The nonlinear dynamic relationship of exchange rates : an analysis of the effect of the financial crisis using Granger causality testing

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An analysis of the effect of the financial crisis using

Granger causality testing

Kevin L. Sipin

Student No. 10523006 Supervisor: Hao Fang

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

Bachelor’s Thesis for the Bachelor’s of Science in Econometrics

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. General dependency is explored between the raw log returns. The nonlinear dependency is studied through causality testing using the consistent Diks-Panchenko test statistic on the pairwise VAR filtered residuals. Finally, the GARCH-DCC filtered VAR residuals are tested to account for volatility spillovers and other events that induce conditional heteroskedasticity. After VAR filtering but before GARCH-DCC filtering, I find significant uni- and bi-directional causal linear and nonlinear relationships between allmost all exchange rate log returns. However, after GARCH-DCC filtering most of these effects vanish, which implies that they are solely the result of volatility inducing events rather than of nonlinear dependency between exchange rates. Finally, this study also suggests that the interdependency between exchange rates has grown stronger during and after the

financial crisis. Received June 29th, 2016

FOREWORD AND ACKNOWLEDGEMENT This thesis is the result of three years of studying Econo-metrics 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 thesis research has therefore been the most

enjoyable assignment 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.

This expedition would not have been quite so much fun, had I not had a supervisor that gave me this freedom. In that regard, I would in particular like to thank Hao Fang for helping me to understand the core concepts that make up the foundation of this thesis, and for his proactive support during my research and writing. His remarks have helped me to really understand the wonderful field of time series dependency analysis. I would also like to thank Umut Can, for introducing me to the world of copulas and thus broadening my view on the concept of interdependency between financial time series. 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 Granger causality . . . 3 3.2 Estimating distributions . . . 4 3.3 Definitions of copulas . . . 5 3.4 Families and properties of copulas . . . 6 3.5 SJC copula . . . 6 3.6 Summary . . . 7

4 Research design 7

4.1 Methodology . . . 7 4.2 Data . . . 7

5 Results and analysis 7

5.1 Causality testing on unfiltered log returns 8 5.2 Causality testing for linear dependency . 8 5.3 Causality testing on VAR-filtered log

returns residuals . . . 9 5.4 Causality testing on GARCH-filtered

VAR residuals . . . 11 5.5 Copula results . . . 12 5.6 Summary . . . 13 6 Conclusions 13 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 changes over the course of the financial crisis. It has been suggested by earlier studies that the 2007– 2009 financial crisis, among others (e.g. the Asian financial crisis), has altered the interdependency between exchange rates.

In this thesis I apply the Diks-Panchenko test to detect Granger causality between the exchange rate time series. I apply this test on, consecutively: raw log returns, VAR filtered residuals and GARCH-DCC filtered VAR residuals. Thus, I explore the first the general dependency structure and then the strictly 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 accompanied 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. 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; Koutmos & Booth, 1995). In addition, returns in financial time series tend to be non-normally distributed (Okimoto, 2008; Patton, 2006; Jondeau & Rockinger, 2006). Research also shows that the dependence structure’s parameters (i.e. measure(s) of association or correlation coefficients) might not be stable over time. For example, there is evidence that the Euro/Deutsche mark versus US Dollar exchange rate has a different dependence structure in the pre-Euro against the post-pre-Euro period (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 tested with Granger causality tests. In the conventional Granger causality tests, a parametric and linear time series model for the conditional mean is assumed. This approach simplifies the testing but has some major drawbacks, namely the assumption on linearity and the limitation to the conditional mean – whilst there might be other variables as well that have a nonlinear influence. For years, the most used test for Granger causality has been a test based on the Hiemstra-Jones test statistic (Hiemstra & Jones, 1994). This test has however its shortcomings, as it is shown to be biased and inconsistent (Diks & Panchenko, 2005, 2006). Baek and Brock (1992) also note that the conventional parametric linear tests for Granger causality have a

low power when confronted with nonlinear dynamics. This obviously poses a problem for the empirical testing of dependence between exchange rates, for all the reason mentioned above (asymmetry, fat tails, etc.). Fortunately, a solution is offered in Diks and Panchenko (2006) which is demonstrated in Bekiros and Diks (2008).

