^{1}

^{2}

^{3}

Dante Mendes Aldrighi

We add to the discussion on the transmission of business cycles, by modeling worldwide banking sector indices cycle synchronization, accounting for the time-varying and frequency-specific behavior of the variables. Based on the multiple coherence, partial coherence, partial phase-difference, and partial gain, we find regions of strong and significant coherency between NAFTA partners, and in the European core: France, Germany, and the United Kingdom. Concerning such trade blocs, we also find strong performance in the period 2010-2012 in all frequencies, a period characterized by the sovereign debt crisis in some European countries.

Nós acrescentamos à discussão sobre a transmissão dos ciclos de negócios, modelando a sincronização dos ciclos dos índices do setor bancário mundial, levando em consideração o comportamento variante no tempo e frequência específica das variáveis. Com base na coerência múltipla, coerência parcial, diferença de fase parcial e ganho parcial, encontramos regiões de coerência forte e significativa entre os parceiros do NAFTA e no núcleo europeu: França, Alemanha e Reino Unido. Em relação a esses blocos comerciais, também encontramos forte desempenho no período 2010-2012 em todas as frequências, período caracterizado pela crise da dívida soberana em alguns países europeu.

According to the extensive literature on the nature of the long-run and of short-run linkages among financial markets and their interaction, international financial integration can be seen able to increase economic efficiency and growth, but it may also influence countries’ vulnerability to contagion.

Concerning the relevance and the complexity,

Theoretically, the reasons the reasons for worrying about this issue can be due to specific transmission channels.

From the empirical point view the context where spillover and contagion events occur make it harder to modelling with traditional time series tools.

Once the motivation for the study of bank contagion is substantiated, we add to the empirical related literature, by modelling the dependence across the international banking sectors into the time-frequency domain. We are the first, to the bets of our knowledge, to use the Continuous Wavelet Transform aiming to measure the local correlation and lead/lag relationship between the international banking sector indexes over different scales (short-lived cycles, medium-lived cycles, and long-lived cycles). While applications in finance have not yet extensively utilized wavelets, there have been some interesting empirical exercises and the list is sure to grow fast. Some of the more recent closely related studies are

This methodology, based on

More specifically, we perform an empirical exercise aiming to test if the synchronization is statistically significant by Monte Carlo simulations. Based on this metric, we fill a dissimilarity matrix which is used to map the countries into a two-dimensional axis in terms of banking cycle synchronization. As preliminary analysis, we compare these banking cycle dissimilarities with geographical physical distances and foreign trade. In our main empirical exercise, we use cross-wavelets and wavelet-phase difference analysis to study in more detail when and at what frequencies each country is synchronized or not. Regarding the data, we use a sample of the main worldwide financial sector indices, which are comprised of the banking, insurance, and financial intermediation companies. Our cross-section considering G-2o economies is composed of financial sector indices of Australia, Brazil, Canada, Germany, France, India, Mexico, UK, USA, and Russia, covering the period from March 30, 2009, to December 31, 2013, 1255 observations. The return on these financial sector indices is stationary, heteroskedastic, leptokurtic and they are not Gaussian but rather driven by probability distribution functions as Johnson SU, Error, Hyperbolic Secant, and Laplace. According to

Summarizing our main findings, banking cycles in Australia and Russia seem independent of all the other cycles. We also find a synchronization between the main European economies: France, Germany, and the United Kingdom, highlighting the French banking cycle’s lowest dissimilarity with the German cycle. This finding complements in some sense the evidence reported in

This paper is structured as the following. Section 2 describes the methodology. We analyze data in section 3. The empirical exercise is reported in section 4, while concluding remarks are offered in the fifth section.

