Business cycle synchronization within the European Union: A wavelet cohesion approach
BBusiness cycle synchronization within the European Union:A wavelet cohesion approach (cid:73)
Lubos Hanus a,b , Lukas Vacha a,b, ∗ a Institute of Economic Studies, Charles University, Opletalova 21, 110 00, Prague, CR b Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, PodVodarenskou Vezi 4, 182 00, Prague, Czech Republic
Abstract
In this paper, we map the process of business cycle synchronization across the EuropeanUnion. We study this synchronization by applying wavelet techniques, particularly thecohesion measure with time-varying weights. This novel approach allows us to study thedynamic relationship among selected countries from a different perspective than the usualtime-domain models. Analyzing monthly data from 1990 to 2014, we show an increasingco-movement of the Visegrad countries with the European Union after the countries beganpreparing for the accession to the European Union. With particular focus on the Visegradcountries we show that participation in a currency union possibly increases the co-movement.Furthermore, we find a high degree of synchronization in long-term horizons by analyzingthe Visegrad Four and Southern European countries’ synchronization with the core countriesof the European Union.
Keywords: business cycle synchronization, integration, time-frequency, wavelets,co-movement, Visegrad Four, European Union
JEL: E32, C40, F15
1. Introduction
One of the most challenging tasks in economics is to identify, understand, and disentanglethe factors and mechanisms that impact the dynamics of macroeconomic variables. Manyquantitative econometric techniques have been developed to study the regular fluctuations (cid:73)
We gratefully acknowledge the financial support from the GAUK, No. 366015 and 588314. This articleis part of a research initiative launched by the Leibniz Community. The authors are also grateful for fundingof part of this research from the European Union Seventh Framework Programme (FP7/2007-2013) undergrant agreement No. 619255. Support from the Czech Science Foundation under the P402/12/G097 (DYME- Dynamic Models in Economics) project is also gratefully acknowledged. We would like to thank Mr. GillesDoufrenot for helpful comments at the 2nd International Workshop on ”Financial Markets and NonlinearDynamics” in Paris. ∗ Corresponding author
Email address: [email protected] (Lukas Vacha)
February 16, 2016 a r X i v : . [ q -f i n . E C ] F e b f macroeconomic indicators and business cycles, e.g., Baxter and King (1999); Hodrickand Prescott (1981); Harding and Pagan (2002). This article investigates business cyclesynchronization over different time horizons. In order to disentangle the desired information,we apply wavelet methodology working in a time-frequency space. The analysis considers thecase of the Visegrad Four, both in terms of the interior relationships among its constituentcountries and in terms of the relationships established within the framework of the EuropeanUnion (EU).It has been more than two decades since the break-up of the Eastern Bloc; followingits disintegration, these countries began their independent economic and political journeys.While undertaking their economic transformations during this time, the Czech Republic,Hungary, Poland, and Slovakia began discussing mutual cooperation. Despite their origi-nally different levels of economic maturity and development, their willingness and regionalproximity led them to establish the Visegrad Four in 1991. One of the chief aims of thisgroup was to help its members to organize their institutions for faster convergence with andintegration into the European Union. In 2004, these four countries became members of theEU, which obliges them to adopt the Euro currency as part of the integration process. Oneof the concerns of successful integration into the European Economic and Monetary Union(EMU) is business cycle synchronization, which is motivated by the theory of OptimumCurrency Area (OCA) (Mundell, 1961). A country joining the OCA gives up its individ-ual monetary policy, which requires a level of integration of macroeconomic variables andpolicies and affects the costs and benefits enjoyed by the nations (De Haan et al., 2008).The common currency can be beneficial for both new and former countries in terms of tradetransaction costs. Otherwise, at the European level, the European Central Bank controlsthose policies that apply to all member states, which may be counter-cyclical for coun-tries with low business cycle synchronization (Kolasa, 2013). On one