This particular study aims to test for the general dependency between exchange rates, which could include nonlinear dependency. Therefore the goal of this paper is to apply the Diks-Panchenko test statistic as proposed in Diks and Panchenko. Another important question is if the structure of the dependency between foreign exchange rates has changed over the course of years or important events, see for example Bekiros and Diks (2008). In this case, I ask if the financial crisis (2007–2009) has significantly altered the dependency structure of the given time series. The central research question then becomes: which log returns of selected exchange rates are Granger causing other log returns, and is this structure stable over time?

To answer this question, a general theoretical and conceptual framework of Granger causality testing is given, based on Diks and Panchenko (2006); Bekiros and Diks (2008); Boero et al. (2010) and Azam (2014). The returns of the EUR/USD, JPY/USD, AUD/USD and CAD/USD exchange rates are filtered by applying a VAR filtering, following Yang (2005) and Chang (2011). Finally, the log returns of the exchange rates are tested with the DP test for nonlinear Granger causality over multiple time periods (see also Bekiros and Diks; Boero et al.; Azam): before, during and after the financial crisis (2007–2009).

This thesis is organized as follows. Section 2 introduces the concepts of Granger causality testing and VAR filtering. Section 3 explains the design of this research, the used methodology and the data. Section 4 presents and analyzes the results. Section 5 summarizes and concludes.

3. THEORETICAL FRAMEWORK

In this section I start with a review the theory of Granger causality and the methods for detection and testing. I also give a concise overview of the ARMA-GARCH framework and VAR filtering. Then finish with the problem of operationalizing these methods to properly determine the returns of exchange rates and in particular the nonlinear dynamics in these returns. 3.1. Granger causality

3.1.1. Definition

Granger (1969) 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

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{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

(1) for some k ≥ 1 and where {(Xt}, {Yt)}, t ∈ Z is a

strictly 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.

3.1.2. Causality detection and testing

In the previous section 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 (1969) 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. Our null

hypothesis then becomes

H0: Yt+1| (FX,t, FY,t) ∼ Yt+1|FY,t (2)

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) (3)

Diks and Panchenko (2006) show, using a stationary bootstrap approach as developed by Politis and Romano (1994), 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)) (5)

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=iIijW 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) (6)

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.

3.2. Estimating distributions

I give a summary of the reduced-form vector (VAR) and ARMA-GARCH filtering and show how these methods can be used to estimate the empirical distributions of the time series.

3.2.1. 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 (7)

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 (8)

Yt= C(l)Xt+ D(l)Yt+ Y,t (9)

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

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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.

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.

3.2.2. 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 (10) σ_t2= α0+ α1r2t−1+ βiσ2t−1 (11)

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.

3.2.3. 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 Engle (2000). This model can be summarized as:

ρ12,t=

Et−1(r1,t, r2,t)

q

Et−1(r1,t2 )Et−1(r22,t)

with r1, r2 two random variables with mean zero.

Now Engle 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) (12)

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 Engle eventually derives the Dynamic Conditional Correlation estimator that is 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.

3.2.4. Estimating empirical distributions

I can use the consistent estimates of the errors in the ARMA(p,q)-GARCH(1,1) and/or GARCH-DCC(1,1) model to estimate the marginal distributions (Boero et al., 2010; Chen & Fan, 2006; Genest & Ghoudi, 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).

To prevent confusion, I would like to point out that these variables are not the same as the time series X, Y and FX,t, FY,t in the beginning of this section.

Following Genest and Ghoudi 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. Both ˆFX(x) and

ˆ

FY(y) will be approximately Unif (0,1) distributed.

3.3. Definitions of copulas

In this section I start with a review 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 University of Amsterdam - Department of Quantitative Economics

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the problem of selecting and building a copula function that has the necessary properties. This copula will then be studied more in depth.

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

C measuring their dependency, and their margins F1, ..., F2specifying their individual characteristics (fat

tails, skewness, etc.). Formally given as

H(y1, ..., yp) = C(F1(y1), ..., Fp(yp)) (13)

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) (14)

where uj = F (yj), which is obtained through

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

In other words, I find the cumulative distribution function (cdf )

FXY(x, y) = C(FX(x), FY(y)) (15)

and the probability distribution function (pdf )

fXY(x, y) = fX(x) · fY(y) · c(FX(x), FY(y)) (16)

which 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). 3.4. Families and properties of copulas

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 and zero tail dependency. In a bivariate case, the Gaussian copula is given by

Cg(u, v|ρ) = Φg(Φ−1(u), Φ−1(v); ρ) = Z Φ−1(u) −∞ Z Φ−1(v) −∞ 1 2π(1 − ρ2₎2₎1₂ × −(s 2_{− 2ρst + t}2₎ 2(1 − ρ2₎ dsdt (17)

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 SJC copula are described in depth.

