventional policies since mid-2013 to eventually restore the conditions for a really common monetary policy in the Euro area.

Keywords: Euro area countries, natural rate of interest, common monetary policy, fragmentation.. JEL classification: C32; E32; E43; E52.

1 Crest, Paris-Saclay University (e-mail: sebastien.fries@ensae-paristech.fr ) 2 Banque de France (e-mail: jean-stephane.mesonnier@banque-france.fr ) 3 Banque de France (e-mail: sarah.mouabbi@banque-france.fr ) 4 HEC Lausanne (e-mail: jean-paul.renne@unil.ch )

Working Papers reflect the opinions of the authors and do not necessarily express the views of the Banque de France. This document is available on the Banque de France Website.

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NON-TECHNICAL SUMMARY5

The persistence of low growth and low long term real interest rates in major developed economies in the aftermath of the Great Recession has prompted a vivid debate about a possible downward shift in the level of the equilibrium, or “natural” real rate of interest (see, e.g., Laubach and Williams, 2015; Hamilton et al., 2015; Holston et al., 2016; Rachel and Smith, 2015). The natural rate of interest is commonly understood as the level of the medium-run real rate that equates investment and savings. It is conceptually an important benchmark for monetary policy. Diverging natural rates of interest across the major economies of a monetary union would therefore represent a major threat for the conduct and efficiency of the common monetary policy. Despite the importance of this issue within the Euro area, to date, measures of the national natural rate of interest in the Member countries are still lacking.

In this paper, we estimate time-varying national natural rates of interest for each of the largest four economies of the Euro area -Germany, France, Italy and Spain- since the inception of the Euro in 1999. We filter these unobserved variables conditionally to a joint stylized model of these four economies and their interactions. We find evidence that national natural rates of interest diverged markedly across core and periphery economies after the Lehman bankruptcy and notably at the height of the Euro area sovereign debt crisis. According to the derived measures of the national real interest rate gap, the difference between the local (one-year) ex ante real rate and the estimated natural rate, the single monetary policy proved relatively neutral in France, Germany and Italy over the post-2009 period (albeit turning briefly expansionary in core countries at the end of 2010 and restrictive in Italy in the second half of 2012), while it became quite disinflationary in Spain in late 2010 and kept tight for several years, in spite of the 3-year Long Term Refinancing Operations (LTROs) at the end of 2011. Eventually, all four real interest rate gaps converged back close to zero over 2014, witnessing a more neutral and consistent single policy stance across the zone as the Eurosystem embarked more decisively on unconventional programs.6

We define here the natural rate as the (one-year) real rate of interest which is consistent with output being at its potential once the effects of transitory demand and supply shocks have dissipated. Having in mind that any difference between output and its potential -i.e., the output gap- shall impinge sooner or later on inflation, this is tantamount to defining the natural rate as the real rate consistent with stable inflation in the medium-run, a definition that obviously echoes the operational objective of many central banks today. As soon as the natural rate of interest moves, reflecting the influence of technology, time preference or other real exogenous shocks, there is then a clear case for a stability-oriented monetary policy to try and track its fluctuations (see, e.g., Barsky et al., 2014).

Over the last fifteen years, various econometric approaches have been proposed to estimate this unobserved variable. Following Laubach and Williams (2003) and Mésonnier and Renne (2007), among others, we follow a semi-structural approach, whereby we use the Kalman filter to estimate the natural rate of interest as an unobserved variable which is conditional to a small dynamic,

5 We thank Jesper Linde (discussant), Julien Matheron and Pierre Sicsic for useful discussions, as well as seminar participants at Banque de France, European Central Bank and IAAE 2016 (Milano). The views expressed herein are those of the authors and do not necessarily reflect those of the Banque de France or the Eurosystem. 6 These measures include explicit forward guidance in mid-2013, the T-LTROs, CBPP and ABSPP in the Fall of 2014, then the EAPP in early 2015, i.e. the long-awaited large scale program of sovereign bond purchases, also dubbed QE.

Banque de France Working Paper #611 ii

backward-looking macroeconomic model of the economy, but without explicit microeconomic foundations. Compared to Mésonnier and Renne (2007), our main innovation is that we extend their framework to a joint model for the largest four economies in the Euro area, instead of the area itself. In order to account for the high degree of economic integration between these four EMU Members, we assume that each national economy is linked to the other three via two channels: a trade channel and a productivity channel. First, the level of the output gap in each country depends on the external demand for national products in the other countries, which we capture by including the lagged foreign output gaps scaled by the share of the foreign partner in the country’s exports. Second, we allow for cross-correlations of the shocks driving the (latent) trends of national potential output growth. All together, we end up having a relatively parsimonious model of a simplified Euro area made up of four countries that we estimate by maximum likelihood.

Our estimation yields time-varying measures of the unobserved, latent variables in the model: the national natural rates of interest and the derived interest rate gaps, the national output gaps and associated rates of potential output growth. We first find that the four national natural rates fluctuated widely over the last fifteen years. Before the crisis, they co-moved heavily with contained cross-sectional dispersion. National natural rates then dropped markedly after the collapse of Lehman Brothers and the ensuing aggravation of the worldwide financial crisis, down to some -3% in Spain, some -6% in Italy and France, and close to -7% in Germany. Although they bounced back rapidly thereafter, their trajectories diverged to some extent from 2010 to the beginning of 2014. National natural rates remained mostly in negative territory during the Euro area sovereign debt crisis. They nevertheless dropped to persistently lower levels in the two Southern economies, reaching a trough in the second half of 2012. National natural rates eventually re-converged in 2014 and stabilized to slightly negative levels, at around -1% in all economies since mid-2015.

Accordingly, estimated national interest rate gaps point to quite differentiated effective monetary policy stances across the four countries during the crisis period, but also to some extent before 2007, with a significantly negative interest rate gap in Spain that accounts for the strongly positive output gap and relatively high inflation in this country during the housing boom of the 2000s. Overall, our estimates suggest that real interest rate gaps ended up being almost neutral in Germany and France over the post-2009 period, but had significantly restrictive effects in Spain since the end of 2010 and up to 2014, and in Italy in 2011-2012.

Banque de France Working Paper #611 iii

RÉSUMÉ : TAUX D’INTÉRÊT NATUREL NATIONAUX ET POLITIQUE MONÉTAIRE UNIQUE DANS LA

ZONE EURO

Nous estimons les taux d'intérêt naturels réels nationaux des quatre plus grandes économies de la zone euro sur la période de 1999 à 2016 dans le cadre d'un modèle semi-structurel and présentons les défis associés pour la politique monétaire unique. Nous trouvons que la dispersion des taux d'intérêt réels d'équilibre à travers la zone euro s'est accrue pendant la crise de la dette souveraine. Cette dispersion a induit une divergence des écarts de taux d'intérêt réels, une mesure synthétique de l'orientation de la politique monétaire, entre d'une part les pays du cœur de la zone et d'autre part les pays du Sud, en particulier l'Espagne. Les écarts de taux réels ont toutefois convergé à nouveau vers zéro au cours de 2014 dans les quatre économies, ce qui suggère que l'intensification des mesures de politique monétaire non-conventionnelle à partir de la mi-2013 a contribué à restaurer les conditions d'une politique monétaire commune dans la zone euro.. Mots-clés : Zone euro, taux d'intérêt naturel, politique monétaire commune, fragmentation.

Banque de France Working Paper #611 iv

I. Introduction

The persistence of low growth and low long term real interest rates in major developedeconomies in the aftermath of the Great Recession has prompted a vivid debate about apossible downward shift in the level of the equilibrium, or “natural” real rate of inter-est (see, e.g., Laubach and Williams, 2015; Hamilton et al., 2015; Holston et al., 2016;Rachel and Smith, 2015). As the natural rate of interest is commonly understood as thelevel of the medium-run real rate that equates investment and savings, theories in supportof this view have pointed to the role of a series of possible factors likely to exacerbatethe current situation of excess savings. Some of these factors are more structural, like thepopulation ageing and the rise in income inequalities in the developed world, others aremore conjonctural, like the private debt overhang and the forced deleveraging of house-holds after the housing bust, as well as the aggressive fiscal retrenchments observed insome economies.1 As this (non-exhaustive) list highlights, several of these factors pos-sibly weighing down on the natural rate are likely to have a differentiated impact acrossthe national economies in the Euro area. The natural rate of interest is conceptually animportant benchmark for monetary policy. Diverging natural rates of interest across themajor economies of a monetary union would therefore represent a major threat for theconduct and efficiency of the common monetary policy. Despite the importance of thisissue within the Euro area, to date, measures of the national natural rate of interest in theMember countries are still lacking.

In this paper, we estimate time-varying national natural rates of interest for each ofthe largest four economies of the Euro area -Germany, France, Italy and Spain- since theinception of the Euro in 1999. We filter these unobserved variables conditionally to ajoint stylized model of these four economies and their interactions. We find evidence thatnational natural rates of interest diverged markedly across core and periphery economiesafter the Lehman bankruptcy and notably at the height of the Euro area sovereign debtcrisis. According to the derived measures of the national real interest rate gap, the dif-ference between the local (one-year) ex ante real rate and the estimated natural rate, thesingle monetary policy proved relatively neutral in France, Germany and Italy over thepost-2009 period (albeit turning briefly expansionary in core countries at the end of 2010and restrictive in Italy in the second half of 2012), while it became quite disinflationaryin Spain in late 2010 and kept tight for several years, in spite of the 3-year Long TermRefinancing Operations (LTROs) at the end of 2011. Eventually, all four real interestrate gaps converged back close to zero over 2014, witnessing a more neutral and consis-tent single policy stance across the zone as the Eurosystem embarked more decisively onunconventional programs.2

1see, e.g., Benigno et al. (2014) and Kocherlakota (2015) on the effects of public debt.2These measures include explicit forward guidance in mid-2013, the T-LTROs, CBPP and ABSPP in theFall of 2014, then the EAPP in early 2015, i.e. the long-awaited large scale program of sovereign bondpurchases, also dubbed QE.

