Conditional dependence structure between oil prices and exchange rates: A copula-GARCH approach

Journal of International Money and Finance 32 (2013) 719–738

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Journal of International Moneyand Finance

journal homepage: www.elsevier .com/locate/ j imf

Conditional dependence structure between oil prices andexchange rates: A copula-GARCH approach

Riadh Aloui a, Mohamed Safouane Ben Aïssa a, Duc Khuong Nguyen b,*

a LAREQUAD & FSEGT, University of Tunis El Manar, B.P 248, El Manar II, 2092 Tunis, Tunisiab ISC Paris School of Management, Department of Finance and Information Systems, 22, Boulevard du Fort de Vaux,75848 Paris Cedex 17, France

JEL classification:C58F37G17Q43

Keywords:CopulasDependence measuresCrude oil priceU.S. dollar exchange ratesCML method

* Corresponding author. Tel.: þ33 140 539 999;E-mail addresses: [email protected] (R. Alo

(D.K. Nguyen).

0261-5606/$ – see front matter � 2012 Elsevier Lthttp://dx.doi.org/10.1016/j.jimonfin.2012.06.006

a b s t r a c t

We study the conditional dependence structure between crude oilprices and U.S. dollar exchange rates using a copula-GARCHapproach. Various copula functions of the elliptical, Archimedeanand quadratic families are used to model the underlying depen-dence structure in both bearish and bullish market phases. Overthe 2000–2011 period, we find evidence of significant andsymmetric dependence for almost all the oil-exchange rate pairsconsidered. The rise in the price of oil is found to be associatedwith the depreciation of the dollar. Moreover, we show thatStudent-t copulas best capture the extreme dependence, and thattaking the extreme comovement into account leads to improve theaccuracy of VaR forecasts. Our main results remain unchangedwhen considering alternative GARCH-type specifications and thecrisis period, but are sensitive to the use of raw returns.

� 2012 Elsevier Ltd. All rights reserved.

fax: þ33 140 539 898.ui), [email protected] (M.S. Ben Aïssa), [email protected]

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R. Aloui et al. / Journal of International Money and Finance 32 (2013) 719–738720

1. Introduction

Crude oil is one of the most important commodities for the global economy today and commonlyregarded as a comparative advantage and a strategic resource. Past studies suggest that oil pricedynamics influences economic activity and equity markets.1 Research on crude-oil price dynamics andits impacts on both real and financial sectors has recently been challenged by several important facts.First, crude oil prices experienced very large fluctuations over the last three decades and became morevolatile than they did over the period from the Second World War to the early 1970s. The greaterinstability in the oil prices initially appeared in the aftermath of the world oil crises of 1973 and 1979.This tendency was strengthened by the collapse of oil prices during the 1980s. More importantly, overthe last five years, oil prices have increased very sharply, rising from about U.S. $42 per barrel at thebeginning of 2005 to their highest level of U.S. $147 per barrel in July 2008. Plourde andWatkins (1998)and Regnier (2007) find that crude oil’s price volatility is substantially higher than that of other energyproducts since the mid-1980s. The unrest in Libya and the ongoing threats to political instability inother Middle East and North Africa countries currently contribute to run up in oil prices,2 which mayseverely slow down the global economic growth. Second, some papers provide evidence of inefficientbehavior of world oil markets, which makes the forecasting of oil price volatility and the oil riskhedging more complicated (Green and Mork, 1991; Shambora and Rossiter, 2007; Arouri et al., 2010).Finally, oil prices are denominated in U.S. dollars, and so their fluctuations in domestic currenciesdepend closely on the dollar exchange rates, which experienced frequent and uncertain changes overrecent years. As an illustration, Fig. 1 shows that the trade-weighted average of the foreign exchangevalue of the U.S. dollar against world major currencies was under downward pressure over the periodJanuary 2002 to August 2008 owing to concerns about the U.S. growing public debt and high currentaccount deficit.3 The depreciation trend was inverted during the most severe episode of the globalfinancial crisis from September 10, 2008 to September 29, 2010. Its ongoing descent since October 2010particularly reflects market concerns about the implications of the U.S. debt, very low interest rates andmonetary expansion policies, known as “quantitative easing 2”.

It is thus obvious from the above discussions that oil and non-oil economies have not seen the oilprice increases in the same manner and that oil traders (consumers and producers) have reasons toworry about the fluctuations in the dollar exchange rates. In this context, modeling and forecasting thecomovements between oil prices and the dollar exchange rates are crucial, not only for market tradingand risk management issues, but also for the proper regulation of foreign exchange markets in alleconomies which operate with floating exchange rate regimes.

As regards theory, Krugman (1983), Golub (1983) and Rogoff (1991), among others, document thepotential importance of oil prices as an explanatory variable of exchange rate movements. Theempirical evidence on the interactions between oil prices and dollar exchange rates is however muchless extensive than that on the oil effects on economic and market activities. Using various datasetsanterior to the global financial crisis, the majority of works in this literature have mainly founda positive link between oil price and dollar value, meaning that an increase in the price of oil isassociated with a dollar appreciation (e.g., Dibooglu, 1996; Amano and van Norden, 1998; Bénassy-Quéré et al., 2007; Chen and Chen, 2007; Basher et al., 2012). For example, Amano and van Norden(1998) find, from an error-correction model (ECM), a stable link between the real effective traded-

1 Hamilton (1983) shows that rising oil prices are responsible for nine out of ten of the U.S. recessions since the Second WorldWar. Other earlier studies in which an analysis of oil–macroeconomy relationships is conducted using various methodologies aswell as different datasets also confirm these findings (Mork, 1989; Hooker, 1996). More recent attempts such as Balke et al.(2002), Zhang (2008) and Cologni and Manera (2009) document both asymmetric and nonlinear links between oil priceshocks and macroeconomic variables. On the other hand, several studies have found that stock market activities are signifi-cantly affected by oil price movements (Sadorsky, 2001; Park and Ratti, 2008; Arouri and Nguyen, 2010; Fayyad and Daly, 2011).The oil’s impact is however sensitively different across economic sectors (e.g., oil versus non-oil industries) and across countries(e.g., net oil-exporting versus net oil-importing ones).

2 The Brent crude oil index closed at U.S. $123.26 on April 29, 2011, or an increase of about 45.32% on a year-to-year basis.3 The countries/area included in the major currencies index are the Euro Area, Canada, Japan, United Kingdom, Switzerland,

Australia, and Sweden.

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011

2040

6080

100

120

140 WTI crude oil

Brent crude oilTrade Weighted U.S. Dollar Index

Fig. 1. Dynamics of the daily Traded Weighted U.S. Dollar Index (major currencies) and crude oil prices.

R. Aloui et al. / Journal of International Money and Finance 32 (2013) 719–738 721

weighted value of the U.S. dollar against currencies of major U.S. trade partners and the U.S. real price ofoil over the period 1972–1993. In addition, the causal effect runs only from oil prices to exchange ratesbut not in the opposite direction, leading to suggest that oil price shocks may be a dominant source ofreal exchange rate fluctuations over the study period. Bénassy-Quéré et al. (2007) document similarcausality from oil to real effective exchange rate of the dollar over a longer period and show that a 10%increase in the oil price coincides with a 4.3% appreciation of the dollar in the long run. By contrast,some studies conclude on the depreciation of the U.S. dollar exchange rates following a rise in the priceof oil (e.g., Narayan et al., 2008; Zhang et al., 2008; Akram, 2009; Wu et al., 2012). We summarize, inTable 1, the key findings of major studies in the related literature in order to offer a comparative view.

The fact that the rise in oil prices is associated with depreciation or appreciation of the U.S. dollarmay be due to the level of oil dependence of different countries. For instance, Lizardo and Mollick(2010)’s cointegration tests and forecasting results show that increases in real oil prices lead toa significant depreciation of the U.S. dollar against the currencies of net oil-exporting countries(Canada, Mexico, and Russia). On the contrary, the value of the dollar relative to currencies of net oil-importing countries, such as Japan, increases when the real oil prices go up. For traded currencieswhose countries are neither net exporters nor significant importers of oil relative to their total trade,such as the U.K. and the European Union, the common tendency is the depreciation of the dollar.

