This section outlines the used methodology to study the contagion effect of migration fear. The appropriate empirical model we estimate such as multivariate GARCH and continuous wavelet transform. Finally, we present our data series with their descriptive statistics.

Methodology

To study the contagion effect of migration fear in our selected countries sample (France, Germany, United Kingdom, and United States), we start by determine the structural changes. Indeed, we use two approaches to make our results robust. So, we use the Zivot and Andrews (1992) (ZA) and Bai and Perron (1998) (BP) tests. The first tests the presence of unit root with one structural break. The second approach estimate L possible structural date breaks in each series. With the ZA estimated date break we divide our full sample into two subsamples. Then, we estimate our appropriate model (M-GARCH) in each subperiod. With DCC-GARCH model we obtain the conditional correlations which gives an idea for the contagion effect between countries fear migration. In addition, to support the results presented by the DCC- GARCH model and to determine the direction of the contagion effect, we use the continuous wavelet transform (CWT). The CWT analyze the dynamic relationship between Fear migration indices and shows coherency and phase difference.

DCC GARCH model

The DCC- GARCH model is given below:

$$H_{t} = D_{t} R_{t} D_{t}$$

where (H_{t}) is conditional variance matrix, (D_{t}) is a (k times k) diagonal matrix having conditional variance (sqrt {H_{t} }).

On it’s diagonal and (R_{t}) is time-varying correlation matrix. The conditional variance (h_{it}) for return series are estimated using univariate GARCH.

$$h_{it} = a_{i} + mathop sum limits_{j = 1}^{{q_{i} }} alpha_{ij} e_{it – j}^{2} + mathop sum limits_{k = 1}^{{p_{i} }} beta_{ik} h_{it – k} ,quad for,i = 1,2, ldots , m$$

where (a_{i}), (alpha_{ij}) and (beta_{ik}) are non-negative and (sumnolimits_{j = 1}^{{q_{i} }} {alpha_{ij} } + sumnolimits_{k = 1}^{{p_{i} }} {beta_{ik} } < 1,) and m is the number of selected sectors.

If, the residual ((e_{t})) and the conditional standard deviation ((sqrt {h_{it} })) are obtained, the conditional standard deviation is expressed by diagonal matrix (D_{t}), which consists ((sqrt {h_{it} })) elements on its diagonals as shown as follow.

$$D_{t} = left[ {begin{array}{*{20}c} {sqrt {h_{11t} } } \ 0 \ vdots \ 0 \ end{array} begin{array}{*{20}c} 0 \ {sqrt {h_{22t} } } \ vdots \ 0 \ end{array} begin{array}{*{20}c} ldots \ ldots \ ddots \ ldots \ end{array} begin{array}{*{20}c} 0 \ 0 \ vdots \ {sqrt {h_{mm,t} } } \ end{array} } right]$$

The standardized residuals (varepsilon_{t}) are used for estimating the symmetric and dynamic correlation matrix (R_{t}).

$$R_{t} = left[ {begin{array}{*{20}c} 1 \ {rho_{12,t} } \ {rho_{13,t} } \ vdots \ {rho_{1m,t} } \ end{array} begin{array}{*{20}c} {rho_{12,t} } \ 1 \ {rho_{23,t} } \ vdots \ {rho_{2m,t} } \ end{array} begin{array}{*{20}c} {rho_{13,t} } \ {rho_{23,t} } \ 1 \ ddots \ ldots \ end{array} begin{array}{*{20}c} ldots \ ldots \ ddots \ ddots \ {rho_{m – 1,m,t} } \ end{array} begin{array}{*{20}c} {rho_{1m,t} } \ {rho_{2m,t} } \ vdots \ {rho_{m – 1,m,t} } \ 1 \ end{array} } right]$$

The element of (H_{t} = D_{t} R_{t} D_{t}) is (left[ {H_{t} } right]_{ij} = sqrt {h_{it} h_{jt} } rho_{ij} ,,where,rho_{11} = 1)

According to Engle (2000), Lim and Masih (2017) and Orskaug (2009), (R_{t} = Q_{t}^{* – 1} Q_{t} Q_{t}^{* – 1})

where (Q_{t}^{* – 1} = left[ {begin{array}{*{20}c} {sqrt {q_{11} } } \ 0 \ vdots \ 0 \ end{array} begin{array}{*{20}c} 0 \ {sqrt {q_{22} } } \ ddots \ ldots \ end{array} begin{array}{*{20}c} ldots \ ldots \ ddots \ 0 \ end{array} begin{array}{*{20}c} 0 \ vdots \ 0 \ {sqrt {q_{mm} } } \ end{array} } right]).

where (Q_{t} = left( {1 – a – b} right)overline{{Q_{t} }} + avarepsilon_{t – 1 } varepsilon ^{prime}_{t – 1 } + bQ_{t – 1})

where (Q_{t}^{*}) is the diagonal matrix of its diagonal elements, and (Q_{t}) is a symetric postive definite conditional correlation matrix, and (overline{{Q_{t} }} = Eleft( {varepsilon_{t} varepsilon_{t} ^{prime}} right)) is unconditional covariance of the standadized residual of univariate GARCH model.