1

2

The concept of a natural rate of interest and its role as a yardstick for monetary policydates back at least to Wicksell (1898[1936]), who defined it as the level of “the [real] rateof interest on loans which is neutral with respect to commodity prices and tends neither toincrease nor to decrease them”. After having vanished from macroeconomics textbooksfor decades, the notion resurfaced as a central concept in the modern monetary policytoolbox at the beginning of the last decade, mostly due to the success of its reinterpre-tation by Woodford (2003) within the canonical New-Keynesian (NK) framework. Inthe NK framework, the natural rate of interest is the equilibrium real rate of interest ina counter-factual version of the economy without any nominal rigidities: the equilibriumrate is therefore equal to the natural one when output is equal to its fully-flexible-priceequivalent.3 In a closely related, but less abstract strand, we define here the natural rateas the (one-year) real rate of interest which is consistent with output being at its potentialonce the effects of transitory demand and supply shocks have dissipated. Having in mindthat any difference between output and its potential -i.e., the output gap- shall impingesooner or later on inflation, this is tantamount to defining the natural rate as the real rateconsistent with stable inflation in the medium-run, a definition that obviously echoes theoperational objective of many central banks today. As soon as the natural rate of interestmoves, reflecting the influence of technology, time preference or other real exogenousshocks, there is then a clear case for a stability-oriented monetary policy to try and trackits fluctuations (see, e.g., Barsky et al., 2014).

Over the last fifteen years, various econometric approaches have been proposed to esti-mate this unobserved variable, ranging from univariate statistical filters extracting uncon-ditional trends of observed real interest rates to fully-fledged DSGE models. FollowingLaubach and Williams (2003) and Mesonnier and Renne (2007), among others, we strikea balance between these two extremes and follow a semi-structural approach, wherebywe use the Kalman filter to estimate the natural rate of interest as an unobserved variablewhich is conditional to a small dynamic, backward-looking macroeconomic model of theeconomy, but without explicit microeconomic foundations. The advantages of followingthis approach are threefold. First, the definition of the natural rate is economically in-tuitive and easy to interpret in terms of modern central banks’ mandates. Notably, ournatural rate tracks smooth changes in potential output growth. Potential growth is in turnpinned down by the linear relationship between the output gap and the lagged real interestgap along the estimated IS curve, while the output gap is itself constrained to be consis-tent with future inflation along the estimated Phillips curve. Second, as we do not needto render explicit or describe the reaction function of the central bank, we can model theobserved ex ante real rate of interest as an exogenous variable. This is particularly conve-nient in the context of recent years in the Euro area, as the conventional monetary policyinstrument, the repo rate of the ECB, has been constrained by its zero lower bound for

3See Woodford (2003) or Gali (2008) for a detailed presentation of the natural rate of interest in the canon-ical NK model.

3

several years now.4 In our specification, we assume that the real rate of interest which isrelevant for the private sector’s expenditures is the one-year ex ante default-free real rate,which we measure as the one-year EONIA swap rate deflated by inflation expectations ofprofessional forecasters. Note that the level of this observed real rate reflects both the levelof the current policy rate and the level of forward short term rates over the next 12 months.It can thus account for the additional degree of monetary accommodation obtained by theimplementation of non-conventional measures at the zero-lower bound (such as explicitforward guidance, long-term refinancing operations or outright asset purchases). Third,a semi-structural approach remains tractable in an open economy, multi-country set-up,even in the case of more than two countries. Compared to Mesonnier and Renne (2007),our main innovation is that we extend their framework to a joint model for the largestfour economies in the Euro area, instead of the area itself. In order to account for thehigh degree of economic integration between these four EMU Members, we assume thateach national economy is linked to the other three via two channels: a trade channel and aproductivity channel. First, the level of the output gap in each country depends on the ex-ternal demand for national products in the other countries, which we capture by includingthe lagged foreign output gaps scaled by the share of the foreign partner in the country’sexports. Second, we allow for cross-correlations of the shocks driving the (latent) trendsof national potential output growth.5 This specification is meant to capture the fact thatthese four economies have faced common shocks over the period, such as the liquiditydry-up of 2007-2008 and the trade collapse of 2009, but also that many national medium-sized and large firms have a regional outreach and that their output relies on transnationalproduction chains within the area. All together, we end up having a relatively parsimo-nious model of a simplified Euro area made up of four countries, that we estimate bymaximum likelihood.

Our estimation yields plausible and significant values for all key parameters, and no-tably the slopes of the Phillips and IS curves. We obtain time-varying measures of theunobserved, latent variables in the model: the national natural rates of interest and thederived interest rate gaps, the national output gaps and associated rates of potential outputgrowth. We first find that the four national natural rates fluctuated widely over the lastfifteen years, a result consistent with the findings of Laubach and Williams (2015) forthe US. Before the crisis, they co-moved heavily, with cross-sectional averages hoveringbetween−0.3% (in March 2003) and 4.2% (in September 2000) and with contained cross-sectional dispersion. However, in all countries, national natural rates dropped markedlyafter the collapse of Lehman Brothers and the ensuing aggravation of the worldwide finan-cial crisis, down to some−3% in Spain, some−6% in Italy and France, and close to−7%

4Since October 2008 and the switch of the ECB’s operational framework to a fixed-rate, full-allotmentprocedure, the effective policy rate has been the deposit facility rate and not the repo rate, which is now 25bp above the former.5Note that we label these shocks trend productivity shocks for simplicity. However, as our model is reduced-form, these shocks may reflect any correlated shocks that imping on trend output growth. Simultaneous oreven coordinated austerity shocks during the euro area sovereign debt crisis are a case in point.

4

in Germany. Although they bounced back rapidly thereafter, their trajectories diverged tosome extent from 2010 to the beginning of 2014. National natural rates remained mostlyin negative territory during the Euro area sovereign debt crisis. They nevertheless droppedto persistently lower levels in the two Southern economies, reaching a trough in the sec-ond half of 2012, at nearly −5% in Spain and around −3.5% in Italy. National naturalrates eventually re-converged in 2014 and stabilized to negative levels, at around −1% inall economies since mid-2015, in line with flat trend output growth across the Eurozone.Accordingly, estimated national interest rate gaps point to quite differentiated effectivemonetary policy stances across the four countries during the crisis period, but also tosome extent before, with a significantly negative interest rate gap in Spain that accountsfor the strongly positive output gap and relatively high inflation in this country during thehousing boom of the 2000s. Overall, our estimates suggest that real interest rate gapsended up keeping almost neutral in Germany and France over the post-2009 period, buthad significantly restrictive effects in Spain since the end of 2010 and up to 2014, and inItaly in 2011-2012.

We contribute to three strands of the literature and current policy debates. First, ourstudy provides another illustration of how to use the concept of natural rate of interest formonetary policy advice. Compared to previous studies, ours is the first to estimate naturalrates for individual Member countries of the Eurozone in a consistent framework.6 Ourfindings shed therefore a new light on the debate about the fragmentation of the areaduring the 2010s and the relevance of a one-size-fits-all monetary policy in this stressedenvironment characterized by the largely asymmetric shocks of the sovereign debt panic.

Second, our estimates of natural rates of interest for four large European economiesalso provide useful evidence to the on-going debate on the secular stagnation hypothesis(see, e.g., Eggertsson and Mehrotra, 2014).7 Although our estimation period is short(since 1999) and we have to assume constant unconditional means of the real interestrates for tractability (which forces related latent variables to mean revert), we find thatnational natural rates remain below their pre-crisis levels, after the crisis.

Last, our study also provides relevant insights for the debate on the interplay betweenstructural reforms and monetary policy in Europe (Pesenti, 2015). When some economiesare hit by a series of negative shocks on their natural interest rates and monetary policy isconstrained by its zero lower bound, structural reforms that have short-term deflationaryeffects can delay the return to growth and even trigger a vicious cycle of disinflation, lowdemand, and further disinflation, which pushes the interest rate gap up and the output gapdown, with possible hysteresis effects (Eggertsson et al., 2014).

The rest of this paper is organized as follows. Section II describes our stylized jointmodel of the largest four Euro area economies. Section III presents the data and details our

6Bouis et al. (2013) provide estimates of time-varying natural rates along the lines of Mesonnier and Renne(2007) for several developed economies but they consider the Euro area as a whole and do not look atnational developments within the EMU.7For a useful survey, see also Rachel and Smith (2015).

5

estimation procedure. Section IV discusses our results. Robustness checks are presentedin section V while section VI concludes.

II. Model

We jointly model the dynamics of the largest four economies of the Euro area usinga parsimonious set-up with unobserved variables that we want to estimate. The four-country area as a whole is modelled as a closed economy, but each of the four membercountries is allowed to interact with the other three and is therefore modelled as an openeconomy under a fixed exchange rate regime.

More specifically, each national economy is summarized by the intersection of a dy-namic supply -or Phillips- curve and a dynamic demand -or IS- curve. Both curvesare backward-looking in nature, as in Rudebusch and Svensson (1999), Laubach andWilliams (2003) and their followers, which keeps the estimation of the unobserved vari-ables of interest tractable with the Kalman filter. Note that, as we do not use our modelfor policy simulations but only for providing an a posteriori assessment of the monetarypolicy stance in these countries, this is not a major shortcoming in theory. Besides, theavailable evidence for Europe suggests that backward-looking Phillips curves fit the databetter than forward-looking ones.8

The Phillips curve in country i therefore reads:

πit = µπi + γiπi,t−1 +βizi,t−1 + ε

πit (1)

where πit denotes monthly (annualized) core HICP inflation and zit denotes the outputgap. We denote by επ

it the residual cost-push shock, which is assumed to be a Gaussianwhite noise with standard deviation σπ

i .In turn, the output gap is a persistent variable which is determined along an IS curve by

a linear relationship with the lagged real interest rate gap:

zit = αizi,t−1 +δi

n

∑j=1

χi jz j,t−1 +λi(it−2−Et−2

[πi,t−2+h

]− r∗i,t−2

)+ ε

zit (2)

In this equation, it represents the one-year nominal interest rate, which we consideras an exogenous variable in the estimation process since we do not model explicitly thereaction function of the ECB. We deflate this nominal rate using survey-based inflationexpectations at the one-year horizon, Et

[πi,t+h

]. Last r∗it is the natural rate of interest that

we want to estimate. The additional term accounts for the strong economic integration ofthe largest four Euro area economies. Intuitively, the output gap in country i depends onthe external demand of domestic goods by its trade partners j within the zone. In the spiritof a global VAR model (see Pesaran et al., 2004), we therefore assume that it is driven bya weighted average of lagged output gaps in partner countries, where the relative weightsreflect the relative shares χi j of these countries in country i’s exports.8See for instance Rudd and Whelan (2007) and Mavroeidis et al. (2014) for more insights on the NKPCdebate.

