Modeling the dependence between foreign exchange rates : an analysis of the effect of the 2007-2009 financial crisis using a copula approach

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rates: An analysis of the effect of the 2007–2009

financial crisis using a copula approach

Kevin L. Sipin

Student No. 10523006 Supervisor: S. Umut Can

Faculty of Economics and Business, University of Amsterdam Email: kevin.sipin@student.uva.nl

Bachelor’s Thesis for the Bachelor’s of Science in Actuarial Sciences This thesis investigates the bivariate dependence structure for four pairs of exchange rates against the US dollar: the Australian dollar (AUD), the Canadian dollar (CAD), the Euro (EUR) and the Japanese yen (JPY) in the periods before, during and after the financial crisis, a total sample period ranging from December 1993 to April 2016. The novelty of this thesis is that I use copula theory to get analytical results for the dependence structure, instead of the commonly used one-dimensional measures of dependence. In particular, important financial time series characteristics as tail dependence and asymmetry are explicitly modeled. After VAR filtering, I find strong nonlinear relationships between allmost all exchange rate log returns. However, after controlling for heteroskedasticity through GARCH-DCC filtering, most of these effects vanish. This suggests that these nonlinearities follow from volatility inducing events in the market, such as volatility spillovers. Also, there seems to be only mild evidence for tail asymmetry, however when the model is restricted to SJC estimation the lower tail dependency is significantly stronger then the upper tail dependency, which is consistent with earlier studies. Further, this study also suggests that this interdependency between exchange rates has grown stronger during the financial crisis and that this increased interdependency between exchange rates has persisted, albeit somewhat

less strong.

Received June 29th, 2016

FOREWORD AND ACKNOWLEDGEMENT This thesis is the result of three years of studying Actuarial Sciences and Econometrics and Operations Research at the University of Amsterdam. Although I may have never seemed like the most eager of students, one must not underestimate the interest that I took in this studies and the enormous benefit that I have everyday because of this studies.

I am greatly indebted to the many great professors and lecturers at the University of Amsterdam, whose professional knowledge and academic work has inspired me. Unfortunately, this never really reflected in my grades. But the field of mathematics and econometrics did change the way that I look at the world and interpret statements, facts and decisions. In a sense, it enabled me to think more clear and abstract about the many concepts that comprise the world that we live in. I might not have put as much effort in my formal studies as it deserved, but I dare to say that I have compensated for this in my extracurricular activities

and the way that I have accomplished my tasks there. I do not enjoy learning from text books, but I do enjoy learning through solving challenges.

My theses have therefore been the most enjoyable assignments that I have done during my studies to date. The freedom, the creativity and the lack of a preset path is what really inspired me to create something good, something of my own of which I could be proud. I think that with this work, that is focused on the application of copulas in financial time series analysis, I have accomplished that goal.

I would in particular like to thank Umut Can for helping me to understand the core concepts of copulas that make up the foundation of this thesis, and for his proactive support during my research and writing. I would also like to thank Hao Fang, who has made clear to me how the classical approach to time series analysis can benefit from applications such as copulas. Finally, I would like to thank Nancy Bruin for her very close reading of my draft papers and her tangible suggestions for improvement.

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CONTENTS

1 Summary 2

2 Introduction 3

3 Theoretical framework 3

3.1 Definitions . . . 3

3.2 Families and properties . . . 4

3.3 Student’s t copula . . . 4

3.4 SJC copula . . . 4

3.5 Estimating distributions . . . 5

3.6 Estimating the copula . . . 6

3.7 Granger causality . . . 6

3.8 Summary . . . 7

4 Research design 7 4.1 Methodology . . . 7

4.2 Data . . . 7

5 Results and analysis 8 5.1 Copula estimation on unfiltered log returns 8 5.2 Copula estimation on VAR residuals . . 8

5.3 Copula estimation on GARCH filtered VAR residuals . . . 9

5.4 Summary . . . 10

6 Conclusions 10

References 14

STATEMENT OF ORIGINALITY

This document is written by Student Kevin Sipin who declares to take full responsibility for the contents of this document.

I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it.

The Faculty of Economics and Business is re-sponsible solely for the supervision of completion of the work, not for the contents.

1. SUMMARY

The broad objective of this thesis is to explore the dependency structure between the AUD/USD, CAD/USD, EUR/USD and JPY/USD exchange rates and to study if and how this dependency structure changed over the course of the 2007–2009 financial crisis. It is reasonable to expect that the 2007–2009 financial crisis alters the dependency structure between exchange rates, such as the Asian financial crisis has (Bekiros & Diks, 2008).

In this thesis I estimate Student’s t copulas and the Symmetrized Joe Clayton copulas to explicitly model the dependency structure of the exchange rate returns time series. I estimate this copulas on, consecutively: raw log returns, VAR filtered residuals and GARCH-DCC filtered VAR residuals. Thus, I first explore the general dependency structure and then the nonlinear dependency structure. By also applying a GARCH-DCC filter, I ensure that the likelihood of having spurious significant results is reduced because the effects of certain events that imply conditional heteroskedasticity (e.g. volatility spillovers) are then strongly reduced.

Based on the results presented in this thesis, I conclude that strictly nonlinear relations (i.e., relations that persist after VAR and GARCH-DCC filtering) are not common. Before the financial crisis, there were no nonlinear causal relations. However, during the financial crisis the interdependency strongly increases. This is especially the case for the lower tails of the log returns distributions: the interdependency is stronger in depreciating markets than in appreciating markets. These relationships loose some strength after the financial crisis, but remain stronger than they were before the financial crisis. Thus, I conclude that all the foreign exchange rates have become more interdependent during and following the financial crisis of 2007–2009.

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2. INTRODUCTION

Everyday, $5.3 trillion is traded on the foreign exchange market. For comparison: the British GDP is valued at $2.8 trillion. Since we transfer on a daily basis almost twice as much money in foreign exchange markets as is produced during an entire year in the United Kingdom, we might want to have a very good understanding of the dynamics of these currency transfers and the risk that is associated with them. This example illustrates the importance of exchange rates in our global economy. In free markets, they are used to maintain the balance of trade and balance of capital. Many economic policies either aim to influence them or depend on them. Therefore, understanding the complex dynamics of exchange rates is of the utmost importance to policy makers and investors. A great deal of the complexity in exchange rates comes from the dependence on many factors, including other exchange rates and underlying economic indicators.

