exchange rates: a vine copula based garch method relationship between oil, stock prices and

8/15/2019 exchange rates: A vine copula based GARCH method Relationship between oil, stock prices and

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1

3 Relationship between oil, stock prices and

4 exchange rates: A vine copula based GARCH

5 method

6

7

8 Riadh Aloui a, Mohamed Safouane Ben Aïssa b,⇑

9 a LAREQUAD & ISGS, University of Sousse, Rue Abedelaziz El Bahi, B.P. 763, 4000 Sousse, Tunisia10 b LAREQUAD & FSEGT, University of Tunis El Manar, B.P. 248, El Manar II, 2092 Tunis, Tunisia

11

1 3a r t i c l e i n f o

14 Article history:15 Available online xxxx

16 Keywords:

17 Vine copulas18 Dependence measures

19 Crude oil price20 Stock index21 Exchange rate22

2 3

a b s t r a c t

24In this paper, we apply a vine copula approach to investigate the

25dynamic relationship between energy, stock and currency markets.

26Dependence modeling using vine copulas offers a greater flexibility

27and permits the modeling of complex dependency patterns for

28high-dimensional distributions. Using a sample of more than

2910 years of daily return observations of the WTI crude oil, the

30Dow Jones Industrial average stock index and the trade weighted

31US dollar index returns, we find evidence of a significant and sym-

32metric relationship between these variables. Considering different

33sample periods show that the dynamic of the relationship between

34returns is not constant over time. Our results indicate also that the

35dependence structure is highly affected by the financial crisis and

36Great Recession, over 2007–2009. Finally, there is evidence to sug-

37gest that the application of the vine copula model improves the

38accuracy of VaR estimates, compared to traditional approaches.

39 2016 Published by Elsevier Inc.

40

41

42

43 1. Introduction

44 Crude oil is one of the most important commodities in the current global world. Over the past

45 decade, the greater instability in energy markets and the persistence of oil prices at higher levels

http://dx.doi.org/10.1016/j.najef.2016.05.002

1062-9408/ 2016 Published by Elsevier Inc.

⇑ Corresponding author. Tel.: +216 58 450 000; fax: +216 71 872 277.

E-mail addresses: [email protected] (R. Aloui), [email protected] (M.S. Ben Aïssa).

North American Journal of Economics and Finance xxx (2016) xxx–xxx

Contents lists available at ScienceDirect

North American Journal of

Economics and Financej o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / ec o fi n

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Please cite this article in press as: Aloui, R., & Ben Aïssa, M. S. Relationship between oil, stock prices and exchange

rates: A vine copula based GARCH method. North American Journal of Economics and Finance (2016), http://dx.doi.

org/10.1016/j.najef.2016.05.002

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46 are largely responsible of the slowing world economic growth (Aydin & Mustafa, 2011; Sanchez,

47 2011). Through the increasing importance of oil price in the economic activity, the study of the rela-

48 tionship between energy, stock and currency markets becomes of greater importance for policy mak-

49 ers, economists and investors.

50 During the last financial crisis, oil prices experienced very large fluctuations as a clear structural

51 change around the second quarter of 2008 is apparent in Fig. 1. In fact, the spot price of crude oil

52 had a very sharp increase, rising from 20$ per barrel at the beginning of 2002 to 147$ per barrel in

53 July 2008, surpassing its 1980 record high in constant prices. Recent unrest in North Africa and the

54 Middle East and fears about the spread of political instability to other major oil producing countries

55 have contributed to higher oil prices and added more instability to energy markets.

56 Economic theory suggests that oil shocks have a significant effect on the stock market activity and

57 exchange rate movements. Huang, Masulis, and Stoll (1996) argue that the impact of crude oil move-

58 ments on stock markets can be completely explained by their effect on current and future real cash

59 flows. Many recent papers found that an increase in oil prices implies a decrease in stock returns

60 (Chiou & Lee, 2009; Miller & Ratti, 2009; Nandha & Faff, 2008; Park & Ratti, 2008). By now, this idea

61 has become widely accepted in the literature and seems to be virtually axiomatic. More recent studies

62 such as Arouri and Nguyen (2010) and Fayyad and Daly (2011) demonstrate that the impact of oil on63 stock markets is sensitively different across economic sectors (e.g., oil versus non-oil industries) and

64 across countries (e.g., net oil-exporting versus net oil-importing ones). According to Bjornland (2009)

65 and Jimenez-Rodriguez and Sanchez (2005), a positive association between oil price movements and

66 stock market returns is expected in the case of an oil exporting country, as the country’s income will

67 increase. It follows an increase in expenditures and investments which in its turn create more employ-

68 ment opportunities and the value of stocks will go up.

