Testing the bivariate distribution of daily equity returns using copulas. An application to the Spanish stock market

Computational Statistics & Data Analysis 51 (2006) 1312 – 1329 www.elsevier.com/locate/csda

Testing the bivariate distribution of daily equity returns using copulas. An application to the Spanish stock market Oriol Roch∗ , Antonio Alegre Dept. Matemàtica Econòmica, Financera i Actuarial, Universitat de Barcelona, Avda. Diagonal 690, 08034 Barcelona, Spain Received 11 February 2005; received in revised form 11 November 2005; accepted 11 November 2005 Available online 5 December 2005

Abstract The problem of the identification of dependencies between time series of equity returns is analyzed. Marginal distribution functions are assumed to be known, and a bivariate chi-square test of fit is applied in a fully parametric copula approach. Several marginal models and families of copulas are fitted and compared with Spanish stock market data. The results show the difficulty in adjusting the bivariate distribution of raw returns, and highlight the effect of a GARCH filtering in the selection of the best fitting copula. © 2005 Elsevier B.V. All rights reserved. Keywords: Copulas; Daily equity returns; Bivariate chi-square statistic; Risk management

1. Introduction The distribution of daily equity returns is still a major problem in financial modelling. Kurtosis and asymmetry are usually reported for the individual returns, whereas recent literature has also highlighted the impact of the dependence between financial variables in the total risk of the portfolio, see e.g., Longin and Solnik (2001) and Ang and Chen (2002). Numerical measures of the risk of the portfolio as VaR, which computes the loss that would be exceeded given a time horizon and a probability level, rely on an accurate specification of the multivariate distribution of asset returns. Otherwise, if such dependencies are not taken into account, the risk of the portfolio as a whole can be seriously underestimated. Embrechts et al. (2001), proposed the application of copula functions in order to model the multivariate distribution of returns. With a copula approach, the modelling of asset returns can be split in two steps: 1. Modelling the marginal distributions. 2. Modelling the dependence structure between marginal distributions. This approach introduces a wide flexibility in the modelling of equity returns since several dependency structures can be generated via copula functions. Such a parameterization enables to model asymmetric dependence structures with joint fat tails. ∗ Corresponding author. Tel.: +34 93 402 1951; fax: +34 93 403 4892.

E-mail address: [email protected] (O. Roch). 0167-9473/$ - see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.csda.2005.11.007

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For simplicity reasons, in most applications the distribution of equity returns is assumed to be a multivariate Gaussian distribution. This assumption generally leads to an underestimation of the portfolio’s risk which can be avoided with alternative hypothesis with respect to the form of the multivariate risk distribution. Models of hedging, asset pricing and portfolio selection could also be improved with a better knowledge of the joint distribution of returns. There are two important issues when trying to describe empirical data with a copula function. First, copula parameters must be estimated. Secondly, the best fitting copula must be chosen among the copulas available. In this context, some authors have applied statistical tests to copula specifications. In Junker and May (2002) a bivariate
2 test is applied. However, as pointed out by Klugman and Parsa (1999), the critical values of the test are not valid since they consider empirical distributions for the margins. Malavergne and Sornette (2003) and Breymann et al. (2003) also deal with the testing of copula specifications. A nonparametric approach is proposed in Fermanian (2005), and a semiparametric approach is developed in Chen et al. (2004), where tests based on the kernel smoothing approach are proposed. In Genest et al. (2006) various goodness of fit tests are proposed, but they are restricted to the arquimedean case. A recent paper similar in spirit to ours is that of Hürlimann (2004), who presents some copula fitting results for bivariate daily cumulative returns between a market index and stocks. However, his analysis is restricted to the class of arquimedean copulas and different marginal distributions are considered. Alternative approaches have also been proposed, e.g., in Huard et al. (2006), where a simple Bayesian method to choose the best fitting copula is introduced. In this paper we present some testing results in a fully parametric copula approach. We apply a bivariate chi-square test against some copula specifications, with marginal distributions parametrically specified. Two models for the marginal distributions are considered. The first model is for the unconditional distribution of raw returns. The distributions employed are allowed to exhibit skewness and kurtosis since this kind of behaviour has been detected in the analyzed data. The second model considers filtered returns by applying an ARMA(1,1)–GARCH(1,1) filter to raw returns. Then the standardized innovations are used in the analysis. The chi-square test also enables us to select the best fitting copula for each pair of time series and to compute the number of pairs for which a copula model cannot be rejected. The remainder of the paper is as follows. Section 2 briefly recalls some concepts about copulas. Section 3 deals with the specification of the marginal distributions. Section 4 describes the estimation methodology, whereas the chi-square testing procedure is explained in Section 5. Section 6 presents the empirical results obtained on the bivariate testing of different copula functions. Data from the Spanish stock market has been used for the analysis. Section 7 summarizes and concludes the paper. 2. Characterization of copulas In this section we briefly recall some well-known results on copula functions. A full treatment of copulas and their properties can be found in Joe (1997) and Nelsen (1999). A copula function is a multivariate distribution function defined on the unit cube [0, 1]d , with uniformly distributed margins. Indeed, a function C : [0, 1]d → [0, 1] is a d-dimensional copula if it satisfies the following properties: (1) For all ui ∈ [0, 1], C (1, . . . , 1, ui , 1, . . . , 1) = ui . (2) For all u ∈ [0, 1]d , C (u1 , . . . , ud ) = 0 if at least one coordinate ui equals zero. (3) C is grounded and d-increasing, i.e., the C-measure of every box whose vertices lie in [0, 1]d is nonnegative. The importance of the copula function relies on the fact that it captures the dependence structure of a multivariate distribution as we will see next. Let X = (X1 , . . . , Xd ) ∈ Rd be a random vector with multivariate distribution F (x1 , . . . , xd ) = P (X1
x1 , . . . , Xd
xd ) ,

x = (x1 , . . . , xd ) ∈ Rd

and continuous margins Fn (xn ) = P (Xn
xn ) , xn ∈ R. According to Sklar’s theorem (Sklar, 1959), there exists a unique copula C : [0, 1]d → [0, 1] of F such that for all x in Rd , F (x1 , . . . , xd ) = P (X1
x1 , . . . , Xd
xd ) 
= C F1 (x1 ) , . . . , Fd (xd ) .

