Output Gap Synchronicity in the Eurozone
An Econometric Analysis of Business Cycle Convergence
Samuël David Nelemans
s2550334
Master’s Thesis Econometrics, Operational Research and Actuarial Studies Rijksuniversiteit Groningen // University of Groningen
Output Gap Synchronicity in the Eurozone
An Econometric Analysis of Business Cycle Convergence
Samuël David Nelemans – s2550344
7th June 2019
Abstract
Output gaps synchronicity of countries within a monetary union can reflect the efficacy of monetary policy within this union. I propose an econometric framework for analysing the intrinsic synchronicity of a region using the majority binomial distribution. I use this framework in three settings: constant synchronicity, country-specific synchronicity and time variant synchronicity. I then apply this framework to the eurozone and the European Union between 1996 and 2018 and find that synchronicity has increased since the introduction of the euro up until the period 2008-2014, after which a decline is observed. During the sampling period, output gap synchronicity seems higher in periods of recession. I also analyse spatial differences in synchronicity and find that the traditional core-periphery distinction does not provide a complete description of European business cycle coherence. Keywords:
1
Introduction
After a period of business cycle convergence in the eurozone, the last decade has seen a pattern of decoupling. European output gap synchronisation seems to have been steadily increasing since the introduction of the euro and peaked sharply during the 2008 financial crisis as all countries experienced a simultaneous recession, yet since 2014 business cycles seem to be diverging again. This may be due to the different speeds of recovery for European national economies, with some countries experiencing a period of economic expansion whilst others are is still recuperating. These findings hold for the original eurozone, the full modern eurozone and for the European Union as a whole.
The importance of output gap coherence to the efficacy of common monetary policy is well documented in the literature (Kenen, 1969; Bergman, 2006; Savva et al., 2010; Crowley and Schultz, 2011). The core of the argument is that in a region with unsyn-chronised business cycles, any monetary policy will be procyclical for some countries and countercyclical for others. On the other hand, some authors have argued that the formation of a monetary union will in itself cause higher synchronisation between regions (Frankel and Rose, 1998; Bergman and Jonung, 2011). Degiannakis et al. (2014) provides
an extensive overview of this literature.
I analyse the evolution of European output gap coherence over time by further developing the synchronicity metric of Mink et al. (2011). This metric distinguishes itself by only considering the signs of output gaps in a region rather than their magnitude, and by using the median output gap as a reference for the region. I develop a new econometric framework for analysing inherent synchronicity by formulating the majority binomial distribution, which is a transformation of the regular binomial distribution. Using this distribution, I can apply maximum likelihood or Bayesian estimation techniques to obtain credible regions and perform statistical tests on the synchronicity metric without bootstrapping. Next, I analyse changes in synchronicity over time using rolling window estimation and formulate a state space model where synchronicity is a latent variable.
In line with the literature, I find that European output gap coherence has been increasing in the first decade after the euro was introduced. However, this pattern was subverted after 2012-2014, when the business cycles of Member States diverged. Furthermore, the differences between individual countries in output gap dependence on the rest of Europe show no clear-cut “core-periphery” divide, with several traditional core countries showing low levels of synchronicity and vice-versa. The results are very similar for the original eurozone members from 2002, the Modern eurozone (2019) and in the European Union as a whole.
2
Related Literature
Although the techniques used by papers to analyse business cycle synchronicity in Europe vary widely, the differences in conclusions seem to arise mainly from the sampling period under analysis. The emerging pattern seems to be in line with the results of this paper: output gap synchronicity has increased since the introduction of the euro until 2010 (Degiannakis et al., 2014; Gogas, 2013; Papadimitriou et al., 2016), after which this
development stagnates or even reverses (Gomez et al., 2017).
It is common in the literature on business cycle synchronicity to consider deviation cycles, that is, deviations from a long-term trend in GDP growth (De Haan et al., 2008). Decomposing GDP into a trend and a cyclical component is a non-trivial matter, for which a wide array of techniques have been developed. Most common in the business cycle synchronicity literature are univariate non-parametric filters such as the ones proposed by Hodrick and Prescott (1997), Baxter and King (1999) and Christiano and Fitzgerald (2003). Despite a large body of criticism on these methods (Hamilton, 2018), they remain popular due to their simplicity and because they yield rather stable results comparable to for instance linear or quadratic trend filters. Although these methods may yield very different business cycle patterns, conclusions regarding synchronisation seem largely unaffected by the specific choice of filter (De Haan et al., 2008).
The disadvantage of these methods is that they require multiple observations over time to be estimated, implying that developments over time can only be analysed by dividing the sample into blocks. Many of these papers test for a difference in synchronicity between two periods using a structural break test or execute a series of rolling window regressions. The window size plays a crucial role here: using a window which is too broad will dilute the visibility of change points, whereas a narrow window will yield idiosyncratic behaviour of the estimator.
One line of research suggests a state space formulation in which synchronicity changes over time (Koopman and Valle E Azevedo, 2008; Savva et al., 2010; Degiannakis et al., 2014). In these models, the interactions between national business cycles in a region are embedded in a structural model with a time-varying correlation structure. Whilst very powerful, such models often require many structural assumptions for their key parameters to be identified.
development of synchronicity over time without employing bootstrapping methods.
Part of the debate on output gap synchronicity in the eurozone has focussed on the core-periphery distinction. Papers such as Lehwald (2013) and Belke et al. (2017b) claim that the eurozone is essentially divided into a core region and a periphery, the latter being often described with the GIIPS (Greece Ireland Italy Portugal Spain) acronym. These papers show evidence that the euro currency has promoted synchronicity within the region’s core, yet that the peripheral countries have been suffering from countercyclical ECB policy. However, Ferroni and Klaus (2015) conduct a multivariate analysis of France, Germany, Italy and Spain, and find that the former three form a coherent economic block. This contradicts the notion of Italy being a peripheral country, or of a simple North-South distinction between European economies. Furthermore, Gomez et al. (2017) uses a network-based clustering approach to show that if clusters are to be determined latently based on business cycle synchronisation, the distinction between core and periphery becomes far less apparent. Although sorting countries into predefined groups and then testing for differences might yield significant results, there is no conclusive evidence that a core-periphery distinction is inherent to the nature of the European economic environment.