Our study also questions the interactions between crude oil prices and U.S. dollar exchange rates.Compared to the extant literature, we first contribute to investigations into the extreme comovementsbetween these variables on the basis of a copula-GARCH approach.We particularly look at themarginaldistributions of oil and exchange rate returns, and test for both the degree and type of their depen-dence at extreme levels conditionally on the possibility of extreme financial events such as the recentglobal financial crisis. By computing the tail dependence coefficients, we are able to examine how oiland exchange rate returns are linked to each other during bearish and bullish markets. Our contri-bution is thus mostly related to that of Wu et al. (2012) who use copula-GARCHmodels to examine thecomovement betweenWTI oil price and U.S. dollar index futures,4 but it rather focuses on the bilateral

4 The USDX represents the futures contract on the trade-weighted value of the U.S. dollar in terms of a basket of six majorforeign currencies, traded on the New York Board of Trade (NYBOT).

Table 1Previous research on the interactions between oil prices and exchange rates.

Studies Purposes Data period Methodology Main findings

The rise in oil prices is associated with the U.S. dollar appreciationDibooglu

(1996)This paper examines whetherinternational differences in realvariables are responsible forPPP deviations.

Quarterly dataover the period1960–1988

Cointegrationand ECM

Real shocks,including realoil price movements,significantly accountfor deviations from PPP.

Amano andvan Norden(1998)

The authors focus on the relativeimportance of real versusmonetary shocks in explainingexchange rate movements.

Monthly data fromFebruary 1972 toJanuary 1993

Causality tests,cointegrationand ECM

The price of oilGranger-causesthe U.S. real effectiveexchange rate andnot vice versa. It isalso the dominantsource of persistentexchange rate shocks.

Bénassy-Quéréet al. (2007)

This paper studies cointegrationand causality between the realprice of oil and that of the dollar

Monthly data fromJanuary 1974 toNovember 2004

Causality tests,cointegrationand vector ECM

A 10% rise in the oilprice coincideswith a 4.3% appreciationof the dollar in thelong run.

Chen and Chen(2007)

This paper investigates thelong-run relationship betweenreal oil prices and real exchangerates.

Monthly data of G7countries fromJanuary 1972 toOctober 2005

Panel cointegrationapproach

World oil prices constitutethe dominant sourceof exchange ratemovements. They arealso robust to differentmeasures of oil prices

Basher et al.(2012)

The authors study the dynamiclink between oil prices, exchangerates and emerging market stockprices.

Monthly data fromJanuary 1988 toDecember 2008

Structural VARmodel

Positive shocks to oilprices tend to depressemerging market stockprices and thetrade-weighted U.S.dollar index in theshort run.

The rise in oil prices is associated with the U.S. dollar depreciationNarayan et al.

(2008)The authors examine therelationship between oil priceand the Fiji–US exchange rate.

Daily data fromNovember 29,2000 to September15, 2006

GARCH andEGARCHmodels

A rise in oil pricesleads to an appreciationof the Fijian dollarvis-à-vis the U.S. dollar.

Zhang et al.(2008)

The authors test the interactionbetween the oil price variationand U.S. dollar exchange rate.

Daily data fromJanuary 4, 2000 toMay 31, 2005

Cointegration, VAR,ARCH models, VaR

The U.S. dollardepreciation was akey factor in drivingup the crude oil price.

Akram (2009) The author investigates whethera decline contributes to highercommodity prices.

Quarterly data from1990 to 2007

Structural VARmodel

A fall in the value ofthe U.S. dollar leads todrive up commodityprices, including crudeoil price.

Wu et al.(2012)

The authors examine theeconomic value of comovementbetween WTI oil priceand U.S. dollar index futures

Weekly data fromJanuary 2, 1990 toDecember 28, 2009

Copula-GARCHmodels

The dependence structurebetween oil andexchange-rate returnsbecomes negative anddecreases continuouslyafter 2003.


U.S. dollar exchange rates and is more general in terms of copula families as well as the selectionprocess of best copula model. Second, the combination of GARCH modeling approach and somemultivariate extreme value copula functions is of particular interest because oil and exchange ratesmight have common extreme variations. Moreover, it enables to capture nonlinearities in the oil–exchange rate relationships as well as some empirical stylized facts of return distributions such asvolatility persistence, fat tail behavior and asymmetric impacts of return innovations on volatility


(Longin and Solnik, 2001; Chan-Lau et al., 2004; Aloui et al., 2011), while avoiding the drawbacks oflinear measures of interdependence (e.g., Pearson correlation and dynamic conditional correlationcoefficients obtained from a DCC–GARCH model). Finally, since we have studied the oil–exchange ratecomovements over a more recent period (January 4, 2000 to February 17, 2011), our results comple-ment those of almost all previous works whose study periods predate the 2007–2009 global financialcrisis by a couple of years. Here it is important to note the unusual and unexpected appreciation of theU.S. dollar against other major currencies during the crisis period.

Using daily time-series of crude-oil spot prices (WTI Cushing and Brent price indices) and nominalexchange rates for the U.S. dollar against five major currencies (Euro, Canadian Dollar, British Pound,Swiss Franc, and Japanese Yen), we mainly find that oil price increases are associated with dollardepreciation for most bilateral exchange rates, as suggested by simple correlations in recent years.The GARCH with Student-t and Skewed Student-t distributions are the most appropriate specifica-tions for modeling the conditional heteroscedasticity of the data. The t-Copula provides the best fitfor the dependence structure of all oil–exchange rate pairs considered, except for the pair of oil andU.S. Dollar/Japanese Yen. More importantly, this dependence structure exists in both bear and bullmarket periods, and is symmetric for all considered pairs, but not the pair of Brent oil and U.S. Dollar/Japanese Yen exchange rate. Our main results regarding the dependence structure are robust to thealternative GARCH specifications and the crisis period, but are sensitive to the use of raw returns. Wetypically document an increase in comovement for some oil–exchange rate pairs over the crisisperiod.

We structure the rest of the article as follows. Section 2 describes the empirical methodology andestimation strategy. Section 3 presents the data used and discusses our empirical results. Section 4reports the results for the robustness check with respect to marginal models and subsampleperiods. Section 5 provides some concluding remarks.

2. Empirical methodology

We first begin with a brief introduction to copula functions, and then present successively theempirical model for the margins of the return distributions, the alternative copula models of condi-tional dependence structure, and the estimation procedure.

2.1. Copula function

Copulas are functions that link multivariate distributions to their univariate marginal functions.They can be defined as follows:

Definition 1. A d-dimensional copula is a multivariate distribution function C with standard uniformmarginal distributions

Theorem 1. Sklar’s theoremLet X1;.;Xd be random variables with marginal distribution F1;.; Fd and joint distribution H, then

there exists a copula C: ½0;1�d/½0;1� such that:

Hðx1;.; xdÞ ¼ CðF1ðx1Þ;.; FdðxdÞÞ (1)

Conversely if C is a copula and F1;.; Fd are distribution functions, then the functionH defined aboveis a joint distribution with margins F1;.; Fd.

Copulas functions provide an efficient way to create distributions that model correlated multivar-iate data. Inasmuch as the measure of interdependence matters, one can construct a multivariate jointdistribution by first specifying marginal univariate distributions, and then choosing a copula toexamine the correlation structure between the variables. Bivariate distributions, as well as distribu-tions in higher dimensions are possible. Additionally, copulas can be used to characterize the depen-dence in the tails of the distribution. Twomeasures of tail dependence related to copulas, known as theupper and the lower tail dependence coefficients, are particularly helpful for measuring the tendencyof markets to crash or boom together.


Let X, Y be random variables with marginal distribution functions F and G. Then the coefficient oflower tail dependence lL is

lL ¼ limt/0þ

PrhY � G�1ðtÞ

��X � F�1ðtÞi

(2)

which quantifies the probability of observing a lower Y assuming that X is itself lower. In the sameway,the coefficient of upper tail dependence lU can be defined as

lU ¼ limt/1�

PrhY > G�1ðtÞ

��X> F�1ðtÞi

(3)

There is a symmetric tail dependence between two variables when the lower tail dependencecoefficient equals the upper one, otherwise it is asymmetric. The tail dependence coefficient providesaway for ordering copulas. Onewould say that copula C1 is more concordant than copula C2 if lU of C1 isgreater than lU of C2.5

2.2. Margin specifications

We investigate the interdependence between crude oil prices and each of the five U.S. dollarexchange rates by combining the copula functions, described above, with a GARCH-type model ofconditional heteroscedasticity. This modeling approach is advantageous in that it offers the possibilityto separately model the margins (GARCH-based model) and association structure of different variables(copula models).