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In general, the copula is affected by some parameters called “copula parameters”, which are represented by the vector  = (1 , . . . , k ) ∈ Rk . The transformations Xn → Fn (xn ) used in the above representation are usually referred to as the probability-integral transformations (to uniformity) and form a standard tool in simulation methodology. A fundamental conclusion of Sklar’s theorem is that in multivariate continuous distribution functions the dependence structure and the margins can be separated, and the dependence structure can be represented by a copula. In a financial context, this key property allows us to fit the marginal distribution of each series of equity returns separately in a first step, and to model the dependence structure between different equity returns in a second step. An important property of copulas is that they are invariant under strictly increasing transformations of the variables. Let X = (X1 , . . . , Xd ) be a vector of continuous random variables with copula C. If g1 , . . . , gd : R → R are strictly increasing on the range of X1 , . . . , Xd , then (g1 (X1 ) , . . . , gd (Xd )) has copula C. This property implies that the dependence structure between the variables is captured by the copula, regardless of the margins. Three fundamental functions related to copulas are M (u1 , . . . , ud ) = min (u1 , . . . , ud ) ,  (u1 , . . . , ud ) = u1 · · · ud , W (u1 , . . . , ud ) = max (u1 + · · · + ud − d + 1, 0) . These functions derive form the Fréchet-Hoeffding bounds, and as a consequence of Sklar’s theorem, it can be proved that for any copula C (u1 , . . . , ud ), W (u1 , . . . , ud )
C (u1 , . . . , ud )
M (u1 , . . . , ud ) , i.e., any copula is constrained between the Fréchet–Hoeffding bounds. The function M (u1 , . . . , ud ) is called the comonotonic copula and represents perfect positive dependence between the variables, whereas the  (u1 , . . . , ud ) function is called the product copula and represents independence between the variables. Unlike the two previously mentioned functions, W (u1 , . . . , ud ) is not a copula for d 3, although is the best possible lower bound for d-copulas. For d = 2, W (u1 , . . . , ud ) is called the countermonotonic copula, and represents perfect negative dependence between the variables. Throughout the paper we shall consider several one-parameter and two-parameter copulas to model bivariate daily equity returns. We now introduce the families of copulas that will be employed in our empirical study. We use the notation C(u, v; ),  ∈  to denote the bivariate parametric families of copulas. Normal copula: The most important copula in financial literature is the normal copula. Let  be the N (0, 1) cumulative distribution function with correlation . The bivariate Normal copula is defined by    −1 (u)  −1 (v) 1 s 2 − 2st + t 2
 C(u, v; ) = ds dt, exp −  2 1 − 2 −∞ −∞ 2 1 − 2 with −1

1, s = −1 (u) and t = −1 (v). t-copula: Another important copula in finance is the t-copula. It belongs to the same class of copulas as the normal copula (elliptical copulas), and add joint fat tails to the normal copula. It has been recommended by several authors such as Mashal and Zeevi (2002) and Breymann et al. (2003). Let T denote the univariate Student’s t cumulative distribution function with  degrees of freedom. The bivariate t-copula with  degrees of freedom and correlation  is given by  C(u, v; , ) =

T−1 (u)  T−1 (v)

−∞

−∞

1

 2 1 − 2



s 2 − 2st + t 2 
1+  1 − 2

−(+2)/2 ds dt,

with −1

1,  > 2, s = T−1 (u) and t = T−1 (v). Note that the normal copula is a limiting case of the t-copula when  → ∞. For a detailed discussion on the t-copula consult Demarta and McNeil (2005). Another class of copulas, the arquimedean copulas, are also used in this paper. Unlike the copulas presented so far, arquimedean copulas are not derived from multivariate distributions using Sklar’s theorem. The bivariate

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arquimedean copulas are generated by C(u, v) = [−1] ( (u) + (v)),

(1)

where : [0, 1] → [0, ∞] is a continuous and strictly decreasing convex function with (1) = 0, and [−1] is the pseudo-inverse of , defined as  (−1) (t), 0
t
(0), [−1] (t) = 0, (0)
t
∞. The function is called the generator of the copula. From now onwards we reference the families of arquimedean copulas with numbers for the sake of a subsequent empirical application, giving more references when possible. Fundamental sources are Nelsen (1999) for one-parameter families of copulas, and Joe (1997) for two-parameter families of copulas. From these sources we have selected the most appropriate copulas for modelling equity returns. Arq. family 1: This family is known as Frank family, and it is probably the one-parameter arquimedean family of copulas most employed in finance. The Frank family is defined as  1 (exp(− u) − 1)(exp(− v) − 1) C(u, v; ) = − ln 1 + , ∈ R\{0}.

exp(− ) − 1 Arq. family 2: This copula can be found in Nelsen (1999) as family number 13, and it is given by

 C(u, v; ) = exp 1 − ((1 − ln u) + (1 − ln v) − 1)1/ , > 0. Arq. family 3: This copula can be found in Nelsen (1999) as family number 17, and it is given by 


 −1/

(1 + u)− − 1 (1 + v)− − 1 C(u, v; ) = 1 + − 1, 2− − 1


∈ R\{0}.

Arq. family 4: This two-parameter family of copulas is also known as the Clayton–Gumbel family, and it is referenced in Joe (1997) as family BB1.  C(u, v; , ) = 1 +




−

u


 1/ − 1 + v − − 1

−1/

,

> 0,  1.