The likelihood of the DCC estimator (see Engle and Sheppard 2001) is:

$$L = – 0.5mathop sum limits_{t = 1}^{T} (Klogleft( {2pi } right) + 2{text{log}}(left| {D_{t} } right|)) + log (left| {R_{t} } right|) + varepsilon_{t} ^{prime}R_{t} varepsilon_{t} )$$

The volatility ((D_{t} )) and the correlation ((R_{t} )) components may vary, thus the estimation process achieved in two steps. Firstly the volatility ((L_{v} )). is maximized:

$$L_{v} = – 0.5mathop sum limits_{t = 1}^{T} (Klogleft( {2pi } right) + 2{text{log}}(|D_{t} |)) + r^{prime}_{t} D_{t}^{ – 2} r_{t} )$$

Then the correlation ((L_{c} )) is maximized

$$L_{c} = – 0.5mathop sum limits_{t = 1}^{T} left( {Klogleft( {R_{t} } right)} right) + varepsilon_{t} ^{prime}R_{t}^{ – 1} varepsilon_{t} – varepsilon_{t} ^{prime}varepsilon_{t} )$$

Wavelet theory and analyse method

Wavelet analysis originated in the mid-1980s as an alternative to the well-known Fourier analysis. Fourier analysis is only suitable for stationary time series. In contrast, wavelet analysis has significant superiority over the Fourier analysis when the time series under study are non-stationary or locally stationary (Roueff & Sachs, 2011). Moreover, wavelet analysis allows us to estimate the spectral characteristics of a time series as a function of time and then extracts localized information in both time and frequency domains (Aguiar-Conraria et al., 2008).

The time series can be expanded into a time frequency space where its time- and (or) frequency-varying oscillations are observed in a highly intuitive way. Often, two classes of wavelet transforms exist: discrete wavelet transforms (DWT) and continuous wavelet transforms (CWT). But, the CWT is more helpful for feature extraction and data self-similarity detection (Loh, 2013). As such, the CWT is widely used in economics and finance (Caraiani, 2012; Rua, 2012).

Given a time series (xleft( t right) in L^{2} left( R right)) and given the mother wavelet (psi left( t right)) the CWT is defined as an inner product of (xleft( t right)) with the family (psi_{tau ,s} left( t right)) of wavelet daughter.

$$W_{x;psi } left( {tau ,s} right) = xleft( t right),psi_{tau ,s} left( t right) = mathop int limits_{ – infty }^{ + infty } xleft( t right)psi_{tau ,s}^{*} left( t right)dt$$

(1)

The asterisk (*) denotes complex conjugation (see Jiang et al., 2015), (psi_{tau ,s}^{*} left( t right)) are complex conjugate functions of the daughter wavelet functions (psi_{tau ,s} left( t right)). With constructing the picture, it shows both the amplitude of any features present in (xleft( t right)) versus the scale and how this amplitude evolves over time. In addition, (tau,and,s) are real values that vary continuously for this, (W_{x;psi } left( {tau ,s} right)) is then named as continuous wavelet transform (more information sees: Daubechies, 1992; Goupillaud et al., 1984; Torrence & Compo, 1998).

To analyze the dynamic relationship between Fear migration indices, we should pay greater attention to the wavelet coherency and phase difference. We start with the wavelet coherency, which can be calculated using the cross-wavelet spectrum and the auto-wavelet spectrums as follows:

$$R_{xy}^{2} left( {tau ,s} right) = frac{{left| {S(s^{ – 1} W_{xy;psi } left( {tau ,s} right)} right|^{2} }}{{Sleft( {s^{ – 1} left| {W_{x;psi } left( {tau ,s} right)} right|^{2} } right)Sleft( {s^{ – 1} left| {W_{y;psi } left( {tau ,s} right)} right|^{2} } right)}}$$

(2)

In this case, it is noted that the wavelet coherency under study is represented as a squared type similar to previous studies (Aguiar-Conraria et al. 2008; Rua, 2012).