In this article we adopt the standard GARCH model developed by Bollerslev (1986), which isprobably the most commonly used financial time series model and has inspired a number of moresophisticated extensions. It allows to not only successfully characterizing some important stylized factsof financial returns such as volatility clustering and time-varying volatility, but also to removing serialdependence in the component time series, as discussed in Grégoire et al. (2008).

Given a time series yt, the GARCH(1,1) model can be written as6

yt ¼ cþ stzts2t ¼ uþ a 32t�1 þ bs2t�1

(4)

where zt is an iid randomvariable with zero mean and variance of one. s2t is the conditional variance ofreturn series at time t, which depends on both past return innovations and past conditional variance.The conditional variance process is positive and stationary if the following conditions hold: u_0, a_0,b_0 and aþ b31. The traditional GARCH model assumes a normal distribution for the innovations.However, it is well known that many financial time series are non-normal and tend to have a fat tailbehavior (Mandelbrot, 1963; Fama, 1965). In order to better capture the fat-tailed characteristic,Bollerslev (1987) suggests the use of the conditional Student-t distribution. Other distributionsproposed in the literature include the skewed-t distribution (Hansen, 1994), the generalized errordistribution (Nelson, 1991) and the skewed generalized error distribution (Theodossiou, 2000). In thisstudy, we consider various distributions for the error term 3t, including the normal, the skewed normal,the Student-t and the skewed Student-t distributions.

2.3. Copula models of conditional dependence structure

The copula models we consider here allow us to investigate both symmetric and asymmetricstructure of extreme dependence between variables. They fall into three families of copulas: elliptical

5 A detailed discussion of copulas functions and their properties may be found in Joe (1997) and Nelsen (1999).6 The optimal lag length for the conditional mean and variance processes of the GARCH model was determined with respect

to the AIC and BIC. The results, consistent with those of previous applied studies which use GARCH-type models for mostfinancial data, can be made entirely available under request addressed to the Corresponding author.


copulas (normal and Student-t), Archimedean copulas (Gumbel, Frank, and Clayton) and quadraticcopulas (Plackett).

For all u, y in [0,1], the bivariate normal copula is defined by

Cðu; yÞ ¼Z f�1ðuÞ

�N

Z f�1ðyÞ

�N

1

2pffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� q2

p exp

� s2 � 2qst þ t2

2�1� q2

� !dsdt (5)

where f represents the univariate standard normal distribution function and q is the linear correlationcoefficient restricted to the interval (�1,1).

The bivariate Student-t copula is defined by

Cðu; yÞ ¼Z t�1

y ðuÞ

�N

Z t�1y ðyÞ

�N

1

2pffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� q2

p 1þ s2 � 2qst þ t2

y�1� q2

� !�yþ22

dsdt (6)

where t�1y ðuÞ denotes the inverse of the CDF of the standard univariate Student-t distribution with y

degrees of freedom.The Gumbel copula of Gumbel (1960) is an asymmetric copulawith higher probability concentrated

in the right tail. It is given by

Cðu; yÞ ¼ exp��hð�lnuÞqþð�lnyÞq

i1=q�; q˛ð1;þNÞ (7)

The Frank copula which appeared in Frank (1979) is defined as

Cðu; yÞ ¼ �1qln�1þ ðexpð�quÞ � 1Þðexpð�qyÞ � 1Þ

expð�qÞ � 1

; q˛ð�N;þNÞ (8)

The Clayton copula which has been introduced by Clayton (1978) and is expressed as

Cðu; yÞ ¼�u�q þ y�q � 1

��1=q; q˛ð0;NÞ (9)

The Plackett copula which has been introduced by Plackett (1965) is expressed as

Cðu; yÞ ¼h1þ ðq� 1Þðuþ yÞ � f1þ ðq� 1Þðuþ yÞg2�4uyqðq� 1Þ�1=2i

2ðq� 1Þ ; q˛ð0;þNÞ � 1 (10)

The elliptical copulas are the most popular in finance literature due to the ease with which they canbe implemented. The normal and the Student-t copulas fall into this family since they are based on anelliptically contoured distribution such as multivariate Gaussian or Student-t distributions. TheGaussian copula is symmetric and has no tail dependence, while the Student-t copula can captureextreme dependence between variables.7 For the normal, Student-t and Frank copulas, q ¼ 0 or q/0leads to independence, while q > 0 and q < 0 lead to positive and negative dependence, respectively.

Unlike elliptical copulas, the Archimedean copulas such as the Gumbel and Clayton copulas are notderived frommultivariate distribution functions and can be used to capture asymmetry between lowerand upper tail dependences. The Clayton copula exhibits greater dependence in the negative tail thanin the positive, whereas the Gumbel copula exhibits greater dependence in the upper tail than in thelower tail. For the Clayton copula, q/0 leads to independence, while q/N leads to perfect positivedependence. For the Gumbel copula, q ¼ 1 and q/N imply independence and perfect positivedependence, respectively. All Archimedean copulas are asymmetric, except for the Frank copula whichenables to capture the full range of dependence for marginals exposed to weak tail dependence.

7 Mashal et al. (2003) show that the empirical fit of the t-copula is generally superior to that of the Gaussian copula.


Similar to the Frank copula, the Plackett copula family is also symmetric and can accommodate all ofthe possible positive and negative dependence. q ¼ 1 implies independence, while q > 1 and q < 1imply positive and negative dependence, respectively.

2.4. Estimation of copula parameters

We estimate the parameters of the copula using a semi parametric two-step estimation method,namely the Canonical Maximum Likelihood or CML (Cherubini et al., 2004). In the first step, we esti-mate the marginals FX and GY nonparametrically via their empirical cumulative distribution functions(ECDF) bFX and bGY defined as

bFXðxÞ ¼ 1n

Xni¼1

1fXi < xg and bGY ðyÞ ¼ 1n

Xni¼1

1fYi < yg (11)

In the implementation, bFX and bGY are rescaled by n=ðnþ 1Þ to ensure that the first order conditionof the log-likelihood function for the joint distribution is well defined for all finite n. We then transformthe observations into uniform variates using the ECDF of each marginal distribution and we estimatethe unknown parameter q of the copula as

bqCML ¼ arg maxq

Xni¼1

lnc�bFXðxiÞ; bFYðyiÞ; q

�(12)

Under suitable regularity conditions, the CML estimator bqCML is consistent, asymptotically normal,and fully efficient at independence. Further details can be found in Genest et al. (1995).

3. Data and results

3.1. Data and stochastic properties

Our dataset contains daily crude oil prices and five U.S. dollar (USD) nominal exchange rates overthe period from January 4, 2000 to February 17, 2011. As our objective is to examine the relationshipbetween oil and foreign exchange markets from the perspective of current market trading activities,nominal data on oil and exchange rates are sufficient. Furthermore, a daily deflator for returns is notavailable.8 Exchange rates correspond to the amount of USD per one unit of each of five majorcurrencies in international trade: the Euro (EUR), the Canadian Dollar (CAD), the British Pound Sterling(GBP), the Swiss Franc (CHF) and the Japanese Yen (JPY). An increase in the nominal exchange ratesconsidered thus reflects a depreciation of the USD relative to foreign currencies. As to oil prices, forcomparison purposes we employ two of the most important benchmarks in crude oil pricing: theWestTexas Intermediate (WTI) crude at Cushing and the Brent crude from the North Sea. The Brent indexactually serves as pricing benchmark for two thirds of the world’s internationally traded crude oilsupplies.

We computed the returns on crude-oil price indices and exchange rates by taking the difference inthe logarithm of the two successive daily prices. The time-paths of return series over the study periodare plotted in Fig. 2. At a glance, we can see that daily returns were fairly stable during the periodpreceding the start of the global financial crisis (January 2000 to the third quarter of 2008), sparked bythe massive failures in the U.S. subprime mortgage and banking sectors. After that, all return seriesexhibited higher instability.

Descriptive statistics and distributional characteristics of return series are reported in Table 2.Returns are skewed and exhibit excess kurtosis. This finding indicates that returns are not normallydistributed. In particular, the probability of extremely negative realizations for the two crude-oil return

8 Narayan et al. (2008) also note that examining the daily behavior of crude oil prices and exchange rates does not requireknowledge of real values.