hand, these policiesmay create difficulties for those countries. On the other hand, countries with low levels ofsynchronization may benefit from being members of the OCA because the business cyclesynchronization appears as an endogenous criterion. This endogeneity of OCA means thatforming a monetary union will make its members more synchronized (Frankel and Rose,1998). The literature regarding EU integration – and particularly that focusing on the eco-nomic integration of the CEE countries – has grown rapidly. Fidrmuc and Korhonen (2006)conduct a meta-analysis of 35 studies involving the synchronization of the EU and CEEcountries and find a high level of synchronization between new member states and the EU.However, only Hungary and Poland among the Visegrad countries reached high synchroniza-tion. Artis et al. (2004) and Darvas and Szap´ary (2008) obtained the same results studying The Eastern Bloc was generally formed of the countries of the Warsaw Pact (as Central and EasternEuropean countries) and the Soviet Union. The Visegrad countries also joined the North Atlantic Treaty Organization in 1999 and applied formembership in the European Union in 1995-1996. The literature focusing on the evolution and determinants of business cycle synchronization betweenCentral and Eastern European (CEE) countries and the EU is extensive, e.g., Darvas and Szap´ary (2008);Artis et al. (2004); Backus et al. (1992). and analyze businesscycle synchronizations within the EU framework, while taking into account the distancesamong regions. These authors show that countries that are closer to one another show highersynchronization. Moreover, the transition countries show high similarity with the EU after2005. Nevertheless, Slovakia, a member of the euro area, shows little significant cohesionwith the EU. With respect to Hungary and the Czech Republic, their business cycles co-move with the EU-12 after 2005. Jim´enez-Rodr´ıguez et al. (2013) also find high correlationsof CEE countries (except for the Czech Republic) with the EU business cycle. However,contrary to this result, they find that these countries exhibit a lower level of concordancewhen a factor model is employed. Crespo-Cuaresma and Fern´andez-Amador (2013) look atsecond moments of business cycles in the EU and they report a significant convergence since90s. Furher, they show there is no decrease in the optimality of the currency area after theEU enlargements.To assess the degree of similarity or synchronization, researchers have searched for ap-propriate tools to capture the relevant information. One of the most popular tools is thePearson correlation coefficient, which simply measures the degree of co-movement in a timedomain. However, market-based economies are structured over different time horizons. Forthis reason, researchers began surveying the behavior of such systems at different frequenciescorresponding to different time horizons, and the interest in frequency domain measures hasgrown. Christiano and Fitzgerald (2003) proposed a model based on a band pass filter thatallowed that desired frequencies of time series to be filtered. Further, Croux et al. (2001)presented a measure of co-movement, the dynamic correlation, based on a spectral analysis,that equals to basic correlation on a band pass filtered time series. Nevertheless, both thetime (static) Pearson correlation and the spectral domain dynamic correlation have severalcaveats. The first loses information about frequency horizons and the latter omits the co-movement dependence in time. The wavelet analysis overcomes such limitations due to itsoperation in both time and frequency domains (Torrence and Compo, 1998). Over the pasttwo decades, wavelet applications have been supported by its another advantage, which isthe localization of the wavelet basis function in time and its bounded support; hence, theanalysis is free from the assumption of covariance-stationarity, from which many filteringmethods suffer (Raihan et al., 2005). The literature presents many studies that successfullyused wavelets that do not necessitate stationary time series, e.g., Aguiar-Conraria et al.(2008) analysing the evolution of monetary policy in the US, Vacha and Barunik (2012)studying energy markets relationships, and Yogo (2008) using wavelet analysis to determine These group consists of Austria, Belgium, Finland, France, Germany, Greece, Ireland, Italy, Luxem-bourg, Netherlands, Portugal, and Spain. We analyze this group and the Visegrad countries. Lamo et al. (2013) offer a short presentation of a filtering methodology applied to the business cyclesin the euro area.