3.5. SJC copula

As mentioned in the previous subsection, 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}−1γ₎1κ (18)

where the upper and lower tail dependency are given by τU ∈ (0, 1) respectively τL_{∈ (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 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|tauU, τL) = 0.5(CJ C(u, v|tauU, τL)

+ CJ C(1 − u, 1 − v|tauL, τU)

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and this copula is used in Sipin and therefore also in this study to capture the dependence between foreign exchange rates.

3.6. Summary

In this section I have first introduced the definitions of the concept Granger causality. Then I gave a quick review of some methods for detection and testing. I showed how one can test for nonlinear Granger causality on the VAR and ARMA-GARCH and GARCH-DCC filtered residuals. I then gave a brief review of some methods that can be used to estimate empirical distribution, so that all the methods can be applied in order to empirically test for the existence of Granger causality in time series. Finally, I introduced copulas and some of their properties.

4. RESEARCH DESIGN

4.1. Methodology

The methodology of the empirical research in this study is primarily derived from Diks and Panchenko (2006) and Bekiros and Diks (2008). 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 the financial crisis (1993–2007, 3459 observations)

• during the financial crisis (2007–2009, 472 observa-tions)

• after the financial crisis (2009–2016, 1792 observa-tions)

The main goal is to determine the dependence structure between the four exchange rates in each of the three time slots using DP (Diks & Panchenko, 2006) testing, and to compare the results to see if there is any change in the dependency structure, as is also done in Bekiros and Diks (2008).

To study the dependence structure, I will apply the three-step framework as in Bekiros and Diks (2008), with a few alterations. First, I apply the nonparametric Diks-Panchenko test to explore the dependence structure between unfiltered exchange rates. Second, I filter pairwise and in four-variate form with a VAR model, and I use the DP test to test for dependence structure in the residuals (i.e. I test for strictly nonlinear dependence). Finally, I use GARCH filtering to control for conditional heteroskedasticity. These results are compared with Sipin (2016), where a copula approach is used to analyze the dependence structure on the same data set, to see if another

functional form of nonlinear dependence structure is detected when another methodology is used.

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 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 ??. The time series of the log returns of this data set are plotted in figure 5. Also, the scatter plots portraying the correlation between exchange rates and their distribution are shown in figure 6. 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 7. The log returns time series are shown in figure 8. The scatter plots showing the correlation of log returns is shown in figure 9.

5. RESULTS AND ANALYSIS

In this section I present the main results of this thesis: the results of the Diks and Panchenko (2006) Granger causality test on exchange rate log returns. First I present the results of the DP test on the unfiltered log returns, to explore the linear dynamic linkages between the exchange rates. In this subsection, I also study the linear dependency between exchange rates. Second, I apply both bivariate (pairwise) and four-variate VAR(1) filtering on the log returns, and the residuals are tested (pairwise) for nonlinear causality. I conclude this section with a test on the GARCH-DCC(1,1) filtered VAR residuals, which are controlled for heteroskedasticity-inducing events, to study the strictly nonlinear causality.

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5.1. Causality testing on unfiltered log returns I start this section with an exploration of the general linkages between exchange rates. My null hypothesis is: H0: Yt+1| (FX,t, FY,t) ∼ Yt+1|FY,t (20)

Or, in words: under the null hypothesis X does not Granger cause Y . I test this by using the DP test, based on a procedure programmed in C-code by Panchenko. Following the guidelines presented in Diks and Panchenko (2006), I choose the bandwidth as = Cn−β with C > 0, β ∈ 1₄,1₃, since the DP test is in that case consistent. The optimal bandwidth does however depend on the correlation of the conditional concentrations of the random variables that generate the time series. This will however take extra time to compute and does only matter for efficiency, not for consistency. I therefore decide to follow Diks and Panchenko and simply use C = 8 and β = 2

7. It might

however occur that for smaller n this leads to absurdly large bandwidths, so in order to account for this effect I truncate the bandwidth in some cases, as Diks and Panchenko do as well: n= max Cn−27, 1.5 (21) This leads to the bandwidths:

• 1993–2016 (n = 5813): = 0.67 • 1993–2007 (n = 3549): = 0.77 • 2007–2009 (n = 472): = 1.38 • 2009–2016 (n = 1792): = 0.94

This ensures that the size α of the test is close to 0.05.