The Fourier analysis can be considered one of the most important bases for the wavelet transform development. The central idea of Fourier analysis it’s that any periodic function can be expressed by an infinite sum of trigonometric functions (Fourier transform). By defining a basis of sines and cosines of different frequencies, the Fourier Transform capture the relative importance of each frequency on the original signals. Given a time series

where

The Fourier Analysis it’s a powerful tool to modelling time series on frequency domain. The function is reversible, which allow back-and-forth between the original and transformed signals, and it gives an effective localization in frequency. So, we can access the power spectra of the signal, which describe the power distribution on different frequency bands. Besides the appealing of Fourier Transform to evaluate financial time series on frequency domain, the function does not allow decompose the time series into different time scales, which limit his applicability to study signals that exhibit bursts of volatility, abrupt regime changes, or non-stationarity, etc. (In and Kim, 2012). To reach a balance between time and frequency, the short-time Fourier transform (STFT) was developed to expand the transformation by frequency and time-shift, it slides a window across the time series and taking the Fourier transform of the windowed series The STFT is given by:

where

To resolve that problem, the Wavelet transform has three additional features over Fourier transform (

Given a time series

where * denotes the complex conjugate,

The factor

The basis function

i) Admissibility: for an integrable function, it means that its average should be zero and the function need be localized in both time and frequency space.

where

ii) Similarity: the scale decomposition should be obtained by the translation and dilation of only one wavelet mother function. This dilation procedure allows an optimal compromise in view of the uncertainty principle: The wavelet transform gives very good spatial resolution in the small scales and very good scale resolution in the large scales (

iii) Invertibility: Once that the energy of the original signal is preserved by the wavelet transform if the admissibility condition is satisfied, we have

iv) Regularity: The wavelet should be concentrated one some finite spatial domain and be sufficiently regular.

v) Vanishing moments: The wavelet should have some vanishing high-order moments. This requirement allows the study of its high-order fluctuations and possible singularities in some high-order derivatives. It means that scale function is smoother.

There are many options of wavelet mother functions to select, which includes Daubechies, Haar, Mexican Hat, Morlet and Meyer wavelets.

where the non-dimensional frequency

By the usual “Fourier” frequency (cycles per unit time) we have that

Finally, because the CWT is applied on finite-length time series, border distortions will occur due the fact that values of the transform at the beginning and the end of the sample are imprecisely computed, which involve artificial padding on the extremes of the sample (the most common is set zero to extend the time series). As larger scales decrease the amplitude near the edges as more zeroes enter the analysis (

The first wavelet measure that we will present it’s the wavelet power spectrum (WPS), which reports the variance distribution of the original time series

To compare the oscillation in energy among a range of bands (or frequency) we define the Global Wavelet Power Spectrum (GPWS), which takes the average of wavelet power spectrum over all times (

To study the dependencies between two original time series

where

where

Hence, the main advantage of the wavelet coherence on XWT is the common measure unit to examine several combinations of signals.

Although the wavelet coherence computes the degree of local linear correlation between two signals, it isn’t reveals patterns of lead-lag relationship neither if the movements are positives or negatives. To deal with these limitations, the phase-difference is commonly used to examine the delays in the flutuactions between the two time-series. Following

Where the smoothed real (

Given the SVD we compute the

To measure a distance between two wavelet spectra reduced to a few components we need to measure the angle between each pair of corresponding segments, defined by the consecutive points of the two vectors, ant take the mean of these values. As the components of lading vectors and leading patterns are complex numbers, we follow

and the distance between two-vectors

where the

Hence, the wavelet spectra distance between

where

To capture the interdependence and/or causality between multiple time series, following

Given a set of

where

The

The

And the

Since that the complex partial wavelet coherency measures the local correlation between the time series

Following

And the

Note that the

The complex partial wavelet coherency

Finally, the complex partial wavelet gain

Major market indices are well known and they use to be composed of stocks of companies from several sectors of the economy. However, by incorporating all these companies, we lose the power of explaining a particular segment of the market. Aiming to deal with this issue, we observe the appearance of sectoral indices, with a proposal to complement the general market indices and providing summary information about a specific sector of the economy. We implement an empirical exercise applying wavelet analysis in a sample of the main worldwide financial sector indices of G20 economies. This sector consists of banks, insurance companies, and other financial intermediation companies and its idiosyncrasies make it more likely to be influenced by contagion and integration. Banks, insurance companies, and other financial companies in major economies are often strongly connected, with interdependence in the short and long term.