2. Methodology
Our analysis aims to address the behavior of the dynamics of a time series at different timehorizons. The Fourier analysis is convenient for observing relations at different frequencies,but it requires time series to be stationary and comes at the cost of losing some informationof the time series when differencing, for instance. Many economic time series might belocally stationary or non-stationary. In other words, using the Fourier transform (FT) makesthe analysis time-invariant and not suitable to provide information about the dynamics4f a process. For this reason, Gabor (1946) developed the short-time Fourier transform(or windowed FT), which is based on applying the Fourier transform on a shorter part ofthe process. Nonetheless, this approach has shortcomings that arise with fixed time andfrequency resolution, and it is impossible to change the resolution at different frequencies.A series of lower or higher frequencies needs lower or higher time resolution, respectively(Gallegati, 2008). As the window width is constant, the resolution is limited, especially forlow frequencies.The wavelet transform has been developed to find a better balance between time andfrequency resolutions. The wavelet functions used for the decomposition are narrow or widewhen we analyze high or low frequencies, respectively (Daubechies, 1992). Thus, a waveletanalysis is suitable to research different types of processes using optimal time-frequencyresolution in great detail (Cazelles et al., 2008). The wavelet analysis is also able to workproperly with locally stationary time series (Nason et al., 2000; Crowley, 2007).The wavelet transform decomposes a time series using functions called mother wavelets ψ ( t ) that are functions of translation (time) and dilation (scale) parameters, τ and s , re-spectively. In many applications, it is sufficient that the mother wavelet has zero mean, (cid:82) ∞−∞ ψ ( t )d t = 0, and that the function has a sufficient decay. These two properties cause thefunction to behave like a wave. Further, the elementary functions called daughter waveletsresulting from a mother wavelet ψ ( τ ) are defined as ψ τ,s ( t ) = 1 (cid:112) | s | ψ (cid:18) t − τs (cid:19) , s, τ ∈ R , s (cid:54) = 0 . (1)The continuous wavelet transform (CWT) of a process, x ( t ), with respect to the wavelet ψ , is defined as a convolution of the given process and the family ψ τ,s , W x ( τ, s ) = (cid:90) ∞−∞ x ( t ) 1 (cid:112) | s | ψ ∗ (cid:18) t − τs (cid:19) d t, (2)where ∗ denotes the complex conjugate. As several wavelet functions are available for CWT,in our analysis we use the Morlet wavelet as the mother wavelet. Use of this wavelet iscommon in the literature due to its well-localized properties in time and frequency (Aguiar-Conraria and Soares, 2011b; Cazelles et al., 2008). The Morlet wavelet is complex, with areal and an imaginary part, which allows us to perform the phase difference analysis. Thesimple definition of the Morlet wavelet is ψ ( t ) = π − e − iω t e − t . (3) It is possible to use methods of evolutionary spectra of non-stationary time series developed by Priestley(1965). However, to run a spectrum over time at different frequencies, larger data are required in order toobtain the same quality time resolution as that obtained using wavelet techniques. And while we studyshort-term fluctuations (such as 2-4 months), we would not obtain such localized information because of thenumber of necessary observations to start with. For more details regarding the conditions that a mother wavelet must fulfill, see, e.g., Mallat (1999) orDaubechies (1992). ω (wavenum-ber) is set equal to 6. With this parameter, we find that a relation between wavelet scales andfrequencies is inverse, f ≈ s . This simplifies further interpretation of the results (Grinstedet al., 2004; Torrence and Compo, 1998).A convenient property of the wavelet transform is that one can reconstruct the originaltime series back from the wavelet transform, x ( t ) = 1 C ψ (cid:90) ∞−∞ (cid:20)(cid:90) ∞−∞ ψ τ,s ( t ) W x ( τ, s )d τ (cid:21) d ss , (4)where C ψ comes from the admissibility condition that allows the reconstruction, C ψ = (cid:82) ∞ | Ψ( f ) | f d f < ∞ , where Ψ( · ) is the Fourier transform of ψ ( · ).Defining the single wavelet power spectrum, | W x ( τ, s ) | , given the wavelet transform, weobtain the measure of the energy of the time series. For a given two time series, x i ( t ) and x j ( t ), the cross-wavelet transform is defined as the product of their transforms W x i x j ( τ, s ) = W x i ( τ, s ) W x j ( τ, s ) ∗ , where ∗ denotes the complex conjugate. In our application, we use the wavelet coherence to quantify the pairwise relationshipof two time series. This coherence in the Fourier analysis is defined as a measure of thecorrelation between the spectra of two time series (Cazelles et al., 2008). The coherencederives from the definition of the coherence as its power of two. With two time series, x i and x j , the wavelet coherence that measures their relationship is defined as (Liu, 1994):Γ( τ, s ) = W x i x j ( τ, s ) (cid:112) W x i ( τ, s ) W x j ( τ, s ) . (5)Given that the coherence is as complex a measure as the wavelet powers are, we preferablyuse the squared wavelet coherence to measure co-movement between two time series, whichis given by: R ( τ, s ) = | S ( s − W x i x j ( τ, s )) | S ( s − | W x i ( τ, s ) | ) · S ( s − | W x j ( τ, s ) | ) , R ∈ [0 , , (6)where S is a smoothing function as S ( W ) = S scale ( S time ( W n ( s )) (Grinsted et al., 2004). Wesmooth the coherence through convolution in both time and frequency domains.Since the wavelet coherence does not have a theoretical distribution, the testing procedureuses Monte Carlo methods to obtain its significance. We follow Torrence and Compo (1998)to assess the statistical significance, which is depicted in figures as a black contour and thelevel of significance is 5%. We use the package developed by Grinsted et al. (2004) to compute the coherence. For further details,please consult Grinsted et al. (2004).