USD/EUR

2000 2002 2004 2006 2008 2010

-0.0

30.

04USD/CAD

2000 2002 2004 2006 2008 2010

-0.0

40.

04

USD/GBP

2000 2002 2004 2006 2008 2010

-0.0

30.

04

USD/CHF

2000 2002 2004 2006 2008 2010

-0.0

40.

04

USD/JPY

2000 2002 2004 2006 2008 2010

-0.0

30.

04

WTI

2000 2002 2004 2006 2008 2010-0

.15

0.20

Brent

2000 2002 2004 2006 2008 2010

-0.1

50.

10

Fig. 2. Daily returns on crude oil and USD exchange rates.


series and the USD/GBP exchange rate is thus higher than that of a normal distribution. The departurefrom normality is confirmed by the Jarque–Bera test and the QQ-plots of the returns against the normaldistribution (Fig. 3). The Ljung–Box Q-statistics of order 12 show the existence of autocorrelation in allthe return series, except for the USD/JPY and Brent series. The Ljung–Box statistics of order 12 appliedto squared returns are highly significant. Results of the Lagrange Multiplier tests point to the presenceof ARCH effects in the return data, thus supporting our decision to use the GARCH-based approach.

Table 2Descriptive statistics and stochastic properties of return series.

Min. Max. Mean (%) Std. dev Skew. Kurt.

USD/EUR �0.038 0.046 0.010 0.006 0.188 2.708USD/CAD �0.043 0.050 0.013 0.006 0.087 5.645USD/GBP �0.039 0.045 �0.000 0.006 �0.048 4.455USD/CHF �0.044 0.054 0.017 0.007 0.234 4.994USD/JPY �0.031 0.046 0.007 0.006 0.464 3.552WTI �0.172 0.213 0.042 0.026 �0.054 5.036Brent �0.199 0.128 0.050 0.024 �0.378 4.577

Q(12) Q2(12) JB ARCH(12)

USD/EUR 24.98þ 276.93þþ 898.14þþ 166.315þþ

USD/CAD 40.82þþ 1137.49þþ 3835.02þþ 450.724þþ

USD/GBP 33.81þþ 924.51þþ 2387.14þþ 392.156þþ

USD/CHF 37.14þþ 295.58þþ 3024.49þþ 232.817þþ

USD/JPY 13.47 185.09þþ 1620.88þþ 134.829þþ

WTI 25.56þ 840.83þþ 3050.73þþ 364.089þþ

Brent 12.99 320.15þþ 2587.80þþ 183.920þþ

Notes: The table displays summary statistics for the daily returns over the period from January 4, 2000 to February 17, 2011.Q(12)and Q2(12) are the Ljung–Box statistics for serial correlation of order 12 in returns and squared returns. ARCH is theLagrange Multiplier test for autoregressive conditional heteroscedasticity. þ and þþ indicate the rejection of the null hypothesisof no autocorrelation, normality and homoscedasticity at the 5% and 1% levels of significance respectively.

USD/EUR

Quantiles of Standard Normal

USD

/EU

R

-2 0 2

-0.0

40.

00.

04

USD/CAD


US

D/C

AD

-2 0 2

-0.0

40.

00.

04

USD/GBP


USD

/GBP

-2 0 2

-0.0

40.

00.

04

USD/CHF


US

D/C

HF

-2 0 2

-0.0

40.

00.

04

USD/JPY


USD

/JPY

-2 0 2

-0.0

20.

02

WTI


WTI

-2 0 2

-0.1

0.1

Brent


Bren

t

-2 0 2

-0.2

0.0

0.1

Fig. 3. QQ-plots of crude oil and USD exchange rate returns against the normal distribution.


3.2. In-sample empirical results

Before copula functions can be used tomeasure the degree of interdependence among variables, wemust choose the most appropriate specifications for modeling the conditional heteroscedasticity.Univariate GARCH models with various distributions for the error term 3t are fitted to the return dataand compared using the usual information criteria such as AIC, BIC and Loglik statistics. The obtainedresults, reported in Table 3, show that GARCH(1,1) models with skewed Student-t distribution areappropriate for three return series (WTI, Brent and USD/CHF), while empirical stylized facts of theremaining series can be satisfactorily modeled by GARCH(1,1) models with Student-t distribution. Notealso that for all the series, the parameter b is close to 0.9 and highly significant, which thus indicatesthat conditional volatility is past-dependent and very persistent over time. The ACFs of the stan-dardized residuals and squared standardized residuals, obtained from the GARCH fit but not presentedhere for concision purposes, indicate that the standardized residuals are approximately i.i.d. They arethus far more suitable to copula estimation than the raw return series.

Given the availability of the estimates of GARCH models, we turn to estimate copula functions foreach pair of oil and exchange rates. For that, we consider the vector of standardized residualsðx1t ; x2t ;.; xntÞTt¼1 from GARCH models and transform it into the vector of uniform variates,ðu1t ;u2t ;.;untÞTt¼1, using the ECDF. Table 4 summarizes the results of the rank correlation coefficientsfor oil and exchange rates dependence. For the overall sample period, both Kendall’s tau and Spear-man’s rho statistics indicate a positive relationship for all considered pairs, except for the USD/JPY-WTIand the USD/JPY-Brent pairs. These findings are globally confirmed by those for the normal period (4January 2000 to 15 July 2008) and the crisis period (16 July 2008 to 17 February 2011). The comove-ment is however substantially higher during the crisis period.

Table 3Maximum likelihood estimation of univariate GARCH models for returns series.

c (�104) AR(1) u (�106) a (�102) b (�10) v Skew.

USD/EUR 1.951*(1.045)

0.008(0.018)

0.149** (0.072) 3.000*** (0.459) 9.669*** (0.048) 10.000*** (1.781) –

USD/CAD 1.143(0.8610)

�0.003(0.019)

0.118** (0.054) 4.655*** (0.673) 9.517*** (0.067) 10.000*** (1.475) –

USD/GBP 0.181(0.943)

0.018(0.019)

0.204** (0.087) 3.799*** (0.612) 9.566*** (0.069) 10.000*** (1.735) –

USD/CHF 2.176(1.223)

�0.052***(0.018)

0.516*** (0.196) 2.547*** (0.552) 9.640*** (0.077) 6.525*** (0.774) 1.055*** (0.026)

USD/JPY �0.811(1.067)

– 0.444*** (0.171) 2.852*** (0.616) 9.610*** (0.084) 6.445*** (0.723)

WTI 8.304**(4.066)

�0.033*(0.018)

6.317*** (2.038) 3.505*** (0.624) 9.537*** (0.079) 7.324*** (0.939) 0.957*** (0.024)

Brent 11.092***(3.932)

– 5.324** (2.190) 3.748*** (0.767) 9.524*** (0.099) 7.669*** (1.009) 0.944*** (0.025)

Notes: The table summarizes the GARCH estimation results. The values in parenthesis represent the standard errors of theparameters. *, **, and *** indicate significance at the 10%, 5% and 1% levels respectively.


The estimates of the dependence parameter q for six copula functions (the Normal, Student-t,Gumbel, Frank, Clayton, and Plackett) are reported in Table 5. They are, as expected, highly significantfor almost all the pairs considered and copula functions. These findings clearly support the hypothesisof significant dependence between oil and exchange rate returns over the study period. Two exceptionsare the USD/JPY-WTI and USD/JPY-Brent pairs where the dependence parameter takes very low values,and is detected by only two copula functions, i.e., Gumbel and Plackett.

In order to check the overall quality of the fit, we use the goodness-of-fit test of Genest et al. (2009),which is based on a comparison of the distance between the estimated and the empirical copula:

Cn ¼ ffiffiffin

p �Cn � Cqn

(13)

The test statistics considered is based on Cramér–Von Mises distances defined as

Sn ¼Z

CnðuÞ2dCnðuÞ (14)

Large values of the statistic Sn lead to the rejection of the null hypothesis that the copula C belongsto a class C0. In practice, we require knowledge about the limiting distribution of Sn which depends onthe unknown parameter value q. To find the p-values associated with the test statistics we usea multiplier approach as described in Kojadinovic and Yan (2011). The highest p-values thus indicatethat the distance between the estimated and empirical copulas is the smallest and that the copula inuse provides best fit to the data.