We report in

Country
Financial Sector Index
Continent
Position in the ranking (GDP, 2013)
Germany
DAX All Banks
Europe
4
^{th}
Australia
ASX 200 Financials
Oceania
12
^{th}
Brazil
IFNC
South America
7
^{th}
Canada
TSX Financials
North America
11
^{th}
United States of America
KBW Bank
North America
1
^{st}
France
CAC financials
Europe
5
^{th}
India
CNX Finance
Asia
10
^{th}
Mexico
BMV
North America
15
^{th}
United Kingdom
NMX 8350
Europe
6
^{th}
Russia
Moscow Exchange Financials Index
Europe
9
^{th}

In principle, whenever econometric or statistical tests are performed, it is preferable to employ a large data set either in the time-series (

We can highlight volatility clusters and higher oscillations, mainly between 2011 and 2012, a period characterized by the sovereign debt crisis in some European countries. The country whose index has the highest cumulative gain in the period in India, with 189%, while DAX All banks in Germany have the worst cumulative gain, only 26%, and the highest drawdown among these indices, 64%. TSX Financials in Canada is the smoothest index considering all volatility measures used here. Its semivariance is only 0.73%, for instance. All series are leptokurtic with a higher intensity for CNX Finance in India, suggesting the frequency of occurring large losses. These skewness and kurtosis are strong evidence that the series does not follow a Gaussian distribution.

A simple but intuitive statistic that signals the expected results in terms of synchronization consists of the correlation, reported in

^{a} This figure plots the nominal net return on financial sector index in terms of the local investor´s currency, based on the daily time series for the end-of-day quote, during the period from March 30, 2009, to December 31, 2013. ^{b} Data source: Bloomberg.

Statistics
CNX Finance
DAX All Bankx
KBW Bank
IFNC
ASX 200 Financials
CAC Financials
BMV
NMX 8350
TSX Financials
Moscow Exchange Financial Index
(India)
(Germany)
(USA)
(Brazil)
(Australia)
(France)
(Mexico)
(U. K.)
(Canada)
(Russia)
Gain
Mean
0.10%
0.04%
0.10%
0.07%
0.05%
0.08%
0.07%
0.07%
0.06%
0.09%
Cumulative
189.57%
26.29%
164.76%
107.04%
77.17%
91.30%
121.28%
84.29%
104.28%
156.61%
Risk
S. D.
1.70%
2.09%
2.12%
1.43%
1.17%
2.16%
1.28%
1.82%
1.03%
1.70%
Semivariance
1.12%
1.46%
1.43%
1.00%
0.82%
1.49%
0.92%
1.22%
0.73%
1.21%
Drawdown
37.60%
64.27%
41.93%
32.09%
30.58%
53.98%
33.88%
43.55%
22.05%
47.43%
Other moments
Asymmetry
1.34
0.26
0.79
0.05
0.04
0.55
-0.11
0.59
-0.05
-0.12
Kurtosis
17.82
5.91
15.83
4.83
4.33
9.73
8.21
9.03
6.65
7.12