Arq. family 5: This family can be found in Joe (1997) as family BB3, and it is defined as 


  1/

C(u, v; , ) = exp − −1 ln exp (− ln u) + exp (− ln v) − 1 , with 1,  > 0. Arq. family 6: This family is referenced in Joe (1997) as family BB7. It is also used in Patton (2004) with the name of Joe–Clayton copula, and it is given by 


−1/ 1/




−

− C(u, v; , ) = 1 − 1 − 1 − (1 − u1 ) + 1 − (1 − v) −1 , with 1,  > 0.
 We also introduce a new family of bivariate two-parameter copulas. From (t) = (1 − ln t) − 1 , and by applying the definition of bivariate arquimedean copula, we can obtain a new family of copulas that satisfies all conditions required for a function to be a copula. The generator is obtained as a transformation of the generator in arquimedean family 2. The same procedure is applied in Junker and May (2002) for the generator of the Frank copula. This method will be useful to analyze the effect of adding a new parameter into a pre-existing one-parameter copula.

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Arq. family 7: This new family of aquimedean copulas is given by  C(u, v; , ) = exp 1 −







 1/ +1 (1 − ln u) − 1 + (1 − ln v) − 1

1/  ,

with > 0,  1. Contour plots of some of these copulas are shown in Fig. 1.

Normal Copula with t4 margins

Normal Copula with normal margins 3

3

2

2

1

1

0

0

-1

-1

-2

-2

-3 -3

-2

-1

0

1

2

3

-3 -3

T Copula with normal margins

3

0

1

2

3

2

3

T Copula with t4 margins

2

1

1

0

0

-1

-1

-2

-2 -2

-1

0

1

2

3

-3 -3

Arquimedean copula 7 with normal margins 3

2

2

1

1

0

0

-1

-1

-2

-2

-2

-1

0

1

2

-2

-1

0

1

Arquimedean copula 7 with t4 margins

3

-3 -3

-1

3

2

-3 -3

-2

3

-3 -3

-2

-1

0

1

2

Fig. 1. Contour plots of different families of copulas with normal margins and Student’s t margins with 4 degrees of freedom.

3

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The fitting of the nine families of copulas presented in this section will be compared for the empirical results. In the next sections, the estimation of the copula parameters and the testing procedure will be detailed. Another important concept related to copulas is the concept of upper tail dependence, defined by U = lim P (Xi > G−1 (u)|Yi > F −1 (u)), u→1

provided that the limit U ∈ [0, 1] exists. In terms of copulas, U can be reformulated as U = lim

u→1

1 − 2u + C(u, u) . 1−u

In a similar way, the concept of lower tail dependence

 L = lim P Xi < G−1 (u)|Yi < F −1 (u) u→0

can be reformulated by L = limu→0 C(u, u)/u. These two concepts measure the dependence in the tails of the bivariate distribution, which is an important property of a copula. The one-parameter families of copulas considered in our analysis have zero value for both upper and lower tail dependence. The normal copula does not have neither lower nor upper tail for  < 1. The
√ dependence  t-copula has symmetric upper and lower tail dependence, √ U = L = 2 (1 − T+1 ) ( + 1) (1 − )/(1 + ) , whereas two-parameters families of copulas have different degrees of asymmetric tail dependence. For more details about this concept see Joe (1997). 3. Marginal models As discussed in the previous section, the advantage of using copulas relies in the fact that marginal distributions and dependence structure can be separated. So, a natural first step is to find an appropriate distribution for each margin. For the case of equity returns, the distribution of the margins is usually assumed to be Gaussian. However, this generalization often leads to an incorrect specification of the parametric distribution, due to the evidence of heavy tails and skewness showed by the returns. It is important that the selected marginal distribution allows for skewness and excess kurtosis, specially when dealing with high-frequency data. A correct selection of the margins is even more important than a correct selection of the copula. For the sake of a comparison, we consider two different types of models. The first model is based on the unconditional distribution of raw returns. Several functional forms of univariate distributions suitable for modelling equity returns have been collected in Appendix A. Note that in some distributions skewness has been introduced in the way of Férnandez and Steel (1998). Since serial dependence and volatility clustering are often present in daily returns, in the second model we apply an ARMA(1,1)–GARCH(1,1) filter. The specification for the log-returns Xt is Xt = + Xt−1 + t−1 + t , 2 2t =  + G2t−1 + Aεt−1

and εt  ∼ G(, ), 2t where 2t is the conditional variance and G(, ) denotes a Hansen’s (Hansen, 1994) skewed t distribution with  degrees of freedom and skewness , which by definition has mean of zero and unit variance. The pdf and the cdf corresponding to the Hansen’s skewed t distribution can be found in Appendix B. Similar models have also been used in Jondeau and Rockinger (2003) and Patton (2004) with time-varying skewness and kurtosis. Another motivation for selecting this model is that the chi square goodness of fit test requires the iid assumption for the margins, and by applying an ARMA–GARCH filter, we can rescale the Xt returns to an iid process (Engle, 1982; Bollerslev, 1986).

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A general approach to estimate the parameters of the margins is the method of maximum likelihood. Given the sample {Xt }Tt=1 we set the log-likelihood function to be L(; x) =

T 

ln f (xt ; ) .

t=1

The maximum likelihood estimator is then the value of  ∈  maximizing L(; x), i.e., ˆ ML = arg max L(; x), where ˆ ML has the property of asymptotic normality (Davidson and MacKinnon, 1993). For large samples, and under certain regularity conditions, it can be shown that

 √
 d T ˆ −  −→ N 0, I −1 () , where I −1 () is the inverse of the information matrix whose ijth element satisfies   1 j L(; x) . Iij () = E − n ji jj Alternatively, a nonparametric approach can be followed by computing the empirical cdf for the margins. In that case, the nth marginal cdf is estimated by Fˆn (x) =

T 1  1{Xtn
x} , T +1

n = 1, . . . , d,

t=1

where Xtn is the t component of the nth data vector and T is the size of the random sample. Note that the T + 1 term is needed to avoid points in the boundary of the unit cube, which would cause the impossibility of estimate the copula parameters by maximum likelihood. Since using a semiparametric approach (nonparametric for the margins and parametric for the copula) would invalidate the results of the chi-square test, we will not use this method in our analysis.