The results of the DP test on the unfiltered log returns are presented in Table 1 (p-values of H0: ’X

is not Granger causing Y’). Based on the results in Table 1, I conclude the following: we see a strong raise in interdependence between exchange rates during and following the financial crisis. Almost all raw returns seem to exhibit some degree of general causality. While over the entire 1993–2016 interval EUR is not caused by any other currency, today EUR is only not caused by JPY. Another interesting result is the causality from JPY to AUD and CAD following the financial crisis.

In this subsection, I have described the most general dependency structure between AUD/USD, CAD/USD, EUR/USD and JPY/USD exchange rates. I have shown that although in the period 1993–2007 the general dependency between exchange rates was already (at least uni-directional) very strong between all currencies except CAD/USD and JPY/USD, this dependency has grown even stronger after the financial crisis. I find statistical evidence that all currencies in this study are Granger causing eachother bi-directional, except EUR which only uni-directional causes JPY. The results of this section have been summarized in Figure 1.

FIGURE 1. Dependency structure of unfiltered log returns

In the next section I explore the nonlinear dynamic linkages, by first removing the autoregressive linear effects through a VAR-filtering procedure and then testing for causality on the residuals.

5.2. Causality testing for linear dependency In this subsection, I estimate a four-variate VAR(1) model:

Yt= α + AYt−1+ t (22)

with α being a vector of intercepts

α =     α1 α2 α3 α4    

and A a matrix of coefficients

A =     a1,1 a1,2 a1,3 a1,4 a2,1 a2,2 a2,3 a2,4 a3,1 a3,2 a3,3 a3,4 a4,1 a4,2 a4,3 a4,4    

and Y a vector of variables

Yt=     Y1,t Y2,t Y3,t Y4,t    

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TABLE 1. 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

p-values are reported for H0: X is not Granger causing Y

and a vector of errors

t=     1,t 2,t 3,t 4,t    

Now I test if the exchange rate log returns Yi are

linear independent, by testing for the exogeneity of the variables Yj, j 6= i. This is done by applying a F -test

on the parameter restrictions that all Yj, j 6= i are equal

to zero. The data is shown in Tables 2 to 5. The results are presented in Table 6.

Based on the results in Table 6, I conclude that linear dependency is present for AUD. I.e.: AUD depends linearly and significantly on the other three exhange rates. On the other hand, during the financial crisis CAD was heavily linearly dependent on the other exchange rates.

5.3. Causality testing on VAR-filtered log returns residuals

In the previous subsection both the general dependency structure and the linear dependency structure between the exchange rates was studied. In this subsection I turn to the problem of detecting the nonlinear dependence between exchange rates.

I filter out the linear effects by fitting the log returns on a vector autoregressive (VAR) model as given by Equation (7). As lag order of the VAR(l) model I have selected l = 1, based on the Hannan-Quinn criterion, Akaike information criterion and Schwarz information criterion (SIC, also called Bayesian information criterion, BIC), see also Table 7.

In this subsection, I first discuss the dependency structure between the bivariate VAR(1)-filtered resid-uals. Then I explore the dependence between the four-variate VAR(1)-filtered residuals.

5.3.1. Pairwise (bivariate) VAR-filtering

Based on the BIC, AIC and HQ criterions, a lag length of lX = lY = 1 is selected. That means that the general

form of the VAR(l) model as presented in Equation (8) and Equation (9) becomes:

Xt= α1,1Xt−1+ α1,2Yt−1+ X,t (23)

Yt= α2,1Xt−1+ α2,2Yt−1+ Y,t (24)

Using R I apply a VAR(1) filtering procedure to the unfiltered time series. The resulting residuals time series (Xand Y) are pairwise tested against eachother.

The results of the DP test on these pairwise (bivariate) VAR(1)-filtered residuals are presented in Table 8 and Figure 2 (p-values of H0: ’X is not Granger causing Y’).

Based on the results in Table 8, I conclude that there is very strong statistical evidence that all the studied exchange rates exhibit bi-directional causality. This nonlinear dependency has grown stronger over the financial crisis, because before the financial crisis CAD was not Granger causing JPY and EUR was not Granger causing CAD. During the financial crisis, JPY stops Granger causing AUD but JPY is increasingly strongly caused by EUR and CAD, which might signal some sort of substitution effect between AUD and JPY.