Table 4Correlation estimates of oil price and exchange rate dependence.

Overall sample Normal period Crisis period

Kendall Spearman Kendall Spearman Kendall Spearman

USD/EUR-WTI 0.105 0.153 0.057 0.084 0.238 0.340USD/CAD-WTI 0.143 0.211 0.085 0.127 0.300 0.424USD/GBP-WTI 0.091 0.133 0.045 0.067 0.217 0.306USD/CHF-WTI 0.081 0.118 0.050 0.074 0.170 0.238USD/JPY-WTI �0.011 �0.016 0.040 0.060 �0.157 �0.229USD/EUR-Brent 0.111 0.163 0.056 0.083 0.267 0.379USD/CAD-Brent 0.168 0.248 0.094 0.141 0.377 0.529USD/GBP-Brent 0.102 0.150 0.044 0.065 0.265 0.378USD/CHF-Brent 0.095 0.139 0.061 0.090 0.197 0.280USD/JPY-Brent �0.011 �0.017 0.035 0.052 �0.147 �0.215

Note: The table summarizes rank correlation estimates for oil and exchange rate dependence over the overall period and the twosubperiods.

Table 5Estimates of the dependence parameters of different copula models.

Normal Student-t Frank Plackett Gumbel Clayton

USD/EUR-WTI 0.153***(0.016)

0.159*** (0.019), y ¼ 10.759*** (2.468) 0.967*** (0.109) 1.652*** (0.089) 1.101***(0.013)

0.160***(0.023)

USD/CAD-WTI 0.207***(0.017)

0.208*** (0.018), y ¼ 28.345* (15.365) 1.250*** (0.112) 1.876*** (0.102) 1.131***(0.014)

0.217***(0.024)

USD/GBP-WTI 0.128***(0.017)

0.130*** (0.019), y ¼ 15.739*** (5.258) 0.784*** (0.110) 1.498*** (0.081) 1.072***(0.013)

0.138***(0.022)

USD/CHF-WTI 0.120***(0.017)

0.124*** (0.020), y ¼ 9.989*** (2.138) 0.751*** (0.108) 1.481*** (0.079 1.077***(0.013)

0.125***(0.022)

USD/JPY-WTI 0.760e�03(0.017)

�0.003 (0.020), y ¼ 11.235*** (2.937) �0.028 (0.107) 0.985*** (0.053) 1.008***(0.009)

0.017(0.019)

USD/EUR-Brent 0.178***(0.017)

0.181*** (0.019), y ¼ 12.437*** (3.343) 1.078*** (0.110) 1.740*** (0.094) 1.114***(0.014)

0.195***(0.023)

USD/CAD-Brent 0.251***(0.017)

0.252*** (0.018), y ¼ 21.834*** (9.489) 1.485*** (0.114) 2.089*** (0.114) 1.165***(0.015)

0.277***(0.015)

USD/GBP-Brent 0.156***(0.017)

0.154*** (0.019), y ¼ 16.272*** (5.611) 0.887*** (0.111) 1.574*** (0.086) 1.086***(0.013)

0.175***(0.024)

USD/CHF-Brent 0.158***(0.017)

0.159*** (0.019), y ¼ 12.412*** (3.304) 0.943*** (0.110) 1.627*** (0.087) 1.098***(0.013)

0.174***(0.024)

USD/JPY-Brent �3.369e�04(0.018)

�0.002 (0.019), y ¼ 27.808* (15.675) �0.039 (0.110) 0.980*** (0.054) 1.010***(0.009)

�0.006(0.019)

Notes: The table summarizes the Copula estimation results. The values in parenthesis represent the standard error of theparameters. *, **, and *** indicate significance at the 10%, 5% and 1% levels respectively.


The results of the goodness-of-fit test are summarized in Table 6. They show that the Student-tcopula is the best one in nine out of ten cases. For the USD/JPY-Brent pair, the results suggest theClayton copula as the best fittingmodel. Notice here that the Student-t copula hasmore observations inthe tails than the Gaussian copulas, while the Clayton copula is more suitable for capturing the lowertail dependence. Combining the findings of the Tables 5 and 6, we see that the dependence betweencrude oil and U.S. dollar exchange rate returns, modeled by Student-t copula models, is positive for allconsidered pairs, except for the USD/JPY-WTI and USD/JPY-Brent where dependence coefficients arenot statistically significant. It turns out that an increase in the price of crude oil is associated witha depreciation of the USD. Our empirical evidence is thus not consistent with the findings of themajority of previous studies using data before the period of the recent global financial crisis (e.g.,Amano and van Norden,1998; Chen and Chen, 2007). The negative relationship between the price of oiland the price of dollar can be explained by the fact that oil is a hedge against rising inflation and servesas a safe haven against growing risk. For example, when the dollar depreciates, consumers withappreciated currencies will find oil less expensive to buy and thus increase their demand for oil, whichin turn leads to the rise in the price of oil. Inversely, oil producers might want to raise oil prices as theirpurchasing power goes down.

Table 6Results for the goodness-of-fit test of different copula functions.


USD/EUR-WTI 0.018 0.039 0.011 0.008 0.010 4.995e�4USD/CAD-WTI 0.189 0.262 0.026 0.025 4.995e�4 4.995e�4USD/GBP-WTI 0.156 0.309 0.144 0.157 0.005 4.995e�4USD/CHF-WTI 0.011 0.024 0.021 0.019 0.016 4.995e�4USD/JPY-WTI 0.033 0.294 0.017 0.012 4.995e�4 0.048USD/EUR-Brent 0.339 0.853 0.041 0.046 0.001 4.995e�4USD/CAD-Brent 0.166 0.200 0.002 0.002 4.995e�4 4.995e�4USD/GBP-Brent 0.109 0.249 0.012 0.005 4.995e�4 0.034USD/CHF-Brent 0.391 0.851 0.115 0.100 0.015 4.995e�4USD/JPY-Brent 0.286 0.320 0.293 0.290 4.995e�4 0.457

Notes: The table presents the p-values of the goodness-of-fit test. Large p-values (bold face numbers) indicate that the copulaprovides best fit to the data.

Table 7Tail dependence coefficients of the best copulas.

Return pairs lL lU Return pairs lL lU

USD/EUR-WTI 0.013 0.013 USD/EUR-Brent 0.009 0.009USD/CAD-WTI 0.136e�03 0.136e�03 USD/CAD-Brent 0.001 0.001USD/GBP-WTI 2.305e�03 2.305e�03 USD/GBP-Brent 0.002 0.002USD/CHF-WTI 0.014 0.014 USD/CHF-Brent 0.008 0.008USD/JPY-WTI 0.004 0.004 USD/JPY-Brent 0 0

Notes: The table presents the upper and lower tail dependence coefficients estimated from the copulas that provide the best fit.


Table 7 reports the values of the lower and the upper tail dependence coefficients, obtained fromthe best fitting copula model. Accordingly, the hypothesis of extreme comovements between oil andexchange rates cannot be rejected for any of the pairs we consider, except for the USD/JPY-Brent. Forexchange rates exhibiting extreme dependence on oil price movements, the dependence structure issymmetric in bear and bull markets because lower and upper tail dependence coefficients are exactlyequal. The strongest comovement with the WTI and Brent crude oil markets is found for USD/CHF andUSD/EUR exchange rates, respectively. The lowest degree of extreme comovement with the oil marketsis obtained for USD/CAD exchange rate.

The above findings are thus consistent with the view that higher oil reserves and productionpositions of a country lead to reduce the extreme dependence level of its foreign exchange markets onfluctuations of the oil prices. In our sample, Canada is among the top world oil producers and importsmuch less oil than euro zone countries. The United Kingdom is also the largest producer of oil in theEuropean Union, and has recently become a net importer of crude oil since 2005. For Switzerland, thiscountry is a net-oil importer without domestic oil production. Moreover, it is common that the USD/CHF exchange rate exhibits strong sensitivity to both Swiss and American economic news as well as tothe prevailing level of risk aversion in themajor global financial markets. For example, if oil price growsup, many Middle Eastern wealthy individuals and businesspeople with large accounts in Switzerlandwill have more dollars on their accounts and may be interested in selling USD/CHF futures contracts toprotect against unfavorable movements in the dollar exchange rates. In a more general way, non-US oilinvestors hedge their positions against movements in the dollar. When oil prices rise (fall), they are

USD/EUR-WTI empirical joint distribution-0.04 -0.02 0.0 0.02 0.04

-0.2

0.0

0.2

USD/EUR-WTI simulated bivariate normal distribution

-0.04 -0.02 0.0 0.02 0.04

-0.2

0.0

0.2

USD/EUR-WTI Monte Carlo simulated bivariate distribution-0.04 -0.02 0.0 0.02 0.04

-0.2

0.0

0.2

Fig. 4. Scatterplot of the joint empirical distribution (top), the joint Gaussian distribution (middle) and the joint Monte Carlosimulated distribution (bottom) for the USD/EUR-WTI returns.