4. Copula calibration Once the marginal models have been selected and estimated, and the data have been transformed using the probabilityintegral transformation, it is embedded into the copula in order to estimate the copula parameters. Given a random sample X = (Xt1 , Xt2 , . . . , Xtd )Tt=1 , it is common to calibrate the copula parameters by the method of maximum likelihood. Assuming known margins, this approach implies the maximization of the function L() =

T 




  ln c F1 xt1 ; ˆ 1 , . . . , Fd xtd ; ˆ d ;  ,

t=1

where c (u1 , . . . , ud ) is the copula density, which can be obtained from c (u1 , . . . , ud ) =

jd C (u1 , . . . , ud ) . ju1 . . . jud

The two-step procedure explained above is known as inference functions for the margins (IFM) and exploits the basic idea of copulas, i.e., the separation between the margins and the dependence structure.  More details about this method

ˆ ˆ are studied in McLeish and Small (1988) and Xu (1996). Let  = ˆ 1 , . . . , ˆ d ,  be the row vector of parameters

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estimated by IFM. The IFM method also verifies the property of asymptotic normality, which results (Joe, 1997)  d √

ˆ −  −→ T  N (0, V ),



      where V = Dg−1 Mg Dg−1 with g() = j1 L1 , . . . , jd Ld , j Lc , Dg = E jg  ()/j and Mg = E g() g() . When the empirical distributions of the margins are used, the two-step procedure is called canonical maximum likelihood method or CML. This method is similar to IFM, except on the fact that the copula can be calibrated without T
specifying the margins. Once the data has been transformed into uniform variates uˆ t = uˆ t1 , . . . , uˆ td t=1 using the empirical distribution, the estimation of the copula parameters implies the maximization of the function L() =

T 


 ln c uˆ t1 , . . . , uˆ td ;  .

i=1

Alternatively, the parameters of the margins and the parameters of the copula can be jointly estimated. The likelihood would result L() =

T 

d T  
  


ln c F1 xt1 ; 1 , . . . , Fd xtd ; d ;  + ln fn xtn ; n .

t=1

t=1 n=1

However, this method could be high time-consuming in case of high dimensional distributions. 5. Testing procedure In empirical applications, it is common to calibrate several copulas and to compare the results obtained with one or more selection criteria. Different criteria have been proposed in the literature in order to select the best fitting copula, e.g., the AIC or by computing distances between each considered copula and the empirical copula (Durrleman et al., 2000). However, those criteria are not sufficient to accept a copula as representative enough. Since we are interested in compare both elliptical and arquimeden copulas, and the aim of the paper is to validate different copula specifications, we use a standard bivariate chi-square goodness of fit (GOF) test. The aim is to test the null hypothesis H0 : C(u, v) = C(u, v; ) for  ∈ , i.e., the unknown copula C(u, v) is a member of the parametric family, against the alternative hypothesis HA : C(u, v)  = C(u, v; )

for  ∈ ,

i.e., the unknown copula C(u, v) is not a member of the parametric family. We assume that observations are independent and identically distributed. Otherwise, the chi-square test does not hold. As we have seen in previous sections, there are different possibilities for modelling the univariate marginal distributions. When a semiparametric approach is applied, transforming the data with the empirical distribution function, the standard chi-square test does not work. On the other hand, if known margins are assumed using parametric models for the individual margins, the standard chi-square test will hold. However, each marginal distribution model must be accepted, which requires some previous estimation and validation procedures. In the sequel, let Aij = [(i − 1)/r, i/r] × [(j − 1)/s, j/s] ⊂ [0, 1]2 , i = 1, . . . r and j = 1, . . . , s, be bins of [0, 1]2 , i.e., the y-axis is divided into r equidistant parts and the x-axis is divided into s equidistant parts. Let Oij be the observed frequency for bin Aij , and let



 Eij ˆ = np ij ˆ be the expected frequency for Aij , where



   dC u, v; ˆ . pij ˆ = Aij

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The chi-square statistic is

2 s r 

  Oij − Eij ˆ

 ,
2 ˆ = Eij ˆ i=1 j =1 where ˆ is the vector of parameters estimated. The test statistic follows, approximately, a chi-square distribution with (rs − 1 − d) degrees of freedom, where rs is the number of nonempty bins and d is the number of estimated

parameters. The critical value will be the (1 − ) 2 2 quantile of the
rs−1−d distribution, being the H0 rejected if
ˆ >
2( ,rs−1−d) . Note that for the chi-square approximation to be valid, the expected frequency should be at least 5. Otherwise, the

 ˆ data must be regrouped so that Eij  > 5 ∀i, j .

6. Empirical analysis In this section we consider daily equity returns for 16 companies of the Spanish stock market from 1/2/1997 to 11/11/2004. The equities selected are: Abertis, Acciona, Acerinox, ACS, Aguas de Barcelona, Altadis, Banco Popular, Bankinter, BBVA, Corp.Alba, Endesa, FCC, NH Hoteles, Repsol, Santander, and Telefónica. The time period practically covers a whole economic cycle. The selected companies are “large caps” within the Spanish stock market, which ensures that the analysis is not affected by liquidity problems. A significant positive dependence between equity returns is expected by the effect of cross participations, investment strategies as index tracking and the simple fact that some companies belong to the same economic sector. Our analysis is restricted to the bivariate case. An accurate bivariate distribution of returns is specially important in order to compute prices of exotic options (Cherubini and Luciano, 2002) and risk measures (Junker and May, 2002). Although we have selected “large caps”, similar analysis could be extended, e.g., to indices (Hu, 2006), exchange rates (Patton, 2005) or “small caps”. Let S = (St1 , St2 , . . . , Std )Tt=0 be the time-series of adjusted equity prices. The returns are defined as  Stn , n = 1, 2, . . . , d, t = 1, . . . , T , Xtn = ln St−1n where d represents the number of equities, T the number of trading days, and being the vector Xt = (Xt1 , . . . , Xtd ) the corresponding to the tth day. The data yields 1968 observations of daily returns for each equity. Prior to the calibration and selection of the copula models it seems essential to√ test the data for independence. A standard procedure, based on Kendall’s tau, requires to compute the statistic T = 3 T |T |/2. The null hypothesis of independence should be rejected if T > −1 (0.975) = 1.96, where T is the Kendall’s tau statistic and T the sample size, which must be large enough. When computing the statistic with our data, T ranges from 5.2533 to 66.5432, which implies that the null hypothesis of independence is clearly rejected for every data pair. If the null hypothesis of independence was accepted, the product copula could be a good model to represent the dependence structure between the variables. In any case, as all the families of copulas considered in the empirical analysis include the product copula as special case or tend to it, the acceptance of the null hypothesis would not deeply affect our analysis. More detailed tests of independence related to copulas can be found in Genest and Rémillard (2004). The testing methodology proceeds as follows: (1) Fitting the univariate marginal distribution of each series of equity returns by maximum likelihood estimation. The distributions considered for the case of raw returns, all of which accept for skewness, have been collected in Appendix A. Among the different tested distributions, we select the one that best fits the data. The selection criterion is the higher log-likelihood value. Note that once a distribution has been estimated and selected, it must be accepted in order to employ the standard chi-square test. In our analysis a Kolmogorov–Smirnov GOF test (Boes et al., 1974) was applied, and all the selected marginal distributions were accepted for a significance level