The modeling and estimating of these exchange rates is further complicated however, as exchange rates tend to show nonlinear and asymmetric behavior (e.g. different behavior in market appreciation and depreciation) (Azam, 2014; Boero, Silvapulle, & Tursunalieva, 2010; Bekiros & Diks, 2008; Patton, 2006; Longin & Solnik, 2001). In addition, because returns in financial time series tend to be non-normally distributed (Okimoto, 2008; Patton, 2006; Jondeau & Rockinger, 2006), it becomes challenging to specify a multivariate distribution for multiple exchange rates. Research also shows that the dependence structure’s parameters (i.e. measure(s) of association) might not be stable over time. For example, there is evidence that the Euro/Deutsche mark/US Dollar exchange rate has a different dependence structure in the pre-Euro against the post-Euro period against the British pound sterling and Japanese yen (Boero et al., 2010; Bekiros & Diks, 2008; Patton, 2006). Knowledge of this change in dependence structure might be of interest for central banks and other policy makers, among other parties that are concerned with international economics, trade and finance.

One of the questions that could be asked is on what other exchange rates a given exchange rate depends, and to what extent. This dependence can be measured by linear correlation, or using multivariate distributions. This study aims to model the general dependence structure, which could include nonlinear dependency. Linear methods should thus be discarded as a method of investigation. Because of their analytic properties and efficiency (Frees & Valdez, 1998), a copula-based modeling approach will be applied. Another important question is if the structure of the dependency between foreign exchange rates has changed over the course of years or important events. In this case, I ask if the financial crisis (2007–2009) has significantly altered the dependency structure between

the given time series. The central research question then becomes: what is the dependency structure between the log returns of selected exchange rates, expressed in a copula model, and is this structure stable over time?

To answer this question, a general theoretical and conceptual framework of copula theory is given, based on Frees and Valdez (1998); Patton (2006); Boero et al. (2010) and Azam (2014). The differences between given families are discussed and an appropiate copula family is chosen for the modeling of the dependence structure. The returns of the EUR/USD, JPY/USD, AUD/USD and CAD/USD exchange rates are filtered by applying a ARMA(p,q)-GARCH(1,1) filtering. Finally,the copulas are fitted to describe the dependency between the log returns of the exchange rates over multiple time periods: before, during and after the financial crisis (2007–2009). This thesis is organized as follows. Section 2 introduces copulas and ARMA(p,q)-GARCH(1,1) filtering. Section 3 explains the design of this research, the used methodology and the data. Section 4 applies the theory of copulas and ARMA-GARCH filtering to study the dependence structure of the aforementioned four exchange rates. Section 5 analyzes these results. Section 6 summarizes and concludes.

3. THEORETICAL FRAMEWORK

In this section I start with a review of the theory of copulas and discuss how a copula with the right properties for this research can be selected. I give a short introduction of copulas, by stating their definition. Then I turn to the problem of selecting and building a copula function that has the necessary properties. This copula will then be studied more in depth. This is followed by a short review of the ARMA-GARCH framework and show how this can be used to model the returns of exchange rates. I finish with a short note on goodness-of-fit testing and its importance in comparing and assessing copulas.

3.1. Definitions

Sklar (1959)’s theorem says that any p-dimensional joint distribution H for some continuous random variables Y1, . . . , Yp can be decomposed into a copula

C describing their dependency, and their margins F1, . . . , Fp specifying their individual characteristics

(fat tails, skewness, etc.). Formally given as

H(y1, . . . , yp) = C(F1(y1), . . . , Fp(yp)) (1)

where C : [0, 1]p _{→ [0, 1]. The copula C is unique if all}

the margins F1, . . . , Fp are continuous. It could also be

stated as

C(u1, . . . , up) = P (U1≤ u1, . . . , Up≤ up) (2)

where Uj = Fj(Yj), which is obtained through

Probability Integral Transformation (see, for example, Bain and Engelhardt (1992)).

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In particular, for a bivariate vector (X, Y ) with joint cdf FXY and marginal cdfs FX and FY, it holds that

FXY(x, y) = C(FX(x), FY(y)) (3)

or, in terms of pdfs,

fXY(x, y) = fX(x) · fY(y) · c(FX(x), FY(y)) (4)

where c denotes the density of the copula C. This clearly shows how using a copula, any bivariate distribution can be written using (i ) the two marginal distributions and (ii ) the copula, which fully describes the dependency structure. Many functional forms of copulas can be found in Joe (1997) and Patton (2006). 3.2. Families and properties

There are many copula families available, each with their own properties (Joe, 1997; Patton, 2006). The most widely used copulas are the Gaussian and t -copula, as they are best fit to describe the dependency for normally distributed marginals. That is, they imply symmetry. In a bivariate case, the Gaussian copula is given by (following Azam (2014)):

Cg(u, v|ρ) = Φg(Φ−1(u), Φ−1(v); ρ) = Z Φ−1(u) −∞ Z Φ−1(v) −∞ 1 2π(1 − ρ2₎1₂ × −(s 2_{− 2ρst + t}2₎ 2(1 − ρ2₎ dsdt (5) with Φg the cdf of standard bivariate normal

distribution, Φ the cdf of the standard normal distribution and ρ ∈ [−1, 1] the correlation parameter. For more on copula generation, see Frees and Valdez (1998).

However, in Section 1 I described two of the main characteristics of financial time series: their (i ) non-normal distribution of residuals and their (ii ) asymmetric tails. I can therefore conclude that both the Gaussian and t -copula are not the most appropiate choices for the modeling of dependence in exchange rate log returns.

I should thus look at a copula that at least accounts for asymmetry in the tails. Possible choices, as researched in Patton (2004), are Clayton’s, Gumbel’s and the Symmetrised Joe-Clayton (SJC) copula. Clayton’s copula has more mass in the negative quadrant, i.e. it is a better fit for greater dependence in joint negative events than for joint positive events (for which it behaves more or less as a Gaussian copula). Gumbel’s copula has more mass in the positive quadrant, thus implying the exact opposite of Clayton’s copula. The SJC copula has two parameters: one for the dependence in the negative quadrant and one for the dependence in the positive quadrant, making it more versatile. Based on Patton (2006); Boero et al. (2010)

and Azam (2014) I select the SJC copula to describe the dependency structure between exchange rates.