6

The remaining equations of the model make explicit the definition and the dynamics ofthe unobserved variables we want to estimate. First, the national output gap in country i issimply the deviation of log (quarterly annualized) real GDP, yit from its potential, denotedy∗it :

yit = y∗it + zit

Second, we define the national natural rate in each country as in Mesonnier and Renne(2007):

r∗it = µri +θiait

where ait is a persistent process that also drives the low-frequency fluctuations in thecountry’s potential growth. Indeed, ∆y∗it , the growth rate of national potential output isdefined as:

∆y∗it = µyi +ait + ε

yit

where µyi is a constant and ε

yit a Gaussian white noise of standard deviation σ

yi .

The intuition behind our specification for the natural rate is provided by the textbookneoclassical growth model, where the real equilibrium rate of interest is an affine functionof the balanced growth path for real (per capita) consumption.9 For simplicity, we referto ait in what follows as the underlying productivity trend in the economy. However, oneshould keep in mind that our approach to estimate the natural rate is reduced form and thatwe cannot disentangle between the various forces which are likely to move the real equi-librium rate of interest in each country (e.g., technology shocks, fiscal shocks, changesin the rate of time preference or in the population growth rate, exogenous changes inhouseholds debt limit etc., depending on the precise theoretical model one has in mind).10

Adding the εyit term to the determinants of potential growth therefore brings some flex-

ibility and allows to accommodate for possible exogenous disturbances that may havedifferentiated effects on the natural rate of interest and potential output growth (as, e.g.,changes in households’ rate of time preference).

This country-specific “productivity trend” is assumed to follow an AR(1) process:

ait = ψai,t−1 + εait

where |ψ| < 1 is the autoregressive coefficient and εait is a Gaussian white noise with

standard deviation σai . Note here that we allow εa

it to be correlated across countries,with correlation matrix Σ, while the other shocks are assumed to be cross-sectionally

9Typically, r∗it = ρi +θic∗it , where ρi is the rate of time preference of households and θi is their risk aversionparameter.10Note here that several of the candidate factors, such as the demographic ones, are very slow moving andthus unlikely to be disentangled from deterministic trends over a relatively short time span of 17 years.

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independent. We therefore explicitly account for correlated productivity shocks acrossEuro area countries.

III. Data and estimation

Our dataset covers the period from January 1999 to June 2016 at a monthly frequencyfor the largest four Euro area economies, Germany, France, Italy and Spain. We usenational series of real GDP and the industrial production index (IPI) from Eurostat, andnational series of the harmonized index of consumer prices excluding energy (core HICP)from the ECB. Real activity data are seasonally adjusted by Eurostat. Since all core HICPseries are not seasonally adjusted in the ECB SDW database, we adjust two of them,the German and Spanish ones, using Tramo-Seats. All growth rates in what follows aredefined in logarithms. Our measure of the one-year risk-free nominal interest rate inthe Euro area (common to all member economies) is the rate of the one-year EONIAswap, taken from Datastream. We derive national ex ante one-year real interest rates fromthis common nominal rate by subtracting national measures of inflation expectations ata one-year horizon. We use national forecasts published by Consensus Economics forthe current and subsequent years, and interpolate them linearly in order to construct theseconstant-horizon inflation forecasts.

Figure 1 displays our dataset. For readability, the upper panel of the figure showsyear-on-year rates of core HICP inflation instead of the monthly (annualized) inflationrates we use for estimation. It reveals that inflation rates were quite heterogeneous acrossthe large Euro area countries before the financial crisis, while they have moved more insynch since then. Indeed, the pre-crisis inflation gap between Spain, the most inflationarycountry, and Germany, the least inflationary country over this period, ranges between 1and 3 percentage points, with a peak reached at the end of 2002. After the global financialcrisis and together with the aggravation of the sovereign debt crisis in Southern Europe,inflation has tended to decline more rapidly in Spain and Italy than in the core economies.Consistently with the view that pre-crisis GDP growth was fueled by sustained excessdemand in Southern countries, this pattern of relative inflation rates is reflected by thepattern of relative growth rates of countries’ real GDP (as shown by the central panelof the figure), with faster growth rates in Spain before 2007, but higher growth rates inGermany after the 2009 trough, France and Italy hovering between the two extremes.Last, the lower panel shows that effective one-year ex ante real interest rates were moredispersed across economies before the subprime crisis than they were thereafter. Thissaid, they globally followed a downward trend from the beginning of the 2000s to theirtrough of 2012, at some −2%, reflecting the long sequence of nominal policy rate cutsafter the outburst of the financial turmoil in the Fall of 2008. Nevertheless, effective realinterest rates rose markedly again in 2013 and 2014, reflecting the continued fall in one-year inflation expectations over this period. Finally, they receded again in 2015 to remaincontained between −2% and −1%, a possible effect of the reinforced forward guidance

8

thanks to the numerous unconventional measures implemented by the Eurosystem in 2014(T-LTRO, ABSPP) and 2015 (QE).

A. Estimation

Our model of the Euro area includes two unobserved variables, ait and zit , and two ob-served variables, monthly inflation πit and quarterly GDP Y Q

it , for each of the largest fourcountries. The model is linear but not closed, as we do not specify the reaction functionof the ECB and take the effective ex ante one-year real interest rate in each country as ex-ogenous. We rewrite the model in state-space form and apply the Kalman filter to recovermeasures of the unobserved variables conditional on parameter estimates that we obtainby maximum likelihood. Note that we estimate measures of the unobserved variableswith monthly frequency, while one of our two observed variables, real GDP, is observedwith quarterly frequency only. Our version of the Kalman filter is flexible enough to ac-commodate such missing observations. Nevertheless, as our sample is relatively small,relying on quarterly GDP data only to measure real activity implies a substantial loss inestimation precision. We therefore increase our information set and add for each countrya third measurement equation linking unobserved monthly GDP growth with the observedgrowth in the industrial production index (IPI)11.

Estimating the complete model with the Kalman filter entails some technical choicesthat we summarize in this section, leaving detailed explanations for the online appendix.We briefly explain below our main choices regarding the calibration of some parametersand the initialisation of the likelihood maximisation algorithm.

1. Calibrated parameters

Some parameters are notoriously difficult to estimate in such a set-up and we calibratethem following notably Mesonnier and Renne (2007) and Bouis et al. (2013), as summa-rized in Table 1. The most important calibrations relate to the process driving the naturalrate of interest r∗it .

First, we calibrate the θi parameter, which links the unobserved natural rate to theunobserved trend productivity growth in country i and is akin to households’ risk aversioncoefficient in a simple model. To the best of our knowledge, no study provides withestimates of this structural parameter for all four countries in a consistent framework.We therefore tried to make an educated guess by combining recent estimates of θ forFrance and information about households’ participation rates into risky assets in Euroarea countries from the 2010 Eurosystem’s Household Finance and Consumption Survey(HFCS). Since the seminal contribution of Friend and Blume (1975), researchers haveindeed commonly implemented a revealed preference strategy and infered households’risk aversion from the share of risky assets, like stocks, in their financial wealth or from theobserved participation rates of households into risky assets. More precisely, we proceed as

11Formally, we use the growth rate of the IPI as an affine proxy of the GDP growth rate: ∆IPIit = µ IPI

i +ϕi∆yit + ε IPI

it . The interested reader can refer to the online appendix for further details.

9

follows. We first consider Arrondel and Savignac (2015)’s recent estimates of households’risk aversion for France, which they infer from a simple two-stage lottery included in theFrench National Statistical Institute’s 2010 Wealth Survey. Their findings suggest a valueof 3.5 for θFR.12 Taking this value as a benchmark, we derive risk aversion values forthe remaining countries by simply assuming that the ratio of local risk aversion to thelocal participation rates into risky assets (i.e. stocks) is constant and then use this ruleof thumb to map the countries’ shares of households holding stocks into country-specificrisk aversion.13 Interestingly, we end up getting quite similar values of θ across countries,ranging from 3.5 in France to 3.7 in Spain and Germany and 3.9 in Italy. Note that thesevalues are very close to the baseline calibration of θ = 4 for the Euro area as a whole inMesonnier and Renne (2007).14 We nevertheless check in the robustness section belowthat our main conclusions remain valid when the θi take other plausible values.

Second, we assume that in each country the variance of the natural interest rate is equalto the sample variance of the effective, ex ante, real interest rate15.

Remaining calibrations relate to, first, the constants (µri )i, (µ

yi )i and (µπ

i )i, and second,the signal-to-noise ratios σ

zi /σ

yi . For each country, the first two constants are respectively

calibrated to the unconditional means of the ex ante national real rate of interest and GDPgrowth. Inflation series are demeaned before estimation, which is tantamount to calibrat-ing (µπ

i )i/(1− γ) = πi, where πi is the unconditional mean of inflation. Furthermore,we impose for all countries σ

zi /σ

yi = 1.5. Last we also calibrate the international trade

weights χi j as the average export share of destination country j over the period from 2000to 2015 in the exports of country i towards the other three countries in the EA4. Never-theless, we check in section V that our main results remain qualitatively unchanged whenwe impose that the key calibrated parameters take alternative, reasonable, values.

2. Initialisation of the optimisation algorithm

The model is heavily parameterised, so that maximum likelihood estimates of the freeparameters may correspond to local optima of the likelihood function. A careful choiceof the initial parameter values used to kick-start the gradient descent algorithm therefore

12The outcome of their estimation under the assumption of risk-averse, CRRA preferences is in fact a rangemeasure in four brackets: 58% of respondents are extremely risk averse (θ > 3.76), 27% very risk averse(2 < θ < 3.76), 10% are moderately risk averse (1 < θ < 2) and only 5% are risk lovers (0 < θ < 1).Assuming a maximum of θ = 5 and taking the average of mid-range values weighted by the shares ofrespondents in each bracket, we find an estimate of θFR = 3.5.13According to the 2010 HFCS, around 10% of German and Spanish households hold stocks, to be com-pared with some 15% of French households and only 5% of Italian ones. We adjust these measures fordifferences in countries’ participation rates in financial assets more generally and construct our proxy forrisk aversion as the probability of not holding shares conditional to the probability of holding financialassets of any kind.14These values are also plausible considering available estimates of risk aversion from several microecono-metric studies for the US and other advanced economies, that range between 0 and 5. See references inMesonnier and Renne (2007) for details.15Formally, we impose for each country i, V(r∗it) = Vsample (rit). Given our specification of the natural

interest rate, this entails a constraint on the standard deviation of (εait), namely that: σa

i =

√(1−ψ2

i )Vsample(rit )

θi.

