- Email: [email protected]
Fitting bivariate cumulative returns with copulas
Computational Statistics & Data Analysis 45 (2004) 355 – 372 www.elsevier.com/locate/csda
Fitting bivariate cumulative returns with copulas Werner H(urlimann Schonholzweg 24, CH-8409 Winterthur, Switzerland Received 7 November 2002
Abstract A copula based statistical method for ,tting joint cumulative returns between a market index and a single stock to daily data is proposed. Modifying the method of inference functions for margins (IFM method), one performs two separate maximum likelihood estimations of the univariate marginal distributions, assumed to be normal inverse gamma mixtures with kurtosis parameter equal to 6, followed by a minimization of the bivariate chi-square statistic associated to an adequate bivariate version of the usual Pearson goodness-of-,t test. The copula ,tting results for daily cumulative returns between the Swiss Market Index and a stock in the index family for an approximate 1-year period are quite satisfactory. The best overall ,ts are obtained for the new linear Spearman copula, as well as for the Frank and Gumbel–Hougaard copulas. Finally, a signi,cant application to covariance estimation for the linear Spearman copula is discussed. c 2002 Elsevier B.V. All rights reserved. Keywords: Copula; Normal inverse gamma mixture; IFM method; Bivariate chi-square statistic; Daily cumulative return; Covariance estimation
1. Introduction The present paper is a sequel and synthesis of work done in H(urlimann (2001). We link empirical results on ,tting univariate daily cumulative returns with the representation of bivariate distributions by copulas to study the statistical ,tting of joint cumulative returns between a market index and a single stock to daily data. In Section 2, the following empirical fact on ,tting univariate distributions to daily cumulative returns is recalled. The normal inverse gamma mixture distribution with E-mail address: [email protected] (W. H(urlimann). c 2002 Elsevier B.V. All rights reserved. 0167-9473/$ - see front matter doi:10.1016/S0167-9473(02)00346-8
356
W. Hurlimann / Computational Statistics & Data Analysis 45 (2004) 355 – 372
kurtosis parameter equal to 6, obtained by mixing the variance of a normal distribution with an inverse gamma prior, usually beats the lognormal and the logLaplace under a chi-square goodness-of-,t test with regrouped data. Therefore, our marginal distributions in bivariate ,tting of cumulative returns are restricted to this analytically tractable two-parameter family of symmetric location-scale distributions. Section 3 contains a short review of the representation of bivariate distributions by copulas. For our purposes, a number of attractive one-parameter families of copulas are retained. They include the copulas (Cuadras and AugEe, 1981; Gumbel, 1960; Hougaard, 1986; Galambos, 1975; Frank, 1979; H(usler and Reiss, 1989; Clayton, 1978). The parameter of these copulas measures the degree of dependence between the margins. In case the widest possible range of dependence should be covered, one is especially interested in one-parameter families of copulas, which are able to model continuously a whole range of dependence between the lower FrEechet bound copula, the independent copula, and the upper FrEechet bound copula. Such families are called inclusive or comprehensive, and include the copulas by Frank and Clayton. Another simple copula with this property is the linear Spearman copula described in details in Section 4. The linear Spearman copula represents a mixture of perfect dependence and independence. If X and Y are uniform (0; 1), Y = X with probability ¿ 0 and Y is independent of X with probability 1 − , then (X; Y ) has the linear Spearman copula with the positive dependence structure. In the statistical literature it has been considered in Konijn (1959) and motivated in Cohen (1960). The chosen nomenclature for this copula suggests that it has piecewise linear sections and that the parameter of dependence coincides with Spearman’s grade correlation coeIcient. Besides the useful fact that this copula is suitable for analytical calculation, it has many important properties. It satis,es two extremal properties, one of which is related to a conjecture by Hutchinson and Lai in Hutchinson and Lai (1990). Of great signi,cance for ,nancial modelling is its simple tail dependence structure. The coeIcient of (upper) tail dependence coincides with the dependence parameter , which implies asymptotic positive dependence in case ¿ 0. This is a desirable property in insurance and ,nancial modelling because data often tend to be dependent in their extreme values. In contrast to this, the ubiquitous Gaussian copula yields always asymptotic independence, unless perfect correlation holds. The main issue of copula ,tting is discussed in Section 5. We apply a method close in spirit to the method of inference function for margins or IFM method studied in McLeish and Small (1988), Xu (1996), and Joe (1997), Section 10.1. This estimation method proceeds by doing two separate maximum likelihood estimations of the univariate marginal distributions, followed by an optimization of the bivariate likelihood as a function of the dependence parameter. In our proposal, we do not maximize the bivariate likelihood. Instead, we determine the dependence parameter, which maximizes the p-value (respectively, minimizes the bivariate chi-square statistic) of a bivariate version of the usual Pearson goodness-of-,t test. The reason for considering a modi,ed method lies in the observation that the IFM method reduces the p-value in some cases rather drastically, leading eventually to a rejection of the model. Our copula ,tting results for daily cimulative returns between the Swiss Market Index and a stock in the index family for an approximate 1-year period are quite satisfactory.
W. Hurlimann / Computational Statistics & Data Analysis 45 (2004) 355 – 372
357
In particular, the analytically tractable linear Spearman copula does very well. The Frank and Gumbel–Hougaard copulas provide competitive best overall ,ts. Finally, Section 6 illustrates our estimation results at a signi,cant application. First, based on a general covariance formula for the linear Spearman copula, derived in Theorem 6.1, we show that for the linear Spearman copula model with the chosen normal inverse gamma mixture margins, the Spearman grade correlation coeIcient coincides with the Pearson linear correlation coeIcient. This allows one to compare the standard product–moment correlation estimator with the estimated dependence parameter from the linear Spearman copula ,tting. We observe a considerable discrepancy between the absolute values of both estimators, but on a relative scale both estimators rank the strength of dependence quite similarly.
2. Fitting univariate cumulative returns There exist many distributions, which are able to ,t the daily cumulative returns on individual stocks and market indices. As the application of the penalized likelihood scoring method in Schwartz (1978) suggests (so-called Schwartz Bayesian Criterion), it is reasonable to restrict the attention to two-parameter distributions, as shown in H(urlimann (2001). The only analytically tractable distributions we retain for comparative purposes are the logonormal, the logLaplace, and a normal inverse gamma mixture with ,xed kurtosis equal to 6. Let X represent the daily cumulative return of a market index or a stock in the index family. If ln X ∼ N (; ) follows a normal distribution, then X has the lognormal distribution FX (x) = ((ln x − )= ), with (x) the standard normal distribution. If the random variable ln X ∼ L(; ) follows a Laplace distribution, one obtains the logLaplace √ 1 (e− x) 2= ; x 6 e ; 2 FX (x) = (2.1) √ 1 − 1 (e− x)− 2= ; x ¿ e : 2 The use of a logLaplace distribution in Finance has been somewhat motivated in H(urlimann (1997). The normal inverse gamma distribution (NI) is a special element of the class of generalized hyperbolic distributions (GH), introduced by BarndorM-Nielsen (1977) in connection with the “sand project” (investigation of the physics of wind-blown sand). Following Eberlein (2001), Section 4, the GH distributions are the variance– mean mixtures of normal distributions constructed as follows. Consider the density of a generalized inverse Gaussian distribution (GIG) with parameters (; ; ) given by 1 1 2 −1 2 + x x exp − ; x ¿ 0; (2.2) fGIG (x; ; ; ) = 2K () 2 x
358
W. Hurlimann / Computational Statistics & Data Analysis 45 (2004) 355 – 372
where K denotes the modi,ed Bessel function of the third kind with index . The parameter space is described by the conditions ¿ 0;
¿0
if ¿ 0;
¿ 0;
¿0
if = 0;
¿ 0;
¿0
if ¡ 0:
(2.3)
Let ’(x) = (x) be the density of the standard normal distribution. Then the density of the generalized hyperbolic distribution with parameters (; ; ; ; ) is given by ∞ 1 x − − y fGIG (y; ; ; 2 − 2) dy: (2.4) fGH (x; ; ; ; ; ) = √ ’ √ y y 0 Since GH distributions are known to be in,nitely divisible by BarndorM-Nielsen and Halgreen (1977), every member of this family generates a LEevy process with stationary independent increments for use in a dynamic context. In ,nancial modelling, the GH distributions are further considered in Eberlein and Prause (1998), Prause (1997,1999) and BarndorM-Nielsen and Shephard (2001). Several special cases have been studied in the literature. For =1 one gets the subclass of hyperbolic distributions (HD) introduced in ,nance by Eberlein and Keller (1995) (see also BarndorM-Nielsen, 1978; BarndorM-Nielsen and Blaesild, 1981; Eberlein et al., 1998; Breckling et al., 2000; Predota, 2002). For =− 12 one gets the subclass of normal inverse Gaussian distributions (NIG) introduced in ,nance by BarndorM-Nielsen (1998). For ¡ 0; ¿ 0; = 0, the GIG distribution (2.2) represents the inverse gamma distribution (I) with density −1 x 2 1 2 fI (x; ; ) = ; x ¿ 0: (2.5) exp − 2 (−) 2 x Setting = = 0; = − ¿ 0; = c ¿ 0 in (2.4), one gets the normal inverse gamma density. In the following, a random variable X with a NI distribution is denoted by X ∼ NI(; c; ). Its density is given by