Table 8Out-of-sample performance.

Model a ¼ 0.10 a ¼ 0.05 a ¼ 0.01 a ¼ 0.005

Normal 0.091 (132) 0.047 (68) (0.023) 34 (0.017) 25T-copula 0.104 (151) 0.045 (65) (0.008) 12 (0.005) 7

Notes: This table reports the VaR backtesting results obtained from the bivariate normal and copula models. A model which issaid to be best suited for calculating VaR refers to the one with the number of exceedances (given in parenthesis) closest to theexpected number of exceedances. The latter are 145, 72, 14 and 7 for 90%, 95%, 99%, and 99.5% confidence levels, respectively.


under-hedged (over-hedged), and in response they sell (buy) dollars against their home currency in thefutures markets. Finally, the result for USD/JPY is not unexpected. Japan’s current account surplus andreliance on nuclear energy help weakening the dependence of the Yen value on changes in the price ofoil, even though this country is the third largest oil importer, following the U.S. and China. The absenceof USD/JPY exchange-rate sensitivity to Brent oil movements can be explained by the fact that Japan’scrude oil imports are almost from OPEC countries.

3.3. Out-of-sample value-at-Risk (VaR) forecasting

To show the implications of the extreme dependence on estimates of VaR forecasts, we propose toillustrate the use of the copula-GARCHmethod in quantifying the risk of an equally weighted portfolio,composed of the USD/EUR exchange rate and WTI index. We first compare the joint empirical distri-bution, the joint Gaussian distribution and the joint Monte Carlo simulated distribution from the bestfitting copula model for the USD/EUR-WTI returns. The results are given in Fig. 4. It is obvious from theplot that the bulk of the joint empirical distribution is reproduced in a reasonable manner by bothGaussian and Monte Carlo simulated distributions. However, the bivariate normal model fails incapturing the isolated points appearing in the empirical distribution. In contrast, the t-copula modeldoes a much better job in reproducing the fat-tail feature of the real data.

A backtesting procedure is applied in order to assess the accuracy of the VaR estimates. First, weestimate thewhole model (copula-GARCH) using data only up to time t0. We then simulate innovationsfrom the copula and transform them into standardized residuals by inverting the marginal CDF of eachseries. Finally, we calculate the returns by reintroducing the GARCH volatility and the conditional meanterms observed in the original return series and we compute the value of the global portfolio. Thisprocedure can be repeated until the last observation and we compare the estimated VaR with theactual next-day value change in the portfolio. The whole process is repeated only once in every 50observations owing to the computational cost of this procedure. However, at each new observation theVaR estimates are modified because of changes in the GARCH volatility and the conditional mean.

Concretely, we start by estimating the model using the first half of the data. Then, we simulate 3000values of the standardized residuals, estimate the VaR and count the number of losses that exceeds theestimated VaR values. In Table 8, we display the out-of-sample proportions of portfolio returns which arelowerthanthe10%,5%,1%and0.5%quantilevalues forboththebivariatenormalmodeland t-copulamodels.

The t-copula provides better VaR forecasts at the 90%, 99% and 99.5% confidence levels. For 95%confidence level, the bivariate normal model offers the closest out-of-sample fit to the expectednumbers of exceedances. Overall, our results thus suggest that the t-copula model outperforms thebivariate normal model for high-quantile VaR measures.

4. Some robustness issues

In this section we examine the robustness of our results to the use of raw returns, to alternativeGARCH-type models for return filtering and to different market conditions. This checking procedure isvery important as it allows us to confirm the suitability of our proposed approach for modeling therelationships between oil and foreign exchange markets.9

9 We are particularly grateful to an anonymous referee for this helpful suggestion.

Table 9Results for the goodness-of-fit test for the raw returns.

Raw returns Normal Student-t Frank Plackett Gumbel Clayton

USD/EUR-WTI 4.995e�4 4.995e�4 4.995e�4 0.001 4.995e�4 4.995e�4USD/CAD-WTI 4.995e�4 4.995e�4 4.995e�4 4.995e�4 4.995e�4 4.995e�4USD/GBP-WTI 0.001 0.003 4.995e�4 0.003 4.995e�4 4.995e�4USD/CHF-WTI 4.995e�4 4.995e�4 4.995e�4 4.995e�4 4.995e�4 4.995e�4USD/JPY-WTI 4.995e�4 4.995e�4 4.995e�4 4.995e�4 4.995e�4 0.001USD/EUR-Brent 0.150 0.739 0.031 0.012 0.003 4.995e�4USD/CAD-Brent 0.006 0.008 4.995e�4 4.995e�4 0.001 4.995e�4USD/GBP-Brent 0.006 0.249 4.995e�4 4.995e�4 0.001 0.116USD/CHF-Brent 0.084 0.605 0.079 0.075 0.010 4.995e�4USD/JPY-Brent 0.277 0.341 0.410 0.410 4.995e�4 0.716

Notes: The table presents the p-values of the goodness-of-fit test for raw returns. Large p-values indicate that the copulaprovides best fit to the data.


4.1. Raw returns versus filtered returns

Malevergne and Sornette (2003) suggest that the dependence structure of a copula is not the samefor raw returns and filtered returns (residuals). It is thus interesting to know how our results presentedin the previous section change when raw returns are used. To do so, we simply estimate the selectedcopula models for the raw returns. Tables 9 and 10 present the results of the goodness-of-fit tests andthe tail dependence coefficients of the best copulas. We see that our copula models are rejected for sixout of ten market pairs at the conventional levels of statistical significance. For the four pairs wherecopula models are relevant, the results are similar to those we obtained when using filtered returns inthree aspects. First, the USD/JPY-Brent pair still exhibits no tail dependence. Second, the same copulafamily is chosen for the remaining pairs with certain degree of extreme dependence. Finally, when it ispresent, the extreme dependence structure is symmetric. The value of the tail dependence coefficientsis, however, much higher for the raw returns than for the filtered returns.

Even though our results are not insensitive to raw returns, we think that the analysis with filteredreturns provides more accurate marginal distributions. Indeed, as shown in Table 2, our return data arestrongly exposed to autocorrelations and ARCH effects. Moreover, the stochastic properties of thefiltered returns are far suitable for further statistical analysis because they are approximately i.i.d.Accordingly, the copula parameter estimation and testing procedure will not suffer from statisticalbiases that arise when treating as i.i.d variables that are not. Roch and Alegre (2006) reach the sameconclusion. Our proposed semiparametric approach (i.e., combination of GARCH-type model andcopula functions) thus creates additional flexibility in that it offers the possibility to combine variousnonlinear models with a rich variety of available copula families.

4.2. Alternative GARCH-type processes

Oil returns and exchange rate returns may exhibit, in addition to serial dependence and volatilityclustering, asymmetry, leverage effects and long memory in conditional volatility. If these additionalfeatures are effectively present, our marginal specifications based on an AR(1)–GARCH(1,1) modelmight not be appropriate. To deal with this issue, we consider three alternative GARCH processes for

Table 10Tail dependence coefficients of the best copulas, applied to raw returns.

Return pairs lL lU Return pairs lL lU

USD/EUR-WTI – – USD/EUR-Brent 0.053 0.053USD/CAD-WTI – – USD/CAD-Brent – –

USD/GBP-WTI – – USD/GBP-Brent 0.035 0.035USD/CHF-WTI – – USD/CHF-Brent 0.028 0.028USD/JPY-WTI – – USD/JPY-Brent 0 0

Note: The table presents the upper and lower tail dependence coefficients, estimated from the copulas that provide the best fitfor the raw returns.

