The Copula Opinion Pooling through a Copula ...

The Copula Opinion Pooling through a Copula-APARCH: An Empirical Application Fernando A. Boeira Sabino da Silva March 30, 2014

Abstract The main goal of this work is to introduce the copula opinion pooling (COP) approach. In the …rst part we combine several techniques used widespread in …nancial econometrics literature. The temporal dependence of each margin is estimated using an ARMA-APARCH structure. As an application of the methodology, Expected Shortfall for a one day forecast horizon is estimated for the portfolio considered. In the last part a brief simulation is performed using the COP procedure to estimate posterior distributions. Keywords and Phrases: Copulas; Copula-APARCH; Copula Opinion Pooling. JEL Classi…cation: C14; C15; C32; C52; G11. Department of Statistics and Graduate Program in Economics, Federal University of Rio Grande do Sul, Porto Alegre, RS, Brazil. e-mail: [email protected]

1

1

Introduction

Multivariate …nancial returns are often assumed to be normally distributed, but in practice, this assumption does not hold. Dependence patterns of …nancial data are often more complex. The assumption of returns normally distributed is very restrictive and do not take account of the stylized facts of the …nancial univariate data such as skewness, heavy tails and conditional heteroscedasticity. The copula approach is more ‡exible and is able to incorporate these patterns of dependence. Standard references on copula theory include Joe (1997) and Nelsen (2006). The Sklar’s theorem (1959) establishes connection between distribution functions and its margins. Let F be a d dimensional distribution function of a random vector X = (X1 ; :::; Xd )T with margins F1 ; :::; Fd . Then there exists a copula C such that for all x = (x1 ; :::; xd )T 2 ( 1; 1)d ; F (x) = C (F1 (x1 ) ; :::; Fd (xd )). C is unique if F1 ; :::; Fd are continuous. Conversely, if C is a copula and F1 ; :::; Fd are distribution functions, then the function F is a joint distribution function with margins F1 ; :::; Fd . C can be interpreted as the multivariate distribution function where the margins of this function are uniform on [0; 1]. In general terms, copula is a function that determines the joint distribution of a random vector in terms of their univariate distributions. The random variables X1 ; :::; Xd will be assumed to be continuous here. The practical implication of Sklar’s theorem is that the modelling of the marginal distributions can be done separated from the dependence modelling. This means that the complexity can be estimated in the margins. Here we work with the bivariate case, where a rich variety of copulas families is available. It is assumed here that the copula dependence parameters are static. The …rst purpose of this working paper is to illustrate the copula approach for modelling the dependence between the returns of two brazilian stocks. modelling multivariate dependence is relevant, since assets interact contemporaneously within a portfolio. Margins are estimated by an ARMA(p,q)APARCH(p,q) model. Asymmetric Power ARCH (APARCH) was introduced by Ding, Granger, and Engle (1993). We choose the best model after plotting the data, identifying the dependence orders of the model and performing diagnostics. In the next stage a copula distribution is estimated based on the pseudo uniform variables obtained from residuals. Finally, Expected Shortfall is estimated via Monte Carlo simulation using parametric information from previous stages. 2

Section 2 presents some methodological aspects of copulas. Section 3 explain how the margins’densities are estimated and the parametric model dependence used to obtain residuals. We also explain how we select the copula function used in this study and present a empirical application. We compare and select the best model based in information criteria and diagnostics. Estimation is carried out by a semi-parametric IFM approach. Section 4 introduces the copula opinion pooling procedure. We show a brief simulation to …nd posterior distributions based on a bayesian framework. Density plots are provided to illustrate the simulations. Section 5 brings some conclusions.