+1=2 c2 1 1 fX (x) = 1 ; (2.6) B( 2 ; ) c c2 + (x − )2 where B(a; b) = (a)(b)=(a + b) is a beta coeIcient. Recall that the location-scale transform Z = (X − )=c has a Pearson type VII density 1 fZ (z) = 1 : (2.7) B( 2 ; )(1 + z 2 )+1=2 √ If = "=2; " = 1; 2; 3; : : :, is an integer, the random variable "Z has a Student t-distribution with " degrees of freedom. The substitution t = z 2 =(1 + z 2 ) shows the identity 2 x 2 dz 1 x =(1+x ) −1=2 = t (1 − t)−1 dt; (2.8) 2 )+1=2 (1 + z 2 0 0
W. Hurlimann / Computational Statistics & Data Analysis 45 (2004) 355 – 372
359
from which one derives the distribution function 2 1 [1 − ( 1 ; ; z 2 )] z 6 0; 2 2 1+z (2.9) FZ (z) = 1 [1 + ( 1 ; ; z2 )] z ¿ 0; 2 2 2 1+z x where (a; b; x) = 0 t a−1 (1 − t)b−1 dt=B(a; b) is a beta density. The variance (if ¿ 1) and kurtosis (if ¿ 2) are equal to 1 E[Z 4 ] −1 2 Z = Var[Z] = : (2.10) =3 ; 2; Z = 2( − 1) −2 Var[Z]2 The kurtosis, which takes values in [3; ∞), is able to model leptokurtic data, a typical feature of observed ,nancial market returns. This model, ,rst suggested by Praetz (1972), has been ,tted to stock returns by Blattberg and Gonedes (1974) and Kon (1984) (see Taylor, 1992, Section 2.8). These authors ,nd maximum likelihood estimates for between 3=2 and 3. With the suggested restriction to two-parameter distributions our proposal is = 3, which yields a kurtosis parameter 2; Z = 6. This choice is motivated by the fact that it yields the less dangerous model among those models for which ∈ [3=2; 3]. For comparison, the kurtosis parameter of the logLaplace is also equal to 6, an additional argument for this choice. As a real-life application we have ,tted the daily cumulative returns of the Swiss Market Index (SMI) as well as some typical stocks using the maximum likelihood method. In our calculations we use the so-called scoring method, where the observed data is grouped into a ,nite number of classes (Hogg and Klugman, 1984, Chapter 3.7 and 4.3, (Klugman et al., 1998). The model NI(; ˆ c; ˆ 3) beats the lognormal and the logLaplace under a chi-square goodness-of-,t test with regrouped data as in H(urlimann (2001), Table 6.5. In view of this comparison result, our marginal distributions in bivariate ,tting of cumulative returns are throughout NI(; c; 3) distributions. Though the obtained ,t is not best possible in the wide class of GH distributions, in particular it is limited to symmetric distributions, our simple two-parameter model is parsimonious and analytically more tractable than the more sophisticated four-parameter NIG distribution and other models derived from the ,ve-parameter GH distributions. 3. Bivariate cumulative returns from copulas Though copulas have been introduced since Sklar (1959) their use in insurance and ,nance is more recent (e.g. Carriere, 1994; Frees et al., 1996; Frees and Valdez, 1998; Li, 1999; Klugman and Prasa, 1999; BouyEe et al., 2000; Embrechts et al., 2002). A lot of background on copulas can be found in Schweizer and Sklar (1983), Hutchinson and Lai (1990); Dall’ Aglio et al., (1991), R(uschendorf et al. (1996), Benes and Stepen (1997), Joe (1997) and Nelsen (1999). In the present paper, we are only interested in the joint cumulative returns between a market index and a single stock, that is in the most important bivariate case. Joe (1997) and Nelsen (1999) provide extensive lists of one-parameter families of copulas C (u; v), where is a measure of dependence, from which we retain the following candidates.
360
W. Hurlimann / Computational Statistics & Data Analysis 45 (2004) 355 – 372
(a) Cuadras–AugEe (1981): C (u; v) = [min(u; v)] [uv]1− ;
0 6 6 1:
(3.1)
(b) Gumbel–Hougaard (Gumbel, 1960; Hougaard, 1986; Hutchinson and Lai, 1990): C (u; v) = exp{−[(−ln u) + (−ln v) ]1= };
¿ 1:
(3.2)
(c) Galambos (1975) (Joe, 1997, p. 142): C (u; v) = exp{[(−ln u)− + (−ln v)− ]−1= }; (d) Frank (1979): 1 C (u; v) = − ln
1+
(e−u − 1)(e−v − 1) e− − 1
¿ 0:
(3.3)
;
= 0:
(3.4)
(e) H(usler–Reiss (1989) (Joe, 1997, p. 143): u 1 1 ln v C (u; v) = exp{ln u [ 1 + 12 ln ( ln ln v )] + ln v [ + 2 ln( ln u )]};
¿0
(3.5)
with (x) the standard normal distribution. (f) Clayton (1978) (Kimeldorf and Sampson, 1975a; Cook and Johnson, 1981): C (u; v) = max([u− + v− − 1]−1= ; 0);
¿ 0:
(3.6)
In all these copula families, the parameter of the joint cumulative distribution F(x; y) = C [FX (x); FY (y)] associated to a random couple (X; Y ) with marginal distributions FX (x); FY (y), measures the degree of dependence between X and Y . The larger is in absolute value, the stronger the dependence. A positive value of indicates a positive dependence. Often one considers one-parameter families of copulas, which are able to model continuously a whole range of dependence between the lower FrEechet bound copula max(u + v − 1; 0), the independent copula uv, and the upper FrEechet bound copula min(u; v). Such families are called inclusive or comprehensive. The extensive list by (Nelsen, 1999, p. 96), contains only two one-parameter inclusive families of copulas, namely Frank (1979) and H(usler and Reiss (1989). Another one is described in the next Section. 4. The linear Spearman copula Consider the linear Spearman copula, which is de,ned as follows. For ∈ [0; 1] one has [u + (1 − u)]v; v 6 u; C (u; v) = (4.1) [v + (1 − v)]u; v ¿ u and for ∈ [ − 1; 0] one has (1 + )uv; C (u; v) = uv + (1 − u)(1 − v);
u + v ¡ 1; u + v ¿ 1:
(4.2)
W. Hurlimann / Computational Statistics & Data Analysis 45 (2004) 355 – 372
361
For ∈ [0; 1] this copula is family B11 in Joe (1977, p. 148). It represents a mixture of perfect dependence and independence. If X and Y are uniform (0; 1); Y = X with probability and Y is independent of X with probability 1 − , then (X; Y ) has the linear Spearman copula. This distribution has been ,rst considered by Konijn (1959) and motivated in Cohen (1960) along Cohen’s kappa statistic (see Hutchinson and Lai, 1990, Section 10.9). For the extended copula, the chosen nomenclature linear refers to the piecewise linear sections of this copula, and Spearman refers to the fact that the grade correlation coe/cient *S by Spearman (1904) coincides with the parameter . This follows from the calculation 1 1 *S = 12 [C (u; v) − uv] du dv = ; (4.3) 0
0
where a proof of the integral representation is given in Nelsen (1991). The linear Spearman copula, which leads to the so-called linear Spearman bivariate distribution, has a singular component, which according to Joe should limit its ,eld of applicability. Despite of this it has many interesting and important properties and is suitable for analytical computation. For the reader’s convenience, let us describe ,rst two extremal properties. Kendall’s , for this copula equals using Nelsen (1991): 1 1 @ @ 1 ,=1−4 (4.4) C (u; v) C (u; v) du dv = *S [2 + sgn(*S )*S ]: @v 3 0 0 @u Invert this to get √ −1 + 1 + 3,; *S = √ −1 − 1 − 3,;
, ¿ 0;
(4.5)
, 6 0:
Relate this to the convex two-parameter copula in FrEechet (1958) de,ned by C; (u; v) = C−1 (u; v) + (1 − − )C0 (u; v) + C1 (u; v); + 6 1;
; ¿ 0; (4.6)
where C (u; v) has been de,ned in (4.1) and (4.2). Since *S = − and , = ( − )(2 + + )=3 for this copula, one has the inequalities √ , 6 *S 6 − 1 + 1 + 3,; , ¿ 0; √ 1 − 1 − 3, 6 *S 6 ,; , 6 0: (4.7) The linear Spearman copula satis,es the following extremal property. For , ¿ 0 the upper bound for *S in FrEechet’s copula is attained by the linear Spearman copula, and for , 6 0 it is the lower bound, which is attained. In case , ¿ 0 a second more important extremal property holds, which is related to a conjectural statement. Recall that Y is stochastically increasing on X , written SI (Y | X ), if Pr(Y ¿ y | X =x) is a nondecreasing function of x for all y. Similarly, X is stochastically increasing on Y , written SI (X | Y ), if Pr(X ¿ x | Y =y) is a nondecreasing function of y for all x. (Note that Lehmann (1966) speaks instead of positive regression
362
W. Hurlimann / Computational Statistics & Data Analysis 45 (2004) 355 – 372
dependence). If X and Y are continuous random variables with copula C(u; v), then one has the equivalences (Nelsen, 1999, Theorem 5.2.10): SI (Y | X ) ⇔
@ C(u; v) is nonincreasing in u for all v; @u
(4.8)
SI (X | Y ) ⇔
@ C(u; v) is nonincreasing in v for all u: @v
(4.9)
The Hutchinson–Lai conjecture consists of the following statement. If (X; Y ) satis,es the properties (4.8) and (4.9), then Spearman’s *S satis,es the inequalities √ (4.10) − 1 + 1 + 3, 6 *S 6 min{ 32 ,; 2, − ,2 }: The upper bound 2, − ,2 is attained for the one-parameter copula in Kimeldorf and Sampson (1975b) (see also Hutchinson and Lai, 1990, Section 13.7). The lower bound is attained by the linear Spearman copula (Konijn, 1959, p. 277). Alternatively, if the conjecture holds, the maximum value of Kendall’s , by given *S is attained for the linear Spearman copula. Note that the upper bound *S 6 (3=2), has been disproved recently in Nelsen (1999, Excercise 5.36). The remaining conjecture −1 + √ 1 + 3, 6 *S 6 2, − ,2 is still unsettled (however, see H(urlimann (2002) for the case of bivariate extreme value copulas). As an important modelling characteristic, let us show that the linear Spearman copula leads to a simple tail dependence structure, which is of interest when extreme values are involved. Recall the coe/cient of (upper) tail dependence of a couple (X; Y ) de,ned by = (X; Y ) = lim− Pr(Y ¿ QY () | X ¿ QX ()); →1
(4.11)
provided a limit in [0; 1] exists (QX (u) = inf {x | Pr(X 6 x) ¿ u} denotes a quantile function of X ). If ∈ (0; 1] then the couple (X; Y ) is called asymptotically dependent (in the upper tail) while if = 0 one speaks of asymptotic independence. Tail dependence is an asymptotic property of the copula. Its calculation follows easily from the relation Pr(Y ¿ QY () | X ¿ QX ()) =
1 − Pr(X 6 QX ()) − Pr(Y 6 QY ()) + Pr(X 6 QX (); Y 6 QY ()) : 1 − Pr(X 6 QX ())
(4.12)
For a linear Spearman couple one obtains (X; Y ) = lim− →1
1 − 2 + C (; ) = lim− (1 − + ) = : 1− →1
(4.13)
Therefore, unless X and Y are independent, a linear Spearman couple is always asymptotically dependent. This is a desirable property in insurance and ,nancial modelling, where data tend to be dependent in their extreme values. In contrast to this, the ubiquitous Gaussian copula yields always asymptotic independence, unless perfect correlation holds (Sibuya, 1961; Resnick, 1987, Chapter 5, Embrechts et al., 2002, Section 4.4).