69 Studying the relationship between energy and currency markets has also received considerable

70 attention in the literature. The importance of oil prices as an explanatory variable of exchange rate

71 movements has been well documented in Krugman (1983), Golub (1983) and Rogoff (1991). In fact,

72 the influence of high oil prices on export competition and price level of a country will lead to frequent

73 and uncertain changes in the exchange rate. Moreover, oil prices are denominated in U.S dollar, and so74 fluctuations in the exchange rate cause changes to the crude oil supply, demand and price. Using dif-

75 ferent datasets, the existing empirical studies have mainly found that the oil price increase is associ-

76 ated with a dollar appreciation (Aloui, Ben Aïssa, & Nguyen, 2013; Ding & Vo, 2012; Wu, Chung, &

77 Chang, 2012). By contrast, some other studies demonstrate a negative relationship between oil prices

78 and the U.S. dollar exchange rates (Narayan, Narayan, & Prasad, 2008; Zhang, Fan, Tsai, & Wei, 2008).

79 The inconsistency in empirical findings can be explained by the distinct features of the investigated

80 countries and the different extent of the used datasets. In this paper, our objective is to investigate

81 whether the relationship between oil, stock and exchange rate is positive, negative or unclear. To over-

82 come the limitation of pair dependence analysis, which is evident in the related literature, we examine

83 the relationship between oil, stock and exchange rate in a multivariate framework. As pointed out by a

84 number of studies, it is important to understand the dependence between several variables interacting85 simultaneously, not in isolation of one another. The omission of one important variable in the

86 extended system can be misleading because the channel through which the two other variables are

87 connected is omitted from the incomplete system.

88 As documented, for example, by Jondeau and Rockinger (2006), Junker, Szimayer, and Wagner

89 (2006) and McNeil, Frey, and Embrechts (2005), the widely used measure of dependence, known as

90 the Pearson correlation coefficient, may not appropriately describe the type of dependence between

91 returns and, consequently, could lead to underestimate the joint risk of extreme events. In order to

92 overcome this problem, the use of the copula methodology may be a very promising solution to char-

93 acterize the multivariate distributions of asset returns. While there is a large literature exploring

94 dependence using bivariate copulas, the choice is much more restricted in the multivariate case.

95 The two most popular choices allowing multivariate dependence to be modeled with a non-96 restricted correlation matrix are the normal and the Student-t copulas. However, these models are

97 restrictive in the tail and they do not allow asymmetric dependence. Recently, Bedford and Cooke

98 (2001) and Bedford and Cooke (2002) introduced vine or pair-copula construction of multivariate dis-

99 tribution. These models are flexible graphical models enabling the extensions to higher dimensions

2 R. Aloui, M.S. Ben Aïssa/ North American Journal of Economics and Finance xxx (2016) xxx–xxx


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100 using a cascade of bivariate copula. The great advantage of vine-copula is that we can select bivariate

101 copulas from a wide range of existing copula families.

102 In this paper, we apply a vine copula approach to shed new light on the dynamic relationship

103 between crude oil price movements, U.S. market stock prices and U.S. dollar exchange rate. Moreover,

104 on the basis of this approach, we attempt to identify changes in the structure of crude oil prices and

105 what implications these change have on the dependence between the three markets. Using daily time-

106 series of crude-oil spot prices, Dow Jones Industrial Average (DJIA) stock index and nominal exchange

107 rate for the trade weighted U.S dollar index, we mainly find a significant and symmetric relationship

108 between these variables. However, this relationship is not constant across sample sub-periods. More

109 importantly, we find that changes in the structure of crude oil returns affect significantly the connec-

110 tion between the considered return series.

111 We structure the rest of the article as follows. Section 2 discusses the economic relationship

112 between oil, stock prices and exchange rates. Section 3 describes the empirical methodology and

113 the estimation strategy. Section 4 presents the used data and discusses our empirical results. Section 5

114 provides some concluding remarks.

115 2. Relationships between oil, stock prices and exchange rates

116 Theoretically, oil price movements affect stock returns in several ways. The value of a company’s

117 stock at any point in time can be measured by making the sum of all expected future cash flows dis-

118 counted back to the present using the discount rate (Huang et al., 1996), meaning that oil price shocks

119 can affect stock returns directly via the expected cash flows or indirectly by impacting the discount

120 rate. Since energy is an essential input to the production process, then higher oil prices lead to the

121 increase of production cost and reduce in the amount of the expected profits for non-oil related

122 companies. At the same time, expected cash flow will drop and company’s stock prices will be123 affected. As the number of the companies with falling stock prices go higher; stock index, reflecting

124 the performance of the whole stock market, will go down. On the other side, oil price increase is

125 expected to raise the overall trade deficit for oil importing countries. A growing trade deficit will gen-

126 erate expectations of future depreciation of the current exchange rate accompanied by higher inflation

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013

2 0

4 0

6 0

8 0

1 0 0

1 2 0

1 4 0

Fig. 1. Dynamics of daily prices of West Texas Intermediate (US Dollar/Barrel).

R. Aloui, M.S. Ben Aïssa / North American Journal of Economics and Finance xxx (2016) xxx–xxx 3


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127 rate. Consequently, the rise in the inflation rate may cause the discount rate to rise and stock prices to

128 fall.