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Table 1 Log-likelihood values at the optimum

Abertis Acciona Acerinox ACS Aguas Bar. Altadis B. Popular Bankinter BBVA Corp. Alba Endesa FCC NH Hoteles Repsol Santander Telefónica

Skewed T -Loc

Kappa

Skewed logistic

Skewed Cauchy

Skewed Laplace

SGED

p-Value

5432.7 5171.9 4819.7 4970.6 5307.5 5017.9 5209.7 4990.3 4758 5038.6 5177.6 4954.7 4894.2 5244.4 4675.2 4652.4

5425.6 5117.7 4815.2 4956.2 5258.8 5008.8 5201.2 4961.9 4733.4 5032.7 5172.8 4952.7 4853.7 5235.1 4658.9 4654.4

5423.6 5114.5 4812.2 4955.5 5296.9 5009.5 5200.8 4960 4731.6 5031.1 5171.1 4949.3 4852.3 5231.8 4655.9 4652.4

5242.4 5031.5 4610.9 4779.1 5103.9 4796.7 5001.1 4826.2 4570.7 4581.2 4973.8 4763.2 4729.1 5041 4487.6 4420.9

5437.2 5171.2 4812.2 4971.5 5301.9 4996.7 5201.6 4994.2 4746.1 5048.3 5171.2 4963.1 4887.4 5235.6 4668.4 4651.6

5441.7 5177.4 4820.9 4973.1 5309.1 5010.1 5210.3 4995.1 4748.4 5051.7 5180.3 4969.1 4887 5241.9 4671.6 4655.2

0.059 0.082 0.826 0.403 0.086 0.686 0.518 0.169 0.317 0.296 0.355 0.471 0.076 0.521 0.112 0.512

= 0.05. For the case of the ARMA(1,1)-GARCH(1,1) filter, we estimate the parameters and then we compute the standardized innovations. (2) Transform each data vector into uniform variates using the probability-integral transformation. (3) For each pair of transformed data vectors, plug the data into a family of copulas and estimate the copula parameters by maximum likelihood as described in Section 4. Repeat this procedure with all the families of copulas considered. (4) Apply the bivariate chi-square test of goodness of fit described in Section 5 for each family of copulas estimated. All the results were obtained using MATLAB software. The results of the univariate marginal fitting for raw returns are reported in Table 1. The values of the log-likelihood at the optimum are compared in order to select the best fitting marginal distribution (higher values are highlighted in bold fonts). The SGED distribution is the most selected, followed by the skewed t-location scale distribution. The skewed Laplace also performs quite well for high-frequency data, as noted in Linden (2001). The worst performance is obtained by the skewed cauchy, which has only been used for the sake of a comparison. Table 1 also reports in the last column the p-value of the Kolmogorov–Smirnov test, which ensures that the marginal fitting is good enough. In Section 5 has been pointed out that the chi-square GOF test requires the iid assumption for the returns. However, this assumption is quite unrealistic for the case of raw returns. In this context some authors ignore the effect of the serial dependence and treat raw returns as if they were independent. The iid assumption for raw returns is made, e.g., in Mashal and Zeevi (2002), Malavergne and Sornette (2003) and implicitly in Hürlimann (2004). With the motivation to obtain iid observations, we consider a second model for the margins by applying an ARMA(1,1)-GARCH(1,1) filter in order to capture the serial correlation and volatility clustering. In Fig. 2 the left graph shows the QQ-plot of the standardized innovations of the equity Endesa against the Hansen’s skewed t distribution with parameters  = 10.41 and = −0.0259, whereas the right graph shows the QQ-plot of the standardized innovations of the equity Telefónica against the same distribution with parameters  = 13.7751 and = 0.02. When applying the ARMA–GARCH filter, it can be easily noted that the kurtosis for the innovations is lower than in the case of raw returns. Table 2 reports the values of the log-likelihood for the filtered returns. A Ljung-Box-Pierce Q-Test is also computed to verify that no serial dependence is present in the standardized innovations when tested for up to 10, 15, and 20 lags of the ACF at a 0.05 level of significance (Fig. 3). The combinatory of the data vectors yields 120 pairs. Each pair is fitted with all the nine copulas described in Section 2. Then, the bivariate chi-square GOF test is applied to each copula estimated. For every test, the unit square [0, 1]2 is divided into bins of equal area, being r = s, and rearranging bins when needed. As the analysis is quite

ENDESA

4 3 2 1 0 -1 -2 -3 -4 -4

-3 -2 -1 0 1 2 3 4 Standardized Innovations Quantiles

Hansen’s skewed t distribution Quantiles

O. Roch, A. Alegre / Computational Statistics & Data Analysis 51 (2006) 1312 – 1329

Hansen’s skewed t distribution Quantiles

1322

TELEFÓNICA

4 3 2 1 0 -1 -2 -3 -4 -4

-3 -2 -1 0 1 2 3 4 Standardized Innovations Quantiles

Fig. 2. QQ-plot of the standardized innovations (empirical quantiles) against a Hansen’s skewed t distribution (theoretical quantiles) for Endesa and Telefónica equities.