EMU reform plans of 2012 and 2015. All these events can be argued to have to some extent impacted the functioning of the eurozone, as explored by Degiannakis et al. (2014). Furthermore, even if for instance the introduction of the euro as a physical currency on the 1st of January 2002 did constitute a one-period shock rather than being one step in a decades-long integration process, given the “sticky” nature of GDP with respect to structural economic developments the introduction of the euro currency in 2002 likely did not constitute an immediate shift in output gap synchronicity. For this reason, testing for a difference average correlation in the decade before and after the introduction of the euro gives no conclusive evidence regarding the efficacy of the European Monetary Union. Instead, analysis of long-term trends in output gap coherence seems preferable, which is why immediate measures such as synchronicity are powerful.
3
Model
Consider a region of n countries, each of which is to some extent economically integrated with the region as a whole. The sign of the output gap of any country in this sample can be considered the outcome of a Bernoulli trial, as can the sign of the median output gap for this region. Let θit denote the probability that a country’s business cycle is synchronised
with the region, defined as the sign of the output gap of country i at time t being equal to the sign of the median output gap for the region at that time. This probability can then be considered a proxy for the level of output gap coherence between a country and the region containing it. I assume that all statistical dependence between the output gap signs of countries as well as any correlation over time is captured by the synchronicity parameter θit. This framework allows an analysis of output gap coherence regardless the phase of
My synchronicity metric resembles the one created by Mink et al. (2011), but differs on a few core aspects. Mink et al. (2011) define a country’s synchronicity φit as a binary
outcome variable equal to 1 or -1 depending on the sign of the output gap at a given moment, and the region’s synchronicity φt as the unweighted average of these individual
synchronicities, which can take any value of the form φt = 2k/n where n is the number
of countries in the region and k ∈ {N|0 ≤ k ≤ n/2}. I will refer to this as observed synchronicity. Contrarily, I define synchronicity θit as a country-specific probability
parameter on the interval [0, 1] ⊂ R, and regional synchronicity θt as a latent parameter
on the interval [0.5, 1] ⊂ R. This change has several advantages: first of all, observed synchronicity is very volatile, which makes analysing longer-term patterns more difficult. Mink et al. (2011) solve this by taking 8-year moving averages of their metric, but this somewhat obscures the impact magnitude of important macroeconomic events. Second, defining synchronicity on a continuum rather than on a discrete set of points between 0 and 1 allows the outcome space of this parameter to be independent of the number of countries in the analysed region, and better fits econometric frameworks such as state space models. Finally, the interpretation of θit as a probability parameter yields an
intuitive link between data and interpretation. Reparametrising 2θt− 1 would again yield
a metric on the outcome space [0,1] as in Mink et al. (2011).
At this point, I distinguish three special cases:
1. θit = θ Constant synchronicity over time, equal for all countries.
2. θit = θi Constant synchronicity over time, varies between countries.
3. θit = θt Time-dependent synchronicity, equal for all countries.
The second case is rather trivial in estimation: the number of periods that country i has the same output gap sign as the regional median during a timespan of T periods can be considered a binomially distributed variable with success probability θi and with T
trials. However, for the first and third case, this matter is more complicated: as the regional reference to determine synchronicity is the median output gap, the number of countries with an equal sign to the reference can never be lower than half the total number of countries. For this reason, I define the majority binomial distribution and use it as a likelihood function to estimate synchronicity for the region as a whole.
Definition: Majority Binomial Distribution
Let Y ∼ b(θ, n) with θ > 12. The majority binomial distribution, denoted MBD(θ, n), is defined as the distribution function of X = max{Y, n − Y }. For any integer k such that n2 ≤ k ≤ n we find:
P (X = k) = n k h θk(1 − θ)n−k + θn−k(1 − θ)ki.
This distribution is very well fit for modelling output gap synchronicity in a region. Considering a region of n countries, each of which has inherent synchronicity θ, the number of countries synchronised with the aggregate business cycle follows a binomial distribution. However, the number of countries observed to be synchronised with the median business cycle is the maximum of the number of countries with a positive output gap and the number of countries with a negative output gap. For this reason, the probability of observing k countries synchronised with the median is equal to the probability of k countries being synchronised plus the probability of n − k countries being synchronised. This justifies the transformation of the binomial distribution to the majority binomial distribution.
different time periods for a single region. Additionally, I employ a Bayesian approach with a flat prior distribution of θ, in which case its posterior distribution is proportional to the likelihood function. I estimate the mean and variance of the posterior by sampling 10001 observations using the rejection method. The parameter θ is restricted to be greater than 1/2 because the likelihood function is always symmetric around this point. Therefore, as the likelihood evaluated at θ is exactly equal to the likelihood at (1 − θ) regardless of the data, leaving θ unrestricted over [0, 1] creates an identification problem.
In the framework where θt varies over time but is given for the region as a whole, I
employ two Bayesian estimation methods. The first is a simple rolling window estimation of θt using a 15-period and a 30-period window size. I chose these numbers because
the average duration of a full euro area business cycle between 1973 and 2013 was approximately 7.5 years, or 30 quarters (Euro Area Business Cycle Dating Committee, 2017). The difference between these window sizes reflects the underlying assumptions regarding how fast inherent synchronicity develops, and any formal test to distinguish model performance for different window sizes would still rely on these assumptions.
Second, I specify a state space model, where the single state space variable θt follows
a random walk bounded on the interval [0.5, 1], with truncated Gaussian increments1.