In the next subsection, the properties of the Student’s t copula and SJC copula are described in depth. 3.3. Student’s t copula

I will give a review of the Student’s t copula based on Demarta and McNeil (2005). Demarta and McNeil show that the t copula represents the dependence structure in a multivariate t distribution. This distribution has a density which is given by

f (X) = Γ v+d 2 Γ v₂ p(πv)d_|Σ| × 1 + (X − µ) 0_Σ−1_{(X − µ)} v −v+d2 (6) where X = (X1, . . . , Xd)0 has a multivariate t

distribution with v degrees of freedom, mean vector µ and positive-definite scatter matrix Σ, such that X ∼ td(v, µ, Σ).

In Equation 2 I showed how copulas can be evaluated. The unique t copula is given by

C_v,Pt (u) = Z t−1v (u1) −∞ · · · Z t−1v (ud) −∞ Γ v+d₂ Γ v₂ p(πv)d_{|P |} × 1 +X 0_P−1_X v −v+d2 dX (7) where t−1_v denotes the quantile function of the standard univariate tv distribution. P is the correlation matrix

as implied by the dispersion matrix Σ. Note that if v → ∞, the t copula converges to the symmetric normal or Gaussian copula, analogous to how the t distribution converges to a normal distribution.

3.4. SJC copula

Contrary to the symmetric t copula, the SJC copula parametrizes the left and the right tail seperately. The SJC copula is based on the Joe-Clayton copula, or “BB7” as Joe (1997) terms it. This is a 2-parameter copula, given by

CJ C(u, v; τU, τL) = 1 − (1 − {[1 − (1 − u)κ]−γ

+ [(1 − v)κ]−γ− 1}−γ1₎1κ ₍₈₎ where the upper and lower tail dependency are given by τU _{∈ (0, 1) respectively τ}L_{∈ (0, 1), and where}

κ = 1

log2(2 − τU)

γ = − 1

log2(τL)

However, this copula tends to report asymmetric de-pendency even if the dede-pendency in both tails is per-fectly symmetric (see Patton (2006)). Patton modifies

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this copula in such a way that it handles symmetry well and is consistent in reporting asymmetry. He terms it “Symmetric Joe-Clayton” and it is given by

CSJ C(u, v|τU, τL) = 0.5(CJ C(u, v|τU, τL)

+ CJ C(1 − u, 1 − v|τL, τU)

+ u + v − 1) (9)

3.5. Estimating distributions

I now give a summary of the Vector Autoregressive model (VAR) and of ARMA-GARCH and GARCH-DCC filtering and show how these methods can be used to estimate the empirical distributions of the time series.

3.5.1. VAR filtering

In this section I take a quick glance at the reduced-form vector, as is also done in Bekiros and Diks (2008). Let Y be the vector of endgenous variables and l the number of lags. Then, a VAR(l) model is given by

Yt= l

X

s=1

AsYt−s+ t (10)

with Yt = [Y1t, . . . , Ylt] the l × 1 vector of endogenous

variables, As tht l × l parameter matrix and t the

residual vector. Bekiros and Diks note that for two stationary time series {Xt} and {Yt} the bivariate VAR

model is given by

Xt= A(l)Xt+ B(l)Yt+ X,t (11)

Yt= C(l)Xt+ D(l)Yt+ Y,t (12)

where A(l), B(l), C(l) and D(l) are all polynomials in the lag operator with their roots outside the unit circle. The test if Y is strictly Granger causing X, the test is reduced to testing that all the coefficients of B(l) are zero. To test if X is strictly Granger causing Y, all the coeffecicients of C(l) should not significantly differ from zero.

The lag lengths of the VAR model can be based on a information criterion, for example the SIC (Schwartz Information Criterion, also called BIC, Bayesian Information Criterion).

Now I have filtered for the linear effects. To effectively model the time series I have to deal with the heteroskedasticity through an ARMA-GARCH or multivariate GARCH-DCC filtering, which is demonstrated in the following sections.

3.5.2. ARMA-GARCH filtering

In order to further define a bivariate distribution, I need to define the marginal functions. As financial time series are often serial correlated and highly volatile, and thus often follow (G)ARCH-like processes, I apply ARMA-(G)ARCH filtering on the exchange rate return time

series first for the derivation of p, q). This ARMA(p,q)-GARCH(1,1) model is given by

Xt= c + t+ p X i=1 ϕiXt−i+ q X i=1 θit−i (13) σ2t = α0+ α1rt−12 + βiσt−12 (14)

where Eqn. 8 is the ARMA(p,q) equation and Eqn. 9 is the GARCH(1,1) equation. Asymptotic theory was built on the idea of margins with i.i.d. observations. This is clearly not the case for most financial time series (serial correlation, clustered volatility), and therefore these basic asymptotic theory results do not hold. However, I can replace these margins with ARMA-(G)ARCH filtered residuals, as is clearly shown in Kim, Silvapulle, and Silvapulle (2007, 2008).

The robustness of the results is also shown in Boero et al. (2010); Kim et al. (2008, 2007). That means that, using ARMA(p,q)-GARCH(1,1) filtering, the filtered exchange rate returns consistent estimates of the errors in the ARMA(p,q)-GARCH(1,1) model.

In the next section I introduce GARCH-DCC filter-ing, which van be used to control for heteroskedasticity in multivariate time series.