Table 11Sensitivity of empirical results to alternative GARCH specifications.

Return pairs EGARCH FIGARCH FIEGARCH

Copula – TDC Copula – TDC Copula – TDC

USD/EUR-WTI t-copula 0.017 t-copula 0.011 t-copula 0.014USD/CAD-WTI t-copula 0.000 t-copula 0.000 t-copula 0.000USD/GBP-WTI t-copula 0.005 t-copula 0.001 t-copula 0.004USD/CHF-WTI t-copula 0.014 t-copula 0.011 t-copula 0.014USD/JPY-WTI t-copula 0.007 t-copula 0.006 t-copula 0.006USD/EUR-Brent t-copula 0.012 t-copula 0.007 t-copula 0.009USD/CAD-Brent t-copula 0.003 t-copula 0.001 t-copula 0.002USD/GBP-Brent t-copula 0.004 t-copula 0.002 t-copula 0.003USD/CHF-Brent t-copula 0.006 t-copula 0.005 t-copula 0.007USD/JPY-Brent Clayton 0.000 Clayton 0.000 Clayton 0.000

Notes: The table presents the best fitting copula model and the tail dependence coefficients (TDC) using three alternative GARCHspecifications (EGARCH, FIGARCH and FIEGARCH). TDCs are symmetric regardless of the filtering models.


the marginals: the Exponential GARCH or EGARCH(1,1) model of Nelson (1991), the FractionallyIntegrated GARCH or FIGARCH(1, d, 1) model of Baillie et al. (1996), and the FIEGARCH(1, d, 1) ofBollerslev and Mikkelsen (1996). The EGARCH, FIGARCH and FIEGARCH allow to capturing respectiveasymmetric and leverage effects, long memory, and all these features of the return series. Table 11reports the best fitting copula models and the associated tail dependence coefficients with respectto different GARCH-type specifications. Overall, the best copula models selected by the alternativeGARCH specifications are fully similar to those chosen by our initial AR(1)–GARCH(1,1) filter. Theextreme dependence structure is also symmetric and in the meantime the values of tail dependencecoefficients are very close to the results we presented in the previous section. Notice that this findingcan be explained by the fact that in our study the marginals are nonparametrically estimated by usingthe CML method.

4.3. Normal versus crisis periods

Since our study period includes the recent financial crisis 2007–2010, the dependence structurebetween oil and foreign exchange markets may be different between normal and crisis periods.Moreover, when studying the links between exchange rate pairs, Malevergne and Sornette (2003)report that statistics of copula models vary with the correlation coefficient. It is thus opportune toexamine the potential impact of crisis period on market interdependencies. To do so, we divide ourstudy period into two subperiods referred to as normal period (4 January 2000 to 15 July 2008) andcrisis period (16 July 2008 to 17 February 2011), respectively. The 15th of July 2008 is chosen asbreakpoint because it marks the end of the continual increasing trend of oil prices since 2000 (Fig. 1) aswell as the beginning of the global economic recession period following the 2007 U.S. subprime crisis.10

Our GARCH-based copula model is then estimated separately for the two subperiods. Not surprisingly,we first find a sharp increase in cross-market comovement over the crisis period compared to thenormal one (Table 4). The Kendall’s tau and Spearman’s rho reach the highest values of 0.377 (USD/CAD-Brent) and 0.529 (USD/CAD-Brent), respectively. However, the pattern of comovement over thecrisis period does not differ from the whole period.

10 As a robustness check for the separation date between the two subperiods, we carried out the Chow forecast test to checkthe stability hypothesis for our return series by choosing 07/15/2008 as the breakpoint. Both forecast statistics (F-stat andLikelihood ratio) reject the null hypothesis of no structural change. The Bai and Perron (2003) testing procedure that examinesthe null hypothesis of no structural change against the alternative of multiple unknown breaks was also applied to all the returnseries over the period from January 2007 to December 2008. Our results indicate one breakpoint for every return series weconsider: Brent (07/11/2008), WTI (07/14/2008), USD/EUR (07/15/2008), USD/CAD (11/07/2007), USD/GBP (07/28/2008), USD/CHF (02/17/2008), and USD/JPY (08/27/2008). More importantly, the 15th of July 2008 is comprised between the 95% confi-dence intervals of all the breakpoints detected by Bai and Perron (2003) test. Note that a constant and a one-period lagged valueof the dependent variable are used as explanatory variables in the linear regression and all the test results can be madeavailable under request addressed to the Corresponding author.

Table 12Copula parameter estimates over the normal and crisis periods.


Normal period: 4 January 2000 to 15 July 2008USD/EUR-WTI 0.089***

(0.021)0.093*** (0.022),y ¼ 12.053*** (3.602)

0.563***(0.124)

1.341***(0.083)

1.059***(0.014)

0.084*** (0.024)

USD/CAD-WTI 0.126***(0.021)

0.126*** (0.021),y ¼ 303.999 (212.572)

0.766***(0.128)

1.467***(0.093)

1.068***(0.015)

0.112*** (0.027)

USD/GBP-WTI 0.128***(0.017)

0.130*** (0.019),y ¼ 15.739*** (5.258)

0.784***(0.110)

1.498***(0.081)

1.072***(0.013)

0.138*** (0.022)

USD/CHF-WTI 0.078***(0.021)

0.078*** (0.022),y ¼ 12.493*** (3.798)

0.458***(0.125)

1.268***(0.079)

1.048***(0.013)

0.085*** (0.024)

USD/JPY-WTI 0.062***(0.021)

0.061*** (0.022),y ¼ 15.860** (6.626)

0.342***(0.123)

1.198***(0.074)

1.034***(0.013)

0.073*** (0.024)

USD/EUR-Brent 0.098***(0.021)

0.099*** (0.022),y ¼ 16.865** (7.003)

0.573***(0.125)

1.343***(0.083)

1.053***(0.014)

0.115*** (0.025)

USD/CAD-Brent 0.144***(0.021)

0.144*** (0.021),y ¼ 113.994 (131.361)

0.841***(0.130)

1.512***(0.097)

1.074***(0.015)

0.145*** (0.027)

USD/GBP-Brent 0.079***(0.021)

0.077*** (0.022),y ¼ 24.915* (14.407)

0.421***(0.126)

1.239***(0.078)

1.033***(0.013)

0.097*** (0.024)

USD/CHF-Brent 0.102***(0.021)

0.102** (0.022),y ¼ 14.228* (4.900)

0.584***(0.125)

1.351***(0.084)

1.056***(0.014)

0.125*** (0.026)

USD/JPY-Brent 0.061***(0.021)

0.060*** (0.022),y ¼ 113.974 (529.786)

0.298**(0.128)

1.160***(0.074)

1.032***(0.013)

0.039*** (0.025)

Crisis period: 16 July 2008 to 17 February 2011USD/EUR-WTI 0.353***

(0.032)0.358*** (0.034),y ¼ 16.427 (10.571)

2.274***(0.245)

3.000***(0.335)

1.267***(0.035)

0.426*** (0.058)

USD/CAD-WTI 0.466***(0.028)

0.466*** (0.030),y ¼ 14.440 (9.131)

2.980***(0.261)

4.063***(0.451)

1.391***(0.041)

0.644*** (0.065)

USD/GBP-WTI 0.328***(0.033)

0.328*** (0.035),y ¼ 13.523* (8.191)

2.012***(0.238)

2.744***(0.302)

1.230***(0.035)

0.408*** (0.058)

USD/CHF-WTI 0.249***(0.036)

0.270*** (0.039),y ¼ 6.176*** (1.718)

1.724***(0.222)

2.472***(0.264)

1.188***(0.031)

0.291*** (0.048)

USD/JPY-WTI �0.192***(0.037)

�0.192*** (0.037),y ¼ 400.001 (2516.892)

�1.229***(0.226)

0.524***(0.058)

1.008***(0.009)

�0.104*** (0.022)

USD/EUR-Brent 0.423***(0.029)

0.424*** (0.030),y ¼ 399.998* (229.197)

2.799***(0.254)

3.720***(0.420)

1.366***(0.039)

0.489*** (0.055)

USD/CAD-Brent 0.587***(0.022)

0.585*** (0.023),y ¼ 30.042 (27.931)

4.017***(0.298)

5.942***(0.677)

1.586***(0.054)

0.848*** (0.068)

USD/GBP-Brent 0.389***(0.031)

0.392*** (0.034),y ¼ 10.228** (4.573)

2.486***(0.247)

3.333***(0.371)

1.308***(0.038)

0.482*** (0.054)

USD/CHF-Brent 0.332***(0.033)

0.339*** (0.036),y ¼ 10.175** (4.761)

2.157***(0.237)

2.923***(0.320)

1.260***(0.035)

0.374*** (0.050)

USD/JPY-Brent �0.198***(0.037)

�0.201*** (0.040),y ¼ 9.532** (4.350)

�1.210***(0.227)

0.531***(0.059)

1.008***(0.009)

�0.118*** (0.028)

Notes: The table summarizes the copula estimation results over the normal and crisis periods. The values in parenthesisrepresent the standard errors of the parameters. *, ** and *** indicate significance at the 10%, 5% and 1% levels respectively.