5.3.2. Four-variate VAR-filtering

The results of the previous paragraphs and subsection suggest that there are many strongly significant general causal relations between the exchange rates. I am now going to test if these relations are nonlinear. Therefore I again apply a VAR(1) filter with all exchange rates, as shown in the following model:

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TABLE 2. 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 6. 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

p-values are reported for H0: ∀ai,j, j 6= i, = 0. 95 percent confidence interval.

TABLE 7. 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

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TABLE 8. VAR filtered Granger causality results (pairwise) Pair X Y X → Y All P1 P2 P3 AUD CAD 0.00001 0.00345 0.00312 0.00112 AUD EUR 0.00000 0.00015 0.00151 0.00010 AUD JPY 0.00003 0.00041 0.01047 0.00011 CAD AUD 0.00000 0.00012 0.00045 0.00067 CAD EUR 0.00000 0.00188 0.00270 0.00213 CAD JPY 0.00577 0.09443 0.00604 0.00003 EUR AUD 0.00023 0.01299 0.00497 0.00813 EUR CAD 0.00152 0.05669 0.00300 0.03721 EUR JPY 0.00050 0.01342 0.00297 0.00091 JPY AUD 0.00143 0.00830 0.08141 0.00558 JPY CAD 0.04112 0.04971 0.00935 0.00220 JPY EUR 0.00274 0.04577 0.01155 0.00970

FIGURE 2. Depedency structure of bivariate VAR-filtered residuals AU Dt= α1,1AU Dt−1+ α1,2CADt−1 + α1,3EU Rt−1+ α1,4J P Yt−1 + AU D,t (25) CADt= α2,1AU Dt−1+ α2,2CADt−1 + α2,3EU Rt−1+ α2,4J P Yt−1 + CAD,t (26) EU Rt= α3,1AU Dt−1+ α3,2CADt−1 + α3,3EU Rt−1+ α3,4J P Yt−1 + EU R,t (27) J P Yt= α4,1AU Dt−1+ α4,2CADt−1 + α4,3EU Rt−1+ α4,4J P Yt−1 + J P Y,t (28)

The resulting residuals (AU D, CAD, EU R, J P Y) are

now tested against eachother. The results (p-values of H0: ’X is not Granger causing Y’) are presented in

Table 9 and Figure 3.

Table 9 supports my earlier findings. The results suggest an even stronger interdependency between the exchange rates. All the causal relations, except CAD→JPY, are significant at at with 95% confidence. Even EUR causing CAD in P1 (1993–2007) and JPY causing AUD in P2 (2007–2009) are now significant. Please note that this might be caused by volatility or other mechanisms. Note, however, that in both cases the interdependence between the exchange rates is significantly higher in P3 than in P1. That is, it seems that the after the financial crisis exchange rates have become significantly more interdependent. The dependency structure is summarized in Figure 3.

In this subsection, I have studied the nonlinear dependency by filtering out the linear dependence through the use of a VAR model. Using the DP test on the resulting time series suggests that there is a significant amount of strictly nonlinear interdependence between the exchange rates, and that this interdependence has grown stronger over the financial crisis. In the next subsection, I apply a GARCH-filter on the VAR residuals that are acquired in this section. In that way I correct for any exchange rate volatility or other mechanisms that might have falsely given the impression of exchange rate interdependence.

5.4. Causality testing on GARCH-filtered

VAR residuals

In the previous subsection I studied the nonlinear dependency structure of the exchange rates. In this subsection I am going to control the data for volatility spillovers and other co-movement mechanisms by using a GARCH-filter. The statistical evidence for interdependency will be strongly reduced if it comes primarily from spillovers and comparable mechanisms. University of Amsterdam - Department of Quantitative Economics

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TABLE 9. 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

FIGURE 3. Dependency structure of four-variate VAR-filtered residuals

In this study, I use a GARCH-DCC(1,1) model (orders derived from Bekiros and Diks (2008)).

Table 10 and Figure 15, 16, 17 and 18 show the results of the DP test and some scatterplots of the GARCH-filtered four-variate VAR residuals. A comparison of Table 9 and Table 10 show that almost all the significant interrelationships have ceased to exist. Based on the results of Table 10 I conclude that volatility effects are causing many relationships to look significant, whilst in fact there is no statistical evidence for their existence. So, the significant nonlinear causality is not persistent after GARCH-DCC filtering.