2

Methodological Aspects of Copulas

Before introducing the concept of copulas we will introduce a result of probability theory that is extremely useful when dealing with copulas, known as the probability integral transform Proposition 1 Let X be a random variable with distribution F and F 1 its inverse, i.e., F 1 ( ) = inf fx jF (x) > g ; 2 (0; 1): If F is continuous then F (X) is uniformly distributed in the interval [0,1].1 The marginal distributions of jointly distributed random variables as well as their dependence are contained in their joint distribution function F (x1 ; :::; xd ) = P (X1 x1 ; :::; Xd xd ) : An elementwise transformation of the random vector X = (X1 ; :::; Xd ) is required such that the resulting variables follow a U (0; 1). Given the assumption that all distribution functions F1 ; :::; Fd are continuous, this projection is given by the probability integral transform. The joint density function, say C, of the transformed random variables is called copula of the vector X, i.e., F (x1 ; :::; xd ) = P (F1 (X1 ) F1 (x1 ) ; :::; Fd (Xd ) = C (F1 (x1 ) ; :::; Fd (xd )) 1

The proof can be found in Casella and Berger (2001).

3

Fd (xd )) (1)

Hence, a copula is the distribution function of a d-element random vector with standard uniform marginal distributions2 . Alternatively, a copula can be interpreted as a function that maps from the d dimensional space [0,1] into the unit interval, i.e., C : [0; 1]d 7! [0; 1]. As already mentioned if the marginal distributions are continuous, then the copula is uniquely de…ned. Therefore, it is possible to employ di¤erent distributions as marginals and capture the dependences among them through a copula function. Nelsen (2006) de…nes a function copula more formally. Theorem 2 (Sklar’s Theorem) Let F be a distribution function d-dimensional with marginals F1 ; :::; Fd . Then there exists a d-copula C such that for all x = (x1 ; :::; xd )T 2 Rd F (x) = C (F1 (x1 ; :::; Fd (xd ))

(2)

If F1 ; :::; Fd are all continuous, then C is uniquely de…ned. Otherwise, C is uniquely determined in Im F1 ::: Im Fd . Conversely, if C is a d-dimensional distribution function with uniform margins and F1 ; :::; Fd are continuous univariate distribution functions for the random variables X1 ; :::; Xd , then the function de…ned by (2) is a d-dimensional distribution function with marginals F1 ; :::; Fd : As a direct corollary of Sklar’s theorem we have the inversion formula C(u1 ; :::; ud ) = F (F1 1 (u1 ); :::; Fd 1 (ud )) (3) where F1 1 ; :::; Fd

1

are quasi-inverse of F1 ; :::; Fd given by Fi

1

(ui ) = inf fx 2 R jFi (x)

ui g

(4)

for all ui 2 [0; 1]: From expression (3) we can notice that the structure of dependence embodied by the copula function can be recovered from the knowledge of the joint distribution F and its marginals Fi : In other words, a copula function allows us to express a joint probability function through their marginals. 2

Although the marginal distributions are uniform by construction, the joint distribution T of (F1 (X1 ) ; :::; Fd (Xd )) is uniform only if all variables belonging to the random vector are independent.

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A function C : [0; 1]d ! [0; 1] is a d-dimensional copula if it satis…es the following properties: (i) For all ui 2 [0; 1], C(1; :::; 1; ui ; 1; :::; 1) = ui ; (ii) For all u 2 [0; 1]d , C(u1 ; :::; ud ) = 0 if at least one of the coordinates is zero; (iii) The volume of every box contained in [0; 1]d is non-negative, i.e, VC ([u1 ; :::; ud ]

[v1 ; :::; vd ])

is non-negative. For d = 2, VC ([u1 ; u2 ] for 0

ui ; vi

[v1 ; v2 ]) = C (u2 ; v2 )

C (u1 ; v2 )

C (u2 ; v1 ) + C (u1 ; v1 )

0

1.

Since C is a continuous distribution function, we can …nd its density c (u1 ; :::; ud ) computing the total derivative of the copula C, i.e., c (u1 ; :::; ud ) =

@C (u1 ; :::; ud ) @u1 @u2 :::@ud

(5)

Obtaining the density is quite useful when one wants to …nd the density function of the random vector (X1 ; :::; Xd ), as follows: f (x1 ; :::; xd ) = c (F1 (x1 ) ; :::; Fd (xd ))

d Q

fi (xi )

(6)

i=1

This last expression is very convenient for estimation purposes as we explain ahead.