Tables 12–14 successively report the copula parameter estimates, the results of the goodness-of-fittests, and the tail dependence parameters for the two subperiods. Table 12 shows that all the estimateddependence parameters are highly significant over both the normal and the crisis periods. They are alsoall positive, except for the USD/JPY-WTI and USD/JPY-Brent pairs with respect to the Normal, Student-t,Frank and Clayton copulas over the crisis period.11 This finding confirms the comovement patterns we

11 The negative link between oil and USD/Yen exchange rate returns over the crisis period is again not unexpected. Indeed,while almost all currencies globally have depreciated against the U.S. Dollar between 2008 and 2010 due essentially to weakcurrent account positions, low foreign exchange reserves and capital outflows, the only exception is the Japanese Yenwhich hasappreciated substantially against the U.S. Dollar. The fact that investors in Japan and abroad unwind their Yen-carry trades byborrowing in the Yen at low interest rates to invest in assets denominated in higher-yielding currencies and repaying their Yenloans contributed largely to the Yen appreciation.

Table 13Results for the goodness-of-fit test of different copula functions for the pre and post-crisis period.


Normal period: 4 January 2000 to 15 July 2008USD/EUR-WTI 0.034 0.087 0.032 0.035 0.189 0.001USD/CAD-WTI 0.274 0.283 0.132 0.172 0.019 4.995e�4USD/GBP-WTI 0.079 0.108 0.179 0.182 0.007 0.014USD/CHF-WTI 0.140 0.198 0.072 0.082 0.301 0.016USD/JPY-WTI 0.070 0.177 0.019 0.015 0.105 0.110USD/EUR-Brent 0.596 0.858 0.309 0.323 0.066 0.481USD/CAD-Brent 0.912 0.914 0.412 0.477 0.017 0.004USD/GBP-Brent 0.362 0.449 0.163 0.193 0.056 0.854USD/CHF-Brent 0.581 0.822 0.361 0.349 0.119 0.236USD/JPY-Brent 0.627 0.647 0.502 0.494 0.923 0.168

Crisis period: 16 July 2008 to 17 February 2011USD/EUR-WTI 0.427 0.565 0.212 0.344 0.009 4.995e�4USD/CAD-WTI 0.510 0.584 0.023 0.106 0.003 4.995e�4USD/GBP-WTI 0.496 0.683 0.088 0.108 0.007 0.001USD/CHF-WTI 0.009 0.067 0.199 0.271 0.013 4.995e�4USD/JPY-WTI 0.081 0.066 0.012 0.005 4.995e�4 0.001USD/EUR-Brent 0.138 0.185 0.024 0.076 0.096 4.995e�4USD/CAD-Brent 0.003 0.004 4.995e�4 4.995e�4 0.025 4.995e�4USD/GBP-Brent 0.491 0.638 0.011 0.049 0.021 4.995e�4USD/CHF-Brent 0.260 0.343 0.301 0.431 0.064 4.995e�4USD/JPY-Brent 0.118 0.167 0.036 0.019 4.995e�4 0.001

Notes: The table presents the p-values of the goodness-of-fit test for the two subperiods. Large p-values (bold face numbers)indicate that the copula provides best fit to the data.


documented in Tables 4 and 5 that for the majority of market pairs, the oil price increases as the U.S.dollar depreciates. One may also note that the degree of dependence varies significantly across marketpairs and is much higher during the crisis period than during the normal one.

The goodness-of-fit test results, and particularly those obtained for the crisis period, also supportour results for the whole study period but to a lesser extent. Effectively, the Student-t copula continuesto provide the best description of the return dependence structure in five and six out of ten marketpairs over the normal and crisis periods, respectively. The Gumbel copula is selected three times overthe normal period but only one time over the crisis period. The Plackett copula is the best dependencemodel for one and twomarket pairs over the normal and crisis periods, respectively. For the remainingcopulas, they are selected only one time depending on the market period, except for the Frank copulawhich is never relevant. In comparison with the results for the whole period, the lower dominance of

Table 14Tail dependence coefficients of the best copulas for the pre and post-crisis period.

lL lU lL lU

Normal period: 01/04/2000 to 07/15/2008 Crisis period: 07/16/2008 to 02/17/2011

USD/EUR-WTI 0 0.076 USD/EUR-WTI 0.010 0.010USD/CAD-WTI 0.00 0.00 USD/CAD-WTI 0.031 0.031USD/GBP-WTI 0.00 0.00 USD/GBP-WTI 0.016 0.016USD/CHF-WTI 0 0.063 USD/CHF-WTI 0.00 0.00USD/JPY-WTI 0.001 0.001 USD/JPY-WTI 0 0USD/EUR-Brent 0.001 0.001 USD/EUR-Brent 0.00 0.00USD/CAD-Brent 0.00 0.00 USD/CAD-Brent 0.00 0.452USD/GBP-Brent 0.00 0 USD/GBP-Brent 0.048 0.048USD/CHF-Brent 0.003 0.003 USD/CHF-Brent 0.00 0.00USD/JPY-Brent 0 0.043 USD/JPY-Brent 0.002 0.002

Note: The table presents the upper and lower tail dependence coefficients estimated from the copulas that provide the best fitfor the two subperiods.


the Student-t copulas over the other copulas during the subperiods is naturally due to the differences inthe stochastic properties of the return data as well as the probability of observing extreme rareobservations.

Finally, the dependence structure is symmetric for almost all market pairs regardless of thesubperiod. It displays asymmetric patterns (upper tail dependence) in three cases over the normalperiod (USD/EUR-WTI, USD/CHF-WTI, and USD/JPY-Brent) and in only one case over the crisis period(USD/CAD-Brent). Compared to the normal period, the tail dependence has increased for four marketpairs: USD/CAD-WTI, USD/GBP-WTI, USD/CAD-Brent, and USD/GBP-Brent. Taken together, our findingssuggest that joint extreme comovements are more likely to occur during times of crisis.

5. Conclusion

This paper investigates the extreme comovement between daily crude oil prices and five U.S. dollarexchange rates using a copula-GARCH approach. We first employ a standard GARCH(1,1) processallowing for different error distributions to model the margins. The adoption of this filtering method ismotivated by the stylized empirical features of our return data including particularly serial dependenceand volatility clustering. Various copula models are then fitted to standardized residuals from themarginal models and their suitability is compared.

Overall, our in-sample results show that the Student-t copula is the best model for modeling theconditional dependence structure of the oil and exchange rate markets. They also provide evidence ofsignificant and symmetric tail dependence for all considered pairs, except for the USD/JPY-WTI andUSD/JPY-Brent. In addition, increases in crude oil prices are found to coincidewith a depreciation of thedollar. From a forecasting perspective, our portfolio simulations illustrate that taking the extremecomovement into account permits to improve the accuracy and precision of the market risk forecasts.Finally, our main results are globally robust to subperiods (normal versus crisis periods) as well as thechoice of alternative GARCH-type specifications allowing for asymmetry, leverage effects and longmemory in the conditional volatility processes, but are sensitive to the use of raw returns. Consistentwith previous studies focusing on financial market comovement, we notice an increase in extremedependence for several oil–exchange rate market pairs in times of crisis.

Acknowledgment

We are grateful to two anonymous referees, Shawkat Hammoudeh, and the co-editor (JoshuaAizenman) for their constructive comments and suggestions. As usual, all remaining errors are ours.

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Conditional dependence structure between oil prices and exchange rates: A copula-GARCH approach