Thus, each period synchronicity follows a truncated normal distribution with the previous synchronicity at its mean, and observed synchronicity is drawn from the majority binomial distribution. This yields the following system:
Observation Equation: Xt∼ MBD(θt, n);
State Equation: θt∼ N{0.5;1}(θt−1, σ2),
1A more general setup in which θ
t follows an AR(1) process yields maximum likelihood estimates of
the autoregressive coefficient very close to 1. This motivates the decision for modelling θtas a bounded
where Xt denotes observed synchronicity at time t for the full region, n is the number of
countries in this region, and N{a;b}(µ, σ2) denotes a normal distribution with mean µ and
variance σ2 truncated at a and b.
This state space formulation satisfies the Markov property, implying that the condi-tional distribution of the state and observation variables on all past observations is equal to their conditional distribution on the previous observation. This allows me to estimate the time series {θt} using a dynamic Bayesian filter. The posterior distribution of θt
can then be expressed in terms of the likelihood function and the posterior distribution as: f (θt|Xt) = f (θt|Xt, Xt−1) ∝ f (Xt|θt, Xt−1)f (θt|Xt−1) = f (Xt|θt) Z f (θt|θt−1)f (θt−1|Xt−1)dθt−1 = MBD(θt, n) Z N{0.5;1}(θt−1, σ2)f (θt−1|Xt−1)dθt−1.
I use an iterative algorithm to sample from the posterior distribution of θt based on a
sample from the posterior distribution of θt−1. First, I consider the probability density
function N{0.5;1}(θt−1, σ2) for all values of θt−1 sampled from its posterior, and take the
average of these density functions. This yields an approximation for the integral term in the equation above, and it serves as the prior density of θt. Multiplying this density
function by the likelihood function corresponding to Xt yields the posterior distribution
of θt, which I then sample 1001 draws from using the rejection method. To initiate my
algorithm with a starting sample, I furthermore assume that θ0 = X1/n almost surely.
Sampling θ0 from a uniform distribution or estimating the starting value by maximum
4
Empirical Analysis
4.1
Retrieving Output Gaps
All data GDP used in this paper was retrieved from Eurostat on April 21st 2019, series namq_10_gdp. I consider seasonally and calendar adjusted real GDP indices from 1996Q1 until 2018Q4. I omit Malta and Croatia from my analysis because I require a balanced panel and GDP data for these countries is only available from the first quarter of 2000. I then consider three regions relevant for my analysis: first of all, 12 original eurozone members (EA12), being Austria, Belgium, Germany, Spain, Finland, France, Ireland, Italy, Luxembourg, the Netherlands, Portugal and Greece. I furthermore consider the modern eurozone (EA19 without Malta), consisting of all countries previously mentioned in addition to Slovenia, Cyprus, Slovakia, Estonia, Latvia and Lithuania. Finally, I consider the European Union as a whole (EU27 without Malta), which also includes Bulgaria, the Czech Republic, Denmark, Hungary, Poland, Romania, Sweden and, up until this point, the United Kingdom.
From each GDP time series, output gaps are retrieved using the Christiano-Fitzgerald filter (Christiano and Fitzgerald, 2003), which according to Koopman and Valle E Azevedo (2008) is “the best performing and most flexible filter available.” I next calculate the observed synchronicity series as in Mink et al. (2011): these consist of three time series displaying the number of countries for the original eurozone, the full modern eurozone and for the European Union as defined above observed to be synchronised with the median of the respective region.
4.2
Constant Synchronicity
maximum likelihood estimates for θ and the log-likelihood at this maximum, as well as the Bayesian posterior mean and standard deviation based on a flat prior. The latter two have been constructed by simulating 10001 draws from the posterior distribution of θ using rejection sampling. Furthermore, Figure 1 plots the posterior density of θ for the three regions under consideration.
Table 1: Estimated Synchronicity for European Regions ˆ
θML θˆBayes St.Dev. Log-likelihood Original Eurozone 0.6804 0.6799 0.0152 -667.6510 Modern Eurozone 0.6630 0.6625 0.0124 -1038.1538 European Union 0.6574 0.6575 0.0101 -1521.2397
Note: The first column gives the maximum likelihood estimate for synchron-icity of the region as a whole. The second column shows the posterior mean of synchronicity, and the third column shows the posterior standard deviation. The fourth column displays the log-likelihood value at the maximum.
Figure 1: Posterior Density of θ 0.5 0.6 0.7 0.8 0.9 1.0 0 10 20 30 40 θ
Density Original EurozoneEurozone EU
Note: Posterior distribution of synchronicity based on a flat prior and the full sample period of observations for the original eurozone, the modern eurozone and the European Union. Note that these functions are proportional to their respective likelihood functions, normalised to be interpretable as probability density functions.
4.3
Country-Specific Synchronicity
Figure 2 shows the average observed synchronicity of individual European countries with their respective regions over time. Furthermore, the leftmost columns of Tables 2, 3 and 4 report the country-specific synchronicity levels. The bottom row of these tables reports average synchronicity over all countries. Note that these estimates differ from the estimates in Table 1: this difference arises from the tables in this section assuming the synchronicity of individual countries being drawn from a binomial distribution. Hence, these average synchronicity coefficients disregard the dependence of the median output gap on the sample of national output gaps, thus yielding upward-biased estimates of true synchronicity.
Figure 2: Synchronicity of European Countries 0.5 0.6 0.7 0.8 Synchroncity
Top left: Original eurozone; Top right: Modern eurozone; Bottom: European Union.
Note: Colours reflect average synchronicity of each country over the full sampling period. Note that these numbers are also reported in Tables 2, 3 and 4 respectively.