2.1

Parametric models: maximum likelihood methods and inference from likelihoods for margins

Often the dependence relation can be obtained through the conditional distributions, i.e., considering past observations. Let W be the past information set and Xi jW for i = 1; :::; n, a random variable with conditional distribution function Fi and joint conditional distribuição function H of X jW , 5

where X = (X1 ; :::; Xn ) and assume that supp(W) = : Assuming these conditions, the Sklar’s theorem for the conditional case assures that there exists a copula C such that for all x 2 Rn and w 2 H (x1 ; :::; xn jw ) = C (F1 (x1 jw ) ; :::; Fn (xn jw ) jw )

(7)

If F1 ; :::; Fn are all continuous, then C is uniquely de…ned. Otherwise, C is uniquely determined in Im F1 ::: Im Fd . Conversely, if C is a n-dimensional distribution function with uniform margins and F1 ; :::; Fn are continuous univariate distribution functions for the random variables X1 ; :::; Xn , then H is a n-dimensional distribution function with marginals F1 ; :::; Fn : We can also …nd the density h (x1 ; :::; xn jw ) similarly as above. Without loss of generality, in bivariate case we have: @ 2 H (x1 ; x2 jw ) h (x1 ; x2 jw ) @x1 @x2 @F1 (x1 jw ) @F2 (x2 jw ) @ 2 C (F1 (x1 jw ) ; F2 (x2 jw ) jw ) = @x1 @x2 @u1 @u2 = f1 (x1 jw ) f2 (x2 jw ) c (u1 ; u2 jw ) (8) where u1 = F1 (x1 jw ) and u2 = F2 (x2 jw ). We can notice that all dependence is in c (u1 ; u2 jw ). If this term is one, then X1 and X2 are independent. Thus, we can express the joint density function as a product of marginals and a copula function. Rewriting the joint density as h (x1 ; x2 jw;

h)

= f1 (x1 jw;

1 ) f2

(x2 jw;

2 ) c (u1 ; u2

jw;

c)

(9)

where h = T1 ; T2 ; Tc is the vector of (static) parameters of the joint density. The log-likelihood funcion is given by T P

t=1

log(h (x1 ; x2 jw;

h ))

=

T P

t=1

+

log(f1 (x1t jw;

T P

t=1

1 ))

log(c (u1t ; u2t jw;

+

T P

t=1 c )):

log(f2 (x2t jw;

2 ))

(10)

Rewriting as l ( h ) = lf1 ( 1 ) + lf2 ( 2 ) + lc ( c ) the maximum likelihood estimator (MLE) is de…ned by bh:M LE = maxl ( h ). Since the maximization 2

procedure can be computationally burdensome one alternative is using margins inference function (IFM) (see Joe and Xu, 1996). According to this 6

approach, we follow two steps. In the …rst step we estimate the margins bi = arg max i

T X

log fi (xit ; w;

i) ;

(11)

i = 1; 2

t=1

In the second step we estimate the copula with the residuals obtained from each marginal, i.e., estimate bc = arg max c

T X t=1

log c F1 x1t w;b1 ; F2 x2t w;b2 jw;

c

i = 1; 2 (12)

Although the likelihood may be lower when compared to the maximum likelihood approach, Joe and Xu (1996) showed that the IFM estimator is asymptotically normal distributed and consistent under suitable regularity conditions.