129 The interaction of inflation and monetary policy exerts also a great influence on interest rates.

130 Assuming that other prices are sticky downwards, higher oil prices allow the rise on the domestic

131 price level which in its turn implies higher inflation and interest rates. In addition, hurdle rate which

132 means the required rate of return that an investment manager demands to undertake a particular pro-

133 ject, will tend to rise leading to a low level of investments. Consequently, profitable projects will be

134 turned down and the stock price will be affected.

135 Several empirical works by Arouri, Lahiani, and Bellalah (2010), Basher and Sadorsky (2006), Boyer

136 and Filion (2007), Hammoudeh, Dibooglu, and Aleisa (2004), Huang et al. (1996), Jones and Kaul

137 (1996), Miller and Ratti (2009), Papapetrou (2001), Park and Ratti (2008) and Sadorsky (2001) have

138 shown that oil price shocks affect stock returns.

139 The relationship between oil prices and exchange rates has also received much attention in liter-

140 ature (see, among others, Amano & van Norden, 1998; Chen & Chen, 2007; Golub, 1983). A frequently

141 given explanation is based on the potential impact of oil shocks in driving term of trade movements,

142 which would therefore justify the effect on the exchange rate. Amano and van Norden (1998) consider

143 a small open economy with two-sectors for tradable (oil) and non-tradable goods (labour). The output144 price of the tradable sector is set on the world markets, while the real exchange rate is determined by

145 the output price in the non-tradable sector. Consequently, an increase in oil prices leads to a decrease

146 in the labor price in order to improve the competitiveness of the tradable sector. Assuming that the

147 production in the non-tradable sector is more energy-intensive, the output price of this sector will

148 tend to rise causing the real exchange rate to appreciate. Another explanation for the link between

149 oil prices and exchange rates focuses on the balance of payments and the international portfolio

150 choices (Golub, 1983). In this approach, a surge in oil prices generates wealth transfers from oil

151 importing economies to oil exporting ones (like OPEC), leading to adjustments in exchange rates.

152 The final impact of oil shocks on exchange rate depends on the distribution of oil imports across oil

153 importing economies and on portfolio choices of both OPEC and oil-importing countries. If wealth

154 reallocation due to oil price increase is the outcome of an excess supply of dollars, then the dollar will155 depreciate. Extending this approach, Krugman (1983) uses a dynamic partial equilibrium framework

156 to model how OPEC uses the accumulated wealth of their oil exports in dollars. Assuming that OPEC

157 will progressively spend its surpluses to import more goods from developing countries, the long-run

158 effect of an oil price hike on the dollar exchange rate will depend on the weight of oil in the U.S. total

159 imports compared to the U.S. weight in OPEC’s imports. On the short run, the effect depends on the U.

160 S. weight in the global oil imports compared to their weight in the dollar-denominated assets held by

161 OPEC.

162 The above discussion highlights the fact that there are strong theoretical arguments for why oil

163 price shocks should affect stock prices and exchange rates. Eventually, empirical analysis is needed

164 to obtain new insights into the relationship between these three variables. Also, It would be of further

165 interest to explore the changing dynamics of this relationship after the last financial crisis.

166 3. Empirical methodology

167 In this section, the vine copula construction method and some useful concepts that are necessary to

168 understand this approach are explained.

169 3.1. Pair-copula construction

170 Introduced first by Joe and Xu (1996) and extended by Bedford and Cooke (2001, 2002), vine

171 copulas are flexible graphical models enabling the extensions to higher dimensions using a cascade172 of bivariate copulas or pairs-copulas. The modeling scheme is based on a decomposition of a

173 multivariate probability density into dðd 1Þ=2 bivariate copula densities, which may be chosen

174 independently from the others to allow for a wide range of dependence structure. Two main classes

175 of pair-copulas have been treated in literature, C- and D-vine copula models. Let us illustrate the



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176 pair-copula construction for three dimensions. Consider three random variables X ¼ ð X 1; X 2; X 3Þ with

177 marginal distribution functions F 1; F 2 and F 3 and corresponding densities. One possible representation

178 of the joint density is

179

f ð x1;

x2;

x3Þ ¼ f 1ð x1Þ f ð x2j x1Þ f ð x3j x1;

x2Þ ð1Þ181181

182 According to the Sklar theorem, we know that the joint density can be decomposed further into

183 univariate marginal densities and a copula density. It follows for the conditional density of x2 given