Table 2: Output Gap Synchronicity per Country – Original Eurozone Full Sample Pre-2008 Post-2008 CEPR Exp. CEPR Rec. Median Positive Median Negative AT 0.6848 0.6875 0.6818 0.6625 0.8333 0.7021 0.6667 BE 0.8152 0.8333 0.7955 0.8250 0.7500 0.8085 0.8222 DE 0.7283 0.6667 0.7955 0.7000 0.9167 0.7447 0.7111 ES 0.6848 0.5625 0.8182 0.6375 1.0000 0.7660 0.6000 FI 0.5870 0.5208 0.6591 0.5750 0.6667 0.6170 0.5556 FR 0.7283 0.6458 0.8182 0.6875 1.0000 0.7872 0.6667 IE 0.6087 0.6042 0.6136 0.6125 0.5833 0.5957 0.6222 IT 0.8152 0.8125 0.8182 0.8250 0.7500 0.8511 0.7778 LU 0.5870 0.5000 0.6818 0.5875 0.5833 0.5106 0.6667 NL 0.7391 0.7292 0.7500 0.7250 0.8333 0.7660 0.7111 PT 0.6739 0.5625 0.7955 0.6375 0.9167 0.6809 0.6667 GR 0.6630 0.6875 0.6364 0.6875 0.5000 0.6383 0.6889 Avg. 0.6929 0.6510 0.7386 0.6802 0.7778 0.7057 0.6796
Note: The first column shows the average proportion over time of each country’s output gap having the same sign as the median output gap. The second column block splits the sample up into the period before and after 2008. The third column block splits the sample in periods of expansion and recession as denoted by the CEPR. The fourth column block splits the sample based on the sign of the relevant median output gap for each row. The bottom row reports the average synchronicity for the original eurozone.
Amongst the most synchronised countries in the original eurozone are Italy and Belgium, followed by Germany. Italy and Germany, large countries at the heart of Europe, would be expected to have a large influence on the economies of surrounding countries, whilst Belgium as a multilingual trade nation is highly dependent on exports to Germany and France. The high synchronicity of Italy in the eurozone reflects its place amongst the core countries rather than in the periphery if such a distinction were to be made.
Table 3: Output Gap Synchronicity per Country – Modern Eurozone Full Sample Pre-2008 Post-2008 CEPR Exp. CEPR Rec. Median Positive Median Negative AT 0.6630 0.6458 0.6818 0.6500 0.7500 0.6889 0.6383 BE 0.7717 0.7500 0.7955 0.7875 0.6667 0.7778 0.7660 DE 0.7283 0.6667 0.7955 0.7125 0.8333 0.7556 0.7021 ES 0.6848 0.5625 0.8182 0.6500 0.9167 0.7778 0.5957 FI 0.5870 0.5208 0.6591 0.5625 0.7500 0.6222 0.5532 FR 0.6630 0.5625 0.7727 0.6250 0.9167 0.7333 0.5957 IE 0.5435 0.5208 0.5682 0.5250 0.6667 0.5333 0.5532 IT 0.7935 0.7292 0.8636 0.8125 0.6667 0.8444 0.7447 LU 0.5435 0.5000 0.5909 0.5250 0.6667 0.4667 0.6170 NL 0.7391 0.7292 0.7500 0.7125 0.9167 0.7778 0.7021 PT 0.6739 0.6458 0.7045 0.6500 0.8333 0.6889 0.6596 GR 0.6630 0.6875 0.6364 0.7000 0.4167 0.6444 0.6809 SI 0.7283 0.6875 0.7727 0.7000 0.9167 0.7111 0.7447 CY 0.6522 0.6042 0.7045 0.6125 0.9167 0.6667 0.6383 SK 0.7500 0.6250 0.8864 0.7375 0.8333 0.7111 0.7872 EE 0.5870 0.5417 0.6364 0.5875 0.5833 0.5556 0.6170 LV 0.5978 0.5833 0.6136 0.6250 0.4167 0.5556 0.6383 LT 0.6957 0.6250 0.7727 0.6750 0.8333 0.7556 0.6383 Avg. 0.6703 0.6215 0.7235 0.6583 0.7500 0.6815 0.6596
Note: The first column shows the average proportion over time of each country’s output gap having the same sign as the median output gap. The second column block splits the sample up into the period before and after 2008. The third column block splits the sample in periods of expansion and recession as denoted by the Centre for Economic Policy Research. The fourth column block splits the sample based on the sign of the relevant median output gap for each row. The bottom row reports the average synchronicity for the modern eurozone.
centre of economic activity, explains some of the disparities, but this notion is again contradicted by Luxembourg.
The low levels of average synchronicity over the sampling period of Luxembourg and Finland could also be explained by the lower pre-crisis national debt levels of these countries (7.7% and 32.7% of national GDP respectively). This gave these countries the freedom to finance economic stimulation measures by issuing new debt without violating the Maastricht criteria, thus limiting the impact of the financial crisis of 2008 on these countries and lowering their synchronicity with the rest of Europe.