6s we work with a finite-length time series and a Fourier transform assumes cyclicaldata, we would obtain a wavelet power spectrum containing errors at the beginning and endof the analyzed periods. One solution to these edge effects is to pad both ends of the timeseries with a sufficient number of zeros. The area affected by zero-padding is called the coneof influence (COI). We indicate the COI in figures as a shaded area having e -folding shape. For more details, consult Cazelles et al. (2008); Torrence and Compo (1998).Moreover, from the cross wavelet transform of two time series, the phase differenceprovides information regarding the relative position of the two series. The phase differencedefined in [ − π, π ] has the form: φ x i ,x j = tan − (cid:18) (cid:61){ W x i x j ( τ, s ) }(cid:60){ W x i x j ( τ, s ) } (cid:19) , (7)where (cid:61){ W x i x j ( τ, s ) } and (cid:60){ W x i x j ( τ, s ) } are the imaginary and real parts of a cross wavelettransform, respectively. The two time series are positively correlated if φ x i ,x j ∈ [ − π/ , π/ x i , leads the second, x j , ifthe phase is in [0 , π/
2] and [ − π, − π/ − π/ ,
0] and [ π/ , π ], the second variableis leading.The assessment of statistical significance is always critical. According to Cazelles et al.(2008), bootstrap methods are used to provide the significance of the power spectrum and thecross-spectrum. Those methods are also used for wavelet coherence. However, testing thesignificance of the phase difference is difficult because there is no “preferred” value becausethe phase may be distributed on the interval of [ − π, π ]. Aguiar-Conraria and Soares (2014)indicate that there are no good statistical tests for the phase difference. They concludewith the support of Ge (2008) that the significance of a phase should be connected withthe significance of the power spectrum or coherence. To obtain the confidence intervals, weuse classical bootstrap techniques. We show the coherence and phase difference of twoartificial time series: x t = sin( t ) + ε, t ∈ [1 , y t = sin( t ) + ε, t ∈ [1 , t ) + ε, t ∈ [101 , sin( t − .