3.5.3. GARCH-DCC filtering

In the previous paragraph I showed how to control for heteroskedasticity in a single time series. However, it is reasonable to also include a form of multivariable GARCH since this makes it possible for condition to events that induce heteroskedasticity in multiple time series. In this paragraph I introduce the Dynamical Conditional Correlation class of models based on ? (?). This model can be summarized as:

ρ12,t=

Et−1(r1,t, r2,t)

q

Et−1(r21,t)Et−1(r22,t)

with r1, r2 two random variables with mean zero. Now

? clarifies the relationship between the conditional correlations (as in the above equation) and the conditional variances, by expressing the returns as the conditional standard deviation times the standardized disturbance:

hi,t = Et−1(r2i,t)

ri,t =phi,ti,t i = 1, 2

where is standardized disturbance ( ∼ N (0, 1)). Then it follows that:

ρ12,t= Et−1(1,t2,t) (15)

And for this conditional correlation, many estimators have been derived. It is beyond the scope of this thesis however, to review these in depth. It should be sufficient to note that ? eventually derives the Dynamic Conditional Correlation estimator that is

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based on a time-varying correlation matrix Rt, which

makes GARCH-DCC an appropiate method to model conditional heteroskedasticity in a multivariate model, such as the financial time series that are under review in this thesis. In the next paragraph I show how the ARMA-GARCH or GARCH-DCC residuals are used to estimate empirical distributions.

3.5.4. Estimating empirical distributions

I can use the consistent estimates of the errors in the ARMA(p,q)-GARCH(1,1) model to estimate the marginal distributions (Boero et al., 2010; Chen & Fan, 2006b; Genest, Ghoudi, & Rivest, 1995). Define X = (x1, . . . , xn) and Y = (y1, . . . , yn) as the residuals

(returns) of two exchange rates and assume that X and Y are i.i.d. variables with continuous marginal distribution functions (FX, FY continuous). Following

Genest et al. and Boero et al. I can then estimate the distributions as ˆ FX(x) = 1 n + 1 n X i=1 I{Xi<x} ˆ FY(y) = 1 n + 1 n X i=1 I{Yi<y}

with I{·}the indicator function. Boero et al. notes that

both ˆFX(x) and ˆFY(y) will be approximately Unif (0,1)

distributed pseudo-observations.

I may now use other methods (e.g. Maximum Likelihood Estimation) to fit the copula model where the marginals are the estimated marginals as derived above.

3.6. Estimating the copula

I end this section with a short note on copula estimation and the selection of the ‘best fitting’ copula.

The copula itself is parametric, and using MLE, TSML, FML or IF I can estimate the parameters of the copula (see Boero et al. (2010); Chen and Fan (2006a) and to lesser extent Chen and Fan (2006b)). As with all MLE procedures, I maximize some (pseudo-)log-likelihood function

n

X

i=1

log c( ˆFX(xi), ˆFY(yi); θ) (16)

where c(·; θ) is the density of the parametrized copula function. I have now built the entire conceptual framework of estimating margins and coupling them through the use of copulas.

Lastly, I am concerned with choosing the ’best fitting’ copula. In order to assess the fit of a copula, there are a few goodness-of-fit tests available (Can, Einmahl, Khmaladze, & Laeven, 2015; Genest, R´emillard, & Beaudoin, 2009; Breymann, Dias, & Embrechts, 2003). Can et al. derives a test based on the null hypothesis

that

R ∈ R = {Rθ: θ ∈ Θ} (17)

where R is a family of copulas, θ the parameter vector and Θ the parameter space. Using a semi-parametric estimate ˆRnof R, and taking the difference ˆηnthis with

Rθˆ, Can et al. shows how numerous tests can be derived.

This is however a very advanced subject and will not be covered in depth in this thesis.

3.7. Granger causality

The final addition to this theoretical framework is a review the concept Granger causality and of methods to detect and formally test dependency between time series based on Granger causality.

3.7.1. Definition

Granger (n.d.) causality is a widely used concept for expressing dependence relations between time series.

Let FX,t, FY,t be all the information contained in

past observations of Xi, Yi, i.e. the set of (all) the

observation of {Xi}, {Yi} up till time t. I then say that

{Xt} is a Granger cause of {Yt} if observations of X

contain additional information on the future values of Y that is not contained in {Yt}. I can formalize this

definition. If ∼ denotes the equivalence in distribution, the formal definition becomes

(Yt+1, ..., Yt+k) | (FX,t, FY,t) (Yt+1, ..., Yt+k) |FY,t

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stationary bivariate time series process. This is the most general definition of Granger causality, as there are no assumptions made on the model (e.g. linearity assumptions).

In the next section, I show how to detect and test for Granger causality in time series based on this definition of general (nonlinear) Granger causality.

3.7.2. Causality detection and testing

In the previous paragraph I have shown what the broadest definition for Granger causality is. It logially follows that a suitable test should test if {Xt} provides

statistically significant additional information on future values of Y .

Granger (n.d.) proposes a test for parametric linear time series models, based on the conditional mean E (Yt+1| (FX,t, FY,t)). I might then test for causality by

comparing the residuals of a fitted AR (autoregressive) model of Yt with those residuals that are obtained by

regressing Yt on infinite past values of both {Xt} and

{Yt}.

In reality the information time series will not be infinite. Let FX,t, FY,t be the information set

Xt−lx+1, ..., Xt respectively Yt−ly+1, ..., Yt. The null hypothesis then becomes

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Bekiros and Diks (2008) note that for a strictly stationary bivariate time series this equation implies that the invariant distribution of the (lx + ly +

1)-dimensional vector Wt = (FX,t, FY,t, Zt) where Zt =

Yt+1. Then I can rewrite Eq. (2) in terms of the joint

distributions, so that Eq. (2) is equal to stating that fX,Y,Z(x, y, z) fY(y) = fX,Y(x, y) fY(y) fY,Z(y, z) fY(y) (20) Diks and Panchenko (2006) show, using a stationary bootstrap approach as developed by ? (?), that this reformulated null hypothesis implies

q ≡ E[fX,Y,Z(X, Y, Z)fY(Y )−fX,Y(X, Y )fY,Z(Y, Z)] = 0

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Tn(n) = n − 1 n(n − 2) X i ( ˆfX,Y,Z(Xi, Zi, Yi) ˆfY(Yi) − ˆfX,Y(Xi, Yi) ˆfY,Z(Yi, Zi)) (22)

where ˆfW(Wi) denotes a local density estimator of a

dW-variate random vector W defined by ˆfW(Wi) =

(2n)−dW(n − 1)−1Pj,j6=iI W

ij where IijW = I(kWi −

Wjk < n) with I(·) the indicator function and n a

bandwith depending on the sample size n (for more on optimal bandwidth choice, see Diks and Panchenko (2006)). And for lx = ly = 1 and n = Cn−β where

C > 0,1₄ < β < 1₃ Diks and Panchenko prove that the test statistic satisfies

√

n(Tn(n) − q) Sn

d

−→ N (0, 1) (23) which can be used for a one sided test, i.e. I reject the null hypothesis when the left-handed side of Eq. (6) is larger than a chosen critical value.