CNX Finance
DAX All Bankx
KBW Bank
IFNC
ASX 200 Financials
CAC Financials
BMV
NMX 8350
TSX Financials
Moscow Exchange Financial Index
Correlation
(India)
(Germany)
(USA)
(Brazil)
(Australia)
(France)
(Mexico)
(U. K.)
(Canada)
(Russia)
CNX Finance
(India)
1.00
0.31
0.18
0.28
0.24
0.29
0.23
0.32
0.20
0.34
DAX All Bankx
(Germany)
0.31
1.00
0.48
0.45
0.29
0.88
0.56
0.73
0.53
0.54
KBW Bank
(USA)
0.18
0.48
1.00
0.50
0.14
0.50
0.68
0.48
0.69
0.34
IFNC
(Brazil)
0.28
0.45
0.50
1.00
0.19
0.47
0.51
0.46
0.52
0.37
ASX 200 F.
(Australia)
0.24
0.29
0.14
0.19
1.00
0.34
0.21
0.29
0.24
0.26
CAC F.
(France)
0.29
0.88
0.50
0.47
0.34
1.00
0.58
0.77
0.55
0.52
BMV
(Mexico)
0.23
0.56
0.68
0.51
0.21
0.58
1.00
0.53
0.58
0.37
NMX 8350
(U. K.)
0.32
0.73
0.48
0.46
0.29
0.77
0.53
1.00
0.50
0.49
TSX Financials
(Canada)
0.20
0.53
0.69
0.52
0.24
0.55
0.58
0.50
1.00
0.41
Moscow E. F. I.
(Russia)
0.34
0.54
0.34
0.37
0.26
0.52
0.37
0.49
0.41
1.00

We also apply

More pronounced episodes of high power and statistically significant were found at shorter periods within the band of 8 - 32 days. Most countries show spikes around 32-64 days band, and someone’s cases of high volatility for lower frequencies (64-256 days) were found too.

For the European countries, the variance (at frequency 8 - 32 days) was concentrated between June 2011 and March 2012, there were too peaks of statistically significant power for Germany and France at longer frequency (128 - 256 days) during the interval. These results probably are related with the fear of contagion effects by sovereign debt crisis, it’s rose the risk aversion especially for the banking sector on Eurozone.

United States and Canada shares statistically significant regions in the beginning of the sample (March 2009 - September 2009) at shorter scales (8 - 32 days), the first one also had high variability at 64-128 days band from June 2009-September 2009, while the wavelet power spectrum reached a peak at 32-64 days band for the former. Weaker signals of high power were found for Mexico in the same interval, without any significant region of power for medium and lower frequencies. The volatility rose in the three countries between June 2011-January 2012 at shorter periods (more pronounced regions were found for Mexico), both Mexico and United shared too high and statistically significant power at 32-64 days band.

The Brics countries in the sample (Brazil, India, and Russia) had many high-power regions at shorter periods distributed between the whole-time intervals. This trend was stronger for Brazil, with spikes episodes statistically significant in every year. Looking at Australia, we conclude that power distribution is close with United States pattern, but without high variance for lower frequencies (more than 64 days).

As, theoretically, the response of these banking indices to external adverse disturbances has associated to increases in volatility levels, we will examine the co-movements across the markets on areas where the returns exhibited high variance. Since the majority of the volatility of international banking returns occurs at frequencies of periods shorter than 256 days, we will discard cycles longer than it in the rest of the paper. By the pattern discussed above, we will denote short-lived cycles for oscillations with frequency between 8 and 32 days, medium-lived cycles for fluctuations between 32~64 days bands and long-lived cycles to movements between 64~256 days.

^{a} This figure plots the nominal net return on financial sector index in terms of the local investor´s currency, based on the daily time series for the end-of-day quote, during the period from March 30, 2009, to December 31, 2013. ^{b} Data source: Bloomberg.

In

To test the null hypothesis that a pairwise dissimilarity index is not synchronized we follow

Following Camacho et al. (2006), we convert the ^{2}^{2} In panel (a) we present the multidimensional scaling map and in panel (b) we present the cluster analysis by the dendogram. Both panels indicate a main group formed by the European countries (especially by France and Germany). Mexico and United States form the core group for America, with Canada sharing some alignment with them. On other hand, the Brics countries are far way one each other, denoting weak financial integration between them. In addition, the Australian banking index is other country that does not share synchronization with the “core” cycles.