184 x1 that

185

f ð x2j x1Þ ¼ f ð x1; x2Þ

f 1ð x1Þ ¼

c 1;2ðF 1ð x1Þ; F 2ð x2ÞÞ f 1ð x1Þ f 2ð x2Þ

f 1ð x1Þ ¼ c 1;2ðF 1ð x1Þ; F 2ð x2ÞÞ f 2ð x2Þ ð2Þ187187

188 For three random variables X 1; X 2 and X 3, we have

189

f ð x3j x1;

x2Þ ¼

f ð x2; x3j x1Þ

f ð x2j x1Þ ¼

c 2;3j1ðF ð x2j x1Þ; F ð x3j x1ÞÞ f ð x2j x1Þ f ð x3j x1Þ

f ð x2j x1Þ

¼ c 2;3j1ðF ð x2j x1Þ; F ð x3j x1ÞÞc 1;3ðF 1ð x1Þ; F 3ð x3ÞÞ f 3ð x3Þ ð3Þ191191

192 Thus, the three-dimensional joint density (1) can be represented in terms of bivariate conditional

193 copulas and marginal densities

194

f ð x1; x2; x3Þ ¼ c 2;3j1ðF ð x2j x1Þ; F ð x3j x1ÞÞc 1;2ðF 1ð x1Þ; F 2ð x2ÞÞc 1;3ðF 1ð x1Þ; F 3ð x3ÞÞ f 1ð x1Þ f 2ð x2Þ f 3ð x3Þ ð4Þ196196

197 For high-dimensional distributions, there are a large number of possible pair-copula decomposi-

198 tions. Therefore, Bedford and Cooke (2001) introduced a graphical model called regular vine to help

199 organize them. In this paper, we concentrate on two special cases of regular vines; the C- and D-

200 vine (Kurowicka & Cooke, 2004). In a canonical vine structure, each tree has a unique node that is con-201 nected to all other nodes of the tree. The intuition behind this is that one variable plays an essential

202 role in the dependency structure, thus all other variables are connected to it. On the other hand, D-

203 vines are uniquely characterized through their first tree which has a path structure. Therefore the

204 order of variables in the first tree defines the complete D-vine tree sequence.

205 3.2. Sequential estimation method

206 Having decided the structure of R-vine to be used and the copula families for each pair and condi-

207 tional pair of variables, the parameters are estimated sequentially starting from the first tree via max-

208 imum likelihood (see, e.g., Czado, Min, Baumann, & Dakovic, 2009). The log-likelihood function for the209 C-vine copula is given by

210

lCV ðhCV juÞ ¼XN

k¼1

Xd1

i¼1

Xdi

j¼1

log½c i;iþ jj1:ði1ÞðF ij1:ði1Þ; F iþ jj1:ði1Þjhi;iþ jj1:ði1ÞÞ212212

213 where hCV denotes the parameter set of the C-vine copula, F jji1:im :¼ F ðukjjuk;i1;...;uk;im Þ and the marginal

214 distributions are uniform. Similarly, the log-likelihood function for the D-vine copula is

215

lDV ðhDV juÞ ¼XN

k¼1

Xd1

i¼1

Xdi

j¼1

log½c j; jþijð jþ1Þ:ð jþi1ÞðF jjð jþ1Þ:ð jþi1Þ; F jþijð jþ1Þ:ð jþi1Þjh j; jþijð jþ1Þ:ð jþi1ÞÞ217217

218 The parameters of the C- and D-vine copulas are estimated using maximum likelihood estimation

219 method.



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220 4. Data and results

221 4.1. Data and stochastic properties

222 In this section, we empirically investigate the relationship between oil, stock and currency markets

223 over the period from January 4, 2000 to May 31, 2013. Daily spot prices on West Texas Intermediate

224 (WTI) are used to represent the energy market, since this benchmark is closely related to other crude

225 oil markers such as those for Brent and Dubai. For stock markets, the Dow Jones Industrial Average

226 (DJIA) index was chosen to represent the US stock market as it is a broad-based stock index. Moreover,

227 exchange rate corresponds to the trade weighted US dollar (TWEXB) index, measuring the movement

228 of dollar against the currencies of a broad group of major U.S. trading partners. Higher values of the

229 TWEXB index indicate an appreciation of the US dollar.

230 For our analysis, we consider log-returns that are computed as r t ¼ lnðP t =P t 1Þ from the original

231 price series. The time-paths of return series over the study period are plotted in Fig. 2. According to

232 the ADF and PP tests, the logarithmics series are all stationary at the 1% significance level. The descrip-

233 tive statistics of the log-return data are presented in Table 1. We can see that the average log-return is

234 positive for the crude oil and stock index. Regarding the variance, the WTI crude oil has the highest

235 variability compared to the other two variables. Furthermore, skeweness values are negative for crude

236 oil and US dollar index and positive for DJIA index indicating that it is more likely to observe large neg-

237 ative returns on crude oil and currency markets. Return series are leptokurtically distributed in view of

238 significant excess kurtosis. These findings clearly show that return series depart from normality and

239 that the probability of extremely negative and positive realizations for our returns is thus higher than

240 that of a normal distribution. The departure from normality is confirmed by the Jarque–Bera test. The

241 Ljung–Box Q-statistics of order 12 show the existence of autocorrelation in all return series. Moroever,

242 the Ljung–Box statistics of order 12 applied to squared returns are highly significant. Finally, the

243 results of the Lagrange Multiplier test for conditional heteroscedasticity point to the presence of ARCH

244 effects in the return data, thus supporting our decision to use a GARCH model to filter the daily