W. Hurlimann / Computational Statistics & Data Analysis 45 (2004) 355 – 372
363
5. Bivariate copula tting Our view of multivariate statistical modelling is that in Joe (1997, Section 1.7). In particular, our bivariate models satisfy the desirable properties suggested in Joe (1997, Section 4.1), and Klugman and Prasa (1999). Firstly, modelling bivariate cumulative returns with the copulas of Sections 3 and 4 yields a closed-form representation for the bivariate distribution, which can be numerically evaluated provided the margins follow NI(; c; 3) distributions. Secondly, the parameter in our copulas, which models the dependence, covers the relevant range of positive dependence lying between the independent copula and the upper FrEechet bound copula. Thirdly, it remains to analyze the eMective ,tting of the chosen copulas to actual data. Statistical inference is done by specifying the estimation method and a bivariate goodness-of-,t test. To estimate the 5 parameters (X ; cX ; Y ; cY ; ) of a copula-based bivariate model F(x; y) = C [FX (x); FY (y)], where FX (x); FY (x) are NI(X ; cX ; 3); NI(Y ; cY ; 3) distributed, we apply a method close in spirit to the method of inference function for margins or IFM method studied in McLeish and Small (1988), Xu (1996), and Joe (1997, Section 10.1). We perform two separate maximum likelihood estimations of the univariate margins, followed by an estimation of the dependence parameter. However, we do not maximize the bivariate likelihood, except for comparison purposes in the Table 5.1. Instead, we determine the dependence parameter, which maximizes the p-value (respectively, minimizes the bivariate chi-square statistic) of a bivariate version of the usual Pearson goodness-of-,t test. We note that an application of the uniform test by Quesenberry (1986) for testing Y given X and X given Y , as proposed in Klugman and Prasa (1999, Section 5), is only satisfactory in one way. Testing the stock return Y given the index return X is accepted, while testing the index return X given the stock return Y is rejected. Let us describe in more details our estimation method. Given n+1 daily observations Ii of a market index and n + 1 daily prices Si of a stock in the index family, let Xi = Ii+1 =Ii ; Yi = Si+1 =Si ; i = 1; : : : ; n, be the daily cumulative returns used for statistical ,tting. We validate the estimation of the marginal distributions using two separate univariate Pearson chi-square tests. Following H(urlimann (2001, Section 6), the raw data Xi is regrouped into 6 classes (v0 ; v1 ]; (v1 ; v2 ]; : : : ; (v5 ; v6 ], where the boundaries vi ’s are chosen such that the number of observations 1 ; 2 ; : : : ; 6 in the corresponding classes are as symmetrically distributed as possible (v3 = 1 appears adequate). The data Yi is regrouped into 6 similar classes (w0 ; w1 ]; (w1 ; w2 ]; : : : ; (w5 ; w6 ] with observations 21 ; 22 : : : ; 26 in each class. One obtains two separate chi-square statistics 3X2 =
6 (i − nfi )2 i=1
fi =
nfi
;
3Y2 =
FX (vi ) − FX (vi−1 ) ; 1 − FX (v0 )
6 (2i − ngi )2 i=1
gi =
ngi
;
FY (wi ) − FY (wi−1 ) : 1 − FY (w0 )
(5.1)
364
W. Hurlimann / Computational Statistics & Data Analysis 45 (2004) 355 – 372
We report the p-values pX ; pY corresponding to 3X2 ; 3Y2 for a chi-square distribution with 3 degrees of freedom. Next, to create a meaningful bivariate chi-square statistic, we look at the number of observations zi; j in the 36 two-dimensional intervals (vi−1 ; vi ]x(wj−i ; wj ]; i; j = 1; : : : ; 6. By Moore (1978, 1986) (Klugman et al., 1998, p. 121) regroup these intervals in 10 larger rectangular interval classes as follows. Recommended is an expected frequency of at least 1% in each class and a 5% expected frequency in 80% of the classes. With n = 250 daily observations over an approximate 1-year period the following rectangular regrouping in 10 classes Ck ; k = 1; : : : ; 10, appears adequate (at least in our examples):
v0
w0
w1
v1 v2 v3 v4 v5 v6
w2
w3
w4
w5
w6
C1
C1
C1
C1
C1
C1
C1
C3
C5
C5
C5
C2
C1
C6
C7
C8
C5
C2
C1
C6
C9
C10
C5
C2
C1
C6
C6
C6
C4
C2
C2
C2
C2
C2
C2
C2
Then, consider the ,tted or expected number of observations fi; j in each of the 36 two-dimensional intervals (vi−1 ; vi ]x(wj−1 ; wj ] given by fi; j = n · F(vi ; wj ) − F(vi−1 ; wj ) − F(vi ; wj−1 ) + F(vi−1 ; wj−1 ) ; i; j = 1; : : : ; 6; F(x; y) = C [FX (x); FY (y)]:
(5.2)
Through summation of zi; j ’s respectively fi; j ’s, one obtains the number of observations Ok , respectively, the expected number of observations Ek , in each class Ck ; k=1; : : : ; 10. The bivariate chi-square statistic is then de,ned by 32 =
10 (Ok − Ek )2 : Ek
(5.3)
k=1
For each copula one obtains numerical values of , which minimize 32 , respectively, maximize the bivariate p-value corresponding to 32 for a chi-square distribution with 4 degrees of freedom. For comparisons, the value of the bivariate negative log-likelihood
W. Hurlimann / Computational Statistics & Data Analysis 45 (2004) 355 – 372
365
Table 1 IFM method for Credit Suisse Group and Novartis stock
Model
pX
pY
X = 1:000656016; cX = 0:030403482; Y = 1:001624799; cY = 0:051511031: 2.65 13.5 599.4 1.97 13.1 600.3 2.48 5 602.9 9.25 16.7 604.4 0.665 31.1 610.8 0.52 66.7 614.2 3.58 5 629.5
74.9 74.9 74.9 75.6 75.9 76.6 76.4
84.9 84.9 84.7 82.9 83.5 83.4 85.1
X = 1:000656016; cX = 0:030403482; Y = 1:000225799; cY = 0:035275623: 2.25 30.7 592.8 6.9 32.5 593 2.23 11.6 594.9 1.57 11 595.1 2.03 9.5 596 0.619 5.5 611.5 0.474 25.7 612.7
75.4 75.6 75.1 75 75.1 75.9 76.7
93.8 93.4 93.4 93.3 93.3 93.5 93.6
Credit Suisse parameters: Gumbel–Hougaard Galambos H(usler–Reiss Frank Cuadras–AugEe Linear Spearman Clayton Novartis parameters: Clayton Frank Gumbel–Hougaard Galambos H(usler–Reiss Cuadras–AugEe Linear Spearman
p-value
(−ln L)min
is also of interest. It is de,ned by −ln L = n ln [1 − FX (v0 ) − FY (v0 ) + F(v0 ; w0 )]; 6 6 − Zi; j ln [F(vi ; wj ) − F(vi−1 ; wj ) − F(vi ; wj−1 ) + F(vi−1 ; wj−1 )]:
(5.4)
i=1 j=1
Following the IFM method described above, one obtains numerical values of , which minimize −ln L under a p-value of at least 5%. Our examples show that the IFM method reduces the p-value of 32 sometimes rather drastically. For this reason, we prefer the proposed bivariate minimum chi-square or maximum p-value estimation method. To illustrate, we start with a comparison of ,t based on the IFM method. Table 1 reports copula ,tting results for daily cumulative returns between the SMI index and a single stock for the 1-year period between 29 September 1998 and 24 September 1999. For the Credit Suisse Group stock the Gumbel–Hougaard copula maximizes the bivariate log-likelihood, the linear Spearman copula yields the highest p-value. For the Novartis stock the Clayton copula maximizes the bivariate log-likelihood, the Frank copula yields the highest p-value. The Table 2 reports comparisons of ,t using the bivariate minimum chi-square estimation method for 6 pairs of daily cumulative returns between the SMI index and a single stock for the same 1-year period. Except for the NestlEe stock, whose ,tted marginal distribution is rejected, we ,nd in most cases satisfactory ,ts, in particular for the SMI stocks Credit Suisse Group, Sulzer, Novartis and UBS. The rejected ,t of the
366
W. Hurlimann / Computational Statistics & Data Analysis 45 (2004) 355 – 372
Table 2 Bivariate maximum p-value method for some stocks in the SMI index
Model
(p-value)max
−ln L
pX
pY
76.6 76 74.7 74.6 75.5 74.5 76.4
83.4 83.3 85.4 85.5 83 85.5 85.1