Aiming to assess if the pairwise synchronization is statistically significant at 5%, we follow this literature by relying on Monte Carlo simulations (1000 times). First, corroborating the

These previous findings suggest an apparent importance of the physical distance on the probability of high co-movement between financial sector indices. As NAFTA and especially Eurozone are trading blocs, the commercial channel is other possibility for pass through of disturbances. Due to this evident pattern, in this subsection we suggest a simple analysis relating these variables. In the upper triangle of

Observing the physical distance between the cities where the respective stock exchanges are located, banking cycle synchronization does not seem to be independent of geographical issues, based on the significative correlation of 0.607, as well as some common patterns for both metrics of distance/difference. The city with the longest average distance concerning the others is Sidney, with more than 14,500 km, while the two cities with the least average distance are London and Paris, 7,89 and 7,154 respectively, the same pattern evidenced based on the dissimilarities. This finding is aligned to

We also compare synchronization with trade between each pairwise of countries (

We summarize this finding by plotting in

We find individual significance at 1% for the intercept and the parameter related to distance, and at 5% for the parameter related to trade, in the direction predicted by the associated fundamentals. These OLS estimation significance are robust in relation to using or not the covariance matrix a la Newey-West. We also find a joint significance and the explanatory power of this simple framework is higher than 0.407.

Dissimilarity
CNX Finance
DAX All Bankx
KBW Bank
IFNC
ASX200 Financials
CAC Financials
BMV
NMX 8350
TSX Financials
Moscow Exchange Financial Index
(India)
(Germany)
(USA)
(Brazil)
(Australia)
(France)
(Mexico)
(U. K.)
(Canada)
(Russia)
CNX Finance
(India)
6570
12537
13773
10156
7009
15645
7191
12488
5028
DAX All Bankx
(Germany)
0.536
6202
9829
16482
478
9559
638
6333
2021
KBW Bank
(USA)
0.484
0.480
7685
15988
5837
3359
5570
550
7510
IFNC
(Brazil)
0.398*
0.507
0.491
13357
9401
7432
9470
8186
11806
ASX 200 F.
(Australia)
0.651
0.516
0.561
0.548
16960
12972
16993
15567
14495
CAC F.
(France)
0.469
0.249*
0.444
0.406*
0.527
9196
344
6000
2486
BMV
(Mexico)
0.493
0.420
0.245*
0.524
0.516
0.393*
8928
3261
10719
NMX 8350
(U. K.)
0.541
0.324*
0.385*
0.563
0.468
0.275*
0.386*
5712
2500
TSX Financials
(Canada)
0.495
0.521
0.377*
0.518
0.552
0.414*
0.347*
0.360*
7484
Moscow E. F. I.
(Russia)
0.545
0.454
0.453
0.538
0.588
0.456
0.472
0.428
0.472

Dissimilarity
CNX Finance
DAX All Bankx
KBW Bank
IFNC
ASX200 Financials
CAC Financials
BMV
NMX 8350
TSX Financials
Moscow Exchange Financial Index
(India)
(Germany)
(USA)
(Brazil)
(Australia)
(France)
(Mexico)
(U. K.)
(Canada)
(Russia)
CNX Finance
(India)
101
265
43
73
44
21
72
23
34
DAX All Bankx
(Germany)
0.536
720
109
65
1067
78
646
78
357
KBW Bank
(USA)
0.484
0.480
301
172
334
2151
512
550
150
IFNC
(Brazil)
0.398*
0.507
0.491
9
47
44
40
29
29
ASX 200 F.
(Australia)
0.651
0.516
0.561
0.548
27
10
62
17
6
CAC F.
(France)
0.469
0.249*
0.444
0.406*
0.527
20
337
38
109
BMV
(Mexico)
0.493
0.420
0.245*
0.524
0.516
0.393*
20
122
6
NMX 8350
(U. K.)
0.541
0.324*
0.385*
0.563
0.468
0.275*
0.386*
118
84
TSX Financials
(Canada)
0.495
0.521
0.377*
0.518
0.552
0.414*
0.347*
0.360*
12
Moscow E. F. I.
(Russia)
0.545
0.454
0.453
0.538
0.588
0.456
0.472
0.428
0.472

^{a} Banking cycle dissimilarity of returns on main worldwide financial sector indices (March 30, 2009, to December 31, 2013). ^{b} Physical distance between the cities where the respective stock exchanges are located. ^{c} Trade balance - exports, FOB to partner countries + imports, CIF from partner countries between the countries in U$$ bi, from 2009:2 to 2013:4.