Table 2 Log-likelihood and LBP Q-test results for the ARMA(1,1)-GARCH(1,1) filtered model Log-likelihood

LBP p-value

Abertis

5575.3

Acciona

5288.3

Acerinox

4940.2

ACS

5102.2

Aguas Bar.

5406.3

Altadis

5117.9

B. Popular

5338.2

Bankinter

5151.8

0.9528 0.8774 0.9025 0.6628 0.8802 0.8462 0.1143 0.0898 0.1817 0.2501 0.3795 0.6059 0.6501 0.4225 0.6276 0.3080 0.3541 0.3651 0.2016 0.3498 0.1140 0.2959 0.2403 0.3943

Log-likelihood

LBP p-value

BBVA

4982

Corp. Alba

5132.9

Endesa

5331

FCC

5102.1

NH Hoteles

4983.4

Repsol

5390.9

Santander

4887.2

Telefónica

4811

0.5102 0.5187 0.5234 0.3594 0.3350 0.4551 0.7241 0.9183 0.9557 0.3694 0.3590 0.4307 0.1311 0.2262 0.2542 0.7946 0.5679 0.4303 0.4299 0.6078 0.8501 0.8762 0.7576 0.5725

sensitive to the selection of the number of bins, the results are reported for different choices of r and s. In Tables 3–5 we examine the goodness of fit of the different copula models. Table 3 shows the percentage of rejection for each family of copulas given a significance level = 0.1, Table 4 given a significance level of = 0.05, and finally, Table 5 given

= 0.01. The analysis have been done for both raw and filtered returns. In the cells, the number in brackets refers to the case of filtered returns. As it was expected, the more flexible two-parameter copulas fit better than one-parameter copulas for the case of raw returns. In particular, note the considerable improvement of arquimedean family 2 when it is extended with a new parameter (case of arquimedean family 7). The normal copula, our benchmark due to its relevance in the financial world, is highly rejected. This result contradicts those of Malavergne and Sornette (2003) and Chen et al. (2004) for the bivariate case. If we attend to the best fitting copula, the t-copula is clearly the one that fits better, although it is rejected

O. Roch, A. Alegre / Computational Statistics & Data Analysis 51 (2006) 1312 – 1329

ACF of the standardized innovations (Endesa)

0.1 0.05 0 -0.05 -0.1 -0.15 -0.2

0

5

10 Lag

15

0.1 0.05 0 -0.05 -0.1 -0.15 -0.2 0

20

5

10 Lag

15

20

ACF of the squared standardized innovations (Telefónica) 0.2 Sample Autocorrelation

0.15 0.1 0.05 0 -0.05 -0.1 -0.15 -0.2

ACF of the squared standardized innovations (Endesa)

0.15

ACF of the standardized innovations (Telefónica)

0.2 Sample Autocorrelation

0.2 Sample Autocorrelation

Sample Autocorrelation

0.2 0.15

1323

0

5

10 Lag

15

20

0.15 0.1 0.05 0 -0.05 -0.1 -0.15 -0.2 0

5

10 Lag

15

20

Fig. 3. ACF of the standardized innovations and the squared standardized innovations for Endesa and Telefónica equities up to lag 20.

Table 3 Percentage of rejections for = 0.1 Family

r =s=8

r = s = 10

r = s = 12

r = s = 14

r = s = 16

Normal copula

90.00 (29.17) 45.00 (14.17) 91.67 (35) 80.83 (20) 90.83 (26.67) 67.50 (15.83) 67.50 (20) 70.00 (26.67) 70.83 (10.83)

89.17 (33.33) 50.83 (27.5) 91.67 (42.5) 93.33 (25) 90.00 (32.5) 60.00 (25.83) 60.83 (28.33) 66.67 (29.17) 70.83 (18.83)

84.17 (25.83) 45.00 (11.67) 81.67 (30.83) 85.00 (16.67) 80.00 (23.33) 55.00 (14.17) 55.83 (19.17) 59.17 (23.33) 59.17 (10)

79.17 (35) 44.17 (25) 87.50 (50) 89.17 (26.67) 80.83 (40.83) 53.33 (24.17) 55.83 (29.17) 56.67 (31.67) 57.5 (19.17)

88.33 (29.17) 53.33 (18.33) 95.83 (34.17) 96.67 (19.17) 94.17 (27.5) 59.17 (18.33) 61.67 (20) 60.00 (22.5) 67.5 (15)

t-copula Arq. family 1 Arq. family 2 Arq. family 3 Arq. family 4 Arq. family 5 Arq. family 6 Arq. family 7

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O. Roch, A. Alegre / Computational Statistics & Data Analysis 51 (2006) 1312 – 1329

Table 4 Percentage of rejections for = 0.05 Family

r =s=8

r = s = 10

r = s = 12

r = s = 14

r = s = 16

Normal copula

82.50 (18.33) 28.33 (6.67) 86.67 (21.67) 73.33 (10.83) 84.17 (14.17) 49.17 (9.17) 53.33 (15) 55.83 (18.33) 62.5 (5)

84.17 (27.5) 33.33 (18.33) 84.17 (32.5) 89.17 (18.33) 82.50 (22.5) 50.83 (17.5) 50.83 (21.67) 52.50 (21.67) 58.33 (10.83)

75.00 (14.77) 31.67 (7.5) 77.50 (20.83) 80.00 (9.17) 74.17 (15) 45.83 (10) 45.83 (11.67) 47.50 (12.5) 47.5 (6.67)

67.50 (21.67) 31.67 (14.17) 77.50 (35.83) 79.17 (15) 75.00 (23.33) 46.67 (15) 48.33 (18.33) 48.33 (20) 49.17 (8.33)