10

matters for the quality of the results. We proceed here in two steps.16 First, we estimatea simplified model for each of the four countries where we shut down the trade channelin the IS equation and assume a diagonal correlation matrix Σ for productivity shocks.For each country, we initialise our parameter estimates with the baseline values of thecorresponding Euro area parameters estimated in Mesonnier and Renne (ibid.). After thisfirst step, we thus get four different sets of parameters, one for each country, and wealso get filtered and smoothed trajectories of the latent variables.17 In a second step, weestimate the full model by opening up the two international contagion channels. Parameterestimates obtained for each country separately in the first step are then used as initialvalues for the second one. There are however additional parameters to initialise, namelythe δi and the non-diagonal elements in the correlation matrix Σ. For this purpose, weuse outputs of the first step: the smoothed trajectories of the national output gaps andof the national productivity processes. We run OLS regressions of the IS equations withestimated output gap series to get an initial value for δi and use the empirical correlationmatrix of the productivity shocks from the first step as an initial value for Σ. 18

IV. Results

Table 3 reports estimation results for the four countries under our baseline model spec-ification. Key parameters, such as the slopes of the Phillips-curve βi and the slopes ofthe IS curve λi turn out to be significant with the expected sign (i.e. a positive βi anda negative λi). On average across countries, these parameters are of similar magnitudeas the corresponding parameters estimated for the Euro area as a whole over the period1979-2004 by Mesonnier and Renne (ibid.).19 However, we find a sizeable heterogeneityacross countries. Indeed, the slope of the Phillips-curve is the largest in France (at 0.29),intermediate in Germany and Italy (close to 0.2) and the smallest in Spain (0.09). Theslope of the IS curve (the interest rate gap semi-elasticity of the output gap) is the high-est in absolute terms in Germany and Italy, where it is twice as large as in France (tehvalue for Spain is in between). The parameters governing the degree of persistence of thenatural rates of interest, ψi, are high but well below unity (between 0.86 and 0.94 acrosscountries) with very tight standard deviations, which vindicates our choice of stationaryprocesses for the ait . The autoregressive parameter in the IS curves, αi is close to unity

16Note however that, once the algorithm has been initialized in a sensible way as described below, theestimation of all parameters and unobservable variables is carried out in one unique step.17The Kalman algorithm provides two types of estimates of the latent factors. The first are filtered estimates,which correspond to the expectations of the latent variables conditional on past and present observations.The second are smoothed estimates, which correspond to the expectations of the latent variables conditionalon all observations (past, present and future).18Note that, since we calibrate the standard deviations σa

i of the national productivity shocks, we cannotuse the standard Cholesky decomposition to estimate the variance-covariance matrix of (εa

1t , . . . ,εant). See

the online appendix for details of the estimation procedure.19These slopes are however somewhat steeper in absolute terms than the slopes estimated by Laubach andWilliams (2003) for the United States over the period 1961-2002 as well as the ones obtained by Holstonet al. (2016) for the Euro area over 1972-2015.

11

and significant in Germany, Italy and Spain. It comes out as equal to unity in France,but this does not necessarily indicate non-stationarity as the output gap in each countryalso reacts strongly to the lagged output gaps of its trade partners within the EA4.20 Thetrade channel (summarized by the size of the δi coefficient) weighs much on the dynamicsof the output gap in Spain, somewhat less in Italy and Germany and is not significantlyactive in France. Last, the autoregressive coefficient in the inflation equation is signifi-cantly negative in Germany, which is not very surprising at monthly frequency, small butpositive in Spain, but not significantly different from zero in the other two countries.

Furthermore, table 4 reports our estimates of the correlation matrix of the national trendproductivity shocks ait . These unobserved variables come out as quite correlated, consis-tently with the strong co-movement of observed GDP growth series across the Euro area’smain economies since 1999. Looking more into details, the correlation coefficient is thehighest (0.99) between Germany and France. It remains high, and above 0.95, betweenItaly and both core countries. In contrast, shocks to the Spanish trend productivity growthappear to comove strongly with Italy’s, while being much less correlated with develop-ments in the core economies (correlation of about between 0.58 and 0.68).

Figure 2 provides an overview of the estimated natural rates of interest for the fourcountries, while figure 3 displays each country’s natural rate separately, together with theassociated 95% confidence bands. Note that confidence intervals account here for to-tal uncertainty, i.e. filtering uncertainty (surrounding estimates of unobserved variablesconditional to given parameters) compounded by estimation uncertainty (surrounding pa-rameter estimates themselves). We compute the corresponding standard deviations usingHamilton (1986)’s Monte Carlo method.21

According to our estimations, the national natural rates of interest in the EA4 economiesfluctuated widely since the start of the Euro, with individual values ranging between some+4% in 2000-2001 and −6% in the last quarter of 2008, at the climax of the subprimecrisis. A comparison of pre-crisis with post-crisis levels shows however that national nat-ural rates were on average higher before the Great Recession. Before 2008 indeed, theyhovered around 2%, while they remained on average well below this threshold between2010 and 2015.22 This finding is in line with views that the trend decline in real interestrates over the last few decades also reflects a downward trend in real equilibrium rates indeveloped economies (see, e.g., Laubach and Williams, 2015; Holston et al., 2016). Last,the dispersion of natural interest rates across the four major economies of the Euro areaincreased markedly after the 2008-2009 recession and remained elevated until the end of2013, when natural rates eventually trended upward in all countries to stabilize −1% inthe first semester of 2016.

20Due to the modeled “trade channel”, stationarity must be assessed here by looking at the dynamics of theVAR(1) in all four output gaps.21See the online appendix for details.22The average of the four national NRIs over 2010-2015 is −1.2%, against 1.9% over 1999-2007

12

Beyond this broad picture, the confidence intervals around each country-specific es-timates shown in figure 3 allow to shed light on episodes when national natural rateswere significantly different from zero. We find that national natural rates were signif-icantly positive in all countries over the first few years of the EMU and in 2006-2007.They plummeted to low negative levels everywhere during the 2008-2009 recession, butresumed rapidly thereafter. After mid-2009 and up to the outburst of the Euro area sov-ereign crisis, national natural interest rates stayed close to zero in all economies. Theydropped significantly below zero everywhere when the sovereign debt crisis escalated inmid-2011, down to some −5% in Spain for instance, and remained significantly negativeuntil the end of 2013 in all countries but Germany, where the national NRI normalized tozero a few quarters earlier. As of June 2016, our estimates point to still slightly negativenational NRIs and close to −1% in all four economies.

Furthermore, a key advantage of our approach is that, since we estimate all four na-tional natural rates jointly within the same model, we can deepen the analysis and alsoidentify episodes of statistically significant divergence of national natural rates acrossthese economies.23 Figure 4 shows bilateral natural rate spreads together with 95% con-fidence intervals. We find that the natural interest rates in France, Germany and Italyhave almost never been statistically different from each other since the start of the EMU,except for the period between the start of the Italian sovereign debt crisis in 2011 andECB President Draghi’s commitment to do “whatever it takes to save the euro” in theSummer of 2012. However, we find that the natural rate spreads of Spain against Franceand Germany have been significantly negative most of the time between the outburst ofthe sovereign crisis in 2010 and the beginning of 2014. Interestingly, we note that suchlarge negative natural rate spreads did never occur before the 2007-2008 crisis, in spite ofthe persistent large negative ex ante real interest rate spreads of Spain vs Germany in theyears 2003-2006.

The natural interest rate is a key benchmark for monetary policy but it is not in itself anindicator of the monetary policy stance. As embedded in our model, the relevant stanceindicator is the real interest rate gap, here the difference in each economy between the one-year ex ante (credit risk-free) real interest rate and its natural counterpart: a positive realrate gap tends to dampen excess demand and drive down the output gap, which ultimatelytransmits into lower inflation. Note that inflation and inflation expectations diverged insome instances across the EA4 economies, driving national ex ante real interest ratesapart although the same short term nominal rate prevailed everywhere. As a consequence,differences between national natural rate estimates did not necessarily translate into equaldifferences between the monetary policy stances faced by each country.

23Computing confidence bounds around spreads of NRI and IRG is possible in this joint framework becausethe Kalman filter also yields the covariance matrix of the latent factors.

13

Figure 5 shows our estimates of the national real interest rate gaps (together with 95%confidence intervals), while figure 7 presents the estimated national output gaps. The fig-ure reveals a persistent negative national rate gap, i.e. an expansionary monetary stance,in Spain over most of the years 1999-2007, which is consistent with the widening pos-itive output gap measured in this country over the same years. Italy also experienced along period of economic expansion before the crisis, but the contribution of the monetarystance to this sustained positive output gap seems to be more limited: while the real rategap was significantly negative in this country in the early years of the Euro, it closed assoon as 2002 and remained not significantly different to zero most of the time and up tothe outburst of the global financial crisis. This suggests that other forces were at playin Italy and contributed to this sustained excess demand, such as an expansionary fiscalpolicy and a dynamic external demand from Euro area partners.

The figures also confirm that the monetary policy stance turned up to be very reces-sionary during the financial crisis of 2008-2009, as the gradual policy rate cuts and theaccompanying liquidity provision measures failed to accommodate rapidly enough theunprecedented drop in national natural interest rates. Interest rate gaps indeed soaredduring the Lehman panic in the last quarter of 2008, hovering around 7% in Germany,against some 6% in France and Italy and 3% in Spain. This involuntary tightening con-tributed to dragging down national output gaps in 2009, well below zero in core coun-tries and Italy, and closer to neutrality in Spain. The slightly negative interest rate gapsachieved in 2010-2011 eventually fueled the post-subprime crisis recovery in the two coreeconomies. However, as the Euro area sovereign crisis escalated in 2011, the deflationaryforces at play in Southern Europe and the falling natural rates combined into imposing asignificantly restrictive monetary stance on the Spanish economy over 2011-2013, whileneutrality was broadly achieved and maintained in the other three countries, with the ex-ception of short-lived restrictive episodes in late 2012 and in 2013 in Italy.

To what extent did the effective monetary policy stance diverge between Spain and theother countries? Again, since we jointly model all four economies, we can estimate thecovariance matrix of unobserved variables and we are therefore able to derive confidenceintervals around our point estimates of the bilateral spreads between national interest rategaps. Figure 6 shows these spreads and confirms that the common monetary policy provedsignificantly more expansionary in Spain than it was in France and Germany, and even inItaly, over the period from 2003 to 2006, but then became significantly tighter over mostof the period from 2011 to 2014. As regards the relative situation of Italy, this countryalso enjoyed a more expansionary monetary policy stance in 2002-2004 than Germany,but it faced a significantly more restrictive one for about one year in 2011-2012.