01) + ε, t ∈ [351 , t + π ) + ε, t ∈ [606 , sin( t ) + ε, t ∈ [901 , . (9)We observe that when the coherence is high and significant, the confidence interval ofthe phase difference is narrow, see Fig. 1. By contrast, for observations around 400-600,we have the sine function with high amplitude and a very noisy time series, which doesnot resemble the sine in this scale. The phase difference of these two is unstable and the We use the Morlet wavelet, and the COI is thus e − -folding. We add 5% noise to each analyzed series. We do the wavelet analysis 1,000 times in a Monte Carlostudy, then we sort the results and determine the 95% confidence interval of the phase difference. − π/ , π/ ρ x i x j ( τ, s ) = (cid:60) ( W x i x j ( τ, s )) (cid:112) | W x i ( τ, s ) | | W x j ( τ, s ) | , (10)where (cid:60) ( W x i x j ( τ, s )) is the real part of the cross-wavelet spectrum of two time series andhas the squared root of two power spectra of the given time series in the denominator. Usingthis measure, Rua and Silva Lopes (2012) developed a wavelet-based measure of cohesion inthe time-frequency.We demonstrate usefulness of the real wavelet-based measure (Eq. 10) that captures thedynamics in time-frequency of two artificial time series in two particular cases. We showthe cohesion with constant weights of white noise, u t ∼ N (0 ,
1) and its lagged values, u t − , u t − , and u t − . For the first 200 observations in Fig. 2, we see the negative correlation equalto minus one in the shortest period, which changes to a positive correlation equal to onein the long-term. If we averaged this part of Fig. 2 over time, the obtained result would8igure 2: Real wavelet-based measure ofco-movement for two series: a t = u t for t = [0 , b t = u t − for t = [0 , b t = u t − for t = [201 , b t = u t − for t = [351 , a t = u t and b t = b t − + u t .be same as the dynamic correlation of Croux et al. (2001). In the second part of Fig. 2,the series have more lags, u t − and u t − , whose relationship is negative with the original u t at longer horizons, which demonstrates the possibility of the well-localized information inthe time-frequency plane. In Fig. 3, we plot the dynamic correlation of the white noise u t with its cumulative sum. Contrary to Fig. 2 for lagged noises, these two series are positivelycorrelated in the short-term and not-correlated in the long-term. Many co-movement measures from the time or frequency domain rely on the bivariatecorrelation. Croux et al. (2001) propose a powerful tool for studying the relationship ofmultiple time series. This measure uses the dynamic pairwise correlation and composes anew measure of cohesion over the frequencies. Let x t = ( x t · · · x nt ) be a multiple time seriesfor n ≥
2, then the cohesion measure in the frequency domain is coh ( λ ) = (cid:80) i (cid:54) = j w i w j ρ x i x j ( λ ) (cid:80) i (cid:54) = j w i w j , (11)where λ is the frequency, − π ≤ λ ≤ π , w i is the weight associated with time series x it , and coh ( λ ) ∈ [ − , ω ij , are attached to the pair The correlation follows a curve from 1 at zero frequency to − π .
9f series, ( i, j ), e.g., for two series we have two weights inversely related. The cohesion existson the interval of [ − ,
1] and we have coh ( τ, s ) = (cid:80) i (cid:54) = j ¯ ω ij ρ x i x j ( τ, s ) (cid:80) i (cid:54) = j ¯ ω ij . (12)Measuring co-movement of multiple time series, the cohesion uncovers important informationabout common dynamics. However, the fixed weights, e.g., population at some time τ , donot consider that data used for weights may also vary over time. Because the developingor emerging countries may have different speed of development, then the importance ofallowing the time-variation of weights appears relevant.We propose a new approach to map a dynamic multivariate relationship using the time-varying weights in the cohesion measure, which is based on Rua and Silva Lopes (2012): coh T V ( τ, s ) = (cid:80) i (cid:54) = j ω ij ( τ ) ρ x i x j ( τ, s ) (cid:80) i (cid:54) = j ω ij ( τ ) , (13)where ω ij ( τ ) is the weight attached to the pair of time series ( i, j ) at given time τ . Cohe-sion allows for the use of different types of weights. For example, using GDP as a weightrepresenting the size of an economy, a country with smaller or larger GDP can have smalleror larger effect on a co-movement than other countries, and in the cohesion, this may leadto a greater dissimilarity of co-movement within the group.
3. Data
To study business cycle synchronization, we use data of the Index of Industrial Pro-duction (IIP) from the database of the Main macroeconomic indicators (OECD, 2015). Fidrmuc and Korhonen (2006) cite many studies where the IIP is broadly used in studyingbusiness cycle synchronization. The dataset period spans from January 1990 to December2014 with seasonally adjusted time series. The dataset includes monthly data of 16 EU coun-tries, of which 13 are EMU members (Austria, Belgium, Germany, Greece, Finland, France,Ireland, Italy, Luxembourg, Netherlands, Portugal, Slovakia, Spain) and 3 countries are not(Czech Republic, Hungary, Poland). In the multivariate analysis, we divide the countriesinto three groups: EU core, Visegrad Four (V4), and PIIGS. Five countries form the PIIGSgroup: four from Southern Europe (Portugal, Italy, Greece, and Spain) and Ireland. As theEU core, we take Belgium, Germany, Finland, France, Luxembourg, and Netherlands.For all countries, we also use the data of the Gross Domestic Product (GDP) in currentprices (Statistical Office of the European Communities, 2016) and GDP at power purchas-ing parity (PPP) per capita (World Bank, 2016) to weight the industrial production in Obtained via Federal Reserve Economic Data, https://research.stlouisfed.org/fred2/ Obtained via EUROSTAT, http://ec.europa.eu/eurostat/web/products-datasets/-/namq_10_gdp , Jan 18, 2016. Obtained via The World Bank http://data.worldbank.org/indicator/NY.GDP.PCAP.PP.CD ,Jan 13, 2016.
