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Journal of Quantitative Economics

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20.04.2017 | Original Article

On the Dynamic Linkages Among International Emerging Currencies

verfasst von: Zouheir Mighri

Erschienen in: Journal of Quantitative Economics | Ausgabe 2/2018

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Abstract

This study examines the interdependence of US dollar exchange rates expressed in five emerging currencies. Focusing on different phases of the global financial and European sovereign debt crises, the aim of this paper is to examine how the dynamics of correlations between emerging exchange markets evolved from January 04, 2000 to July 11, 2014. To this end, we adopt a dynamic conditional correlation model into a multivariate Fractionally Integrated Asymmetric Power ARCH framework, which accounts for long memory, power effects, leverage terms and time varying correlations. The empirical findings indicate a general pattern of decrease in exchange rates correlations across the phases of the global financial crisis and the European sovereign debt crisis, suggesting the depreciation against US dollar and different vulnerability of the currencies. Moreover, our analysis supports the existence of a general pattern of increase in dynamic correlations across several phases of the two crises, indicating the existence of a “contagion effect”.

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Engle (2002) derives a different form of DCC model. The evolution of the correlation in DCC is given by: $Q_t =(1-\alpha -\beta )\bar{{Q}}+\alpha Z_{t-1} +\beta Q_{t-1},$ where $Q=(q_{ijt} )$ is the $N\times N$ time-varying covariance matrix of $z_t ,\bar{{Q}}=E[z_t z_t ^{\prime }]$, denotes the $n\times n$ unconditional variance matrix of $z_t $, while $\alpha $ and $\beta $ are nonnegative parameters satisfying $(\alpha +\beta )<1$. Since $Q_t $ does not generally have units on the diagonal, the conditional correlation matrix $R_t $ is derived by scaling $Q_t $ as follows: $R_t =(diag(Q_t ))^{-1/2}Q_t (diag(Q_t ))^{-1/2}$.

http://www.ecb.int/ecb/html/crisis.en.html.

http://www.reuters.com/article/2010/08/25/eurozone-crisis-events-idUSLDE67O0YD20100825.

Constancio (2012), Kalbaska and Gatkowski (2012), and Arghyrou and Kontonikas (2012), among others, use a similar timeline for the European sovereign debt crisis.

In MS-DR model, the lags of the dependent variable are added in the same way as other regressors. An example is:

$$\begin{aligned} y_t =v(s_t )+\alpha y_{t-1} +X_t ^{{\prime }}\beta +\varepsilon _t \text { where } \varepsilon _t \rightarrow N(0,\sigma ^{2}) \end{aligned}$$

$s_t$ is the random variable denoting the regime. If there are two regimes, we could also write:

Regime 0: $y_t =v(0)+\alpha y_{t-1} +X_t ^{{\prime }}\beta +\varepsilon _t $
Regime 1: $y_t =v(1)+\alpha y_{t-1} +X_t ^{{\prime }}\beta +\varepsilon _t $

which shows the regime dependent intercept more clearly.

http://www.federalreserve.gov.

The $z_t $ random variable is assumed to follow a student distribution (Bollerslev 1987) with $v>2$ degrees of freedom and with a density given by:

$$\begin{aligned} D(z_t ,v)=\frac{\Gamma \left( v+\frac{1}{2}\right) }{\Gamma \left( \frac{v}{2}\right) \sqrt{\pi (v-2)}}\left( 1+\frac{z_t ^{2}}{v-2}\right) -(v+1)/2 \end{aligned}$$

where $z_t =\varepsilon _t /\sigma _t ,\Gamma (v)=\int _0^\infty {e^{-x}x^{v-1}dx} $ is the gamma function and v is the parameter that describes the thickness of the distribution tails. The Student distribution is symmetric around zero and, for $v>4$, the conditional kurtosis equals $3(v-2)/(v-4)$ which exceeds the normal value of three. For large values of v, its density converges to that of the standard normal.