Eventually, all interest rate gaps converged to a neutral stance at the end of our sample,with point estimates between -0.8% and -0.2% in June 2016, but none of these estimatesare significantly different from zero. The return to a common monetary stance across allfour major Euro area economies can therefore be viewed as vindicating the success of the

14

unconventional monetary policies implemented since 2013 in fighting the fragmentationof monetary and financial conditions in the area. This said, our point estimates of nationaloutput gaps in June 2016, which range from significantly negative values of −4.3% inSpain and −2.8% in Italy to a non-significative −1.2% in France and a slightly positivegap of 0.5% in Germany, suggest that more macroeconomic stimulus was still needed atthat time at least in Southern economies, while the substantial heterogeneity in economicconditions and fiscal space may limit the room for coordinated action.

Last, Figure 8 shows the estimated potential output growth rates (expressed as yearlyaverages) together with the actual rates of real output growth. In line with other studies(Holston et al., 2016, e.g., ), our results point to a slowdown of potential output growthin Spain, Italy and France since the early 2000s. In contrast, the estimated trend outputgrowth remains quite stable in Germany over the last decade and a half. In 2016, we findthat estimated potential output growth hovers around 1% in France, Germany and Spain,while it remains stuck close to 0% in Italy.

V. Robustness

To what extent should we believe in our estimates? Elements in support of our find-ings are twofold. First, our estimates of unobserved variables are of reasonable order ofmagnitudes. Figure 9 shows a measure of the Euro area natural rate of interest whichwe compute as the GDP-weighted average of our four national estimates.24 As the figureshows, this Euro area NRI derived from our national ones compares well with alterna-tive estimates from other studies. Unsurprinsingly, the average of our national NRIs isclose to the two measures of the area NRI, denoted MR and BRRWC on the figure, thatare based on the model of Mesonnier and Renne (2007), which is formally very close toours.25 Note however that Bouis et al. (2013) estimate their Euro area NRI conditional totheir estimated potential output being very close to the level of potential output estimatedby the OECD using a production function approach. Last, Holston et al. (2016), whichimplement a variant of the semi-structural model of Laubach and Williams (2003), alsoget an estimated NRI for the Euro area that strongly comoves with ours. Although theydo not obtain a sharp drop in the Euro area NRI at the height of the financial crisis, theynevertheless also find that the natural rate of interest in the euro has trended downwardover the last decade and remains in negative territory at sample end.

As far as output gap measures are concerned, they also compare well with estimatesfrom other sources. For instance, measures of the output gaps available at the date ofthis writing from the OECD, the IMF and the European Commission for the year 2016range between −0.6% and +0.4% for Germany, between −2.0% and −1.5% for France,24We compute for each calendar year nominal GDP weights of each of the 4 countries with respect to thenominal GDP of EMU4 as a whole and use these annual weights to aggregate the national series into an“EMU4” NRI.25Mesonnier and Renne (2007) estimate their model over 1979-2005 and define inflation as headline HICPinflation, while Bouis et al. (2013) also estimate their model on more than three decades and look at inflationin terms of the GDP deflator.

15

−2.5% and−1.6% for Italy and−2.8% and−1.5% in Spain. Although these values maydiffer to some extent with our point estimates at sample end, ours being more optimisticas regards core economies and less as regards Southern ones, they remain consistent withour estimated 95% confidence intervals around these estimates.26. Interestingly, as fig-ure 10 shows, annual changes in our estimated output gaps are also strongly correlatedwith the respective national ESI indicators computed by Eurostat, although one shouldremember that the ESIs are not used for estimation. The ESIs are composite confidenceindicators based on surveys conducted among firms in various sectors and consumers.They are basically computed as averages of seasonally adjusted balances of positive andnegative individual answers to questions related to, e.g., observed and expected changesin turnover, production or expenses over the last three months.27 As a consequence, wemay expect them to be correlated with any meaningful measure of the change in the outputgap, which is the difference between observed and trend output growth.

Second, we checked for the robustness of our main findings to changes in some impor-tant specifications and calibrations. More precisely, we look at the impact of changes inthe calibration of the “risk aversion” parameters θi, which largely determine the magni-tude of fluctuations of the natural rate estimates, as suggested by Mesonnier and Renne(2007).28 Tables 5 and 6 present parameter estimates for three different calibrations ofthe θi parameters: 3, the baseline (between 3.5 and 3.9 depending on the country) and 5.Key parameters look quite robust to these changes. Unsurprisingly then, these alternativecalibrations have little bearing on our main conclusions regarding the national interestrates and rate gaps, as shown in figure 11.29

Third, we also relaxed the assumption regarding the lags in the transmission of realinterest rate gaps to activity. Whereas national rate gaps are assumed to impinge on therespective output gaps after two months in the baseline, we also look at the results that weobtain when this transmission is supposed to take 1 month or 3 months instead. Tables 7and 8 compare parameter estimates, while figure 12 shows the consequences for estimatedreal interest rate gaps and our main message regarding the effectiveness of the singlemonetary policy. Again, we find that our conclusions are qualitatively robust to thesechanges.

Last, we looked at the effects of alternative calibrations for the signal-to-noise ratios,where σ

zi /σ

yi = 1, 2 or 4, instead of 1.5 (the baseline). Tables 9 and 10 present parameter

26Our upper bound is −2.8% for Spain and −1.0% for Italy, while our lower bounds are −3.0% for Franceand −2.1% for Germany as of June 2016.27A complete description of the methodology is available from Eurostat at:http://ec.europa.eu/economy finance/db indicators/surveys/documents/bcs user guide en.pdf.28Mesonnier and Renne (2007) find in their closely related model that the choice of this θ parameter mattersmuch more for the qualitative results than the calibration of the signal-to-nois ratios (σ z

i /σyi )i.

29For completeness, the figure also shows NNRI gaps for lower values of θ : 1 or 2. These estimated gapsdiverge somehow from the baseline. However, these low-θ calibrations yield weird results, such as non-stationary estimated dynamics and as a consequence very large, implausible values for national output gapsat sample end (e.g. some −7% in Spain and less than −4% in Germany in June 2016), which convinces usto dismiss them as irrelevant.

16

estimates, while figure 13 shows the estimated real interest rate gaps. Our main conclu-sions are still qualitatively robust to changes in this ratio.

VI. Conclusion

To conclude, we use a parsimonious set-up to jointly estimate national time-varyingnatural rates of interest for the largest four economies of the Euro area over 1999-2016.This paper is the first to estimate natural rates for individual Member countries of the Euroarea in a consistent framework. Our findings thus shed light on the debate about the frag-mentation of the area during the Euro area sovereign crisis in the first half of this decadeand the relevance of a one-size-fits-all monetary policy in this stressed environment.

Indeed, we find evidence of an increased dispersion of real equilibrium rates acrossmajor Euro area economies during the Euro area sovereign crisis, which translated into aquite restrictive monetary policy stance in Southern economies, notably in Spain, whilethe policy stance remained neutral or slightly accomodative in core countries. Accord-ing to our estimates, real interest rate gaps had nonetheless converged towards zero inall four economies as of the end of 2014, suggesting that it took the acceleration of un-conventional policies since mid-2013 to eventually achieve a neutral policy stance in allmajor economies of the area and restore the conditions for an effective common monetarypolicy.

Beyond, we also find that diverging national interest rate gaps across economies partici-pating in the EMU already showed up during the boom phase of the first half of the 2000s.With the benefit of hindsight, the single monetary policy stance was then clearly too acco-modative in Spain, whose economic activity remained buoyant because of the pre-crisiscredit and housing boom, and possibly in other Southern economies as well. This clearlycalls for other, national but coordinated policies to complement the single monetary policywhen asymmetric shocks drive national natural interest rates apart. Macroprudential poli-cies and structural reforms, which should both help to better match the supply of savingsand productive investment needs, are obvious candidates.

REFERENCES 17

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TABLE 1. Calibrated parameters

θr µr µy σ z/σ y σa

Spain 3.7 -0.079 1.803 1.5 0.100Italy 3.9 0.262 0.276 1.5 0.161

France 3.5 0.573 1.351 1.5 0.200Germany 3.7 0.472 1.305 1.5 0.225

TABLE 2. Intra-Euro area trade weights

SP IT FR DESP 0.0 2.2 4.7 2.9IT 1.9 0.0 3.6 4.2FR 2.4 2.4 0.0 4.5DE 2.1 3.3 5.2 0.0

Note. This matrix collects the χi j parameters in the IS curve equations of each country i and reads asfollows: exports of country i (in lines) towards country j (in columns) as a ratio of the GDP of country i(%).

TABLE 3. Parameter estimates I

Spain Italy France GermanyParameter value t-student value t-student value t-student value t-student

γ 0.38 5.69 -0.06 -0.87 0.00 0.06 -0.43 -6.66β 0.09 3.55 0.21 4.89 0.29 2.83 0.23 2.16σπ 1.57 20.22 1.32 20.06 1.39 20.09 1.70 19.42α 0.97 78.94 0.98 39.17 1.00 39.54 0.98 47.25δ 0.67 2.80 0.35 0.94 0.03 0.14 0.40 1.90λ -0.13 -7.11 -0.16 -5.94 -0.09 -5.05 -0.17 -6.16ψ 0.94 70.92 0.90 30.00 0.90 24.06 0.86 15.96

ϕIPI 1.70 4.42 2.88 9.02 3.34 5.97 2.37 9.45µIPI -3.84 -3.00 -1.93 -2.08 -5.07 -3.90 -1.52 -1.51σIPI 15.47 20.11 13.31 15.47 15.22 15.88 13.67 13.21σy 0.48 5.86 0.95 7.01 0.79 7.54 1.62 7.16

TABLE 4. Parameter estimates II: correlation matrix of national NRIshocks (Σ)

Σ tΣ

SP IT FR DE SP IT FR DESP 1 0.802 0.684 0.578 10.19 6.72 4.60IT · 1 0.984 0.951 · 79.33 35.70FR · · 1 0.991 · · 93.07DE · · · 1 · · ·

TABLE 5. Parameter estimates for three different calibrations of θi: 3,baseline (country-specific, between 3.5 and 3.9) and 5.