I have now shown how to develop a consistent test for Granger causality. In the next subsection I show how to obtain an actually testable time series. I do that by showing how an ARMA-GARCH or GARCH-DCC filtering procedure can be applied to estimate marginal distributions.

3.8. Summary

In this section I have first introduced the definitions of the concept copula. Then I gave a quick review of some families and their properties. In particular, I zoomed in on the Symmetrized Joe Clayton copula as this provides a very flexible framework in which tail asymmetry can be modeled. I then showed some filtering procedures to obtain residuals on which nonlinear dependency can be tested: the VAR and ARMA-GARCH filtered residuals. I concluded with a short overview of some methods that can be used to estimate empirical distributions and copulas, in such a way that the aforementioned concepts can be operationalized in order to model the dependency in time serie, and I have introduced Granger causality testing as a means of detecting interdependency between time series.

4. RESEARCH DESIGN

4.1. Methodology

The methodology of the empirical research in this study is primarily derived from Azam (2014); Boero et al. (2010) and Patton (2006). As a preliminary note, notice that where “time series” is said, this might refer to the “exchange rate log returns” time series, instead of the exchange rate values itself.

I consider four exchange rates: the EUR/USD, JPY/USD, AUD/USD and CAD/USD, from December 1993 to April 2016. This dataset will be divided into three subsamples: prior to, during, and after the financial crisis (2007–2009). The main goal is to fit on each of the exchange rate log returns a copula, for each of the three time slots, and to compare the results to see if there is any change in dependency structure between foreign exchange rate log returns, as is also done in Bekiros and Diks (2008).

To study this, I first select a copula famiily that handles the characteristics of financial time series well. For instance, evidence (Patton, 2006; Okimoto, 2008) suggests that the exchange rate dependency may not be the same for appreciations as for deprecations. This means that the copula must be able to handle a non-normal distribution of errors and asymmetry in the tails. That is, I should allow for asymmetric dependence structures. This leads to a selection of possible copulas, such as the “Symmetric Joe-Clayton” copula, as proposed by Patton based on the Joe-Clayton “BB7” copula (Joe, 1997).

Second, the exchange rate time series will be filtered with a VAR model, analogous to ARMA-filtering as is also done by Azam (2014); Patton (2006); Longin and Solnik (2001). The residuals (returns) are stored.

Third, I use these residuals to estimate the marginal distributions ˆFX(x) and ˆFY(y), following the procedure

of (Boero et al., 2010; Genest et al., 1995). I now obtain results about the nonlinear dependency structure. By also applying GARCH-DCC filtering, I can control for conditional heteroskedasticity (e.g. volatility spillovers).

Fourth, still following Genest et al., I apply maximum likelihood estimation on the (pseudo) log-likelihood function that contains the copula parameters.

Finally, I combine all these results to fit the estimated copulas on the time series.

4.2. Data

The original data set consists of daily exchange rates for the Australian dollar (AUD), Chinese yuan (CNY), Russian ruble (RUB), Canadian dollar (CAD), Euro (EUR) and Japanese yen (JPY) against the United States dollar (USD). It should be noted that before January 1st 1999 the EUR/USD exchange rate is actually the German Mark (DEM)/USD exchange rate, at which the official conversian rate of 1 EUR = 1.95583

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DEM is applied so that the time series is ’normalized’ to current EUR prices.

The full data sample is taken from 13th April 1966 up to 13th April 2016, collected via Thomson Reuters’ Datastream. However, I restrict this research to only the free-floating currencies, thus looking at the EUR/USD, JPY/USD, AUD/USD and CAD/USD exchange rates. I take the longest possible time series that contains all exchange rates: 31st December 1993 up to 13th April 2016.

The time series of this original data set are plotted in figure 1. The time series of the log returns of this data set are plotted in figure 2. Also, the scatter plots portraying the correlation between exchange rates and their distribution are shown in figure 3. It is clearly visible that research into the non-free floating exchange rates (CNY/USD, RUB/USD) will most likely not lead to any results.

The same plots are also shown for the time series that were eventually selected for this research, over the full time period of this study (EUR/USD, JPY/USD, AUD/USD, CAD/USD; 1993–2016). Exchange rates (with JPY/USD normalized) are shown in Figure 4. The log returns time series are shown in Figure 5. The scatter plots showing the correlation of log returns is shown in Figure 6.

5. RESULTS AND ANALYSIS

In this section I present the main results of this thesis, which are the results of estimating copula on the exchange rate log returns. I want to explicitly state here that the results of the causality tests come from Sipin (2016), a study that is conducted on exactly the same data set. First, I present the results of the copula estimations on the unfiltered log returns to describe the most general dependency structure. Second, I apply four-variate VAR(1) filtering to the log returns. On these VAR residuals, SJC and t copulas are estimated. This should describe the nonlinear dependency structure. I conclude this section with an estimation on the GARCH-DCC(1,1) filtered VAR residuals, to control for events that induce conditional heteroskedasticity (e.g. volatility spillovers).

5.1. Copula estimation on unfiltered log re-turns

I start this section with an exploration of the most general dependency structure between exchange rates. I fit both a Student’s t copula and a SJC copula on the unfiltered log returns. This fitting is done through a MATLAB procedure that was developed for Patton (2004) and Patton (2006) and that is made available by the author1. The results are presented in Table 1 and Figures 15 to 20.