4. Results
At first, we analyze the business cycles of the V4 on an intra-group basis to disclosesimilarities in their pairwise co-movements and the development of particular relationshipswithin the group. The V4’s cooperation began in the early 90s, and the countries sharehigher coherence at the beginning of the transition for 1-2 year period during 3-4 years,see Fig. 4. Another common feature among the V4 countries is a weak relationship ofall pairs at short-term business cycles, from 2 months to 1 year during whole 25 years.Only Hungary and Poland co-move significantly around 2010 for periods shorter than oneyear. The important result is that all pairs show a high degree of synchronization overthe 2-4 year period beginning around 1998 for Hungary with Slovakia, 1999 for the CzechRepublic with both Poland and Hungary, and 2000 for Poland with Slovakia. In addition,the Czech Republic and Slovakia have been synchronized over a 2-year business cycle periodthrough the whole sample period with a small decrease around 2000. Hungary and Polandshow a high degree of synchronization at all business cycle frequencies, 2-5 years. Theirco-movement covers the largest part of the frequency spectrum at the beginning of thetransition period and during the last 5 years.We find a short time of higher synchronization for the first 3-5 years in 1-2 year periods.However, we see that all countries have almost zero co-movement around 1995, except forHungary and Poland. This low degree of similarity may be caused by Slovakia’s cold-shouldered participation in the political discussions during 1993-1997 that translated intothe business cycles with a delay. Another possible explanation relevant for all countries canbe that after a few years of formally intensive cooperation the monetary and fiscal policiesstarted diverging. For example, during the late 1990s, the Czech Republic went throughdifficult stabilization years (Antal et al., 2008). These diverging economic situations mightcause some asynchrony in business cycle behavior over both the short- and long-terms. Thislong-term low synchronization may also come from the low level of convergence of othermacroeconomic variables (Kutan and Yigit, 2004).Further, we are interested in the co-movement of each country of the V4 within the frame-work of the European Union. In the pairwise analysis, we take Germany as a representativeof the EU. Germany is often used as a reference country (Fidrmuc and Korhonen, 2006) for In the text, we use notions as ”1-2 year period”, which correspond business cycles at this frequency(period length), thus 1-2 year business cycle. , π ] interval, which also shows that the V4 business cycles lead the cycles of Germany13
990 1995 2000 2005 2010−pi−pi/20pi/2pi Czech Republic − Germany (1−4 years) P ha s e d i ff e r en c e P ha s e d i ff e r en c e P ha s e d i ff e r en c e P ha s e d i ff e r en c e Figure 6: Phase differences. The black solid line is the true phase difference of two timeseries. The blue solid line is the 95% bootstrapped confidence interval. For each phasedifference, its distribution is provided.over the long-term period during the 2006-2010 period. One possible explanation for thisfinding can be that recessions or rebounds in the productions of the V4 contries happensooner compared to Germany. For example, in case of Slovakia, we have the significantleading position of Germany during 2000-2004. Showing that the bootstrapped confidenceintervals, we demonstrate that this technique is relevant and desired for conclusion aboutphase differences between time series.