Germany France Italy Spainθ = 3 θ = basel. θ = 5 θ = 3 θ = basel. θ = 5 θ = 3 θ = basel. θ = 5 θ = 3 θ = basel. θ = 5

γ-0.40 -0.43 -0.44 -0.00 0.00 -0.01 -0.07 -0.06 -0.06 0.35 0.38 0.39

(-6.29) (-6.66) (-6.86) (-0.01) (0.06) (-0.09) (-1.03) (-0.87) (-0.87) (5.19) (5.69) (5.90)

β0.01 0.23 0.25 0.27 0.29 0.28 0.19 0.21 0.20 0.10 0.09 0.08

(0.24) (2.16) (2.79) (3.24) (2.83) (3.16) (4.88) (4.89) (4.90) (4.10) (3.55) (3.35)

σπ

1.75 1.70 1.69 1.38 1.39 1.38 1.31 1.32 1.32 1.56 1.57 1.58(20.35) (19.42) (19.66) (20.22) (20.09) (20.15) (20.18) (20.06) (20.05) (20.17) (20.22) (20.24)

α1.01 0.98 0.98 0.99 1.00 0.99 0.98 0.98 0.98 0.97 0.97 0.97

(111.80) (47.25) (45.00) (48.02) (39.54) (38.79) (62.54) (39.17) (36.10) (85.01) (78.94) (80.08)

δ0.37 0.40 0.32 0.25 0.03 -0.01 0.52 0.35 0.24 0.75 0.67 0.59

(2.02) (1.90) (1.57) (1.98) (0.14) (-0.04) (2.54) (0.94) (0.60) (3.37) (2.80) (2.52)

λ-0.17 -0.17 -0.18 -0.08 -0.09 -0.10 -0.14 -0.16 -0.16 -0.12 -0.13 -0.14

(-6.36) (-6.16) (-6.16) (-5.41) (-5.05) (-5.25) (-6.75) (-5.94) (-6.13) (-6.87) (-7.11) (-7.52)

ψ0.81 0.86 0.86 0.86 0.90 0.91 0.87 0.90 0.91 0.94 0.94 0.94

(11.52) (15.96) (17.69) (15.83) (24.06) (28.45) (21.31) (30.00) (34.17) (61.24) (70.92) (73.34)

ϕIPI2.38 2.37 2.37 3.37 3.34 3.35 2.89 2.88 2.88 1.70 1.70 1.70

(9.30) (9.45) (9.51) (6.04) (5.97) (6.00) (9.01) (9.02) (9.04) (4.41) (4.42) (4.43)

µIPI-1.52 -1.52 -1.52 -5.09 -5.07 -5.07 -1.93 -1.93 -1.93 -3.84 -3.84 -3.84

(-1.50) (-1.51) (-1.51) (-3.93) (-3.90) (-3.91) (-2.08) (-2.08) (-2.08) (-3.00) (-3.00) (-3.00)

σIPI13.79 13.67 13.62 15.14 15.22 15.20 13.32 13.31 13.29 15.48 15.47 15.47

(13.45) (13.21) (13.17) (15.45) (15.88) (15.78) (15.60) (15.47) (15.43) (20.12) (20.11) (20.10)

σy1.54 1.62 1.64 0.81 0.79 0.79 0.94 0.95 0.95 0.47 0.48 0.49

(6.24) (7.16) (7.46) (7.49) (7.54) (7.44) (7.23) (7.01) (6.98) (5.92) (5.86) (5.77)

Note. Student-t statistics in parenthesis.

TABLE 6. Estimated correlation matrix of national NRI shocks for threedifferent calibrations of θi: 3, baseline (country-specific, between 3.5 and3.9) and 5.

Σ tΣ

SP IT FR DE SP IT FR DE

θ = 3

SP 1 0.789 0.658 0.559 9.72 6.08 4.31IT · 1 0.982 0.950 · 69.90 34.74FR · · 1 0.992 · · 96.17DE · · · 1 · · ·

θ = baseline

SP 1 0.802 0.684 0.578 10.19 6.72 4.60IT · 1 0.984 0.951 · 79.33 35.70FR · · 1 0.991 · · 93.07DE · · · 1 · · ·

θ = 5

SP 1 0.807 0.697 0.589 10.46 7.09 4.79IT · 1 0.985 0.952 · 83.53 36.93FR · · 1 0.990 · · 90.57DE · · · 1 · · ·

TABLE 7. Parameter estimates for three different lags of the real interestgap in the IS equations: 1 month, 2 months (baseline), 3 months.

Germany France Italy SpainLag 1 Lag 2 Lag 3 Lag 1 Lag 2 Lag 3 Lag 1 Lag 2 Lag 3 Lag 1 Lag 2 Lag 3

γ-0.43 -0.43 -0.43 0.01 0.00 0.00 -0.06 -0.06 -0.06 0.38 0.38 0.38

(-6.71) (-6.66) (-6.67) (0.08) (0.06) (0.03) (-0.87) (-0.87) (-0.87) (5.70) (5.69) (5.69)

β0.23 0.23 0.23 0.29 0.29 0.29 0.21 0.21 0.21 0.09 0.09 0.09

(2.32) (2.16) (2.18) (2.76) (2.83) (2.90) (4.85) (4.89) (4.88) (3.55) (3.55) (3.56)

σπ

1.70 1.70 1.70 1.39 1.39 1.38 1.32 1.32 1.32 1.57 1.57 1.57(19.51) (19.42) (19.41) (20.09) (20.09) (20.11) (20.06) (20.06) (20.04) (20.22) (20.22) (20.22)

α0.99 0.98 0.99 1.00 1.00 1.00 0.97 0.98 0.98 0.97 0.97 0.97

(49.09) (47.25) (50.42) (40.26) (39.54) (39.40) (35.33) (39.17) (39.96) (82.48) (78.94) (74.05)

δ0.39 0.40 0.43 0.02 0.03 0.06 0.35 0.35 0.37 0.65 0.67 0.67

(1.91) (1.90) (2.09) (0.11) (0.14) (0.28) (0.92) (0.94) (0.98) (2.74) (2.80) (2.62)

λ-0.16 -0.17 -0.17 -0.08 -0.09 -0.09 -0.15 -0.16 -0.15 -0.13 -0.13 -0.14

(-6.01) (-6.16) (-6.69) (-5.08) (-5.05) (-5.80) (-5.65) (-5.94) (-6.47) (-7.16) (-7.11) (-7.19)

ψ0.87 0.86 0.81 0.90 0.90 0.87 0.91 0.90 0.88 0.95 0.94 0.94

(19.87) (15.96) (11.16) (29.55) (24.06) (16.16) (35.82) (30.00) (20.45) (77.79) (70.92) (61.41)

ϕIPI2.35 2.37 2.35 3.18 3.34 3.35 2.85 2.88 2.92 1.65 1.70 1.74

(9.16) (9.45) (9.37) (5.53) (5.97) (5.96) (8.79) (9.02) (9.30) (4.26) (4.42) (4.52)

µIPI-1.47 -1.52 -1.51 -4.83 -5.07 -5.07 -1.92 -1.93 -1.93 -3.75 -3.84 -3.85

(-1.45) (-1.51) (-1.50) (-3.64) (-3.90) (-3.90) (-2.06) (-2.08) (-2.11) (-2.92) (-3.00) (-3.02)

σIPI13.78 13.67 13.63 15.47 15.22 15.20 13.40 13.31 13.16 15.52 15.47 15.43

(13.18) (13.21) (12.90) (16.54) (15.88) (15.75) (15.56) (15.47) (15.16) (20.13) (20.11) (20.09)

σy1.64 1.62 1.66 0.78 0.79 0.79 0.96 0.95 0.96 0.48 0.48 0.48

(7.38) (7.16) (6.94) (7.47) (7.54) (7.71) (7.03) (7.01) (6.89) (5.71) (5.86) (5.95)

Note. Student-t statistics in parenthesis.

TABLE 8. Estimated correlation matrix of national NRI shocks for threedifferent lags of the interest gap in the IS equations: 1 month, 2 months(baseline), 3 months

Σ tΣ

SP IT FR DE SP IT FR DE

Lag 1

SP 1 0.793 0.677 0.567 9.80 6.62 4.46IT · 1 0.985 0.951 · 79.20 35.18FR · · 1 0.990 · · 87.80DE · · · 1 · · ·

Lag 2

SP 1 0.802 0.684 0.578 10.19 6.72 4.60IT · 1 0.984 0.951 · 79.33 35.70FR · · 1 0.991 · · 93.07DE · · · 1 · · ·

Lag 3

SP 1 0.813 0.692 0.587 10.82 6.95 4.69IT · 1 0.982 0.948 · 72.03 31.89FR · · 1 0.991 · · 84.32DE · · · 1 · · ·

TABLE 9. Parameter estimates for three different calibrations of σz/σy: 1,1.5 (baseline) and 2.

Germany France Italy Spainσzσy

= 1 σzσy

= 1.5 σzσy

= 2 σzσy

= 1 σzσy

= 1.5 σzσy

= 2 σzσy

= 1 σzσy

= 1.5 σzσy

= 2 σzσy

= 1 σzσy

= 1.5 σzσy

= 2

γ-0.44 -0.43 -0.42 -0.01 0.00 0.01 -0.06 -0.06 -0.06 0.38 0.38 0.38

(-6.84) (-6.66) (-6.26) (-0.08) (0.06) (0.13) (-0.90) (-0.87) (-0.86) (5.65) (5.69) (5.70)

β0.25 0.23 0.18 0.32 0.29 0.27 0.21 0.21 0.20 0.09 0.09 0.09

(2.67) (2.16) (1.14) (2.51) (2.83) (3.06) (4.69) (4.89) (5.05) (3.57) (3.55) (3.55)

σπ

1.69 1.70 1.72 1.38 1.39 1.39 1.32 1.32 1.32 1.57 1.57 1.57(19.56) (19.42) (18.56) (19.93) (20.09) (20.14) (20.02) (20.06) (20.06) (20.20) (20.22) (20.22)

α0.99 0.98 0.98 1.00 1.00 1.00 0.98 0.98 0.97 0.97 0.97 0.97

(46.99) (47.25) (46.67) (31.19) (39.54) (44.70) (36.03) (39.17) (44.78) (77.56) (78.94) (80.36)

δ0.45 0.40 0.35 0.07 0.03 0.01 0.33 0.35 0.36 0.65 0.67 0.69

(2.15) (1.90) (1.53) (0.32) (0.14) (0.03) (0.85) (0.94) (1.05) (2.60) (2.80) (2.95)

λ-0.17 -0.17 -0.18 -0.09 -0.09 -0.09 -0.16 -0.16 -0.16 -0.14 -0.13 -0.13

(-6.24) (-6.16) (-6.02) (-5.10) (-5.05) (-4.94) (-5.90) (-5.94) (-6.00) (-7.17) (-7.11) (-7.04)

ψ0.85 0.86 0.86 0.90 0.90 0.90 0.90 0.90 0.90 0.94 0.94 0.94

(16.05) (15.96) (15.85) (24.35) (24.06) (24.30) (29.99) (30.00) (29.20) (71.60) (70.92) (69.43)

ϕIPI2.37 2.37 2.37 3.36 3.34 3.34 2.88 2.88 2.89 1.70 1.70 1.70

(9.46) (9.45) (9.42) (5.99) (5.97) (5.97) (8.99) (9.02) (9.04) (4.42) (4.42) (4.41)

µIPI-1.52 -1.52 -1.51 -5.07 -5.07 -5.08 -1.93 -1.93 -1.93 -3.84 -3.84 -3.84

(-1.51) (-1.51) (-1.50) (-3.91) (-3.90) (-3.90) (-2.08) (-2.08) (-2.08) (-3.00) (-3.00) (-3.00)

σIPI13.67 13.67 13.69 15.17 15.22 15.24 13.32 13.31 13.29 15.47 15.47 15.48

(13.25) (13.21) (13.10) (15.56) (15.88) (16.01) (15.53) (15.47) (15.42) (20.10) (20.11) (20.11)

σy1.93 1.62 1.35 0.96 0.79 0.66 1.12 0.95 0.80 0.58 0.48 0.40

(7.27) (7.16) (6.64) (7.25) (7.54) (7.66) (6.96) (7.01) (7.11) (5.86) (5.86) (5.85)

Note. Student-t statistics in parenthesis.