245 returns.246 We first filter the returns using the GJR-GARCH model proposed by Glosten, Jagannathan, and

247 Runkle (1993) which has several advantages over standard GARCH model.1 The aim is to obtain

248 approximately i.i.d series suitable for copula estimation, while controlling the effects of conditional

249 heteroskedasticity and asymmetry. The resulting filtered returns are then transformed into uniform vari-

250 ates by applying the probability integral transform to each marginal distribution. The scatterplot of the

251 copula data (uniform variates) and the corresponding contour plots with standard normal margins are

252 displayed in Fig. 3. The estimated Kendall’s tau are equals to 0.083, 0.16 and 0.082 for the WTI-

253 DJIA, WTI-TWEXB and DJIA-TWEXB pairs respectively. It follows that the dependence is negative for

254 two pairs: WTI-TWEXB and DJIA-TWEXB and positive for only one pair, the WTI-DJIA.

255 Dißmann, Brechmann, Czado, and Kurowicka (2013) suggest selecting the vine structure using

256 maximum spanning trees with absolute values of pairwise Kendall’s taus as weights. Using this tree257 selection algorithm suggests to choose the variable WTI as root in the C-vine. The node order of the

258 first tree is determined as the following: WTI, TWEXB and DJIA. In the next step, adequate pair-

259 copula families associated with the C-vine structure selected in the previous step have to be identified.

260 We select a copula family among the Gaussian, Student-t, Clayton, Gumbel, Frank, Joe, BB1, BB6, BB7,

261 BB8, Survival Clayton, Survival Gumbel, Survival Joe, Survival BB1, Survival BB6, Survival BB7 and Sur-

262 vival BB8 copula, which cover a wide range of dependence structures.2 For pairs with negative depen-

263 dence the choice of a copula model is limited to the Gaussian, Student-t, Frank and rotated version of the

264 Clayton, Joe, BB1, BB6, BB7 and BB8 copulas. The selection of bivariate copula models is based on the AIC

265 and the BIC information criterions corrected for the numbers of parameters used in the models (Manner,

266 2007; Brechmann, 2010). Since the choice of copula models in the first tree have a greatest influence on

1 The optimal lag length for the conditional mean and variance processes of the GJR-GARCH model was determined with respect

to the AIC and BIC.2 Following Joe (1997), the two-parameters copulas namely Clayton–Gumbel, Joe–Gumbel, Joe–Clayton and Joe–Frank are

simply referred as BB1, BB6, BB7 and BB8, respectively.



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267 the global fit of the R-vine model, we apply two goodness of fit tests based on scoring approach of Vuong

268 (1989) and Clarke (2007). Both tests are likelihood-ratio-based tests for model selection using the Kull-

269 back–Leibler information criterion. The results of the scoring test, the AIC and the BIC suggest choosing

270 the Student-t copula for all the pairs of the first tree. Furthermore, we also look at the k-function intro-271 duced by Genest and Rivest (1993) to check the adequacy of the selected bivariate copula family. Com-

272 paring empirical to theoretical k-functions in Fig. 4, we can see that the Student-t copula fits the

273 empirical data of the two pairs WTI-TWEXB and WTI-DJIA remarkably well.

WTI crude oil

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013

- 0 .

1 5

0 .

0 5

DJIA index

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 - 0 .

0 8

0 .

0 8

USD trade-weighted index

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 - 0 .

0 2 0

0 .

0 1 5

Fig. 2. Daily returns on crude oil, stock index and USD trade-weighted index.

Table 1

Descriptive statistics.

Min Mean Max Std Dev Skewness Ex. kurtosis

Panel A

WTI 0.171 3.770e04 0.164 0.025 0.267 4.697

DJIA 0.082 9.367e05 0.105 0.012 0.006 7.422

TWEXB 0.023 3.397e05 0.017 0.003 0.034 4.089

Q ð12Þ Q 2(12) J-B ARCHð12Þ

Panel B

WTI 40.796⁄ 1051.918⁄ 3149.051⁄ 413.988⁄

DJIA 61.078⁄ 3306.721⁄ 7766.256⁄ 976.909⁄

TWEXB 32.968⁄

1316.654⁄

2356.926⁄

492.051⁄

Notes: The table displays summary statistics for daily crude oil, DJIA stock index and TWEXB returns. The sample period is from

January 4, 2000 to May 31, 2013. Q(12) and Q^2(12) are the Ljunk–Box statistics for serial correlation in returns and squared

returns for order 12. JB is the empirical statistic of the Jarque–Bera test for normality. ARCH is the Lagrange multiplier test for

autoregressive conditional heteroskedasticity.⁄ Indicates the rejection of the null hypotheses of no autocorrelation, normality and homoscedasticity at the 1% level of

significance.