In this section, we present our main empirical results.

To assess the unconditional local correlation between the banking sector indices and to examine the delay between the same we estimate the wavelet coherency and the phase-difference, respectively. The significance of coherency is obtained by the same method of the dissimilarity pairwise, performing Monte-Carlo simulations and extracting critical values by an ARMA (1,1) model to each of the series. How the phases are angular measures (Zar, 1996), to compute the critical values of the phase-difference, first we compute the average phase difference within the short-lived cycles (8~32 days), medium-lived cycles (32~64 days) and long-lived cycles (64~256 days) and then the confidence intervals are computed by consider values as extreme as two endpoints of the confidence interval for the circular mean at each point in time. For wavelet coherency the 5% significance level region is identified with a black contour and for phase difference the limits of the confidence intervals are drawn with black dashed lines.

We follow the same procedure for all 45 pairwise available, given 10 banking indices used here: we plot the wavelet coherency on the left and phase-difference on the right. Concerning the coherency, it ranges from low coherency in blue to high coherency in red and the respective cone of influence is shown with a black line, designating the 5% significance level. Concerning the phase-difference, we also plot it with plus and minus two standard deviations. In this subsection, we discuss the countries that we have identified as belonging to a bloc or group, based on the previous finding.

In

^{a} The coherency ranges from low (blue) to high (red) values and the respective cone of influence is shown with a black line, designating the 5% significance level. ^{b} We also plot the phase-difference with plus and minus two standard deviations.

We find a somewhat similar pattern for the UK and Germany, but with less intensity. Between 2010 and 2013 Germany and France exhibits strong coherency at medium-lived cycles (32~64 days), after the first quarter of 2011 the phase-difference analysis indicates that France leads the Germany cycle in some intervals. When we look at long-term frequency (64~256 days) we observe a big area of high coherency between the fourth quarter of 2009 until 2012, again France has been leading the Germany fluctuations during 2012. The France has been leading the United Kingdom cycle for medium-term cycle between the aftermath of 2010 and 2012 and between 2012 for the long-lived cycle.

Concerning oscillations between 32~64 days the estimates suggest small areas of coherency between Germany and United Kingdom, with one unique episode of significant region that covers the whole frequency band in 2013. At long term frequency, the countries share regular regions of strong coherency across the sample with phases completely aligned (phase-difference statistically equal zero). Finally, discarding the region affected by edge effects (COI), the banking indices moving in phase

The following Euro sovereign debt crises period demarks a rising of high coherency between the countries from the medium-term cycles reaching the short-term cycles in the second quarter of 2011. The phase-difference relation between the countries is erratic around zero, indicating common response of the indices to adverse shocks at short-term. The American banking sector have big areas of high coherence with Canada only for medium-lived and long-lived cycles. Most of the strong co-movement begins in late of 2010 at long-term horizon and it spreads to 32~128 days band since the second quarter of 2011 until the end of interval. It’s worth noting a small area of high coherency at short-term frequency at the end of 2011. Looking the phase-difference into intervals of high coherency, it’s clear that USA leads Canada for long-term frequencies and the opposite occurs for medium-term frequencies.

Moreover, there are regions of strong coherency between the high-frequency cycles only for the USA and Mexico in 2011. The American cycles for the one-month to six-month frequency-band have regions of strong coherence in the beginning and at the end of the time with Canada and Mexico. With a frequency of six months or more, the cycles are coherent principally between Canada and the USA during the first half of the time and between Canada and Mexico in the second half. To summarize the lower frequency pattern, these economies of NAFTA bloc behave as if they were highly synchronized all the time and we find some isolated out of phase behavior with the other partners only for the Canadian higher frequency cycles.