TABLE 10. Estimated correlation matrix of national NRI shocks for threedifferent calibrations of σz/σy: 1, 1.5 (baseline) and 2.

Σ tΣ

SP IT FR DE SP IT FR DE

σz/σy = 1

SP 1 0.803 0.690 0.581 10.14 6.81 4.59IT · 1 0.985 0.951 · 81.84 35.17FR · · 1 0.990 · · 87.62DE · · · 1 · · ·

σz/σy = 1.5

SP 1 0.802 0.684 0.578 10.19 6.72 4.60IT · 1 0.984 0.951 · 79.33 35.70FR · · 1 0.991 · · 93.07DE · · · 1 · · ·

σz/σy = 2

SP 1 0.802 0.681 0.577 10.36 6.73 4.64IT · 1 0.983 0.950 · 78.79 36.18FR · · 1 0.991 · · 98.12DE · · · 1 · · ·

FIGURE 1. Data

Note. Upper panel: monthly year-on-year core inflation rate. Middle panel: quarterly GDP growth rate(annnualized). Lower panel: ex ante one-year real rate of interest. Germany: stars, France: squares, Italy:dots, Spain: triangles.

FIGURE 2. Smoothed trajectories of the national natural interest rates.

Note. Germany: stars, France: squares, Italy: dots, Spain: triangles.

FIGURE 3. Smoothed trajectories of the national natural interest rates with95% confidence intervals.

FIGURE 4. Bilateral spreads of natural interest rates with 95% confidence intervals.

Note. The shaded areas indicate periods where the spreads are statistically significant at the 95% level.

FIGURE 5. Smoothed trajectories of the national interest rate gaps with95% confidence intervals.

FIGURE 6. Bilateral differences of national natural interest rate gaps with95% confidence intervals.

Note. The shaded areas indicate periods where the spreads are statistically significant at the 95% level.

FIGURE 7. Smoothed trajectories of the national output gaps with 95%confidence intervals.

FIGURE 8. Potential and actual output growth.

Note. Estimated potential output growth and actual output growth are expressed as the growth rates ofannual averages in percent (i.e., cumulated flows of the last four quarters compared to the cumulated flowsof the preceding four quarters).

FIGURE 9. Alternative estimates of the Euro area natural rate of interest.

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FMMR 4 EMU (w. av.) HLW 2016 MR 2007 BRRWC 2013 HJM 2016

Note. FMMR 4EMU is the average of the national NRI estimated for the largest for countries in this paper,weighted by the share of each country in the nominal GDP of EMU4. HLW 2016 is the Euro area NRIestimated by Holston et al. (2016). MR 2007 is the baseline Euro area NRI as estimated in Mesonnier andRenne (ibid.). BRRWC 2013 refers to the baseline Euro area NRI estimated in Bouis et al. (2013). Last,HJM 2016 is the DSGE estimate of Haavio et al. (2016)

FIGURE 10. Changes in national output gaps (yoy), vs Eurostat’s Eco-nomic Sentiment Index (ESI).

Note. All variables have been centered and standardized. ESI : solid lines. Output gaps: dashed lines.Variations in the output gaps are measured over periods of twelve months (i.e. ∆zt := zt − zt−12, where zt isthe logarithm of the output gap).

FIGURE 11. Robustness: national real interest rate gaps for θi = 1, 2, 3,4, 5 and θbaseline.

Note. Dotted line: θi = 1, solid line: θi = 2, dash-dotted line: θi = 3, 4, 5, shaded area: 95% confidenceintervals of the baseline estimate with σz/σy = 1.5, the IRG lagged by two months and θi = θbaseline.

FIGURE 12. Robustness: national real interest rate gaps for different lagsof the real interest rate gap (IRG) in the IS equation (1,2 or 3).

Note. Solid line: IRG lagged by 1 months, dotted line: IRG lagged by 3 months, shaded area: 95%confidence intervals of the baseline estimate with σz/σy = 1.5, the IRG lagged by two months and θi =θbaseline.

FIGURE 13. Robustness: national real interest rate gaps for σz/σy = 1, 2,3, 4.

Note. Solid line: σz/σy = 1, dotted line: σz/σy = 2, dashed line: σz/σy = 4, shaded area: 95% confidenceintervals of the baseline estimate with σz/σy = 1.5, the IRG lagged by two months and θi = θbaseline.

[Not for publication, online appendix]

A. State space form of the model

A. Dealing with quarterly GDP in a monthly model of potential growth

We aim to perform a frequency decomposition of quarterly GDP into a trend, the outputgap, a cyclical component, and a high frequency shock. This decomposition is made moredifficult because we observe quarterly GDP only every three months (at the end of eachquarter), while the time period in our model is one month.

We detail in this appendix how we circumvent this problem in practice. Let us assumethat we could observe monthly GDP. We could then write:

∆yit = ∆y∗it +∆zit

∆yit = µyi +ait + zit− zi,t−1 + ε

yit (3)

where ∆yit is the change in log GDP over one month. This would provide us with asecond measurement equation alongside the Phillips Curve in a state-space formulationand would allow us to filter the unobserved components ait and zit using the IS curve andthe AR(1) specification of the productivity process as the main transition equations.

However, the growth rate of GDP is available at a quarterly frequency only. Let usdefine monthly GDP growth ∆yit (in logs) so that the (observed) quarterly growth rate ofthe GDP, ∆yQ

it , is the average of monthly growth rates over the three months in a givenquarter:

∆yQit =

13(∆yit +∆yi,t−1 +∆yi,t−2)

Hence, by substituting for ∆yit , we get:

∆yQit = µ

yi +

13(ait +ai,t−1 +ai,t−2)+

13(zit− zi,t−3)+

13

(ε

yit + ε

yi,t−1 + ε

yi,t−2

)Where now ∆yQ

it is the observed data. The Kalman filter can be adapted so as to dealwith missing observations in the measurement equations. However, having less data forthe same number of parameters to be estimated will mechanically lead to a higher uncer-tainty of parameter estimates and of the filtered trajectories of the latent variables. This isall the more a concern than our sample is relatively small: it includes about 200 monthlyobservations for each country, but only about 70 observations as far as GDP is concerned.

To solve this dimensionality problem, we augment the model with an additional mea-surement equation where we use the growth rate in the monthly Industrial ProductionIndex (IPI) as a proxy for monthly GDP growth, ∆y:

∆IPIit = µ

IPIi +ϕi∆yit + ε

IPIit

where ε IPIit

iid∼ N (0,(σ IPIi )2) and µ IPI

i , ϕi and σ IPIi > 0 are constants to be estimated.

Substituting ∆yit by its expression from (3), we get:

∆IPIit = ϕiµ

yi +µ

IPIi +ϕi

(ait + zit− zi,t−1 + ε

yit)+ ε

IPIit

B. State-space form of the model

For any country i, let si, pi and qi be the number of lags to be included in the systemrespectively for the inflation in the PC (with parameters (γi1, . . . ,γisi)), the autoregressivecomponent of the output gap in the IS equation(with parameters (αi1, . . . ,αipi))) and theoutput gaps in the PC (with parameters (βi1, . . . ,βiqi)). Let also li be the single lag orderof the interest rate gap to be included in the IS equation. In practice, we use the samenumber and order of lags for all the countries.

• Measurement equations

∆yQit = µ

yi +

13(ait +ai,t−1 +ai,t−2)+

13(zit− zi,t−3)+

13

(ε

yit + ε

yi,t−1 + ε

yi,t−2

)πit = µ

πi +

si

∑j=1

γi jπi,t− j +qi

∑j=1

βi jzi,t− j + επit

∆IPIit = ϕiµ

yi +µ

IPIi +ϕi

(ait + zit− zi,t−1 + ε

yit)+ ε

IPIit

• Transition equations

ait = ψiai,t−1 + εait

ai,t−1 = ai,t−1

ai,t−2 = ai,t−2

εyit = ε

yit (Strong white noise)

εyi,t−1 = ε

yi,t−1 (Past value)

εyi,t−2 = ε

yi,t−2 (Past value)

zit =pi

∑j=1

αi jzi,t− j +δi

n

∑j=1

χi jz j,t−1 +λi(it−li−Et−li

[πi,t−li+h

])−λiθiai,t−li−λiµ

ri + ε

zit

zi,t−1 = zi,t−1

...

zi,t−pi = zi,t−pi

where pi = max(pi,qi).

επ1t , . . . ,ε

πnt ,ε

y1t , . . . ,ε

ynt ,ε

z1t , . . . ,ε

znt are all assumed to be pairwise independent. On the

contrary, εa1t , . . . ,ε

ant are correlated and we denote Σ the corresponding variance-covariance

matrix of the Gaussian vector (εa1t , . . . ,ε

ant) and (ρi j)i j the correlation coefficients.

We assume that expected inflation at the future date t+h conditional to the informationavailable at t is measured without error by the Consensus forecast. Formally, this implies:

Et[πi,t+h

]= π

ci,t|t+h

For each country i, the system can be written under the following matrix formulation:{Yit = AiZit + BiX1

it + m1i + ε

Yit

Zit =CiZi,t−1 + DiX2it + m2

i + εZit

where Yit is the vector of observed variables and Zit is the vector of state variables:

Yit =

∆yQit

πit

∆IPIit

Zit =

ait

ai,t−1

ai,t−2

εyit

εyi,t−1

εyi,t−2

zit...

zi,t−pi

X1

it and X2it are vectors containing the exogenous and predetermined variables:

X1it =

πi,t−1...