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274 Having selected adequate copula families for all variable pairs, we estimate the corresponding cop-

275 ula parameters using the sequential method. A preliminary bivariate independence test based on Ken-

276 dall’s tau (Genest & Favre, 2007) is performed to identify possible independent conditional variable

277 pairs. If the p-value of the test is larger than 5% then the independence copula is chosen. Otherwise

278 the sequential estimation method is left unchanged. To improve the estimation results, the parame-

279 ters obtained from the sequential method are used as starting values to determine the corresponding

280 MLE estimates. Results of the parameters estimation are displayed in Table 2. We can see that all esti-

281 mated parameters of the conditional and unconditional copulas are significant at 1% significance level.

282 The strongest negative dependence is between WTI and TWEXB as shown by Kendall’s tau value. The283 tail dependence estimates show that, in the Student-t copula, the unconditional pair WTI-DJIA shows

284 strong dependence in the tails. The dependence in the tail is also symmetric for the three pairs. Sim-

285 ilarly, we fitted a D-vine copula model to the return data and reported the results in Table 3. As noted

286 above, the Student-t copula provide the best fit for all conditional and unconditional pairs. The C- and

287 D-vine structures share one unconditional pair in common, WTI-TWEXB and produce identical Ken-

288 dall’s tau and tail dependence estimates for this pair. The dependence is also negative between TWEXB

289 and DJIA. For the tail dependence measures, the D-vine specification substantially shows stronger

290 lower and upper tail dependence in the conditional pair WTI-DJIA—TWEXB. Thus, we can conclude

291 that given the TWEXB as the condition reduce by more than half the lower and upper tail dependence

292 between WTI and DJIA, i.e., the information of currency market can help investors reduce significantly

293 the tail dependence between stock and oil markets.294 In order to compare the two fitted vine-copula models, we calculate the loglikelihood, AIC, BIC and

295 p-values for Vuong (1989) test in Table 4. According to the loglikelihood, Akaike and Bayesian Infor-

296 mation criteria, the C-vine copula model produces better fit, with little difference between the two

Fig. 3. Pairs plot of the copula data formed from the transformed standardized residual returns with scatter plots (top, right)and the corresponding contour plots (bottom, left).



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0.0 0.2 0.4 0.6 0.8 1.0

− 0 .

6

− 0 .

4

− 0 .

2

0 .

0

WTI−TWEXB

v

λ

( v )

0.0 0.2 0.4 0.6 0.8 1.0

− 0 .

4

− 0 .

2

0 .

0

WTI−DJIA

v

λ ( v )

Fig. 4. The empirical lambda function (black line) and its confidence bands (dashed lines) corresponding to independence and

comonotonicity (k ¼ 0) are presented together with the fitted lambda functions for the Student-t copula (grey line) for the two

pair of the first tree.

Table 2

C-vine copula estimation results.

Copula Parameters (SE) Kendall’s s Tail dependence

hTWEXBWTI Student-t 0.242 m ¼ 12:702 0.156 kU ¼ kL ¼ 0:033e02

ð0:017Þ⁄ (3.039)⁄

hDJIAWTI Student-t 0:127 m ¼ 7:978 0.081 kU ¼ kL ¼ 2:707e02

(0.018)⁄ (1.245)⁄

hDJIATWEXBjWTI Student-t 0:114 m ¼ 14:964 0.072 kU ¼ kL ¼ 0:038e02

(0.018)⁄ (4.407)⁄

Notes: The table summarizes the C-vine copula estimation results over the overall sample. The values in parenthesis represent

the standard error of the parameters.⁄ Indicates significance at the 1% level.

Table 3

D-vine copula estimation results.


hWTI TWEXB Student-t 0:242 m ¼ 12:702 0.156 kU ¼ kL ¼ 0:033e02

(0.017)⁄ (3.039)⁄

hTWEXBDJIA Student-t 0:135 m ¼ 9:539 0.086 kU ¼ kL ¼ 0:36e02

(0.018)⁄ (1.862)⁄

hWTI DJIAjTWEXB Student-t 0:093 m ¼ 9:991 0.059 kU ¼ kL ¼ 1:166e02(0.018)⁄ (1.884)⁄

Notes: The table summarizes the D-vine copula estimation results over the overall sample. The values in parenthesis represent




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297 specified vine structures. Under the null hypothesis that the C- and D-vine copula models are statis-

298 tically equivalent, the Vuong test, failed in distinguishing between the two models. We conclude that

299 the both vine specifications are suitable for describing the multivariate dependence between returns

300 and can provide additional insights due to their specific structures.

301 4.2. Structural changes in crude oil

302 In this section, we will investigate whether structural changes have occurred in the dynamic

303 relationship between oil, stock and currency markets. We only deal with the C-vine copula which is

304 simpler and more comprehensive. Since crude oil is selected as root variable for all unconditional

305 pair-copulas in the C-vine, we attempt to identify changes in the structure of crude oil prices and

306 when these changes occur. Moreover, we try to identify what implications these change have on

307 the dependence between the three variables of interest.