2.2

Semiparametric and nonparametric estimation

The quality of ML and IFM estimators depend on choosing suitable marginal models. Another option is to estimate the dependence parameters by a semiparametric approach. In …rst stage, the univariate margins Fi non-parametrically, i.e., by the empirical distribution functions Fbi or their scaled versions. In the second stage the copula parameters are estimated using maximization of the contribution to the log-likelihood function from the dependence structure in the data represented by a copula function C, i.e. bc = arg max c

T X t=1

log c Fb1 (x1t ) ; Fb2 (x2t ) ;

c

i = 1; 2

In the nonparametric approach both stages are estimated nonparametrib (u1 ; :::; ud ) of a d dimensional copula C (u1 ; :::; ud ) is cally. An estimator C typically given by an empirical analogue of the inversion formula, i.e., b 1 ; :::; ud ) = Fb(Fb 1 (u1 ); :::; Fb 1 (ud )) C(u 1 d

(13)

where Fb is a nonparametric estimator of the d dimensional distribution function F and Fb1 1 ; :::; Fbd 1 are nonparametric estimators of the pseudo-inverses 7

n o Fbi 1 (s) = inf x 2 R Fbi (x) s ; of the univariate margins F1 ; :::; Fd . Usually, Fb is taken to be the empirical d dimensional distribution function T 1X b I (X1t F (x1 ; :::; xd ) = T t=1

x1 ; :::; Xdt

xd )

(14)

. Fermanian et al. establish consistency and asymptotic normality of the empirical copula process for general copulas C with continuous partial derivatives. They also show that under regularity condition, asymptotic normality also holds for smoothed copula processes like b 1 ; u2 ) = Fb(Fb1 1 (u1 ); Fb2 1 (u2 )) C(u

(15)

that are constructed using nonparametric kernel estimators T 1X b K F (x1 ; x2 ) = T t=1

X1t x1 X2t x2 ; ht ht

(16)

where K can be some bivariate kernel function with usual regularity conditions and ht > 0 are a sequence of bandwidths that satis…es ht ! 0 as T ! 1.

2.3

Tail Dependence and Copula Families

There are several copulas that can be chosens in order to capture dependence tails and stylized facts. The concept of tail dependence can be especially useful when dealing with co-movements of two assets. De…nition 3 (Tail Dependence) If lim+ P (U1

"!0

exists and if

L

"; U2

") = lim+ "!0

C ("; ") = "

L

2 (0; 1]; then the copula C has lower tail dependence. Similarly,

2 +C( ; ) = U !1 !1 1 exists and U 2 (0; 1]; then the copula C has upper tail dependence. lim P (U1 > ; U2 > ) = lim

8

1

Therefore, tail dependence can be interpreted as the probability of a series exhibit an extreme value at the same time that another series show an extreme value in the same direction. Financial time series often present these co-movements. Often gaussian and t-student copulas are used in Finance. Both are elliptical copulas. The dependence structure associated to elliptical copulas is the Pearson correlation coe¢ cient that is suitable for symmetric distribuitions. In many cases, the shapes of theoretical distributions are not elliptical and thus we need to …nd out the dependence that is more compatible with the dependence empirically veri…ed. In addition, the mathematical properties of copulas should be exploited in inference procedures and particularly in a stochastic simulation framework (see Nelsen, 2006 for more details). Another important family of copulas is known as archimedean. Depending on the functional form, this family allows one to handle with a wide range of dependence structures. It also has good mathematical properties that are useful in inference procedures. Studies about …nancial markets show that the density of …nancial univariate data are often asymmetrical and have heavy tails. Thus, the concept of tail dependence is often useful to identify these stylized facts3 . Instead of using the Pearson correlation coe¢ cient to measure dependence we often use nonparametric measures of correlation, as Kendall’s tau and spearman’s rho. Another family to be mentioned is known as extreme value copulas (see Demarta (2002) and Embretchs et al. (1999) for details).

3

Empirical Application

3.1

Data and Descriptive Statistics

In our empirical analysis we use two stocks that are traded in BOVESPA (São Paulo Stock Exchange): PETR4 and VALE5. The data sets are the daily 3

Archimedean copulas are more ‡exible than elliptical copulas, allowing one to use different coe¢ cients of dependence and to handle with tail dependence, for example. Moreover, because a linear combination of two Archimedean copulas is an Archimedean copula (see Nelsen 2006), one can also create a mixture of copulas in order to account for the stylized facts.