614.3 593 592.8 612.9 594.9 595.1 596.5
76.6 75.6 75.3 76 75.1 75 75
93.5 93.4 93.8 93.4 93.4 93.3 93.3
X = 1:000656016; cX = 0:030403482; Y = 1:0016621; cY = 0:0477182: Linear Spearman 0.403 38 Clayton 2.42 25.3 Frank 7.43 14.5 Gumbel–Hougaard 2.38 14.1 Galambos 1.69 14 H(usler–Reiss 2.34 13.8 Cuadras–AugEe 0.55 5.5
629.6 648.2 633 626.9 629 638.6 634
77.1 76.7 76.7 76.8 76.8 76.8 76.7
52.3 49.6 54.2 52.6 52.6 52.5 54
X = 1:000656016; cX = 0:030403482; Y = 1:00096518; cY = 0:03362619: Cuadras–AugEe 0.5 34 Gumbel–Hougaard 2.11 32.1 Linear Spearman 0.377 31.5 Galambos 1.42 31.2 H(usler–Reiss 2.01 29.7 Frank 6.34 29.3 Clayton 2.03 2.5
604.1 579.9 606.1 579.5 578.9 584.9 592.2
79.1 80.8 78.8 80.8 80.9 75.6 75.8
0 0 0 0 0 0.1 0
X = 1:000656016; cX = 0:030403482; Y = 1:0014062; cY = 0:0466025: Linear Spearman 0.264 69.7 Gumbel–Hougaard 1.77 64.9 Galambos 1.09 64.9 H(usler–Reiss 1.62 64.8 Frank 4.72 50.8 Cuadras–AugEe 0.382 46.2 Clayton 1.45 24.3
649.1 636.9 637.8 639.9 641.8 649.9 634.9
76.8 77.1 77.1 77 76.3 76.7 76.2
46.7 47.9 47.9 48 48.7 47.8 42.7
X = 1:000656016; cX = 0:030403482; Y = 1:001624799; cY = 0:051511031: 0.495 71.8 614.5 0.64 36.9 611.3 3.05 28.8 601.4 2.37 28.2 602.8 10.6 26.4 605.7 3.15 26.4 609.5 3.83 5.7 632.2
Credit Suisse group parameters: Linear Spearman Cuadras–AugEe Gumbel–Hougaard Galambos Frank H(usler–Reiss Clayton Novartis parameters: Linear Spearman Frank Clayton Cuadras–AugEe Gumbel–Hougaard Galambos H(usler–Reiss
X = 1:000656016; cX = 0:030403482; Y = 1:000225799; cY = 0:035275623: 0.399 56.4 6.9 32.5 2.18 31.1 0.547 15.6 2.23 11.6 1.56 11.1 2.17 10.3
UBS parameters:
Nestl:e parameters:
Sulzer parameters:
W. Hurlimann / Computational Statistics & Data Analysis 45 (2004) 355 – 372
367
Table 2 (continued) Model Swisscom parameters: Cuadras–AugEe Frank Linear Spearman Gumbel–Hougaard Galambos H(usler–Reiss Clayton
(p-value)max
X = 1:000656016; cX = 0:030403482; Y = 1:00071313; cY = 0:05098643: 0.151 15.9 2.09 12.8 0.094 9.1 1.22 8.5 0.47 7.9 0.84 7.7 0.36 2.5
−ln L
pX
pY
700.6 695.7 703.7 696.3 696.6 697 701.9
75.6 75.3 75.2 75.7 75.9 75.9 72
18.2 18.1 18.3 18 18 17.9 18.4
NestlEe marginal distribution may be due to the fact that the extreme values of the daily returns were rather moderate between −6% and 6%. The ,t of the Swisscom stock is not rejected but less satisfactory. Rather surprisingly, the highest bivariate p-value is obtained for the very tractable linear Spearman copula (except for the Swisscom stock). Except for the Cuadras–AugEe (UBS) and Clayton copulas (Credit Suisse Group and Swisscom), the p-values seem suIciently high. In our view, the linear Spearman, Frank and Gumbel–Hougaard copulas provide the best overall ,ts for the analyzed pairs. Despite of the obtained good ,t for the linear Spearman copula, we do not claim that the proposed model explains the joint distribution inferred by the data. Indeed, from a modelling viewpoint, the linear Spearman copula with positive dependence is a mixture of the independent copula and the FrEechet upper bound. This implies that the data can be split into two populations, one independent and the other with perfect positive dependence. Therefore, a scatter plot should reveal a sub-population of data, where one variable is a monotonically increasing function of the other. This property does not seem to be exhibited easily if at all. 6. Covariance estimation with the linear Spearman copula As an application of our estimation results, let us compare standard empirical values of the linear correlation coeIcient with the estimated linear correlation coeIcient from a ,tted copula. This is of interest because for heavy tailed data the standard product– moment correlation estimator has a very bad performance, and there is a need for more robust estimators (e.g. Lindskog, 2000). In view of the importance of correlation measures in modern ,nance (portfolio theory, CAPM models), this is a highly relevant issue. Extreme, synchronized rises and falls of indices and stocks occur infrequently, but more often than what is predicted by a bivariate normal model. Since the linear Spearman copula has a simple tail dependence structure (end of Section 4) and our copula ,tting was quite satisfactory, we focus on the evaluation of the covariance for this copula.
368
W. Hurlimann / Computational Statistics & Data Analysis 45 (2004) 355 – 372
Theorem 6.1. Let (X; Y ) be distributed as F(x; y) = C [FX (x); FY (y)], where C (u; v) is the linear Spearman copula, and the continuous and strictly increasing marginal distributions are de;ned on the open supports (aX ; bX ); (aY ; bY ). For an arbitrary di
(y)FY (y) = 0;
lim
(y) E[X ]FY (y) −
y→aY
y→bY
y
aY
FX−1 [FY (t)] dFY (t) = 0:
(RA)
Then one has the covariance formula Cov[X; (Y )] = sgn()E (FX−1 FY (Y ) − E[X ]) (Y ) ; where one sets FY (y)
=
FY (y);
¿ 0;
FW Y (y);
¡0
(6.1)
(6.2)
with the abbreviation FW Y (y) = 1 − FY (y). Proof. Let us ,rst derive the regression function E[X | Y = y]. The conditional distribution of X given Y = y equals for ¿ 0 FX (x) + FW X (x); x ¿ FX−1 [FY (y)]; @C [FX (x); FY (y)] = F(x | y) = (6.3) @v x ¡ FX−1 [FY (y)] (1 − )FX (x); and for ¡ 0
F(x | y) =
(1 + )FX (x);
x ¡ FX−1 [FW Y (y)];
FX (x) − FW X (x);
x ¿ FX−1 [FW Y (y)]
Through calculation one obtains the regression formula ∞ 0 E[X | y] = [1 − F(x | y)] d x − F(x | y) d x 0
−∞
=
E[X ] − (E[X ] − FX−1 [FY (y)]);
¿ 0;
E[X ] + (E[X ] − FX−1 [FW Y (y)]);
60
(6.4)
(6.5)
(6.6)
which is a weighted average of the mean E[X ] and the quantile FX−1 [FY (y)], respectively, FX−1 [FW Y (y)]. To obtain the stated covariance formula, one uses the well-known formula in HoeMding (1940) and Lehmann (1966, Lemma 2), to get the expression ∞ ∞ Cov[X; (Y )] = [F(x; y) − FX (x)FY (y)[ (y) d x dy −∞
=
∞
−∞
−∞
∞
−∞
FY (y)[F(x | Y 6 y) − FX (x)] (y) d x dy
W. Hurlimann / Computational Statistics & Data Analysis 45 (2004) 355 – 372
=
∞
−∞
+ =
0 ∞
−∞
FY (y) ∞
0
−∞
F(x | Y 6 y) d x −
FW X (x) d x −
0
∞
0
−∞
369
FX (x) d x
[1 − F(x | Y 6 y) d x]
(y) dy
(E[X ] − E[X | Y 6 y])FY (y) (y) dy:
(6.7)
Furthermore, from (6.6) one obtains E[X ] − E[X | Y 6 y] = sgn()(E[X ] − E[FX−1 [FY (Y )]Y 6 y]): Inserted in (6.7), a partial integration yields by E[X ]FY (y) − Cov[X; (Y )] = sgn()
y
aY
aY
(6.8)
FX−1 [FY (t)] dFY (t)
(y) dy
bY y −1 FX [FY (t)] dFY (t) (y) E[X ]FY (y) − aY aY = sgn() b y + (y)(FX−1 [FY (y)] − E[X ]) dFY (y) aY
; (6.9)
which implies (6.1) by the regularity assumption. The application of this result to margins from a symmetric location-scale family is simple. Corollary 6.1. Under the assumptions from Theorem 6.1 suppose that FX (x)=FZ (x − X =cX ); FY (x) = FZ (x − Y =cY ); X = E[X ]; Y = E[Y ]; and FW Z (−z) = FZ (z). Then one has cX Cov[X; (Y )] = Cov[Y; (Y )]: (6.10) cY Proof. The result follows from (6.1) noting that FX−1 [FY (y)] = X + sgn()(cX =cY ) (y − Y ). Example 6.1. In case X ∼ NI(X ; cX ; ); Y ∼ NI(Y ; cY ; ); ¿ 1, one has cX = 2( − 1)Var[X ]; cY = 2( −√1)Var[Y ]. For (y) = y the regularity assumption (RA) holds, hence Cov[X; Y ] = Var[X ]Var[Y ]. In this special situation Spearman’s correlation coeIcient coincides with Pearson’s linear correlation coeIcient. It is therefore possible to compare the standard product–moment correlation estimator with the estimated obtained from the linear Spearman copula ,tting. Results for some stocks are found in Table 3. The discrepancy between both estimators is considerable. However, on a relative scale both estimators rank the strength of dependence similarly.