308 To test whether crude oil data contain one or more structural break, we considered the Bai and

309 Perron (2003) test allowing us to test for an unknown number of breakpoints at unknown dates.

310 The results of the test indicate that there are five potential breaks at the following dates:

311 18/01/2002, 26/10/2004, 18/01/2007, 12/02/2009 and 08/04/2011.3 In fact, oil price spikes after major

312 world events such as the Afghanistan war in 2002, the Great Recession over2007–2009, and the European

313 Debt crisis in 2011. To examine the potential impact of oil shocks on market interdependencies, we

314 divide our study period into six sub-periods as follow: from 4 January 2000 to 17 January 2002, from

315 18 January 2002 to 25 October 2004, from 26 October 2004 to 17 January 2007, from 18 January 2007

316 to 11 February 2009, from 12 February to 7 April 2011 and from 8 April 2011 to 31 May 2013. The

317 C-vine copula model is then estimated separately for each sub-period under the assumption that WTI

318 is the root variable. Estimation results are reported in Table 5.

319 The reported results show that the tail dependence, the level and the structure of dependence are

320 changing between pre-crisis and post-crisis periods. In fact, it can be seen that all pairs are indepen-

321 dents during the first sub-period. After the financial crisis of 2009, we notice that all conditional and

322 unconditional pairs became significantly dependents.323 In particular, the dependence between WTI and TWEXB is significantly negative for all sub-periods

324 except the first one. The negative dependence between the two variables reaches its lower level, dur-

325 ing the post-global financial crisis, over 2009–2011. The Rotated Gumbel copula (90) provides the

326 best description of the dependence structure over this sub-period. Recall that the rotation by 90

327 and 270 degrees allows for the modeling of negative dependence.

328 The dependence between WTI and DJIA became apparent after the financial crisis of 2009. Kendall’s

329 tau values are positives and equal to 0.318 and 0.358. It turns out that an increase in the price of crude

330 oil is associated with an appreciation of the stock prices. We note that this pair show the highest

331 degree of tail dependence during bear and bull markets. This is not surprising as we expect that the

332 tail dependence increases during periods of extreme turbulence.

Table 4

Comparison of the C-vine and D-vine.

C-vine D-vine

LogLik 185.68 184.42

AIC 359.363 356.839

BIC 322.583 320.059

Vuong test 0.427

Notes: The table reports the loglikelihood value, the AIC, the BIC and p-value of the Vuong test for

the C-vine and D-vine copula models.

3 A structural break is considered significant if its F-statistic scaled by the number of varying regressors is higher than the Bai–

Perron critical value. Note that a constant and a one-period lagged value of the dependent variable are used as explanatory

variables in the linear regression and all the test results can be made available under request addressed to the Corresponding

author.



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333 The conditional pair DJIA-TWEXB—WTI shows a relatively small degree of dependence and fluctu-

334 ates within a range of 0.191 and 0.062. The tail dependence coefficients are equal to zero in most

335 cases. We conclude that the information of oil market can help investors reducing significantly the tail

336 dependence between stock and currency markets.

337 4.3. Value at risk

338 In this section, we illustrate the use of the C-vine copula model in quantifying the risk of an equally339 weighted portfolio, composed of the WTI, TWEXB and DJIA returns. We indeed consider the Value-at-

340 Risk (VaR) as the portfolio’s market risk measure and estimate it using Monte Carlo simulations.

341 Value-at-Risk is one of the most popular measures for market risk assessment, defined by the maxi-

342 mum loss in a portfolios value with a given probability over a given time period.

Table 5

Sub-periods estimation results.