Until now, we have used the bivariate analysis, and we have omitted the assessment ofsynchronization of more than two time series. In this part, we investigate the multivariaterelationship of countries in the EU. The proposed measure of cohesion with time-varyingweights allows us to assess the co-movement within the groups of countries. In contrast tothe coherence, the cohesion may be negative, which means it can capture a counter-cyclicalco-movement among the time series. We employ the time series of Gross Domestic Product(GDP) in current prices and GDP at purchasing power parity (PPP) per capita to weightthe economic activity. The choice of GDP as weights is particularly to quantify the size ofcountries’ economies. Compared to the fixed weights of Rua and Silva Lopes (2012), thetime-varying weights take into account different developments of countries. In the multivariate analysis, we analyze a period of 17 years spanning from 1997M1-2014M3 due to thelack of data for some countries. .2.1. Does the size of economies affect business cycle cohesion? A broader picture about the business cycle synchronization in the region of the V4countries and Germany can be studied using the cohesion measure. Since the beginningof the transition, we have observed a significant growth of nominal GDPs of the Visegradcountries, see Table 1. Slovakia’s nominal GDP grew by almost 300%, the nominal GDPsof Czech Republic, Hungary and Poland have more than doubled by that time, whereasthe nominal GDP of Germany increased only by 52%. The time-varying weights allowus take into account gradual changes of proportions between economies and reflect thisdevelopment in their business cycle synchronization. The evolving size of economies canproject and emphasize possible convergence or divergence among economies. Further, welook at the GDPs at power purchasing parity per capita where, at smaller extent, the samehappen. GDPs at PPP per capita more than doubled for the Visegrad countries and forGermany it grew by 75%, almost doubled.Table 1: Change of GDPs1997 vs. 2014 Germany Czech Rep. Hungary Poland Slovakia∆ GDP level (in %) 52 169 133 196 290∆ GDP at PPP (in %) 75 102 110 165 151
Note:
The values show by how many per cents the GDPs of given countries havegrown between 1997 and 2014.This motivates our new approach to show clearly that the nature of weighting in suchmeasure is crucial. The cohesion puts a weight to each pair in the multivariate analysis withrespect to all variables. For instance, when considering Germany and Poland within thegroup weighted by nominal GDP, their pair has the largest effect on cohesion. Although,their pairwise co-movement is weak and thus, it lowers the multivariate cohesion at manypoints with respect to weights. In contrary, from previous analysis we know that the CzechRepublic and Hungary strongly co-move with Germany in the long-term frequencies, thusthe effect on the cohesion can be larger because the presence of Germany in the pair.To illustrate the advantage of dynamic weights, we compare four cases of weighting:equal weights – each country has equal size, fixed weight corresponding to one moment –1997Q1 (Eq. 12), and then two cases of employing time-varying weights (Eq. 13). Further,the distinction was made and we show the differences between cohesions that employingequal, fixed, and time-varying weights yields different results, Fig. 7 and 8.In Fig. 7, we observe high cohesion of 2-4 year business cycles of the Visegrad countriesand Germany. Further, the cohesion is quite mild (around zero) at one-half to 1 yearbusiness cycle period. There appear several areas of positive cohesion at shorter periods,as well as, few small areas of negative cohesion distributed over periods from a quarter to1 year. In the case when the nominal GDP is used, we observe that the negative areas ofcohesion are not that sharp as when fixed weights were used. This refers that changes ofproportions between economies lower counter-cyclicalities in cohesion of the Visegrad Four15igure 7: Wavelet cohesion of the Visegrad Four and Germany using different type of weights:equal (top-left), fixed (top-right), time-varying GDP at PPP (bottom-left), and time-varyingGDP level (bottom-right).Figure 8: Differences between cohesions from Figure 7: equal vs. fixed (left), equal vs.time-varying (center), and fixed vs. time-varying (right).and Germany. Comparing the two GDP weights, we conclude that proportions betweencountries when considering GDP at PPP per capita get closer to the situation when countrieshave equal position, thus equal effects on synchronization. It is shown in figures on the left For example, when we would analyze commodity prices and taking their volume as weights, the twofigures of fixed (sum of volumes) and time-varying weights (daily volume) showed greater differences thanin our case.