370
W. Hurlimann / Computational Statistics & Data Analysis 45 (2004) 355 – 372
Table 3 Linear correlation estimators for the linear Spearman copula
Stock
Product–moment estimator
Fitted estimator
Credit Suisse Group UBS NestlEe Novartis Sulzer Swisscom
0.829 0.708 0.777 0.798 0.663 0.285
0.495 0.403 0.377 0.399 0.264 0.094
Acknowledgements The author is grateful to the referees for corrections and helpful comments. References BarndorM-Nielsen, O.E., 1977. Exponentially decreasing distributions for the logarithm of particle size. Proc. Royal Soc. London Ser. A 353, 401–419. BarndorM-Nielsen, O.E., 1978. Hyperbolic distributions and distributions on hyperbolae. Scandinavian J. Statist. 5, 151–157. BarndorM-Nielsen, O.E., 1998. Processes of normal inverse Gaussian type. Finance Stochastics 2, 41–68. BarndorM-Nielsen, O.E., Blaesild, P., 1981. Hyperbolic distributions and rami,cations: contributions to theory and application. In: Taillie, D., Patil, G., Baldessari, B. (Eds). Statistical Distributions in Scienti,c Work, Vol. 4. Reidel, Dordrecht, pp. 19 – 44. BarndorM-Nielsen, O.E., Halgreen, O., 1977. In,nite divisibility of the hyperbolic and generalized inverse Gaussian distributions. Z. Wahrscheinlichkeitstheorie und verwandte Gebiete 38, 309–312. BarndorM-Nielsen, O.E., Shephard, N., 2001. Modelling by LEevy Processes for Financial Econometrics. In: BarndorM-Nielsen, O.E., Mikosch, T., Resnick, S. (Eds.), LEevy Processes: Theory and Applications. Birkh(auser, Boston, pp. 283–318. Benes, V., Stepen, J. (Eds.), 1997. Distributions with Given Marginals and Moment Problems. Kluwer Academic Publishers, Dordrecht. Blattberg, R.C., Gonedes, N.J., 1974. A comparison of the stable and Student distributions as statistical models for stock prices. J. Bus. 47, 244–280. BouyEe, E., Nikeghbali, A., Riboulet, G., Roncalli, T., 2000. Copulas for Finance. A Reading Guide and some Applications. Groupe de Recherche OpEerationnelle, Bercy-Expo, Paris, preprint. Breckling, J., Eberlein, E., Kokic, P., 2000. A tailored suit for risk management: hyperbolic model. In: Franke, J., H(ardle, W., Stahl, G. (Eds.). Measuring Risk in Complex Stochastic Systems. Lecture Notes in Statistics, Vol. 147, pp. 189 –200 (www.xplore-stat.de/ebooks/ebooks.html). Carriere, J., 1994. Dependent decrement theory. Trans. Soc. Actuaries XLVI, 45–74. Clayton, D., 1978. A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence. Biometrika 65, 141–151. Cohen, J., 1960. A coeIcient of agreement for nominal scales. Edu. Psychol. Meas. 20, 37–46. Cook, R.D., Johnson, M.E., 1981. A family of distributions for modeling non-elliptically symmetric multivariate data. J. Roy. Statist. Soc. Ser. B 43, 210–218. Cuadras, C.M., AugEe, J., 1981. A continuous general multivariate distribution and its properties. Comm. Statist.—Theory Methods Ser. A 10 (4), 339–353. Dall’ Aglio, G., Kotz, S., Salinetti, G. (Eds.), 1991. Advances in Probability Distributions with Given Marginals. Kluwer Academic Publishers, Dordrecht.
W. Hurlimann / Computational Statistics & Data Analysis 45 (2004) 355 – 372
371
Eberlein, E., 2001. Application of generalized hyperbolic LEevy motions to ,nance. In: BarndorM-Nielsen, O.E., Mikosch, T., Resnick, S. (Eds.), LEevy Processes: Theory and Applications. Birkh(auser, Boston, pp. 319–336. Eberlein, E., Keller, U., 1995. Hyperbolic distributions in ,nance. Bernoulli 1, 281–299. Eberlein, E., Prause, K., 1998. The generalized hyperbolic model: ,nancial derivatives and risk measures. Preprint 56, Freiburg Center for Data Analysis and Modelling. Eberlein, E., Keller, U., Prause, K., 1998. New insights into smile, mispricing and value at risk: the hyperbolic model. J. Bus. 71, 371–406. Embrechts, P., McNeil, A., Straumann, D., 2002. Corrrelation and dependency in Risk Management: properties and pitfalls. In: Dempster, M.A.H. (Ed.), Risk Management: Value at Risk and Beyond. Cambridge University Press, Cambridge, pp. 176–223. Frank, M.J., 1979. On the simultaneous associativity of F(x; y) and x+y−F(x; y). Aequationes Mathematicae 19, 194–226. FrEechet, M., 1958. Remarques au sujet de la note prEecEedente. C. R. l’Acad. Sci. Paris. SEer. I Math. 246 2719 –2220. Frees, E.W., Valdez, E.A., 1998. Understanding relationships using copulas. North American Actuarial J. 2, 1–25. Frees, E.W., Carriere, J., Valdez, E.A., 1996. Annuity valuation with dependent mortality. J. Risk Insur. 63, 229–261. Galambos, J., 1975. Order statistics of samples from multivariate distributions. J. Amer. Statist. Assoc. 70, 674–680. Gumbel, E.J., 1960. Distributions des valeurs extrˆemes en plusieurs dimensions. Publ. Inst. Statist. Univ. Paris 9, 171–173. HoeMding, W., 1940. Massstabinvariante Korrelations-theorie. Schriften Math. Instituts. Univ. Berlin 5, 181–233. Hogg, R., Klugman, S., 1984. Loss Distributions. Wiley, New York. Hougaard, P., 1986. A class of multivariate failure time distributions. Biometrika 73, 671–678. H(urlimann, W., 1997. Is there a rational evidence for an in,nite variance asset pricing model? Proceedings of the Seventh International AFIR Colloquium, Cairns, Australia. H(urlimann, W., 2001. Financial data analysis with two symmetric distributions. (First Prize G. Benktander ASTIN Award Competition, 2000). ASTIN Bull. 31, 187–211. H(urlimann, W., 2002. Hutchinson–Lai’s conjecture for bivariate extreme value copulas. Statist. Probab. Lett., 61 (2), 191–198. H(usler, J., Reiss, R.-D., 1989. Maxima of normal random vectors: between independence and complete dependence. Statist. Probab. Lett. 7, 283–286. Hutchinson, T.P., Lai, C.D., 1990. Continuous Bivariate Distributions, Emphasizing Applications. Rumsby, Sydney, Australia. Joe, H., 1997. Multivariate Models and Dependence Concepts. In: Monographs on Statistics and Applied Probability, Vol. 73. Chapman & Hall, London. Kimeldorf, G., Sampson, A., 1975a. One-parameter families of bivariate distributions with ,xed marginals. Comm. Statist. 4, 293–301. Kimeldorf, G., Sampson, A., 1975b. Uniform representations of bivariate distributions. Comm. Statist. 4, 617–627. Kon, S.J., 1984. Models of stock returns—a comparison. J. Finance 39, 147–165. Konijn, H.S., 1959. Positive and negative dependence of two random variables. SankhyEa 21, 269–280. Klugman, S.A., Prasa, R., 1999. Fitting bivariate loss distributions with copulas. Insur.: Math. Econom. 24, 139–148. Klugman, S., Panjer, H., Willmot, G., 1998. Loss Models. From Data to Decisions. Wiley, New York. Lehmann, E.L., 1966. Some concepts of dependence. Ann. Math. Statist. 37, 1137–1153. Li, D.X., 1999. On default correlation: a copula function approach. Risk Metrics Group. Lindskog, F., 2000. Linear correlation estimation. Preprint, RiskLab ETH Z(urich, www.math.ethz/ ∼lindskog. McLeish, D.L., Small, C.G., 1988. The Theory and Applications of Statistical Inference Functions. In: Lecture Notes in Statistics, Vol. 44. Springer, New York.