1_{Code can be found on Andrew Patton’s personal website,}

http://public.econ.duke.edu/ ap172/code.html

Based on the results in Table 1 and Figures 15 to 20, I conclude the following: the general interdependency structure becomes stronger during the financial crisis (2007–2009) for all exchange rate pairs. Interestingly, in almost all cases the symmetric Student’s t copula is preferred over the asymmetric SJC copula, based on a log-likelihood estimation2. However, the measure of association that defines the strength of the dependence in the tails (τLfor the lower tail (depreciating exchange rates) and τU for the upper tail (appreciating exchange rates)) is almost always stronger for the lower tails than for the upper tail, in concordance with earlier studies (see for example Patton (2006); Boero et al. (2010); Azam (2014)).

5.2. Copula estimation on VAR residuals In the previous subsection, I studied the most general dependency structure. In this subsection I apply four-variate VAR-filtering, to filter out the linear dependency structure:

X1,t= α1,1X1,t−1+ . . . + α1,4X4,t−1+ X1,t (24) ..

.

X4,t= α4,1X1,t−1+ . . . + α4,4X4,t−1+ X4,t (25) The lag length l = 1 of this VAR(l) model is determined on basis of both the AIC and the SIC, see Table 2. The linear coefficients are shown in Tables 5 to 8 and a test on H0: ∀ai,j, j 6= i, = 0.

The dependency between the VAR residuals X1,t, . . . , X4,t is pairwise estimated. The results of the copula estimation on the VAR filtered residuals are shown in Table 3, Figures 7 to 10 and Figures 21 to 26. I conclude that the nonlinear dependency between the exchange rate returns has increased during the financial crisis and that for some exchange rate pairs (AUD/CAD, EUR/JPY, JPY/CAD) this increased dependency has persisted even after the financial crisis. Just as with the copulas that were estimated on the unfiltered log returns, we can see here that the lower tail dependency (τL_{) is in most cases stronger than}

upper tail dependency (τU_{), thus once again supporting}

the studies of Patton; Boero et al. and Azam. An interesting change compared to the results of the previous subsection is that the best fitting copula for the AUD/CAD exchange rate in the post-crisis (2009– 2016) time frame is now the Student’s t copula instead of the SJC copula. This implies that the VAR residuals are less asymmetrically distributed than the unfiltered log returns.

I will now restrict my findings to only the linear dependency that is significant (a = 0.05) significant

2_{Please note that I do not have used AIC or SIC/BIC as}

a selection criterion, as Patton remarks that rankings of the estimated copulas will be the same using LL, AIC or SIC estimation since the length of the time series is so large that the number of parameters to be estimated does not make a significant difference

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TABLE 1. Estimated copulas on unfiltered log returns Pair 1993-2007 2007-2009 2009-2016 t τL _τU _t _τL _τU _t _τL _τU AUD/EUR 0.1493 0.2165 0.1636 0.3470 0.4848 0.4940 0.2106 0.3941 0.2770 AUD/JPY 0.1493 0.2165 0.1636 0.0858 0.0000 0.0000 0.1086 0.0043 0.0442 AUD/CAD 0.0535 0.2011 0.1866 0.4373 0.5343 0.4933 0.3314 0.5396 0.4805 EUR/JPY 0.1312 0.2895 0.1982 0.1348 0.0607 0.0000 0.1285 0.1365 0.1085 EUR/CAD 0.0582 0.0713 0.0651 0.2911 0.4211 0.3262 0.1528 0.3308 0.2310 JPY/CAD 0.0039 0.0129 0.0079 0.0257 0.0000 0.0000 0.0598 0.0000 0.0001

In bold: SJC copula preferred above Student’s t copula

TABLE 2. Lag lengths for log returns time series Lags lX= lY AIC BIC HQ 1 -41.21660 -41.19359 -41.20859 2 -41.21406 -41.17264 -41.19965 3 -41.21321 -41.15339 -41.19240 4 -41.21435 -41.13612 -41.18714 5 -41.21353 -41.11690 -41.17992 6 -41.21220 -41.09716 -41.17218 7 -41.20846 -41.07501 -41.16203 8 -41.20684 -41.05499 -41.15402 9 -41.20688 -41.03662 -41.14765 10 -41.20600 -41.01733 -41.14036 11 -41.20619 -40.99912 -41.13415 12 -41.20431 -40.97883 -41.12587 13 -41.20246 -40.95857 -41.11761 14 -41.20138 -40.93908 -41.11013 15 -41.20138 -40.92068 -41.10373 16 -41.19864 -40.89953 -41.09458 17 -41.19476 -40.87724 -41.08430 18 -41.19333 -40.85741 -41.07647 19 -41.19334 -40.83901 -41.07007 20 -41.18990 -40.81716 -41.06023

(see Table 9). Table 5 to 8 show that JPY is always negatively caused by CAD and positively caused by AUD. This might indicate the role of JPY as a ‘anchor currency’ to AUD. On the other hand, it seems to be an substitute to EUR. First, I will look at P1 (1993–2007). In P1 we see that AUD and CAD are significantly linearly dependent on other time series. Table 6 shows that AUD follows EUR strongly, whilst CAD does not. On the other hand, CAD returns seem to move contrary to the returns in the other time series that are under consideration. This might signal that CAD is bought when other currencies depreciate and vice versa: CAD is a ‘safe-haven currency’. I will now study what happened during the 2007–2009 financial crisis. In P2 (2007–2009) CAD is the only time series that is significantly linearly dependent. We see that CAD moves together with EUR and contrary to JPY. Moving together with EUR instead of contrary to EUR is a difference from the behavior in P1. This might for instance indicate that CAD and EUR are both ‘safe-haven currencies’ in P2. Finally, I will study what happened with the linear dependency after the financial crisis. The only significantly caused time series in P3

(2009–2013) is AUD. AUD moves together with CAD, EUR and JPY. Earlier, in 1993–2009, AUD moved contrary to JPY.

In this subsection, I have studied the strictly nonlinear interdependency between the exchange rates by filtering out the linear dependency structure using a VAR(1) model. The filtered-out linear dependency structure has also been reviewed. In the next subsection, I apply a GARCH-DCC filter to control for any volatility that might falsely give the impression of interdependency, such as given by volatility spillovers.