Table 4: Output Gap Synchronicity per Country – European Union Full Sample Pre-2008 Post-2008 CEPR Exp. CEPR Rec. Median Positive Median Negative AT 0.6739 0.6458 0.7045 0.6500 0.8333 0.6957 0.6522 BE 0.7174 0.6250 0.8182 0.7125 0.7500 0.7174 0.7174 DE 0.7826 0.7500 0.8182 0.7625 0.9167 0.8043 0.7609 ES 0.7174 0.6042 0.8409 0.6750 1.0000 0.8043 0.6304 FI 0.5761 0.5208 0.6364 0.5625 0.6667 0.6087 0.5435 FR 0.7609 0.6875 0.8409 0.7250 1.0000 0.8261 0.6957 IE 0.4891 0.4375 0.5455 0.4750 0.5833 0.4783 0.5000 IT 0.7609 0.6875 0.8409 0.7625 0.7500 0.8043 0.7174 LU 0.5109 0.4583 0.5682 0.5000 0.5833 0.4348 0.5870 NL 0.6630 0.6458 0.6818 0.6375 0.8333 0.6957 0.6304 PT 0.6413 0.5625 0.7273 0.6000 0.9167 0.6522 0.6304 GR 0.6304 0.6042 0.6591 0.6500 0.5000 0.6087 0.6522 SI 0.7174 0.6875 0.7500 0.7000 0.8333 0.6957 0.7391 CY 0.7065 0.6875 0.7273 0.6625 1.0000 0.7174 0.6957 SK 0.7391 0.6667 0.8182 0.7375 0.7500 0.6957 0.7826 EE 0.5978 0.5833 0.6136 0.6125 0.5000 0.5652 0.6304 LV 0.5652 0.5000 0.6364 0.6000 0.3333 0.5217 0.6087 LT 0.6848 0.6250 0.7500 0.6750 0.7500 0.7391 0.6304 BG 0.5435 0.3542 0.7500 0.5125 0.7500 0.5652 0.5217 CZ 0.7283 0.6458 0.8182 0.7125 0.8333 0.6739 0.7826 DK 0.6522 0.6875 0.6136 0.6375 0.7500 0.6522 0.6522 HU 0.6957 0.6667 0.7273 0.6875 0.7500 0.6739 0.7174 PO 0.5761 0.5833 0.5682 0.5625 0.6667 0.5435 0.6087 RO 0.7609 0.7917 0.7273 0.7500 0.8333 0.7826 0.7391 SE 0.6848 0.6667 0.7045 0.6875 0.6667 0.6957 0.6739 GB 0.6522 0.6042 0.7045 0.6500 0.6667 0.6739 0.6304 Avg. 0.6626 0.6146 0.7150 0.6500 0.7468 0.6664 0.6589
Note: The first column shows the average proportion over time of each country’s output gap having the same sign as the median output gap. The second column block splits the sample up into the period before and after 2008. The third column block splits the sample in periods of expansion and recession as denoted by the Centre for Economic Policy Research. The fourth column block splits the sample based on the sign of the relevant median output gap for each row. The bottom row reports the average synchronicity for the European Union.
instance, Germany, Italy and France all have a large automobile industry, which could result in greater dependence of national GDP on European economic conditions.
The next columns in Tables 2, 3 and 4 report the observed synchronicity of Member States for specific parts of the sampling period. I make this split in three ways: first, a split between the periods 1996Q1-2007Q4 and 2008Q1-2019Q1. Next, a split between periods of economic expansion and contraction as denoted by the Centre for Economic Policy Research (CEPR) (Euro Area Business Cycle Dating Committee, 2017). This agency has defined two periods of recession since 1995: one from 2008Q2 until 2009Q2, and one from 2011Q3 until 2013Q2. Finally, I split the sample into periods where the median output gap of the respective region used for synchronicity calculations is positive or negative. I report these differences mainly as a reference for the next section, to shed some light on which regions possibly cause observed changes in synchronicity over time. However, as these numbers are both country and time specific yielding rather observation series, many observed differences may be sampling irregularities, which is why I refrain from a formal analysis here.
4.4
Time-dependent Synchronicity
To investigate how European output gap synchronicity may have changed over the past two decades, Figure 3 reports the Bayesian posterior mean and credible intervals for syn-chronicity from rolling window estimation, with window sizes of 15 and 30 quarters.
Figure 3: Rolling Window Synchronicity Estimation
0.5
0.6
0.7
0.8
0.9
Original Eurozone −− Window Size 15
Synchronicity 2000Q1 2003Q1 2006Q1 2009Q1 2012Q1 2015Q1 2018Q1 0.5 0.6 0.7 0.8 0.9
Original Eurozone −− Window Size 30
2004Q1 2007Q1 2010Q1 2013Q1 2016Q1 2019Q1 0.5 0.6 0.7 0.8 0.9
Modern Eurozone −− Window Size 15
Synchronicity 2000Q1 2003Q1 2006Q1 2009Q1 2012Q1 2015Q1 2018Q1 0.5 0.6 0.7 0.8 0.9
Modern Eurozone −− Window Size 30
2004Q1 2007Q1 2010Q1 2013Q1 2016Q1 2019Q1 0.5 0.6 0.7 0.8 0.9
European Union −− Window Size 15
Last Quarter of Rolling Window
Synchronicity 2000Q1 2003Q1 2006Q1 2009Q1 2012Q1 2015Q1 2018Q1 0.5 0.6 0.7 0.8 0.9
European Union −− Window Size 30
Last Quarter of Rolling Window
2004Q1 2007Q1 2010Q1 2013Q1 2016Q1 2019Q1
Apparent from this figure is that synchronicity clearly rose between 2007 and 2010, a period marked by the financial crisis and the subsequent recession experienced throughout Europe, and started falling again in 2014. This contradicts the popular notion that European economies diverged directly after the Greek sovereign debt crisis of 2010. One possible explanation for the post-2014 divergence is that at this point, the stronger economies of Europe were well recovered from a period of economic turmoil whilst less resilient countries were still suffering from recession.
In the left side graphs of Figure 3, another peak in synchronicity can be observed right after the introduction of the euro currency. This peak is however smoothed out when increasing the window size. Thus, this peak in synchronicity after the euro introduction was either very short-lived or simply a statistical anomaly. However, for the shorter window size, this early peak extends beyond the credible intervals of the low periods surrounding it. Thus, assuming that synchronicity is a quick-moving variable, the low and brief peak in synchronicity after the introduction of the euro likely occurred. Note that this peak is even more pronounced when analysing the full European Union rather than the eurozone. This may be due to the lowered volatility of the synchronicity statistic in larger regions, producing more stable time series and more pronounced peaks. Nevertheless, it confirms the hypothesis that the introduction of the euro also served to increase synchronicity for non-eurozone EU Member States.
difference in average synchronicity. These result should, however, be analysed critically: Figure 3 implies that synchronicity changes multiple times during the sampling period, and identifying a single break in the sample could easily give the wrong impression.