81.67 (18.33) 40.83 (10.83) 88.33 (25.83) 90.00 (10) 81.67 (19.17) 51.67 (10) 54.17 (14.17) 54.17 (15) 54.17 (5.83)

t-copula Arq. family 1 Arq. family 2 Arq. family 3 Arq. family 4 Arq. family 5 Arq. family 6 Arq. family 7

Table 5 Percentage of rejections for = 0.01 Family

r =s=8

r = s = 10

r = s = 12

r = s = 14

r = s = 16

Normal copula

65.83 (7.5) 9.17 (1.67) 68.33 (10) 42.50 (4.17) 65.00 (5) 25.83 (2.5) 27.50 (4.17) 30.00 (4.17) 31.67 (1.67)

64.17 (12.5) 16.67 (4.17) 74.17 (20) 76.67 (10) 66.67 (13.33) 27.50 (5) 29.17 (9.17) 29.17 (11.67) 33.33 (3.33)

56.67 (8.33) 18.33 (3.33) 61.67 (10.83) 64.17 (6.67) 53.33 (6.67) 25.83 (5) 26.67 (7.5) 28.33 (7.5) 29.17 (3.33)

50.00 (8.33) 10.00 (4.17) 63.33 (13.339) 65.00 (3.33) 55.83 (6.67) 23.33 (4.17) 24.17 (6.67) 24.17 (9.17) 25.53 (2.5)

50.83 (8.33) 16.67 (5) 68.33 (13.33) 71.67 (5) 57.50 (7.5) 21.67 (5.83) 25.00 (6.67) 25.83 (7.5) 26.67 (3.33)

t-copula Arq. family 1 Arq. family 2 Arq. family 3 Arq. family 4 Arq. family 5 Arq. family 6 Arq. family 7

nearly 30% of the times at = 0.05. Other two-parameter families do not achieve the degree of fitting of the t-copula: nearly 50% of the data pairs cannot be well represented with these families at = 0.05. These results highlight the difficulty in fitting the bivariate distribution of daily equity raw returns. If we attend to the results for filtered returns, it can be seen that all the families of copulas perform much better than in the case of raw returns. This improvement is even more pronounced for copulas with just one parameter, included the normal copula. The percentage of rejections for one-parameter copulas respect two-parameter copulas reduces after the ARMA–GARCH filter, and the null hypothesis cannot be rejected for the most of the data pairs. For example, given = 0.05, the normal copula is not rejected near 20% of the times, and the t-copula near 10% of the times, which represents a significant improvement respect to the case of raw returns. It should be taken into account that the analysis with filtered returns is more accurate that the analysis for raw returns due to the bias produced in the parameter estimation and in the testing methodology that arises when treating as iid variables that are not.

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Table 6 Hill estimator m = 100

Min Max Mean Std ˆ > 0.9 ˆ > 0.8 ˆ > 0.7

m = 150

m = 200

Raw returns

Filtered returns

Raw returns

Filtered returns

Raw returns

Filtered returns

0.4259 0.9768 0.6457 0.0825 3 5 23

0.3928 0.7592 0.5941 0.0641 0 0 12

0.4268 0.9773 0.647 0.0719 1 5 21

0.4399 0.751 0.5983 0.0521 0 0 4

0.4843 0.9822 0.6482 0.0633 1 3 22

0.4717 0.7523 0.6007 0.048 0 0 4

Although the null hypothesis that data comes from a theoretical distribution is contrasted taken into account the whole distribution, we can focus on the tails to see the effect of the ARMA–GARCH filter into the GOF test. At this point we follow Draisma et al. (2004) for estimating the coefficient  for the upper tail. A value of  < 1 implies asymptotic independence, whereas a value  = 1 implies asymptotic dependence. Other estimators are proposed in Ledford and Tawn (1996) and Peng (1999). Given an iid sample (Xi , Yi )ni=1 , consider the random variable (n)

Ti

=

n+1 n+1 ∧ , X n + 1 − Ri n + 1 − RiY

i = 1, . . . , n,

where RiX and RiY denotes the rank of Xi and Yi , respectively. The coefficient  can now be estimated with the Hill estimator (n) m Tn,n−i+1 1  log ˆ = (n) m Tn,n−m i=1 (n)

which uses the m + 1 largest order statistics of Ti . The ˆ has been computed for all the 120 pairs of data vectors. For the sake of brevity, we do not report here the whole results, but some descriptive statistics for different choices of m, that are collected in Table 6. The values of ˆ are smaller for filtered returns than for raw returns, which implies that the joint probability of extreme values is smaller for filtered returns. The number of pairs with a high level of ˆ clearly reduces after filtering, being 0 for ˆ > 0.8. The better performance of one-parameter copulas for filtered returns might be explained in part because the ARMA–GARCH filter has removed the tail dependence. Remember from Section 2 that the one-parameter copulas used in this analysis have no tail dependence. The effect on the reduction of extremal dependence provoked by the GARCH filter is also noted and discussed in Poon et al. (2004), where it is shown how volatility is a major contributing factor to the extremal dependence between time series. The chi-square GOF test can also be used as a criterion for choosing a copula to represent the sample data. Generally, the lower the chi-square statistic, the better the copula represents the data. However, as in this case we compare statistics distributed with different degrees of freedom, it is better to use the p-value of the contrasts in order to select the best fitting copula. For each pair of data vectors we compare the fitting of the nine copulas, and the selected will be that with higher p-value. Table 7 summarizes the number of times a copula has been selected according to this criterion. The t-copula is the most selected one for the case of raw returns. Although other bivariate copulas have been selected, the t-copula clearly outperforms other copula specifications. Very different results are obtained after applying the ARMA–GARCH filter. Our new arquimedean copula number 7 is the most selected, followed by the t-copula and the one-parameter arquimedean copula number 2 (recall form Section 2 that arquimedean family number 7 is an extension from arquimedean family number 2). The normal copula is selected is some cases, as other one-parameter families of arquimedean copulas. As commented before, this better performance might be explained in part for the effect of the ARMA–GARCH filter in the tails of the distribution. Items for future research are the extensions of these results to the multivariate case.