FIGURE 4. Depedency structure of GARCH-filtered VAR-filtered residuals

5.5. Copula results

In the previous subsection I have tested for the general, linear and nonlinear causality in the log returns of exchange rates. In this final section, I compare the results to some copula results as estimated on the same data set by Sipin. These results are presented in Table 11 and Figures 19 to 30.

The first conclusion that I draw based on Table 11 is that during the financial crisis (2007–2009) the nonlinear interdependency between exchange rates has increased: the measure of association of the Student’s t copula has increased for all pairs. Also, during the financial crisis the Symmetrized Joe-Clayton copula is estimated to have a lower tail dependence, whilst upper

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TABLE 10. 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

TABLE 11. 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

tail dependence is absent. This is fully in concordance with earlier findings of Azam; Boero et al. and Patton, among others.

I am in particular interested in how the estimated copulas of AUD/EUR, JPY/CAD and EUR/JPY have changed over the financial crisis. Sipin finds that the measure of assocation for the Student’s t copula for AUD/EUR increases during the crisis and then decreases, as is shown in Table 6: 0.0011, 0.0078, 0.0005. JPY/CAD shows a similar yet even more extreme pattern: 0.0000, 0.0116, 0.0001. Finally, EUR/JPY also shows the same pattern: 0.0009, 0.0049, 0.0002. In conclusion, there is not many strictly nonlinear dependency left after controlling for conditional heteroskedasticity, however we can clearly see in the estimated τL shape parameter of the SJC copula that there is evidence that during the financial crisis the (lower tail) dependency between exchange rates has increased.

5.6. Summary

In this section I have first studied the general dependency structure of exchange rates, based on the raw log returns. Next, I removed the linear dependency through the use of a VAR filter. Then, I controlled the VAR residuals with a GARCH filter to account for volatility spillovers and other events that induce conditional heteroskedasticity. Finally, I compared these results to some the results of copula estimation

on the same data.

The data suggests at first that a strong nonlinear interdependency exists between the studied exchange rates, that is also in most cases bi-directional. The statistical evidence also suggests that the interdependency has grown stronger over the financial crisis. However, after controlling for conditional volatility, the interdependency between exchange rates has indeed significantly increased, but way less strong than the VAR residuals suggest: prior to the financial crisis there was no significant strictly nonlinear causality (after filtering for conditional volatility). However, during the financial crisis we see that EUR causes AUD. This effect vanishes after the financial crisis and is substituted by AUD causing EUR, CAD causing JPY and EUR causing JPY. I therefore conclude that most nonlinear dependency has to be caused by volatilty inducing events (e.g. spillovers), yet that there are some currencies that show (strictly) nonlinear interdependency.

6. CONCLUSIONS

In this thesis I have investigated which exchange rates exhibit causal relations and if the strength of these relations is stable over time, for four of the most traded currencies in the world: the Australian dollar (AUD), the Canadian dollar (CAD), the Euro (EUR) and the Japanese yen (JPY). This thesis shows first and foremost that the nonlinear linkages between University of Amsterdam - Department of Quantitative Economics

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these foreign exchange rates exists, in the current day between AUD and EUR, EUR and JPY and CAD and JPY. Additionally, the statistical evidence in this study suggests that all the foreign exchange rates have become more interdependent during and after the financial crisis. That is, during the financial crisis the interdependency grew strongly. There is some evidence that suggests that the Canadian and Australian dollar are each others substitute in times of great volatility.

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. 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. For the AUD/EUR pair, this relationship is the strongest. This and the other relationships loose some strength after the financial crisis, but remain stronger than they were before the financial crisis. Thus, 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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FIGURE 6. Log returns time series

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FIGURE 8. Exchange rates against dollar, 1993–2016

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FIGURE 10. Correlation of log returns, 1993–2016

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

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FIGURE 15. Correlation of GARCH-filtered VAR residuals, 1993–2016

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FIGURE 20. Copulas on VAR residuals, 1993–2007 (ii)

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FIGURE 25. Copulas on GARCH filtered VAR residuals, 1993–2007 (i)

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FIGURE 26. Copulas on GARCH filtered VAR residuals, 1993–2007 (ii)

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