Table 5: Likelihood Ratio Test Statistics
2002Q1 2008Q1 2010Q3 CEPR Median Original Eurozone θ Low 0.6061 0.6202 0.6646 0.6647 0.6663 θ High 0.7018 0.7339 0.7069 0.7741 0.6939 Test Stat. 6.2307∗∗ 13.2092∗∗∗ 1.8723 6.6745∗∗∗ 0.8201 Modern Eurozone θ Low 0.5502 0.5950 0.6398 0.6485 0.6486 θ High 0.6909 0.7220 0.7021 0.7495 0.6770 Test Stat. 16.6776∗∗∗ 24.5047∗∗∗ 6.2153∗∗ 8.5708∗∗∗ 1.3312 European Union θ Low 0.5148 0.5981 0.6328 0.6431 0.6528 θ High 0.6890 0.7138 0.6970 0.7466 0.6621 Test Stat. 32.2647∗∗∗ 31.8258∗∗∗ 9.8646∗∗∗ 13.1490∗∗∗ 0.2122 ∗p<0.1;∗∗p<0.05;∗∗∗p<0.01
Note: Split-sample parameter estimates and likelihood ratio test statistics for structural breaks and two-regime models for synchronicity. The first three columns show test statistics for a structural break at the introduction of the euro as a physical currency, the onset of the Lehman crisis and the Greek sovereign debt crisis. The fourth column splits the sample up in periods of expansion and retraction as denoted by the Centre for Economic Policy Research. The fifth column splits the sample based on the sign of the relevant median output gap for each row. Assuming an asymptotic
χ21 distribution of the test statistic, critical values for all tests are 2.706 at the 90% level, 3.841 at the 95% level and 6.635 at the 99% level.
4.5
State Space Model
Finally, I estimate the trajectory of European synchronicity over time using a state space model. As the variance of θt− θt−1 is subject to the same ambiguity as the rolling window
size earlier, I estimate three models for each region, for which the standard deviation of a one-period increment in latent synchronicity is given by σ = 0.05, σ = 0.02, and σ = 0.01 respectively2. Table 6 reports the mean and standard deviation over time of
the retrieved series, as well as the log-likelihood for the full time series. Note that the standard deviations in this table are the total standard deviations of latent synchronicity over the full sampling period, rather than the one-step standard deviations reflected by σ. Figures 4 and 5 report the trajectory of estimated latent synchronicity over time for σ = 0.02 and for σ = 0.05 and σ = 0.01 respectively.
Table 6: Maximum Likelihood Synchronicity from State Space Model σ = 0.05 σ = 0.02 σ = 0.01 Original Eurozone Mean 0.6696 0.6560 0.6583 Std. Dev. 0.0935 0.0933 0.0885 Log-lik 62.9765 61.7823 60.1705 Modern Eurozone Mean 0.6719 0.6603 0.6578 Std. Dev. 0.1001 0.0765 0.0692 Log-lik 61.2915 60.6280 60.5113 European Union Mean 0.6644 0.6538 0.6368 Std. Dev. 0.1066 0.0869 0.0857 Log-lik 60.6624 59.6165 58.5905
Note: Mean and standard deviation over time of the maximum likeli-hood estimate for synchronicity from a state space model. Every third row reports the total log-likelihood of the model at the maximum-likelihood filtered time series, given by the sum of log likelhoods for each individual timeperiod. The columns reflect the assumed one-period standard deviation of latent synchronicity.
These numbers paint a very similar picture to the rolling window estimations. A small brief peak can be observed for synchronicity after the introduction of the euro in all regions, as well as two larger peaks around 2010 and 2014. Contrary to the rolling
2Unfortunately, estimating σ through maximum likelihood results in an arbitrarily small value for σ,
Figure 4: State Space Model Synchronicity and Bayesian Credibility Intervals for σ = 0.02 0.5 0.6 0.7 0.8 0.9 1.0 Original Eurozone θt 1996Q1 1998Q1 2000Q1 2002Q1 2004Q1 2006Q1 2008Q1 2010Q1 2012Q1 2014Q1 2016Q1 2018Q1 0.5 0.6 0.7 0.8 0.9 1.0 Modern Eurozone θt 1996Q1 1998Q1 2000Q1 2002Q1 2004Q1 2006Q1 2008Q1 2010Q1 2012Q1 2014Q1 2016Q1 2018Q1 0.5 0.6 0.7 0.8 0.9 1.0 European Union θt 1996Q1 1998Q1 2000Q1 2002Q1 2004Q1 2006Q1 2008Q1 2010Q1 2012Q1 2014Q1 2016Q1 2018Q1 Posterior Mean Observations 80% Credible 95% Credible
Figure 5: State Space Model Synchronicity and Bayesian Credibility Intervals for σ = 0.02 0.5 0.6 0.7 0.8 0.9 1.0 σ =0.05 θt 1996Q1 1999Q1 2002Q1 2005Q1 2008Q1 2011Q1 2014Q1 2017Q1 0.5 0.6 0.7 0.8 0.9 1.0 σ =0.01 θt 1996Q1 1999Q1 2002Q1 2005Q1 2008Q1 2011Q1 2014Q1 2017Q1 Original Eurozone 0.5 0.6 0.7 0.8 0.9 1.0 θt 1996Q1 1999Q1 2002Q1 2005Q1 2008Q1 2011Q1 2014Q1 2017Q1 0.5 0.6 0.7 0.8 0.9 1.0 θt 1996Q1 1999Q1 2002Q1 2005Q1 2008Q1 2011Q1 2014Q1 2017Q1 Modern Eurozone 0.5 0.6 0.7 0.8 0.9 1.0 θt 1996Q1 1999Q1 2002Q1 2005Q1 2008Q1 2011Q1 2014Q1 2017Q1 0.5 0.6 0.7 0.8 0.9 1.0 θt 1996Q1 1999Q1 2002Q1 2005Q1 2008Q1 2011Q1 2014Q1 2017Q1 Posterior Mean Observations 80% Credible 95% Credible European Union
Note: Estimated time series of latent synchronicity in a state space model. The solid black lines indicate the posterior mean synchronicity at each point in time, whilst the dashed blue and red lines indicate the bounds of the 80% and 95% credible Bayesian interval for synchronicity respectively. The green crosses correspond to the observed synchronicity values. The graphs on the left report the models for which the standard deviation of θt− θt−1 is equal to σ = 0.05, whilst the right side showcases the estimated time
series for σ = 0.01. The graphs in the top row report synchronicity for the original eurozone, whilst the middle row shows the modern eurozone and the bottom graphs describe synchronicity for the European Union.