πi,t−pi

X2it =

(it−li

πci,t−li|t−li+h

)

m1i and m2

i are drift terms:

m1i =

µyi

µπi

ϕ IPIi µ

yi +µ IPI

i

m2i =

0R6

−λiµri

0R pi

εY

it and εZit are independent multivariate Gaussian white noises:

εYit =

0επ

it

εESIit

εZit =

εait

00ε

yit

00ε

zit

0R pi

Finally, the remaining matrices Ai, Bi, Ci and Di are coefficients matrices which couldbe time-varying in the general case, but here are considered constant:

Ai =

13

13

13

13

13

13

13 0 0 −1

3 0 · · · 00 0 0 0 0 0 0 βi1 · · · · · · βipi

ϕi 0 0 ϕi 0 0 ϕi −ϕi 0 0 0 0 0

Bi =

0 · · · 0γi1 · · · γipi

0 · · · 0

Ci =

ψi 0 0 0 0 0 0 0 0 01 0 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0 00 0 0 0 1 0 0 0 0 0

−λiθi1li=1 −λiθi1li=2 −λiθi1li=3 0 0 0 αi1 · · · αipi 00 0 0 0 0 0 1 0 0 0...

......

......

... 0 . . . 0...

0 0 0 0 0 0 0 0 1 0

where 1li= j = 1 if li = j and 0 otherwise.

Di =

0 00 00 00 00 00 0λi −λi

0 0...

...0 0

where Ci and Di have the same number of rows as the state vector Zit , that is 7+ pi.

Omitting for a moment the term δi ∑nj=1 χi jz j,t−1, the complete system including all

the countries of interest also admits a compact matrix formulation. We define the new

vectors standing for the observed and the state variables by stacking the (Yit)i∈J1,nK and(Zit)i∈J1,nK:

Yt =

Y1t...

Ynt

Zt =

Z1t...

Znt

m1, m2, εY

t and εZt are defined similarly. Finally, for the remaining matrices (Ai)i∈J1,nK,

(Bi)i∈J1,nK, (Ci)i∈J1,nK and (Di)i∈J1,nK, we denote generically:

M =

M1 0. . .

0 Mn

Hence, the complete system can now be written as:

{Yt = AZt + BX1

t + m1 + εYt

Zt =CZt−1 + DX2t + m2 + ε

Zt

Let us now introduce the term that we previously omitted, the matrix formulation ofthe system has to be slightly adapted. We denote JZ the square matrix of same size asthe dimension of the state vector, Zt , full of zeros except the 7th diagonal term (the onecorresponding to the position of zt in the state vector) equal to 1, and δ = diag(δ1, ...,δn)

is a hyperparameter that needs to be estimated. It can be shown that the system now reads:

{Yt = AZt + BX1

t + m1 + εYt

Zt =CZt−1 + DX2t + m2 + ε

Zt +((δ χt)⊗ JZ)Zt−1

Hence:

{Yt = AZt + BX1

t + m1 + εYt

Zt = (C+(δ χ)⊗ JZ)Zt−1 + DX2t + m2 + ε

Zt

B. Initialisation of the Kalman Filter

One important point using the Kalman Filter, is to properly choose the initial first andsecond order moments –including all covariances– of the state vector. Recall that for eachcountry i, this state vector can be divided in three blocks:

ait

ai,t−1

ai,t−2

εit

εi,t−1

εi,t−2

zit

...zi,t−pi

Moments of (ait)it Since the dynamic of the productivity process is assumed to be anAR(1), we can analytically derive the unconditional moments, which are the natural can-didates for the initial point.Moments of (εy

it)it We already know that the first order moment of the elements of thisblock is zero and their variances are equal to σ

yi

2. Besides, since our specification as-sumes these shocks as independent of all other processes and across time, we get straight-forwardly that all the covariances are nil.Moments of (zit)it Regarding the mean and autocovariance of (zit)it, and cross-countrycovariances Cov(zit ,z j,t−k), we opt for the empirical approach performed in Mesonnierand Renne (2007): we first extract proxies of the output gaps of each country by apply-ing the Hodrick-Prescott filter on GDP data, and we then compute the sample mean andcovariance using these proxies.

C. Building confidence intervals of the factors

A useful output of the Kalman Filter is the covariance matrix of the factors, which allowus to build confidence intervals around the trajectories of the filtered and smoothed statevariables. As pointed out by Hamilton (1986), those confidence intervals are conditionedon the knowledge of the parameters of the model, which are in fact estimated by maxi-mum likelihood. He provides a method to build global confidence intervals that take intoaccount the uncertainty around the parameter estimates. Having performed the estimationand computed the covariance matrix of the parameter estimates, we would then generateM sets of parameters from the asymptotic distribution of our estimators:

θiiid∼N (θ , V)

where θ is the maximum likelihood estimate and V is the corresponding covariancematrix computed using the Hessian of the likelihood at θ . For each set of parameters θi,we run the Kalman Filter and get both the covariance matrix of the factors Pi

t (filtered orsmoothed), and the trajectories of these factors Zi

t , at all dates t. Hamilton (ibid.) showsthat the global variance around the factors at any date t can be approximated by:

Pt =1M

M

∑i=1

Pit +

1M

M

∑i=1

(Zi

t− Zt) t(Zi

t− Zt)

where: Zt =1M

M

∑i=1

Zit .

D. A useful mapping for estimating the correlation matrix

We need to estimate a correlation matrix:

Σ =

1 · · · ·ρ21 1 · · ·ρ31 ρ32

. . . · ·... . . . 1 ·

ρn1 . . . . . . ρn,n−1 1

Such matrix must verify the following conditions:

• Diagonal of ones• Off-diagonal entries between -1 and 1• Symmetric positive semi-definite

For any symmetric positive-semi-definite matrix M, we know from the Cholesky de-composition that there is a unique lower-triangular matrix, say U , with positive diagonalelements such that:

M =U tU

Let us denote:

U =

y11 0 0 0 0y21 y22 0 0 0

y31 y32. . . 0 0

... . . . yn−1,n−1 0yn1 . . . . . . yn,n−1 ynn

Estimating the matrix U would allow us to systematically obtain a symmetric positive

semi-definite Σ. However, in order for the resulting Σ to verify the two other conditions,the entries of U cannot be chosen independently. In particular, from a direct computation,it can be noticed that we would always have y11 = 1. Starting from any,

X =

x21 0 0 0

x31 x32. . . 0

... . . . 0xn1 . . . . . . xn,n−1

where xi j ∈ R for all i and j, we construct recursively a matrix U : for any line of

i ∈ J2,nK and any column j ∈ J2, i−1K

y11 = 1

yi1 = f (xi1)

yi j = f (xi j)

√√√√1−j−1

∑l=1

y2il

yii =

√√√√1−i−1

∑l=1

y2il

where f : x 7−→ 1− ex

1+ ex .

It can be shown that the matrix U constructed in such way implies that U tU verifies theproperties listed above. Reciprocally, any matrix Σ verifying those same properties canbe decomposed in such way.

The function

g : Rn(n−1)

2 −→ S+n (R)

X 7−→ Σ =U tU

is continuously differentiable. For the estimation stage, only the knowledge of g isrequired in order to constraint the optimisation algorithm to find an optimum in the setof correlation matrices. However, to obtain the covariance matrix of the estimates (ρi j),given the covariance matrix of the estimates of (xi j), we use the delta method whichrequires the knowledge of the Jacobian of the function g. It can be derived analytically.Let us proceed in two stages: we first derive the Jacobian of a : X 7−→U , and then weturn to the Jacobian of b : U 7−→ Σ.

• Jacobian of a : X 7−→UFor any line i ∈ J2,nK and any column j ∈ J2, i−1K of the matrix U :

∂yi1

∂xi1= f ′(xi1)

∂yi j

∂xip=− f (xi1)

j−1∑

l=1yil

∂yil

∂xip√1−

j−1∑

l=1y2

il

, 1≤ p < j

∂yi j

∂xi j= f ′(xi j)

√√√√1−j−1

∑l=1

y2il

∂yii

∂xip=−

i−1∑

l=1yil

∂yil

∂xip√1−

i−1∑

l=1y2

il

, 1≤ p≤ i

Else:

∂yi j

∂xmp= 0, i f m 6= i, 1≤ p≤ n

∂yi j

∂xip= 0, i f j < i, j < p≤ n

• Jacobian of b : U 7−→ Σ

Expanding for each entry the computation Σ =U tU , we have:

∀i ∈ J2,nK,∀ j ∈ J1, i−1K, ρi j =j

∑l=1

yily jl

Therefore, for any (m,n) ∈ J2,nK× J1,n−1K:

∂ρi j

∂xmn=

j

∑l=1

[yil

∂y jl

∂xmn+ y jl

∂yil

∂xmn

]

Documents de Travail

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601. R. S.J. Koijen; F. Koulischer; B. Nguyen and M. Yogo, “Quantitative Easing in the Euro Area: The Dynamics of

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602. O. de Bandt and M. Chahad, “A DGSE Model to Assess the Post-Crisis Regulation of Universal Banks” September 2016

603. C. Malgouyres, “The Impact of Chinese Import Competition on the Local Structure of Employment and Wages:

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604. G. Cette, J. Lopez and J. Mairesse, “Labour market regulations and capital labour substitution” October 2016

605. C. Hémet and C. Malgouyres, “Diversity and Employment Prospects: Neighbors Matter!” October 2016

606. M. Ben Salem and B. Castelletti-Font, “Which combination of fiscal and external imbalances to determine the long-run dynamics of sovereign bond yields?” November 2016

607. M. Joëts, V. Mignon and T. Razafindrabe, “Does the volatility of commodity prices reflect macroeconomic

uncertainty?” November 2016

608. M. Bussière, G. Gaulier and W. Steingress, “Global Trade Flows: Revisiting the Exchange Rate Elasticities” November 2016

609. V.Coudert and J. Idier, “An Early Warning System for Macro-prudential Policy in France” November 2016

610. S. Guilloux-Nefussi, “Globalization, Market Structure and Inflation Dynamics” December 2016

611. S.Fries, J-S. Mésonnier, S. Mouabbi, and J-P. Renne, “National natural rates of interest and the single monetary

policy in the Euro Area” December 2016

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