Finally, Mexico and Canada show stable regions of coherency only at long-term frequencies. In the first area (2009Q2-2010Q2) it is Canada that leads the Mexican cycle, and in the second area (2011Q2-2013Q1) the phases are aligned.

Concerning emerging economies (

^{a} The coherency ranges from low (blue) to high (red) values and the respective cone of influence is shown with a black line, designating the 5% significance level. ^{b} We also plot the phase-difference with plus and minus two standard deviations.

^{a} The coherency ranges from low (blue) to high (red) values and the respective cone of influence is shown with a black line, designating the 5% significance level. ^{b} We also plot the phase-difference with plus and minus two standard deviations.

In the previous subsection, we find out regions of strong coherency between NAFTA partners, as well as considering the European core: France, Germany, and the United Kingdom. Aiming to better understand the relationship in each of such blocs, we propose the use of the multiple coherence, partial coherence, as well as partial phase-difference and partial gain. The first one measures the degree of adjustment of the explanatory variables on the dependent variable in the time-frequency domain. The other measures calculate the relationship between the fluctuations of two markets, controlling the influence of a third party on the oscillations in the time-frequency space. According to

^{a} The coherency ranges from low (blue) to high (red) values and the respective cone of influence is shown with a black line, designating the 5% significance level. ^{b} We also plot the phase-difference with plus and minus two standard deviations.

In short-term (8~32 days) we find a regular pattern of absence of synchronization from the second quarter of 2011 to the first quarter of 2012. For mid-run frequencies (32 ~128 days), the multiple coherence is continuously significant from 2010 to 2013 in some locations. At the long-term horizon (128 ~256 days), the range of significant decreases to 2011 until the third quarter of 2012. Therefore, we can find a strong performance in the period 2010-2012 in all frequencies, characterized by greater uncertainties in the market due to the sovereign debt crisis in some European countries. Controlling the effects of the US index, we highlight a broader decrease in significant coherency area between Mexico and Canada. The sovereign debt crisis period is correlated with a small area of high and significant coherence spanning 32~128 days. At long run frequency (128~256 days), the partial coherence is significant in a narrow area after the first quarter of 2011.

The partial coherence between Mexico and Canada is overall lower than the unconditional case. It shows greater coherence in the medium-term horizon, and absence of synchronization in the long term. In the short term, there is a region of coherence between the second half of 2011 and early 2012, the partial phase-difference is set to zero, indicates strong contemporaneous co-movements among the Countries. On the other hand, for long-term frequencies (0.5 ~ 1 year), the partial coherency is significant between 2010 and first-half of 2012, with Mexico leading Canada, but during 2011 the partial phase-difference is set to (0,

The partial coherence between Mexico and USA repeats the pattern of the previous relationship, but with greater intensity in terms of synchronization. At higher frequencies (0.05 ~ 0.1 year), is predominant peaks with low duration from 2010-2012, the partial phase-difference are located in the interval (

Now, we analyze the multiple coherencies between France versus Germany and the UK (

There is a lack of high coherency in the short-term and we identify two regions with strong co-movements at a lower frequency. The first one covers the period between the end of 2009 and the first half of 2010 (frequency band 0.65 ~ 1.2 years), while the most important covers the region of the year 2012 (frequency equal to or greater than 1 year). The medium-term banking cycles show statistical significance between the first and third quarter of 2010 (0.2 ~ 0.3 year), and a second period with weaker convergence in the year 2011.