European Union. This reconfirms the idea that European output-gap synchronicity is shared across the EU and that the effects of the monetary union transgress beyond the official borders of the eurozone.
5
Conclusion
This paper has shown that output gap synchronicity in the EU has been rather high during the period 2008-2014, but has since been reverting back towards levels observed before the financial crisis of 2008. This development can be observed in the original eurozone, in the modern eurozone and in the European Union as a whole. Although the business cycles of the original eurozone Member States seem more aligned than of the eurozone as a whole, the difference between the eurozone and the EU seems very small. This suggests that adoption of the euro by an individual Member State does not necessarily impact its business cycle convergence with the rest of Europe, although the eurozone as a whole may have promoted output gap synchronicity throughout the EU. Furthermore, the traditional distinction between the core and periphery countries in business cycle coherence analysis fails to uphold if the periphery is defined by the GIIPS countries.
The analysis in this paper is subject to several limitations. First of all, as noted by Mink et al. (2011) and Belke et al. (2017a), only accounting for the sign of output gaps gives a very limited picture of business cycle synchronicity. Two countries may be in nearly opposite phases of their business cycles yet still share a common output gap sign. A proper investigation of this topic would need to take into account both the signs and amplitudes of output gaps, as well as the direction in which the output gap is developing.
better reflect whether a country needs stimulating or contractionary monetary policy, and synchronicity would reflect the number of countries which need the same type of policy.
It could be interesting to look at different subclusters of Member States (e.g. regional clusters) and analyse how synchronised these are both within themselves and between one another. A problem which such an analysis must circumvent is that synchronicity can only be meaningfully compared between two regions if both have the same number of countries in it. Furthermore, my geographical analysis showed that larger countries in terms of their economy tend to have higher synchronicity levels. Therefore, rather than looking at Member States one could use first-level NUTS regions to analyse synchronicity, as this would yield a sample of similarly sized regions and likely more reliable results in terms of economic mechanics. However, using a country-based synchronicity measure better reflects the process of determining monetary policy in Europe, in which every Member State has approximately equal weight.
References
Artis, M., M. Marcellino, and T. Proietti (2004). Dating business cycles: A methodological contribution with an application to the euro area. Oxford Bulletin of Economics and Statistics 66 (4), 537–565.
Baxter, M. and R. G. King (1999). Measuring business cycles: Approximate band-pass filters for economic time series. Review of Economics and Statistics 81 (4), 575–593.
Belke, A., C. Domnick, and D. Gros (2017a). Business cycle desynchronisation: Amplitude and beta vs. co-movement. Intereconomics 52 (4), 238–241.
Belke, A., C. Domnick, and D. Gros (2017b). Business cycle synchronization in the EMU: Core vs. periphery. Open Economies Review 28 (5), 863–892.
Bergman, U. M. (2006). How similar are European business cycles? In G. L. Mazzi and G. Savio (Eds.), Growth and Cycle in the Eurozone, pp. 124–135. St. Martin’s Press.
Bergman, U. M. and L. Jonung (2011). Business cycle synchronization in Europe: Evidence from the Scandinavian currency union. The Manchester School 79 (2), 268–292.
Christiano, L. J. and T. J. Fitzgerald (2003). The band pass filter. International Economic Review 44 (2), 435–465.
Crowley, P. M. and A. Schultz (2011). Measuring the intermittent synchronicity of mac-roeconomic growth in Europe. International Journal of Bifurcation and Chaos 21 (04), 1215–1231.
Darvas, Z. and G. Szapáry (2008). Business cycle synchronization in the enlarged EU. Open Economies Review 19 (1), 1–19.
Degiannakis, S., D. Duffy, and G. Filis (2014). Business cycle synchronization in eu: A time-varying approach. Scottish Journal of Political Economy 61 (4), 348–370.
Euro Area Business Cycle Dating Committee (2017). A slow but steady euro area recovery.
Ferroni, F. and B. Klaus (2015). Euro area business cycles in turbulent times: Convergence or decoupling? Applied Economics 47 (34-35), 3791–3815.
Frankel, J. A. and A. K. Rose (1998). The endogenity of the optimum currency area criteria. The Economic Journal 108 (449), 1009–1025.
Gächter, M., A. Riedl, D. Ritzberger-Grünwald, et al. (2012). Business cycle synchroniz-ation in the euro area and the impact of the financial crisis. Monetary Policy & the Economy 2 (12), 33–60.
Gogas, P. (2013). Business cycle synchronisation in the European Union. OECD Journal: Journal of Business Cycle Measurement and Analysis 2013 (1), 1–14.
Gomez, D. M., H. J. Ferrari, B. Torgler, and G. J. Ortega (2017). Synchronization and diversity in business cycles: A network analysis of the European Union. Applied Economics 49 (10), 972–986.
Hamilton, J. D. (2018). Why you should never use the hodrick-prescott filter. Review of Economics and Statistics 100 (5), 831–843.
Harding, D. and A. Pagan (2006). Synchronization of cycles. Journal of Economet-rics 132 (1), 59–79.