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Table 7 Number of times a copula has been selected Family

r =s=8

r = s = 10

r = s = 12

r = s = 14

r = s = 16

Normal copula

0 (7) 93 (31) 0 (6) 1 (22) 0 (6) 9 (17) 5 (1) 6 (1) 6 (29)

0 (4) 92 (25) 0 (4) 0 (24) 0 (4) 12 (17) 6 (2) 4 (0) 6 (40)

0 (7) 94 (16) 0 (3) 0 (19) 0 (2) 10 (23) 5 (13) 5 (1) 6 (36)

0 (11) 85 (26) 0 (2) 0 (27) 0 (4) 9 (11) 10 (3) 4 (1) 12 (35)

0 (11) 73 (22) 0 (2) 0 (26) 0 (2) 16 (13) 12 (4) 8 (2) 11 (38)

t-copula Arq. family 1 Arq. family 2 Arq. family 3 Arq. family 4 Arq. family 5 Arq. family 6 Arq. family 7

Another interesting result is obtained when considering the number of pairs that could be accepted given all nine considered copulas. That result is a crucial result when trying to explore the possibilities of copulas for modelling equity returns. For the case of raw returns and s = t = 12, we found that in the 60% of the pairs, the H0 could not be rejected at a significance level = 0.1, the 69% at a significance level of = 0.05 and the 83% at a significance level of = 0.01. For the case of filtered returns the results are much better: 93.33%, 95.83% and 97.5%, respectively. For few pairs of filtered returns have not been possible to find an appropriate distribution according to this methodology. 7. Conclusions In this paper we have analyzed different models of dependence structures for equity returns within a copula approach. The bivariate distribution of time series has been tested with Spanish stock market data. Different models for the marginal distributions and a total of nine copulas have been fitted and compared. Using a fully parametric chi-square goodness of fit test, we have found difficulties in order to find appropriate bivariate distributions for raw returns. The results have been much satisfactory for the case of filtered returns, where just few pairs of time-series could not be fitted properly according to this test procedure. The better adjustment of filtered return may be in part explained for the reduction of the tail dependence provoked by an ARMA(1,1)–GARCH(1,1) filter. New solvency regulations and the fast growth of multivariate instruments have turned out the problem of dependency between risks into a crucial issue. The knowledge of the multivariate joint distribution function, and the dynamic dependence structure of equity returns over time keep as an important line of future research. Acknowledgements The authors would like to express their thanks to the Spanish Ministerio de Educación y Cultura for the project Becas FPU, which has partially supported this paper, as well as to both referees for their scrutiny. Appendix A The Appendix A collects some functional forms of the distributions used in the univariate marginal fitting. Skewness has been allowed in some distributions introducing inverse scale factors in the positive and the negative orthand. This approach was introduced in Férnandez and Steel (1998).

O. Roch, A. Alegre / Computational Statistics & Data Analysis 51 (2006) 1312 – 1329

1327

The cdf will be denoted by F, and the pdf by f. a. Skewed logistic: f (x) =

2  + −1

  x I[0,∞) + g(x)I(−∞,0) , g 

where

exp

x−





g(x) =

2 ,   1 + exp x− 

 > 0.

b. Skewed t-location scale:   2 x f (x) = g I[0,∞) + g(x)I(−∞,0) ,  + −1  where −(+1)/2  (( + 1)/2)  + ((x − )/)2 g(x) = √ ,   (/2)

 > 0,  > 0.

c. Four-parameter Kappa from Hosking (1994):

1/ h F (x) = 1 − h[1 − k(x − )/ ]1/k , f (x) = −1 (1 − k(x − )/ )(1/k)−1 [F (x)]1−h ,

> 0.

d. Skewed Cauchy: 2 f (x) =  + −1

  x g I[0,∞) + g(x)I(−∞,0) , 

where g(x) =

1 1 , 2  + (x − )2

> 0.

e. Skewed Laplace: More details about this distribution can be found in Linden (2001). ⎧ b1 ⎪ exp (−b1 |x − |) x > 0, ⎨ 2 f (x) = ⎪ ⎩ b2 exp (−b2 |x − |) x
0, 2 with b1 , b2 > 0. f. Skewed generalized error distribution (SGED). Introduced in Theodossiou (2002), it is an asymmetric version of the generalized error distribution  C 1 k f (x) = exp − , |x −  + |  [1 − sign(x −  + ) ]k k k  > 0, k > 0, −1 < < 1,

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O. Roch, A. Alegre / Computational Statistics & Data Analysis 51 (2006) 1312 – 1329

where

 −1 1 k  , 2 k  1/2  −1/2 3 1  S( )−1 , = k k

C=

 = 2 AS( )−1 , with

 S( ) = 1 + 3 2 − 4A2 2 ,   −1/2  −1/2 3 1 2  .  A= k k k

Appendix B The density of Hansen’s skewed t distribution is defined by ⎧   −(+1)/2 ⎪ 1 bx + a 2 ⎪ ⎪ bc 1 + , x < − a/b, ⎪ ⎪ ⎨ −2 1− g(x; , ) =  ⎪  2 −(+1)/2 ⎪ ⎪ bx + a 1 ⎪ ⎪ , x  − a/b, ⎩ bc 1 + −2 1+ where  ∈ (2, ∞), ∈ (−1, 1), and a = 4 c

−2 , −1

b2 = 1 + 3 2 − a 2 , c= √

1 (( + 1)/2) . (/2) ( − 2)

The skewed t cumulative distribution function is defined by   ⎧ bx + a  ⎪ ; , ⎪ ⎨ (1 − ) · F −2 1− G(x; , ) =     ⎪  bx + a ⎪ ⎩ 1 − + (1 + ) · F ; v − 0.5 , 2 −2 1−

y < − a/b, y  − a/b,

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