For a Student-t distribution, the log-likelihood is given as: $L_{student} =T\{log \Gamma (\frac{v+1}{2})-log \Gamma (\frac{v}{2})-\frac{1}{2}log [\pi (v-2)]\}-\frac{1}{2}\sum _{t=1}^T {[\log (h_t) +(1+v)log (1+\frac{z_t ^{2}}{v-2})]} $

where T is the number of observations, v is the degrees of freedom, $2<v\le \infty $ and $\varGamma (.)$ is the gamma function.

The lag orders (1, d, 1) and (0, 0) for FIAPARCH and ARMA models, respectively, are selected by AIC and BIC information criteria. The results are available from the author upon request.

According to Fernandez and Steel (1998) and Lambert and Laurent (2001), and provided that $\upsilon >2$, an innovation process $z_t $ is said to be standardized skewed Student’s tdistributed, ie, the innovation $z_t \sim SKST\left( {0,1,\xi ,\upsilon } \right) $,if:

$$\begin{aligned} f\left( {z_t /\xi ,\upsilon } \right) =\left\{ {{\begin{array}{l} {\frac{2}{\xi +\frac{1}{\xi }}sign\left( {\xi \left( {sz_t +m} \right) /\upsilon } \right) \,if\, z_t <-m/s} \\ {\frac{2}{\xi +\frac{1}{\xi }}sign\left( {\left( {\left( {sz_t +m} \right) /\xi } \right) /\upsilon } \right) \,if\, z_t \ge -m/s} \\ \end{array} }} \right. \end{aligned}$$

where $g\left( {\cdot /\upsilon } \right) $ is the symmetric (unit variance) Student’s t density and $\xi $ is the asymmetry parameter.

The parameters m and $s^{2}$ are respectively the mean and the variance of the non-standardized skewed Student’s t-distribution:

$$\begin{aligned} m= & {} m\left( {\xi ,\upsilon } \right) =\frac{{\Gamma }\left( {\left( {\upsilon -1} \right) /2} \right) \sqrt{\upsilon -1}}{\sqrt{\pi }{\Gamma }\left( {\upsilon /2} \right) }\left( {\xi -\frac{1}{\xi }} \right) \\ s^{2}= & {} \left( {\xi ^{2}+\frac{1}{\xi ^{2}}-1} \right) -m^{2} \end{aligned}$$

The log-likelihood function of a standardized (zero mean and unit variance) skewed Student’s t-distribution is given by: $L_{SKST} =T\left\{ {ln{\Gamma }\left( {\frac{\upsilon +1}{2}} \right) -ln{\Gamma }\left( {\frac{\upsilon }{2}} \right) -\frac{1}{2}ln\left[ {\pi \left( {\upsilon -2} \right) } \right] +ln\left( {\frac{2}{\xi +\frac{1}{\xi }}} \right) +ln\left( s \right) } \right\} -\frac{1}{2}\sum \nolimits _{t=1}^T \left[ {\ln \left( {h_t } \right) +\left( {1+\upsilon } \right) ln\left( {1+\frac{\left( {sz_t +m} \right) ^{2}}{\upsilon -2}\xi ^{-2I_t }} \right) } \right] $

where $I_t =\left\{ {{\begin{array}{l} {1\,if\,z_t \ge -m/s} \\ {-1\,if\,z_t <-m/s} \\ \end{array} }} \right. $ with T denotes the number of observations, $\upsilon $ is the degrees of freedom, $2<\upsilon \le \infty $, and $\varGamma \left( . \right) $ is the gamma function.

Lambert and Laurent (2001) found that the skewed Student-t density is more appropriate for modeling the financial time series than symmetric densities.

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Titel: On the Dynamic Linkages Among International Emerging Currencies
verfasst von: Zouheir Mighri
Publikationsdatum: 20.04.2017
Verlag: Springer India
Erschienen in: Journal of Quantitative Economics / Ausgabe 2/2018
Print ISSN: 0971-1554
Elektronische ISSN: 2364-1045
DOI: https://doi.org/10.1007/s40953-017-0088-1

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