5.3. Copula estimation on GARCH filtered VAR residuals

In the previous subsection I estimated a VAR model to remove as much of the linear dependency as possible. I studied the residuals which showed quite strong nonlinear interdependency (Table 3). In this section, I will fit a GARCH-DCC model on the VAR residuals to control for events that induce heteroskedasticity. Applying GARCH-DCC filtering will reduce the probability of obtaining spurious

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TABLE 3. Estimated copulas on four-variate VAR filtered residuals Pair 1993-2007 2007-2009 2009-2016 t τL _τU _t _τL _τU _t _τL _τU AUD/EUR 0.1777 0.3093 0.2378 0.3411 0.4789 0.4929 0.0220 0.0201 0.0001 AUD/JPY 0.1433 0.2109 0.1066 0.0727 0.0000 0.0000 0.0685 0.0558 0.0003 AUD/CAD 0.0388 0.1954 0.2064 0.4417 0.5408 0.4853 0.0584 0.0823 0.0873 EUR/JPY 0.0955 0.2367 0.1532 0.1300 0.0503 0.0000 0.1281 0.2929 0.1498 EUR/CAD 0.0456 0.1184 0.1106 0.2648 0.4215 0.3153 0.0080 0.0000 0.0000 JPY/CAD 0.0007 0.0391 0.0353 0.0172 0.0000 0.0000 0.0176 0.0000 0.0000

TABLE 4. Estimated copulas on GARCH-DCC filtered VAR residuals

Pair 1993-2007 2007-2009 2009-2016 t τL _τU _t _τL _τU _t _τL _τU AUD/EUR 0.0011 0.0000 0.0000 0.0078 0.0140 0.0000 0.0005 0.0000 0.0000 AUD/JPY 0.0088 0.0000 0.0000 0.0974 0.0012 0.0000 0.0042 0.0000 0.0000 AUD/CAD 0.0000 0.0000 0.0000 0.0906 0.0023 0.0000 0.0003 0.0000 0.0000 EUR/JPY 0.0009 0.0000 0.0000 0.0049 0.0033 0.0000 0.0002 0.0000 0.0000 EUR/CAD 0.0019 0.0000 0.0000 0.0228 0.0015 0.0000 0.0002 0.0000 0.0000 JPY/CAD 0.0000 0.0000 0.0000 0.0116 0.0010 0.0000 0.0001 0.0000 0.0000

In bold: SJC copula preferred above Student’s t copula

significant results due to volatility.

The results of the copula estimation on the GARCH-DCC filtered VAR residuals are shown in Table 4, Figures 11 to 14 and Figures 27 to 32.

When compared to Table 3, Table 4 clearly shows that conditional volatility produces a lot of the perceived nonlinear dependency. As in the previous sections, I estimated both a t copula and a SJC copula. The estimates of the measures of assocation in Table 4 show that for GARCH-DCC filtered VAR residuals, upper tail dependency is virtually non-existent for the SJC copula (τU _{= 0). It is interesting to note that}

the lower tail dependency measure of assocation, τL_{, is}

for P1 (1993–2007) equal to zero for all pairs, and also for P3 (2009–2016). However, during the 2007–2009 financial crisis it shows clear estimates for lower tail dependency. This supports earlier conclusions made by Azam (2014) and Boero et al. (2010). The estimates for the measure of association for the t copula show that the dependency increases strongly during the financial crisis. Overall, the AUD/JPY pair seems to have the strongest interdependency. This might be explained by currency carrying: AUD is a ‘high-beta currency’, a relatively volatile currency, whilst JPY is a currency with a historically low interest. Thus: when AUD is appreciating, extra borrowing of JPY might occur as the result of ‘carry trade’. This might cause the JPY to appreciate as well, explaining the co-movement between AUD and JPY. Another interesting result is that after the financial crisis EUR did not exhibit many strong co-movement with the other studied return time series, whilst before the financial crisis EUR showed strong co-movement with CAD and AUD. In conclusion, EUR has lost some nonlinear interdependency, while JPY became more nonlinearly dependent on other return time series. I compare this results with the Granger causality tests results from Sipin (2016) as shown in Table 10,

11 and 12. Table 12 shows that strictly nonlinear interdependency, after controlling for conditional heteroskedasticity, is not very common. This supports the conclusions that I based on the copula estimates.

5.4. Summary

In this section I have studied the dependency structure between exchange rate returns by estimating t copulas and SJC copulas. First, I studied the most general dependency structure of exchange rate returns by estimating on the returns data. Then, I differentiated between linear dependency (coefficients of VAR(1) model) and nonlinear dependency (VAR residuals). Finally, I controlled for conditional volatility by applying a GARCH-DCC(1,1) filtering.

I compared the estimated copulas for the time periods before the financial crisis (1993–2007), during the financial crisis (2007–2009) and after the financial crisis (2009–2016). The estimated copulas suggest at first that there is a strong nonlinear interdependency between the studied exchange rate returs. However, after controlling for conditional volatility, most of this interdependency vanishes. Using Granger causality tests however, I found some significant (a = 0.05) nonlinear relationships between exchange rate returns.

6. CONCLUSIONS

In this thesis I have studied what the dependency structure between looks like, and if this structure has been stable over time, for the log returns of four of the most traded exchange rates in the world: the Australian dollar (AUD), Canadian dollar (CAD), Euro (EUR) and Japanese yen (JPY) against the US dollar (USD). In particular, I studied if during and after the financial crisis (2007–2009) the interdependency between the log returns of exchange rates changed.

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I find that the interdependency between the exchange rates increased during the financial crisis, and that this effect - somewhat less strong - persists after the financial crisis. Based on the results presented in this thesis, I conclude that strictly nonlinear relations (i.e., relations that persist after VAR and GARCH-DCC filtering) are not common. Before the financial crisis, there was no statistical evidence for nonlinear causal relations (Sipin, 2016). For AUD/CAD, EUR/JPY and JPY/CAD the dependency as estimated in the form of the copula measure of association increases, whilst AUD/JPY becomes less dependent. Therefore, I conclude that there might be some kind of substitution of ‘anchor currency’ where EUR and CAD substitute the AUD against the JPY. I also show that the nonlinear returns, in concordance with earlier studies, indeed show asymmetry in the tails of their (estimated) distributions. However, when selecting copulas using maximum likelihood estimation, the symmetric Student’s t copula is almost always preferred to the asymmetric Symmetrized Joe-Clayton copula.