16f Fig. 7 that the cohesion weighted by GDP per capita does not differ much in comparisonto the one with equal weights.We show that the time-varying weights, representing sizes of the economies, have sub-stantial effect for business cycle synchronization measure. In our particular case, the absolutemaximum in the difference between fixed and time-varying weights results is approximately0.2, which comprises 10% of the scale [ − , We compute the multivariate relationship quantities for three groups of countries: theV4, the EU core, and the PIIGS countries. We analyze those groups individually as wellas in combination. We concentrate on the V4 itself and within the framework of theEU. Examining the synchronization of the PIIGS countries with the EU core, we directlycompare already integrated countries with those in the process of integration.With regard to the V4, in Fig. 9, we show that the degree of synchronization withinthis region is high in the long-term period. The strong co-movement of 2-4 year periodslasts from 1997 to 2014; beginning in 2008, it spreads over a 0.5-5 year period. The short-term synchronization, up to 1 year, provides some insights that countries often co-move atthat frequency but only for short periods of time. This short-term fluctuation co-movementappears as well as for groups of EU core and PIIGS countries in Fig. 10, which may reflectcommon reactions to events in the markets.Comparing the multiple relationships of V4 and EU core, EU core and PIIGS, and V4and PIIGS countries in Fig. 9, we observe similar patterns of co-movement over the long-term. This result of is in line with Rua and Silva Lopes (2012), who find a large cohesionof the long-term dynamics. Regarding the Visegrad counties within the EU, in the periodafter 2000, the strongest relationship is among the V4 countries and the EU core comparedto other groups.Further, we investigate the relationship of all 16 selected countries and the sub-samplesof the EU core and the PIIGS countries. Surprisingly, the analysis reveals low cohesion inthe short-term, 0.5-1 year, although it is quite high in the long-term. Regarding the EUcore, there is more commonality in short-term dynamics. Further for all groups, we notethat the cohesion is high for long-term business cycles. In the second half of the sample,the relationship has increased for the shorter periods. With respect to the case of PIIGScounties, their cohesion appears to be along the same lines, but these countries do not exhibitany co-movement over the 1-2 year period. Notably, we have seen that the cohesion of theV4 with the EU core is stronger than that of the 16 European countries. For example, inSlovakia, there was a gradual increase of co-movement with Germany after accession andafter the Euro adoption. In terms of synchronization, the V4 could benefit from joining theEMU, but we do not know the specific sources of the higher co-movement. By combination we mean the co-movement of all countries from both groups with respect to theirweights.
5. Concluding remarks
Business cycle synchronization is a central question of economic integration and thus itneeds a rigorous examination. We have overcome the problems of traditional measures, suchas operation in time or frequency domain only and of the necessity of stationary time series,by using wavelet methodology. In this paper, we have proposed the multivariate measure ofco-movement with time-varying weights called wavelet cohesion. This wavelet-based measureallows for precise localization of information in time and frequency.We have investigated the impact of V4 cooperation, which has one of its aims to convergefaster towards the EU. We have found short but high co-movement for the first years of theircooperation until the economic turbulences of the late 90s. During the 1995-1999 period,their business cycles for 2-5 year periods show very low levels of synchronization.Further, we have studied the business cycle synchronization of the V4 with Germany. Theresults confirmed some already known but interesting patterns. Slovakia’s synchronizationwith the EU was poor before its accession to the EU but gets stronger after 2005, whichsupports the theory of the endogeneity of the OCA and the adoption of Euro. We haverevealed that the highest coherence is between Germany and both the Czech Republic and18igure 10: Wavelet cohesion of 16 European countries (left), EU core (center), and PIIGScountries (right). We use Gross Domestic Product as weights for this analysis. The shadedarea is the cone of influence.Hungary beginning in 2000. By contrast, the degree of synchronization of the business cyclesof Poland and Germany is the lowest among V4.Employing a multivariate measure, we have uncovered relationships in both time and fre-quency domains for multiple time series. Regarding the V4, the EU core countries, and/orthe PIIGS countries, we show that there is a very weak synchronization of short-term dynam-ics,less than 1 year, among those countries. Conversely, we have found that cohesion withinthe EU is high after 2000 in 2-5 year periods. Finally, we have found high co-movement ofthe long-term business cycles of the V4 and of the EU core countries over the whole sample.This supports that countries tend to have similar approaches to their long-term dynamicsor policies.
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