4 January 2000 to 17 January 2002

hTWEXBWTI Indep. – – – –

hDJIAWTI Indep. – – – –

hDJIATWEXBjWTI Indep. – – – –

18 January 2002 to 25 October 2004

hTWEXBWTI Rotated Gumbel ð90Þ -1.078 – -0.072 kL ¼ kU ¼ 0

(0.026)⁄

hDJIAWTI Indep. – – –

hDJIATWEXBjWTI Survival Gumbel 1.066 – 0.062 kL ¼ 0:084; kU ¼ 0

(0.025)⁄

26 October 2004 to 17 January 2007

hTWEXBWTI Rotated Gumbel ð270Þ 1.108 0.098 kL ¼ kU ¼ 0

(0.032)⁄

hDJIAWTI Rotated Gumbel ð270Þ 1.086 – 0.079 kL ¼ kU ¼ 0

(0.031)⁄

hDJIATWEXBjWTI Indep. – – – –

18 January 2007 to 11 February 2009

hTWEXBWTI Gaussian 0.307 – 0.198 kL ¼ kU ¼ 0

(0.035)⁄

hDJIAWTI Indep. – – –

hDJIATWEXBjWTI Frank 0.537 – 0.059 kL ¼ kU ¼ 0

(0.266)⁄

12 February 2009 to 8 April 2011

hTWEXBWTI Rotated Gumbel ð90Þ 1.500 – 0.333 kL ¼ kU ¼ 0

(0.050)⁄

hDJIAWTI Student-t 0.479 6.057 0.318 kL ¼ kU ¼ 0:158

(0.034)⁄ (1.591)⁄

hDJIATWEXBjWTI Rotated BB7 ð90Þ 1.175 0.276 0.191 kL ¼ kU ¼ 0

(0.056)⁄ (0.062)⁄

8 April 2011 to 31 May 2013

hTWEXBWTI Student-t 0.334 7.535 0.217 kL ¼ kU ¼ 0:003

(0.043)⁄ (2.726)⁄

hDJIAWTI BB1 0.532 1.231 0.358 kL ¼ 0:347; kU ¼ 0:244

(0.104)⁄ (0.065)⁄

hDJIATWEXBjWTI Frank 1.334 – 0.146 kL ¼ kU ¼ 0

(0.262)⁄

Notes: The table summarizes the C-vine copula estimation results over the four subperiods. The values in parenthesis represent




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343 Methodologically, our procedure for computing the VaR requires the following steps. First, we esti-

344 mate the whole model (GJR-GARH + C-vine copula) using a window of 1697 observations. Then, we

345 simulate 10.000 random trials of uniform variates from the C-vine copula and transform them into

346 standardized residuals by inverting the marginal CDF of each series. Finally, we reintroduce the auto-

347 correlation and heteroskedasticity observed in the original return series, compute the value of the con-348 sidered portfolio and estimate the VaR. This procedure is repeated until the last observation, and we

349 compare the estimated VaR with the actual next-day change in the portfolio’s value. The estimation

350 and simulation from the C-vine copula are repeated only once in every 50 observations owing to

351 the computational cost of this procedure. However, at each new observation, the VaR estimates are

352 modified because of changes in the GJR-GARCH parameters.

353 For comparison purposes, we also estimate the VaR using four other approaches: the multivariate

354 gaussian copula, the multivariate Student-t copula, the normal and the historical simulation methods.

355 For the normal and historical simulation methods, the model parameters are updated for every obser-

356 vation. The results for the backtesting are reported in Table 6. Note that a model is said to be best sui-

357 ted for calculating VaR is the one with the number of exceedances closest to the expected number of

358 exceedances. We can see that The C-vine copula provides better VaR forecasts at the 99% confidence

359 level followed by the multivariate Student-t and Gaussian copulas. We also use the Kupiec (1995) pro-

360 portion of failures (POF) test to check the robustness of the VaR estimates. According to the backtest

361 results, all copula based models perform well for the considered portfolio. However, the accuracy of

362 the VaR estimates from the historical simulation and normal methods are rejected, since the p-

363 values are inferior to the 5% significance level. Overall, our results thus confirm the relevance of the

364 C-vine copula model.

365 5. Conclusion

366 In contrast to previous literature, this paper investigates the multivariate dependence between367 crude oil, exchange rate and stock returns using a vine copula approach. We first employ a

368 GJR-GARCH model with skewed-t distribution to filter the return series and construct their marginal

369 distributions. The C- and D-vine copulas are then fitted to filtered returns and their suitability is

370 compared. After detecting structural change in crude oil returns, the C-vine copula model is estimated

371 separately for each sub-period to identify what implications these changes have on the dependence

372 between the three markets. We finally show the implications of the empirical findings for risk man-

373 agement issues related to an equally weighted oil, stock and exchange rate portfolio within a VaR

374 framework.

375 Our results show that both vine specifications are well suited for modeling the multivariate depen-

376 dence between the return series over the entire sample period. It can also be concluded that an

377 increase of crude oil price is associated with a depreciation of exchange rate and an appreciation of 378 stock market prices. Furthermore, given the information of exchange rate can help investors and port-

379 folio managers reducing significantly the extreme dependence between oil and stock returns. The sub-

380 period estimation results show that the level and the structure of dependence are not constant over

381 time. More importantly, the multivariate dependence between the considered return series is highly

Table 6

VaR backtesting results

Expected number Exceedences 1 a Kupiec test

Normal 16.97 47 0.01 5.098e-09

Historical Simulation 16.97 27 0.01 0.041

C-vine copula 16.97 22 0.01 0.163

Gaussian copula 16.97 24 0.01 0.106

Student-t copula 16.97 23 0.01 0.106

Notes: The table reports the VaR backtesting results obtained from the C-vine copula, Gaussian copula, Student-t copula, the

historical simulations and the normal method. It also presents the p-values for the Kupiec (1995) test of unconditional coverage.



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382 affected by the financial crisis and Great Recession, over 2007–2009. We finally find that the C-vine

383 copula model leads to more accurate VaR forecasts than the traditional VaR approaches.

384 Future extensions of this work could focus on a much more flexible analysis of the high dimen-

385 sional dependence by allowing the vine copula parameters to be dynamic and switch between differ-

386 ent regimes. Our methodology can also be used in the context of computing optimal portfolio

387 allocation weights and optimal hedging ratios.

388 Acknowledgement

389 We are grateful to two anonymous referees for their constructive comments and suggestions. As

390 usual, all remaining errors are ours.

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