Based on both the copula estimates and the Granger causality test results I conclude that the studied foreign exchange rates have become more interdependent during and following the financial crisis of 2007–2009.

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TABLE 5. Four-variate VAR(1) coefficients, 1993–2016

α AU Dt−1 CADt−1 EU Rt−1 J P Yt−1 Trend

AU Dt -6.221422e-05 -6.254640e-02 9.418235e-02 2.306479e-02 -1.342586e-02 1.456295e-08

CADt -4.192192e-05 -8.615429e-03 7.830324e-03 -2.208036e-03 -1.315082e-02 1.266241e-08

EU Rt -6.969701e-05 -3.212755e-02 1.466412e-02 2.915196e-02 -2.677780e-02 2.357973e-08

J P Yt -4.732149e-05 1.959560e-02 -3.025338e-02 -1.224360e-02 4.577141e-03 1.469088e-08

AU Dt 1.218469e-04 2.416821e-04 7.696146e-02 1.550150e-0 -1.034468e-02 -1.002768e-07

CADt 2.006178e-04 -7.548346e-03 2.692676e-03 -1.315830e-02 -2.977599e-03 -1.483939e-07

EU Rt 9.158271e-05 -2.020954e-02 2.561304e-02 2.332733e-02 -3.590723e-02 -8.157119e-08

J P Yt -1.263991e-05 3.086057e-02 -3.365358e-02 -1.696954e-02 2.851910e-02 1.478142e-08

AU Dt 2.586469e-04 -2.982204e-01 1.953279e-01 2.654613e-01 -2.653488e-01 -1.047145e-06

CADt 1.595426e-04 -5.768780e-02 5.311115e-02 1.044684e-01 -1.238593e-01 -6.274798e-07

EU Rt -3.628519e-04 -6.693039e-02 3.684824e-02 1.357127e-01 -8.850089e-02 1.115530e-06

J P Yt -0.000815706 0.017211218 -0.065825921 0.069698896 -0.047838720 0.000001607

AU Dt -3.512491e-04 -2.345794e-02 9.612199e-02 4.093066e-02 9.535727e-03 5.504593e-07

CADt 7.201107e-05 -9.996717e-03 5.542286e-02 1.870651e-02 3.387024e-04 1.318684e-08

EU Rt -3.648198e-04 -2.414089e-02 1.728850e-02 3.745046e-02 -3.188878e-02 5.706781e-07

J P Yt -4.977622e-06 3.979285e-02 -3.219785e-02 -1.781105e-02 5.121297e-02 -1.808789e-08

TABLE 9. Granger causality test, linear dependency

All P1 P2 P3

AUD 0.1508 0.0000 0.1868 0.0053

CAD 0.0918 0.0022 0.0121 0.9158

EUR 0.1601 0.0632 0.4384 0.2791

JPY 0.0002 0.1155 0.0688 0.3552

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TABLE 10. Unfiltered Granger causality results (pairwise) Pair X Y X → Y All P1 P2 P3 AUD CAD 0.00003 0.00443 0.00328 0.00062 AUD EUR 0.92513 0.00018 0.00205 0.00020 AUD JPY 0.00009 0.00040 0.00424 0.02214 CAD AUD 0.00000 0.00014 0.00060 0.00038 CAD EUR 0.49275 0.00284 0.00221 0.00007 CAD JPY 0.01024 0.07169 0.00218 0.01371 EUR AUD 0.85774 0.01258 0.00456 0.00266 EUR CAD 0.54751 0.05983 0.01125 0.00053 EUR JPY 0.89423 0.01379 0.00207 0.03817 JPY AUD 0.00209 0.01231 0.08777 0.04846 JPY CAD 0.04512 0.05289 0.01802 0.03657 JPY EUR 0.73543 0.03846 0.01275 0.05960

TABLE 11. VAR filtered Granger causality results (four-variate) Pair X Y X → Y All P1 P2 P3 AUD CAD 0.00002 0.00500 0.00118 0.00096 AUD EUR 0.00000 0.00016 0.00154 0.00004 AUD JPY 0.00006 0.00067 0.02271 0.00008 CAD AUD 0.00000 0.00014 0.00042 0.00053 CAD EUR 0.00000 0.00224 0.00191 0.00219 CAD JPY 0.00908 0.07427 0.00747 0.00004 EUR AUD 0.00059 0.02078 0.00974 0.00747 EUR CAD 0.00188 0.04385 0.00574 0.02117 EUR JPY 0.00106 0.00900 0.00339 0.00090 JPY AUD 0.00193 0.01536 0.19154 0.00844 JPY CAD 0.05570 0.04348 0.01817 0.00288 JPY EUR 0.00311 0.04507 0.04717 0.00767

TABLE 12. GARCH-DCC(1,1) filtered VAR(1)-filtered residuals Pair X Y X → Y All P1 P2 P3 AUD CAD 0.13285 0.08937 0.60197 0.20731 AUD EUR 0.01834 0.05732 0.58798 0.00681 AUD JPY 0.09450 0.09518 0.71500 0.10772 CAD AUD 0.03554 0.16251 0.26868 0.10457 CAD EUR 0.00582 0.31950 0.45038 0.17064 CAD JPY 0.21891 0.13782 0.37105 0.00966 EUR AUD 0.08203 0.23204 0.03627 0.25455 EUR CAD 0.48304 0.30224 0.63431 0.65356 EUR JPY 0.08926 0.12202 0.13494 0.04101 JPY AUD 0.06339 0.26385 0.25169 0.40974 JPY CAD 0.13521 0.16315 0.32139 0.60579 JPY EUR 0.43166 0.85059 0.75514 0.43347

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FIGURE 7. Correlation of VAR-filtered log return residuals, 1993–2016

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FIGURE 11. Correlation of GARCH-filtered VAR residuals, 1993–2016

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FIGURE 15. Copulas on unfiltered log returns, 1993–2007 (i)

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FIGURE 16. Copulas on unfiltered log returns, 1993–2007 (ii)

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FIGURE 27. Copulas on GARCH filtered VAR residuals, 1993–2007 (i)

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FIGURE 28. Copulas on GARCH filtered VAR residuals, 1993–2007 (ii)

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