After smoothed by a smoothing operator S, the squared wavelet coherency gives a quantity between 0 and 1 in a time–frequency space. It is represented by colors in wavelet coherency plots, with red corresponding to a strong correlation and blue corresponding to a weak correlation. In this way, wavelet coherency allows for a three-dimensional analysis that can simultaneously consider the time and frequency components as well as the strength of correlation. Therefore, it helps us to distinguish the local correlation between our time series and to identify structural changes over time and the short-run and long-run relations across frequencies (Loh, 2013).

Because the wavelet coherency is squared, we cannot distinguish between positive and negative correlations. Therefore, we need the phase difference tool to present positive or negative suggestions for correlations and lead-lag relationships between series. Therefore, following Bloomfield et al (2004), the phase difference between x(t) and y(t) is defined as follows:

$$phi_{xy} = tan^{ – 1} left| {left( {frac{{Ileft{ {S(s^{ – 1} W_{xy;psi } left( {tau ,s} right))} right}}}{{Rleft{ {Sleft( {s^{ – 1} W_{xy;psi } left( {tau ,s} right)} right)} right}}}} right)} right|,quad with,varphi_{xy} in left[ { – pi , pi } right]$$

(3)

where I and R are the imaginary and real parts of the smoothed cross-wavelet transform, respectively. According to Voiculescu and Usoskin (2012) and Aguiar-Conraria and Soares (2013), we can easily convert the phase difference into the instantaneous time lag between x(t) and y(t) as the following:

$$left( {{Delta }t} right)_{xy} = frac{{phi_{xy} }}{2pi f}$$

where (2pi f) is the angular frequency with respect to the time scale.

In our following work, the phase differences are represented as arrows in the wavelet coherency plots. Arrows pointing to the right mean that x(t) and y(t) are in phase (or positively related), while arrows pointing to left mean that x(t) and y(t) are out of phase (or negatively related if up or down). Arrows pointing to other directions mean lags or leads between them. It is noteworthy that phase differences can also be suggestive of causality between x(t) and y(t) (Grinsted et al., 2004; Tiwari et al., 2013).

In our present study, we use the Wavelet transform analyze thanks to its ability to decompose the micro and macroeconomic time series, whereas data can also be presented in their time scale components. Most time series techniques interpret data in short run and long run time frames, while in reality, it could not be explained precisely how long is the long and how short is the short.

Thus, wavelet coherence analysis gives an idea of the direction of the effects by indicating the leading variable and the lagging one. So, this method offers a useful analysis in the economic, financial, political and sociological field which presents the source of the effect and the destination of repercussion.

Data and descriptive statistics

Considering the availability of data and seeing that France, Germany, United Kingdom, and the United States are the countries attracting job seekers and seekers of liberty and luxury living, they are the most affected by the waves of migration. For this our study sample is composed by these four countries.

We use quarterly data on Fear migration index for France, Germany, United Kingdom, and united states which will be noted respectively FR-Fear, GER_Fear, UK_Fear and, USA_Fear.

The data series are downloaded from the following website: www.policyuncertainty.com.

To construct the Migration Fear Indices, Baker et al (2015) define the following term sets:

  • Migration (M): “border control”, Schengen, “open borders”, migrant, migration, asylum, refugee, immigrant, immigration, assimilation, “human trafficking”

  • Fear (F): anxiety, panic, bomb, fear, crime, terror, worry, concern, violent

These term sets are translated into German and French with the assistance of native speakers. Finally, Baker et al (2015) count the number of newspaper articles with at least one term from each of the M and F term sets, and then divide by the total count of newspaper articles (in the same calendar quarter and country). Figure 1 present periodic properties for these four series. To explore more details about the periodic properties, we subsequently divide the full samples of these four series into two subsamples as shown in Fig. 1 (low mean and variance, high mean, and high variance) by estimating one date break based on Bai–Perron and Zivot–Andrews approaches. Furthermore, descriptive statistics are simply used here to identify their main features within each subsample.

Fig. 1
figure 1

Time series plots of FR-Fear, GER_Fear, UK_Fear and USA_Fear

When computing the descriptive statistics and matrix of correlation as indicated respectively in Tables 1 and 2, we notice that the mean in the skewness coefficients are positive for all Fear series of the countries, which indicate right-skewed distributions. For the kurtosis coefficients, all are greater than 3, indicating that the Fear series index in a leptokurtic distribution. Moreover, Jarque–Bera tests show that all series are non-normally distributed.

Table 1 Descriptive statistics of the full sample
Table 2 Correlation matrix in the full sample

Table 2 presents results of correlation matrix of Fear series index of the selected countries (French, Germany, United Kingdom, and USA). All values are positive which shows that Fear series move in the same direction. So, the correlation between variables, mean, and variance analysis indicate the possible existence of contagion effect. Results showed in Tables 1 and 2 justified the uses of DCC- GARCH empirical analysis.

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