The partial coherence between France and Germany (controlling for the UK) indicates three regions of strong coherence at the horizon of medium-term (quarters 2012: 2-2012: 3) and long-term (quarters 2010: 2 and 2010: 2 - 2012: 4). At medium-term, the cycles move in-phase, with Germany leading, and the partial gain increased during the range, getting close to 0.5 in the third quarter of 2012. At a frequency band located in 128 ~ 256 days, the first region statistically significant is associated with the partial phase-difference between

The partial coherence between France and the United Kingdom exhibits an important region of strong synchronization of long-term cycles after the beginning of 2010. More specifically, France leads the UK throughout the year 2011, while the UK is leading France from the second quarter of 2012 until the middle of 2013. In the last interval, the coefficient of partial gain increased, reaching a value close to unity in the first quarter of 2013. In other words, our results indicate strong co-movements between the series, with the cycle of UK contagious the France index. In the short and medium-term the statistically significant regions do not exceed one quarter, while the partial phase-difference presents an irregular pattern, which suggests the absence of synchronization.

^{a} The coherency ranges from low (blue) to high (red) values and the respective cone of influence is shown with a black line, designating the 5% significance level. ^{b} We also plot the phase-difference with plus and minus two standard deviations.

Banking crises are costly, and a great deal of prudential effort is undertaken to avoid them. For instance,

Regardless of the cause of the shocks in the banking system - due to fraud and internal irregularities in an institution or a consequence of macro fundamentals -, the contagion effect of bank failure is seen by much of the literature on international finance as a relevant and worrying issue to be addressed. In practice, policymakers aim to avoid banking crises, and although they can to some extent control domestic conditions, internationally transmitted crises are difficult to tackle, which motivates banking crises as one of the main reasons for worldwide bank regulation mainly from the ’80s. In this sense,

Moreover, this literature on business cycle synchronization is usually related to the optimal currency areas because it is seen as a necessary condition: a country with an asynchronous business cycle must face difficulties in a monetary union. We claim that it is also relevant to relate cycle synchronization to discussion on trade, as a useful tool for a suggestion of admission of new commercial partners or even in evaluating the benefits of such commercial arrangements.

In this context of banking contagion among economies in a trade bloc, considering NAFTA, we find that the presence of successive areas with significance statistics in the multiple coherence denotes a good overall fit in the model, reporting that the Canadian and American financial indices are useful to explain Mexico’s financial index in the time-frequency location. Controlling the effects of the US index, we highlight a broader decrease in significant coherency area between Mexico and Canada. The partial coherence between Mexico and USA repeats the pattern of the previous relationship, but with greater intensity in terms of synchronization. In sum, US index anticipates fluctuations of the counterpart Mexican in the cycles of 0.1 ~ 0.3 year (medium frequency), and cycles of 0.5 ~ 1.0 year (long-term) during the period of instability in the market, with no contagion in another range.

Observing main Eurozone economies, the partial coherence between France and Germany (controlling for the UK) indicates three regions of strong coherence at the horizon of medium-term (quarters 2012: 2-2012: 3) and long-term (quarters 2010: 2 and 2010: 2 - 2012: 4), while the partial coherence between France and the United Kingdom exhibits an important region of strong synchronization of long-term cycles after the beginning of 2010, with the cycle of UK contagious the France index.

Here we claim that potentially there is a gain for regulators, researchers, and policymakers to consider the functioning of transmission of business cycles between each pair of banking systems or even considering a small group of countries. It seems useful to identify which banking system can act as a leader in the group of synchronized countries. In this context, our main findings can shed light on this discussion on business cycle synchronization and trade, as a useful tool for a suggestion of admission of new commercial partners or even in evaluating the benefits of such commercial arrangements.

C63; G21; O16.

To perform the wavelet-based results, we used the ASTollbox2018 for Matlab by M. Joana Soares and L. Aguiar-Conraria. To access: https://sites.google.com/site/aguiarconraria/joanasoares-wavelets/the-astoolbox.

The transformation allows the dissimilarities were represented in a small number of dimensions that preserves, approximately, the original values (for technical details, see Timm, 2002 and Camacho et al. 2006).