372
W. Hurlimann / Computational Statistics & Data Analysis 45 (2004) 355 – 372
Moore, D., 1978. Chi-square tests. In: Hogg, R. (Ed.), Studies in Statistics, Vol. 19. Mathematical Association of America, pp. 453– 463. Moore, D., 1986. Tests of chi-squared type. In: D’Agostino, R., Stephens, M. (Eds.), Goodness-of-Fit Techniques. Marcel Dekker, New York, pp. 63–95. Nelsen, R.B., 1991. Copulas and association. In: Dall’Aglio, G., Kotz, S., Salinetti, G. (Eds.), Advances in Probability Distributions with Given Marginals. Kluwer, Academic Publishers, Dordrecht, pp. 51–74. Nelsen, R.B., 1999. An Introduction to Copulas. Springer, New York. Praetz, P.D., 1972. The distribution of share price changes. J. Bus. 45, 49–65. Prause, K., 1997. Modelling ,nancial data using generalized hyperbolic distributions. Preprint 48, Freiburg Center for Data Analysis and Modelling. Prause, K., 1999. The generalized hyperbolic model: estimation, ,nancial derivatives, and risk measures. Dissertation, University of Freiburg. Predota, M., 2002. The hyperbolic model: Option pricing using approximation and Quasi-Monte Carlo methods. Dissertation, www.finanz.math.tugraz.at/∼predota. Technical University of Graz. Quesenberry, C., 1986. Some transformation methods in goodness-of-,t. In: D’Agostino, R., Stephens, M. (Eds.), Goodness-of-Fit Techniques. Marcel Dekker, New York, pp. 235–277. Resnick, S.I., 1987. Extreme Values, Regular Variation and Point Processes. Springer, Berlin. R(uschendorf, L., Schweizer, B., Taylor, M.D. (Eds.), 1996. Distributions with Fixed Marginals and Related Topics. Institute of Mathematical Statistics, Hayward. Schwartz, G., 1978. Estimating the dimension of a model. Ann. Statist. 6, 461–464. Schweizer, B., Sklar, A., 1983. Probabilistic Metric Spaces. North-Holland, Amsterdam. Sibuya, M., 1961. Bivariate extreme statistics. Ann. Math. Statist. 11, 195–210. Sklar, A., 1959. Fonctions de rEepartition et leurs marges. Publ. l’Inst. Statist. l’Univ. Paris 8, 229–231. Spearman, C., 1904. The proof and measurement of association between two things. Amer. J. Psychol. 15, 72–101. Taylor, S.J., 1992. Modeling Financial Time Series (3rd reprint). Wiley, New York. Xu, J.J., 1996. Statistical modelling and inference for multivariate and longitudinal discrete response data. Ph.D. Thesis, University of British Columbia.
Fitting bivariate cumulative returns with copulas Werner H(urlimann Schonholzweg 24, CH-8409 Winterthur, Switzerland Received 7 November 2002
Abstract A copula based statistical method for ,tting joint cumulative returns between a market index and a single stock to daily data is proposed. Modifying the method of inference functions for margins (IFM method), one performs two separate maximum likelihood estimations of the univariate marginal distributions, assumed to be normal inverse gamma mixtures with kurtosis parameter equal to 6, followed by a minimization of the bivariate chi-square statistic associated to an adequate bivariate version of the usual Pearson goodness-of-,t test. The copula ,tting results for daily cumulative returns between the Swiss Market Index and a stock in the index family for an approximate 1-year period are quite satisfactory. The best overall ,ts are obtained for the new linear Spearman copula, as well as for the Frank and Gumbel–Hougaard copulas. Finally, a signi,cant application to covariance estimation for the linear Spearman copula is discussed. c 2002 Elsevier B.V. All rights reserved. Keywords: Copula; Normal inverse gamma mixture; IFM method; Bivariate chi-square statistic; Daily cumulative return; Covariance estimation
1. Introduction The present paper is a sequel and synthesis of work done in H(urlimann (2001). We link empirical results on ,tting univariate daily cumulative returns with the representation of bivariate distributions by copulas to study the statistical ,tting of joint cumulative returns between a market index and a single stock to daily data. In Section 2, the following empirical fact on ,tting univariate distributions to daily cumulative returns is recalled. The normal inverse gamma mixture distribution with E-mail address: [email protected] (W. H(urlimann). c 2002 Elsevier B.V. All rights reserved. 0167-9473/$ - see front matter doi:10.1016/S0167-9473(02)00346-8
356
W. Hurlimann / Computational Statistics & Data Analysis 45 (2004) 355 – 372
kurtosis parameter equal to 6, obtained by mixing the variance of a normal distribution with an inverse gamma prior, usually beats the lognormal and the logLaplace under a chi-square goodness-of-,t test with regrouped data. Therefore, our marginal distributions in bivariate ,tting of cumulative returns are restricted to this analytically tractable two-parameter family of symmetric location-scale distributions. Section 3 contains a short review of the representation of bivariate distributions by copulas. For our purposes, a number of attractive one-parameter families of copulas are retained. They include the copulas (Cuadras and AugEe, 1981; Gumbel, 1960; Hougaard, 1986; Galambos, 1975; Frank, 1979; H(usler and Reiss, 1989; Clayton, 1978). The parameter of these copulas measures the degree of dependence between the margins. In case the widest possible range of dependence should be covered, one is especially interested in one-parameter families of copulas, which are able to model continuously a whole range of dependence between the lower FrEechet bound copula, the independent copula, and the upper FrEechet bound copula. Such families are called inclusive or comprehensive, and include the copulas by Frank and Clayton. Another simple copula with this property is the linear Spearman copula described in details in Section 4. The linear Spearman copula represents a mixture of perfect dependence and independence. If X and Y are uniform (0; 1), Y = X with probability ¿ 0 and Y is independent of X with probability 1 − , then (X; Y ) has the linear Spearman copula with the positive dependence structure. In the statistical literature it has been considered in Konijn (1959) and motivated in Cohen (1960). The chosen nomenclature for this copula suggests that it has piecewise linear sections and that the parameter of dependence coincides with Spearman’s grade correlation coeIcient. Besides the useful fact that this copula is suitable for analytical calculation, it has many important properties. It satis,es two extremal properties, one of which is related to a conjecture by Hutchinson and Lai in Hutchinson and Lai (1990). Of great signi,cance for ,nancial modelling is its simple tail dependence structure. The coeIcient of (upper) tail dependence coincides with the dependence parameter , which implies asymptotic positive dependence in case ¿ 0. This is a desirable property in insurance and ,nancial modelling because data often tend to be dependent in their extreme values. In contrast to this, the ubiquitous Gaussian copula yields always asymptotic independence, unless perfect correlation holds. The main issue of copula ,tting is discussed in Section 5. We apply a method close in spirit to the method of inference function for margins or IFM method studied in McLeish and Small (1988), Xu (1996), and Joe (1997), Section 10.1. This estimation method proceeds by doing two separate maximum likelihood estimations of the univariate marginal distributions, followed by an optimization of the bivariate likelihood as a function of the dependence parameter. In our proposal, we do not maximize the bivariate likelihood. Instead, we determine the dependence parameter, which maximizes the p-value (respectively, minimizes the bivariate chi-square statistic) of a bivariate version of the usual Pearson goodness-of-,t test. The reason for considering a modi,ed method lies in the observation that the IFM method reduces the p-value in some cases rather drastically, leading eventually to a rejection of the model. Our copula ,tting results for daily cimulative returns between the Swiss Market Index and a stock in the index family for an approximate 1-year period are quite satisfactory.
W. Hurlimann / Computational Statistics & Data Analysis 45 (2004) 355 – 372
357
In particular, the analytically tractable linear Spearman copula does very well. The Frank and Gumbel–Hougaard copulas provide competitive best overall ,ts. Finally, Section 6 illustrates our estimation results at a signi,cant application. First, based on a general covariance formula for the linear Spearman copula, derived in Theorem 6.1, we show that for the linear Spearman copula model with the chosen normal inverse gamma mixture margins, the Spearman grade correlation coeIcient coincides with the Pearson linear correlation coeIcient. This allows one to compare the standard product–moment correlation estimator with the estimated dependence parameter from the linear Spearman copula ,tting. We observe a considerable discrepancy between the absolute values of both estimators, but on a relative scale both estimators rank the strength of dependence quite similarly.