Hodrick, R. J. and E. C. Prescott (1997). Postwar US business cycles: An empirical investigation. Journal of Money, credit, and Banking, 1–16.
Inklaar, R. and J. De Haan (2001). Is there really a European business cycle? A comment. Oxford Economic Papers 53 (2), 215–220.
and A. Swoboda (Eds.), Monetary Problems of the International Economy. Chicago: University of Chicago Press.
Koopman, S. J. and J. Valle E Azevedo (2008). Measuring synchronization and convergence of business cycles for the euro area, UK and US. Oxford Bulletin of Economics and Statistics 70 (1), 23–51.
Lehwald, S. (2013). Has the euro changed business cycle synchronization? Evidence from the core and the periphery. Empirica 40 (4), 655–684.
Massmann, M. and J. Mitchell (2005). Reconsidering the evidence. Journal of Business cycle Measurement and analysis 2004 (3), 275–307.
Mink, M., J. P. A. M. Jacobs, and J. de Haan (2011). Measuring coherence of output gaps with an application to the euro area. Oxford Economic Papers 64 (2), 217–236.
Papadimitriou, T., P. Gogas, and G. A. Sarantitis (2016). Convergence of European business cycles: A complex networks approach. Computational Economics 47 (2), 97–119.
Savva, C. S., K. C. Neanidis, and D. R. Osborn (2010). Business cycle synchronization of the euro area with the new and negotiating member countries. International Journal of Finance & Economics 15 (3), 288–306.
Appendix A
Distribution Properties
The probability density function of the majority binomial distribution is given by:
P (X = k) = n k h θk(1 − θ)n−k + θn−k(1 − θ)ki. I next define γ = dn/ 2−1e X i=0 (n − 2i) n i θi(1 − θ)n−i= E n − 2Y Y < n 2 P Y < n 2 .
Some trivial algebraic manipulation then yields:
E[X] = nθ + γ;
Var[X] = nθ(1 − θ) − γ(n(2θ − 1) + γ).
These expressions are verified by simulated data. Figure A.1 plots the mean and variance at a function of θ. Note that γ > 0, implying that the expected value of the majority binomial distribution is greater than for the binomial distribution with the same parameters, whilst the variance is lower.
Appendix B
Simulation Results
Figure A.1: Mean and Variance of the Majority Binomial Distribution 0.5 0.6 0.7 0.8 0.9 1.0 5 6 7 8 9 10
Expected value for n=10
θ E [k ] 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.5 1.0 1.5 2.0 2.5 Variance for n=10 θ V ar [k ] 0.5 0.6 0.7 0.8 0.9 1.0 12 16 20 24
Expected value for n=25
θ E [k ] 0.5 0.6 0.7 0.8 0.9 1.0 0 1 2 3 4 5 6 Variance for n=25 θ V ar [k ]
Note: The top graphs show the mean and variance of the majority binomial distribution respctively as a function of θ for n = 10, whilst the bottom graphs report these moments for n = 25. The black lines indicate the mean and variance for the classical binomial distribution, whereas the red line plots these moments for the majority binomial distribution.
Table B.1: Estimated Bias and Variance of Estimators
E[θˆML− θ] Var[ˆθML] E[θˆBayes− θ] Var[ˆθBayes]
n = 12; θ = 0.6 -0.2226 0.0507 -0.7314 0.0558 n = 12; θ = 0.8 -0.0214 0.0163 -0.0776 0.0162 n = 18; θ = 0.6 -0.1144 0.0219 -0.3405 0.0262 n = 18; θ = 0.8 -0.0050 0.0086 -0.0412 0.0085 n = 26; θ = 0.6 -0.2031 0.0158 -0.2939 0.0173 n = 26; θ = 0.8 0.0163 0.0064 -0.0083 0.0064
Note: Estimated bias and variance in percentage points of maximumm likelihood estimator and posterior mean, based on 500 simulated samples. Each sample contains 92 iid observations.
Table B.2: Likelihood Ratio Test Simulations p>0.1 p>0.05 p>0.01 n = 12; θ = 0.6 T1 = 50 0.086 0.042 0.008 T1 = 60 0.100 0.058 0.012 T1 = 70 0.110 0.052 0.010 T1 = 80 0.108 0.038 0.004 n = 12; θ = 0.8 T1 = 50 0.120 0.064 0.018 T1 = 60 0.118 0.058 0.010 T1 = 70 0.122 0.072 0.014 T1 = 80 0.108 0.052 0.008 n = 18; θ = 0.6 T1 = 50 0.110 0.046 0.012 T1 = 60 0.098 0.042 0.010 T1 = 70 0.114 0.050 0.004 T1 = 80 0.092 0.048 0.000 n = 18; θ = 0.8 T1 = 50 0.088 0.046 0.008 T1 = 60 0.080 0.044 0.008 T1 = 70 0.092 0.038 0.006 T1 = 80 0.122 0.062 0.012 n = 26; θ = 0.6 T1 = 50 0.080 0.042 0.010 T1 = 60 0.098 0.042 0.014 T1 = 70 0.100 0.044 0.018 T1 = 80 0.092 0.038 0.008 n = 26; θ = 0.8 T1 = 50 0.086 0.040 0.006 T1 = 60 0.082 0.048 0.002 T1 = 70 0.086 0.028 0.008 T1 = 80 0.120 0.064 0.016 Average 0.101 0.048 0.009
Note: Rejection rates of the likelihood ratio test based on 500 samples simulated under the null hypothesis. Each sample contains 92 iid observations. Every first row of a block splits the sample into a part of 50 and a part of 42 observations, the other rows use 60, 70 and 80 observa-tions for the first half respectively. The bottom row takes the average rejection rate over all tests. The columns re-port rejection rates of tests at the 10%, 5% and 1% levels respectively. Critical values are based on a χ2