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closing prices covering the period from January, 1st, 1999 to January, 09th, 2014, totalizing 3718 observations. Figure 1 provides plots for the sample considered. In the next step we compute the log returns of both stocks. One of the observations was taken from the sample due to be clearly an outlier (01/15/1999). It was the day in that Brazil adopted a ‡oating rate system. In this day the BOVESPA Stock Index was increased by 33.4%. Figure 2 contains daily returns and a box plot of returns for the sample considered. Table 1 shows some summary statistics for the sample considered. We can notice that the distributions are leptokurtic, i.e., the occurrence of extreme events is more likely compared to the normal distribution as we can see by the box plot and the kurtosis coe¢ cient and that the extreme returns are closely in time (volatility clustering). However, the empirical distributions of returns are barely skewed to the left, i.e., it does not seem that negative returns are much more likely to occur than positive ones, especially for VALE5. We can also notice the contemporaneous comovements between the series, i.e., extreme observations in one return series are often accompanied by extremes in the other return series. A risk model that does not capture the time series characteristic of …nancial market data adequately will not be useful for risk management or for deriving risk measures. For empirical data, the following stylized facts are usually found: (1) time series data of returns, in particular daily return series, are in general not independent and identically distributed; (2) the absolute or squared returns are highly autocorrelated; (3) the volatility of returns is not constant with respect to time. Figures 3 to 6 illustrate these points. The ACF and PACF hint a slight …rst order autocorrelation. The quantile–quantile (QQ) plot shows departures from normal distribution as expected. The autocorrelations and partial autocorrelations of the absolute returns are clearly signi…cantly di¤erent from zero and taper o¤ only slowly. Figures 7 and 8 provide plots of cross correlations of returns and absolute returns. Returns themselves are barely cross-correlated between markets and taper o¤ quickly (only contemporaneous cross-correlation is strong). However, signi…cant cross-correlation are evident for their absolute counterpart. Finally, the correlations based upon a moving window of 252 observations are exhibited in Figure 9. For PETR4-VALE5 the correlation ranges between 0.035 and 0.867.

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3.2

Modelling the Marginal Distributions

The ACFs and PACFs of the returns and absolute (or squared) returns of PETR4 and VALE5 show some evidence of serial correlation in the conditional mean and variance. In order to capture these characteristics we will parameterize the marginal distribution of returns according to an ARMAGARCH model. Since the data has signs of hevay tails, we will assume that the errors may have a student t distribution, possibly skewed as seen in the last subsection. To capture the leverage e¤ect4 , present in …nancial time series, we also consider asymmetric GARCH speci…cations. Performed a preliminary analysis, we estimate models that reproduce the characteristics observed in order to select the candidate apparently most appropriate according to information criteria5 We pre-selected candidates based in graphical inspection. After that we perfomed statistical tests in order to verify the adjustment quality to the data. The best …t found for both returns is a MA(1) for the conditional mean equation and an APARCH(1,1) for the conditional variance equation with t distribution innovation process for both stocks. Table 2 contains the estimates found.6 Standardized Residual Tests are also provided. All coe¢ cientes are signi…cantly di¤erent from zero at 5% (except for VALE5) and the stability condition 1 + 1 < 1 holds. We also do not reject the null hypothesis of no autocorrelation left in the standardized and squared 4

Volatility tends to increase after a price drop. We use BIC since the bayesian criteria is more likely to select the correct model than AIC asymptotically. 6 We consider the following notation here: 5

rt =

+

m X

i rt

i+

i=1

n X

i "t j

+ "t

j=1

for the ARMA(m,n) mean equation and "t zt t

= zt

t,

tv , = w+

p X i=1

where

i

i (j"t i j

represent the leverage e¤ect,

1