2. Fitting univariate cumulative returns There exist many distributions, which are able to ,t the daily cumulative returns on individual stocks and market indices. As the application of the penalized likelihood scoring method in Schwartz (1978) suggests (so-called Schwartz Bayesian Criterion), it is reasonable to restrict the attention to two-parameter distributions, as shown in H(urlimann (2001). The only analytically tractable distributions we retain for comparative purposes are the logonormal, the logLaplace, and a normal inverse gamma mixture with ,xed kurtosis equal to 6. Let X represent the daily cumulative return of a market index or a stock in the index family. If ln X ∼ N (; ) follows a normal distribution, then X has the lognormal distribution FX (x) = ((ln x − )= ), with (x) the standard normal distribution. If the random variable ln X ∼ L(; ) follows a Laplace distribution, one obtains the logLaplace √ 1 (e− x) 2= ; x 6 e ; 2 FX (x) = (2.1) √ 1 − 1 (e− x)− 2= ; x ¿ e : 2 The use of a logLaplace distribution in Finance has been somewhat motivated in H(urlimann (1997). The normal inverse gamma distribution (NI) is a special element of the class of generalized hyperbolic distributions (GH), introduced by BarndorM-Nielsen (1977) in connection with the “sand project” (investigation of the physics of wind-blown sand). Following Eberlein (2001), Section 4, the GH distributions are the variance– mean mixtures of normal distributions constructed as follows. Consider the density of a generalized inverse Gaussian distribution (GIG) with parameters (; ; ) given by 1 1 2 −1 2 + x x exp − ; x ¿ 0; (2.2) fGIG (x; ; ; ) = 2K () 2 x
358
W. Hurlimann / Computational Statistics & Data Analysis 45 (2004) 355 – 372
where K denotes the modi,ed Bessel function of the third kind with index . The parameter space is described by the conditions ¿ 0;
¿0
if ¿ 0;
¿ 0;
¿0
if = 0;
¿ 0;
¿0
if ¡ 0:
(2.3)
Let ’(x) = (x) be the density of the standard normal distribution. Then the density of the generalized hyperbolic distribution with parameters (; ; ; ; ) is given by ∞ 1 x − − y fGIG (y; ; ; 2 − 2) dy: (2.4) fGH (x; ; ; ; ; ) = √ ’ √ y y 0 Since GH distributions are known to be in,nitely divisible by BarndorM-Nielsen and Halgreen (1977), every member of this family generates a LEevy process with stationary independent increments for use in a dynamic context. In ,nancial modelling, the GH distributions are further considered in Eberlein and Prause (1998), Prause (1997,1999) and BarndorM-Nielsen and Shephard (2001). Several special cases have been studied in the literature. For =1 one gets the subclass of hyperbolic distributions (HD) introduced in ,nance by Eberlein and Keller (1995) (see also BarndorM-Nielsen, 1978; BarndorM-Nielsen and Blaesild, 1981; Eberlein et al., 1998; Breckling et al., 2000; Predota, 2002). For =− 12 one gets the subclass of normal inverse Gaussian distributions (NIG) introduced in ,nance by BarndorM-Nielsen (1998). For ¡ 0; ¿ 0; = 0, the GIG distribution (2.2) represents the inverse gamma distribution (I) with density −1 x 2 1 2 fI (x; ; ) = ; x ¿ 0: (2.5) exp − 2 (−) 2 x Setting = = 0; = − ¿ 0; = c ¿ 0 in (2.4), one gets the normal inverse gamma density. In the following, a random variable X with a NI distribution is denoted by X ∼ NI(; c; ). Its density is given by
+1=2 c2 1 1 fX (x) = 1 ; (2.6) B( 2 ; ) c c2 + (x − )2 where B(a; b) = (a)(b)=(a + b) is a beta coeIcient. Recall that the location-scale transform Z = (X − )=c has a Pearson type VII density 1 fZ (z) = 1 : (2.7) B( 2 ; )(1 + z 2 )+1=2 √ If = "=2; " = 1; 2; 3; : : :, is an integer, the random variable "Z has a Student t-distribution with " degrees of freedom. The substitution t = z 2 =(1 + z 2 ) shows the identity 2 x 2 dz 1 x =(1+x ) −1=2 = t (1 − t)−1 dt; (2.8) 2 )+1=2 (1 + z 2 0 0
W. Hurlimann / Computational Statistics & Data Analysis 45 (2004) 355 – 372
359
from which one derives the distribution function 2 1 [1 − ( 1 ; ; z 2 )] z 6 0; 2 2 1+z (2.9) FZ (z) = 1 [1 + ( 1 ; ; z2 )] z ¿ 0; 2 2 2 1+z x where (a; b; x) = 0 t a−1 (1 − t)b−1 dt=B(a; b) is a beta density. The variance (if ¿ 1) and kurtosis (if ¿ 2) are equal to 1 E[Z 4 ] −1 2 Z = Var[Z] = : (2.10) =3 ; 2; Z = 2( − 1) −2 Var[Z]2 The kurtosis, which takes values in [3; ∞), is able to model leptokurtic data, a typical feature of observed ,nancial market returns. This model, ,rst suggested by Praetz (1972), has been ,tted to stock returns by Blattberg and Gonedes (1974) and Kon (1984) (see Taylor, 1992, Section 2.8). These authors ,nd maximum likelihood estimates for between 3=2 and 3. With the suggested restriction to two-parameter distributions our proposal is = 3, which yields a kurtosis parameter 2; Z = 6. This choice is motivated by the fact that it yields the less dangerous model among those models for which ∈ [3=2; 3]. For comparison, the kurtosis parameter of the logLaplace is also equal to 6, an additional argument for this choice. As a real-life application we have ,tted the daily cumulative returns of the Swiss Market Index (SMI) as well as some typical stocks using the maximum likelihood method. In our calculations we use the so-called scoring method, where the observed data is grouped into a ,nite number of classes (Hogg and Klugman, 1984, Chapter 3.7 and 4.3, (Klugman et al., 1998). The model NI(; ˆ c; ˆ 3) beats the lognormal and the logLaplace under a chi-square goodness-of-,t test with regrouped data as in H(urlimann (2001), Table 6.5. In view of this comparison result, our marginal distributions in bivariate ,tting of cumulative returns are throughout NI(; c; 3) distributions. Though the obtained ,t is not best possible in the wide class of GH distributions, in particular it is limited to symmetric distributions, our simple two-parameter model is parsimonious and analytically more tractable than the more sophisticated four-parameter NIG distribution and other models derived from the ,ve-parameter GH distributions. 3. Bivariate cumulative returns from copulas Though copulas have been introduced since Sklar (1959) their use in insurance and ,nance is more recent (e.g. Carriere, 1994; Frees et al., 1996; Frees and Valdez, 1998; Li, 1999; Klugman and Prasa, 1999; BouyEe et al., 2000; Embrechts et al., 2002). A lot of background on copulas can be found in Schweizer and Sklar (1983), Hutchinson and Lai (1990); Dall’ Aglio et al., (1991), R(uschendorf et al. (1996), Benes and Stepen (1997), Joe (1997) and Nelsen (1999). In the present paper, we are only interested in the joint cumulative returns between a market index and a single stock, that is in the most important bivariate case. Joe (1997) and Nelsen (1999) provide extensive lists of one-parameter families of copulas C (u; v), where is a measure of dependence, from which we retain the following candidates.
360
W. Hurlimann / Computational Statistics & Data Analysis 45 (2004) 355 – 372
(a) Cuadras–AugEe (1981): C (u; v) = [min(u; v)] [uv]1− ;
0 6 6 1:
(3.1)
(b) Gumbel–Hougaard (Gumbel, 1960; Hougaard, 1986; Hutchinson and Lai, 1990): C (u; v) = exp{−[(−ln u) + (−ln v) ]1= };
¿ 1:
(3.2)
(c) Galambos (1975) (Joe, 1997, p. 142): C (u; v) = exp{[(−ln u)− + (−ln v)− ]−1= }; (d) Frank (1979): 1 C (u; v) = − ln
1+
(e−u − 1)(e−v − 1) e− − 1
¿ 0:
(3.3)
;
= 0:
(3.4)
(e) H(usler–Reiss (1989) (Joe, 1997, p. 143): u 1 1 ln v C (u; v) = exp{ln u [ 1 + 12 ln ( ln ln v )] + ln v [ + 2 ln( ln u )]};
¿0
(3.5)
with (x) the standard normal distribution. (f) Clayton (1978) (Kimeldorf and Sampson, 1975a; Cook and Johnson, 1981): C (u; v) = max([u− + v− − 1]−1= ; 0);
¿ 0:
(3.6)
In all these copula families, the parameter of the joint cumulative distribution F(x; y) = C [FX (x); FY (y)] associated to a random couple (X; Y ) with marginal distributions FX (x); FY (y), measures the degree of dependence between X and Y . The larger is in absolute value, the stronger the dependence. A positive value of indicates a positive dependence. Often one considers one-parameter families of copulas, which are able to model continuously a whole range of dependence between the lower FrEechet bound copula max(u + v − 1; 0), the independent copula uv, and the upper FrEechet bound copula min(u; v). Such families are called inclusive or comprehensive. The extensive list by (Nelsen, 1999, p. 96), contains only two one-parameter inclusive families of copulas, namely Frank (1979) and H(usler and Reiss (1989). Another one is described in the next Section. 4. The linear Spearman copula Consider the linear Spearman copula, which is de,ned as follows. For ∈ [0; 1] one has [u + (1 − u)]v; v 6 u; C (u; v) = (4.1) [v + (1 − v)]u; v ¿ u and for ∈ [ − 1; 0] one has (1 + )uv; C (u; v) = uv + (1 − u)(1 − v);
u + v ¡ 1; u + v ¿ 1:
(4.2)
W. Hurlimann / Computational Statistics & Data Analysis 45 (2004) 355 – 372
361
For ∈ [0; 1] this copula is family B11 in Joe (1977, p. 148). It represents a mixture of perfect dependence and independence. If X and Y are uniform (0; 1); Y = X with probability and Y is independent of X with probability 1 − , then (X; Y ) has the linear Spearman copula. This distribution has been ,rst considered by Konijn (1959) and motivated in Cohen (1960) along Cohen’s kappa statistic (see Hutchinson and Lai, 1990, Section 10.9). For the extended copula, the chosen nomenclature linear refers to the piecewise linear sections of this copula, and Spearman refers to the fact that the grade correlation coe/cient *S by Spearman (1904) coincides with the parameter . This follows from the calculation 1 1 *S = 12 [C (u; v) − uv] du dv = ; (4.3) 0
