exchange rates: a vine copula based garch method relationship between oil, stock prices and
TRANSCRIPT
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8/15/2019 exchange rates: A vine copula based GARCH method Relationship between oil, stock prices and
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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
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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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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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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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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.
Copula Parameters (SE) Kendall’s s Tail dependence
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
the standard error of the parameters.⁄ Indicates significance at the 1% level.
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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.
Copula Parameters (SE) Kendall’s s Tail dependence
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
the standard error of the parameters.⁄ Indicates significance at the 1% level.
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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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