0
where a proof of the integral representation is given in Nelsen (1991). The linear Spearman copula, which leads to the so-called linear Spearman bivariate distribution, has a singular component, which according to Joe should limit its ,eld of applicability. Despite of this it has many interesting and important properties and is suitable for analytical computation. For the reader’s convenience, let us describe ,rst two extremal properties. Kendall’s , for this copula equals using Nelsen (1991): 1 1 @ @ 1 ,=1−4 (4.4) C (u; v) C (u; v) du dv = *S [2 + sgn(*S )*S ]: @v 3 0 0 @u Invert this to get √ −1 + 1 + 3,; *S = √ −1 − 1 − 3,;
, ¿ 0;
(4.5)
, 6 0:
Relate this to the convex two-parameter copula in FrEechet (1958) de,ned by C; (u; v) = C−1 (u; v) + (1 − − )C0 (u; v) + C1 (u; v); + 6 1;
; ¿ 0; (4.6)
where C (u; v) has been de,ned in (4.1) and (4.2). Since *S = − and , = ( − )(2 + + )=3 for this copula, one has the inequalities √ , 6 *S 6 − 1 + 1 + 3,; , ¿ 0; √ 1 − 1 − 3, 6 *S 6 ,; , 6 0: (4.7) The linear Spearman copula satis,es the following extremal property. For , ¿ 0 the upper bound for *S in FrEechet’s copula is attained by the linear Spearman copula, and for , 6 0 it is the lower bound, which is attained. In case , ¿ 0 a second more important extremal property holds, which is related to a conjectural statement. Recall that Y is stochastically increasing on X , written SI (Y | X ), if Pr(Y ¿ y | X =x) is a nondecreasing function of x for all y. Similarly, X is stochastically increasing on Y , written SI (X | Y ), if Pr(X ¿ x | Y =y) is a nondecreasing function of y for all x. (Note that Lehmann (1966) speaks instead of positive regression
362
W. Hurlimann / Computational Statistics & Data Analysis 45 (2004) 355 – 372
dependence). If X and Y are continuous random variables with copula C(u; v), then one has the equivalences (Nelsen, 1999, Theorem 5.2.10): SI (Y | X ) ⇔
@ C(u; v) is nonincreasing in u for all v; @u
(4.8)
SI (X | Y ) ⇔
@ C(u; v) is nonincreasing in v for all u: @v
(4.9)
The Hutchinson–Lai conjecture consists of the following statement. If (X; Y ) satis,es the properties (4.8) and (4.9), then Spearman’s *S satis,es the inequalities √ (4.10) − 1 + 1 + 3, 6 *S 6 min{ 32 ,; 2, − ,2 }: The upper bound 2, − ,2 is attained for the one-parameter copula in Kimeldorf and Sampson (1975b) (see also Hutchinson and Lai, 1990, Section 13.7). The lower bound is attained by the linear Spearman copula (Konijn, 1959, p. 277). Alternatively, if the conjecture holds, the maximum value of Kendall’s , by given *S is attained for the linear Spearman copula. Note that the upper bound *S 6 (3=2), has been disproved recently in Nelsen (1999, Excercise 5.36). The remaining conjecture −1 + √ 1 + 3, 6 *S 6 2, − ,2 is still unsettled (however, see H(urlimann (2002) for the case of bivariate extreme value copulas). As an important modelling characteristic, let us show that the linear Spearman copula leads to a simple tail dependence structure, which is of interest when extreme values are involved. Recall the coe/cient of (upper) tail dependence of a couple (X; Y ) de,ned by = (X; Y ) = lim− Pr(Y ¿ QY () | X ¿ QX ()); →1
(4.11)
provided a limit in [0; 1] exists (QX (u) = inf {x | Pr(X 6 x) ¿ u} denotes a quantile function of X ). If ∈ (0; 1] then the couple (X; Y ) is called asymptotically dependent (in the upper tail) while if = 0 one speaks of asymptotic independence. Tail dependence is an asymptotic property of the copula. Its calculation follows easily from the relation Pr(Y ¿ QY () | X ¿ QX ()) =
1 − Pr(X 6 QX ()) − Pr(Y 6 QY ()) + Pr(X 6 QX (); Y 6 QY ()) : 1 − Pr(X 6 QX ())
(4.12)
For a linear Spearman couple one obtains (X; Y ) = lim− →1
1 − 2 + C (; ) = lim− (1 − + ) = : 1− →1
(4.13)
Therefore, unless X and Y are independent, a linear Spearman couple is always asymptotically dependent. This is a desirable property in insurance and ,nancial modelling, where data tend to be dependent in their extreme values. In contrast to this, the ubiquitous Gaussian copula yields always asymptotic independence, unless perfect correlation holds (Sibuya, 1961; Resnick, 1987, Chapter 5, Embrechts et al., 2002, Section 4.4).
W. Hurlimann / Computational Statistics & Data Analysis 45 (2004) 355 – 372
363
5. Bivariate copula tting Our view of multivariate statistical modelling is that in Joe (1997, Section 1.7). In particular, our bivariate models satisfy the desirable properties suggested in Joe (1997, Section 4.1), and Klugman and Prasa (1999). Firstly, modelling bivariate cumulative returns with the copulas of Sections 3 and 4 yields a closed-form representation for the bivariate distribution, which can be numerically evaluated provided the margins follow NI(; c; 3) distributions. Secondly, the parameter in our copulas, which models the dependence, covers the relevant range of positive dependence lying between the independent copula and the upper FrEechet bound copula. Thirdly, it remains to analyze the eMective ,tting of the chosen copulas to actual data. Statistical inference is done by specifying the estimation method and a bivariate goodness-of-,t test. To estimate the 5 parameters (X ; cX ; Y ; cY ; ) of a copula-based bivariate model F(x; y) = C [FX (x); FY (y)], where FX (x); FY (x) are NI(X ; cX ; 3); NI(Y ; cY ; 3) distributed, we apply a method close in spirit to the method of inference function for margins or IFM method studied in McLeish and Small (1988), Xu (1996), and Joe (1997, Section 10.1). We perform two separate maximum likelihood estimations of the univariate margins, followed by an estimation of the dependence parameter. However, we do not maximize the bivariate likelihood, except for comparison purposes in the Table 5.1. Instead, we determine the dependence parameter, which maximizes the p-value (respectively, minimizes the bivariate chi-square statistic) of a bivariate version of the usual Pearson goodness-of-,t test. We note that an application of the uniform test by Quesenberry (1986) for testing Y given X and X given Y , as proposed in Klugman and Prasa (1999, Section 5), is only satisfactory in one way. Testing the stock return Y given the index return X is accepted, while testing the index return X given the stock return Y is rejected. Let us describe in more details our estimation method. Given n+1 daily observations Ii of a market index and n + 1 daily prices Si of a stock in the index family, let Xi = Ii+1 =Ii ; Yi = Si+1 =Si ; i = 1; : : : ; n, be the daily cumulative returns used for statistical ,tting. We validate the estimation of the marginal distributions using two separate univariate Pearson chi-square tests. Following H(urlimann (2001, Section 6), the raw data Xi is regrouped into 6 classes (v0 ; v1 ]; (v1 ; v2 ]; : : : ; (v5 ; v6 ], where the boundaries vi ’s are chosen such that the number of observations 1 ; 2 ; : : : ; 6 in the corresponding classes are as symmetrically distributed as possible (v3 = 1 appears adequate). The data Yi is regrouped into 6 similar classes (w0 ; w1 ]; (w1 ; w2 ]; : : : ; (w5 ; w6 ] with observations 21 ; 22 : : : ; 26 in each class. One obtains two separate chi-square statistics 3X2 =
6 (i − nfi )2 i=1
fi =
nfi
;
3Y2 =
FX (vi ) − FX (vi−1 ) ; 1 − FX (v0 )
6 (2i − ngi )2 i=1
gi =
ngi
;
FY (wi ) − FY (wi−1 ) : 1 − FY (w0 )
(5.1)
364
W. Hurlimann / Computational Statistics & Data Analysis 45 (2004) 355 – 372
We report the p-values pX ; pY corresponding to 3X2 ; 3Y2 for a chi-square distribution with 3 degrees of freedom. Next, to create a meaningful bivariate chi-square statistic, we look at the number of observations zi; j in the 36 two-dimensional intervals (vi−1 ; vi ]x(wj−i ; wj ]; i; j = 1; : : : ; 6. By Moore (1978, 1986) (Klugman et al., 1998, p. 121) regroup these intervals in 10 larger rectangular interval classes as follows. Recommended is an expected frequency of at least 1% in each class and a 5% expected frequency in 80% of the classes. With n = 250 daily observations over an approximate 1-year period the following rectangular regrouping in 10 classes Ck ; k = 1; : : : ; 10, appears adequate (at least in our examples):
v0
w0
w1
v1 v2 v3 v4 v5 v6
w2
w3
w4
w5
w6
C1
C1
C1
C1
C1
C1
C1
C3
C5
C5
C5
C2
C1
C6
C7
C8
C5
C2
C1
C6
C9
C10
C5
C2
C1
C6
C6
C6
C4
C2
C2
C2
C2
C2
C2
C2
Then, consider the ,tted or expected number of observations fi; j in each of the 36 two-dimensional intervals (vi−1 ; vi ]x(wj−1 ; wj ] given by fi; j = n · F(vi ; wj ) − F(vi−1 ; wj ) − F(vi ; wj−1 ) + F(vi−1 ; wj−1 ) ; i; j = 1; : : : ; 6; F(x; y) = C [FX (x); FY (y)]:
(5.2)
Through summation of zi; j ’s respectively fi; j ’s, one obtains the number of observations Ok , respectively, the expected number of observations Ek , in each class Ck ; k=1; : : : ; 10. The bivariate chi-square statistic is then de,ned by 32 =
10 (Ok − Ek )2 : Ek
(5.3)
k=1
For each copula one obtains numerical values of , which minimize 32 , respectively, maximize the bivariate p-value corresponding to 32 for a chi-square distribution with 4 degrees of freedom. For comparisons, the value of the bivariate negative log-likelihood
W. Hurlimann / Computational Statistics & Data Analysis 45 (2004) 355 – 372
365
Table 1 IFM method for Credit Suisse Group and Novartis stock
Model
pX
pY
X = 1:000656016; cX = 0:030403482; Y = 1:001624799; cY = 0:051511031: 2.65 13.5 599.4 1.97 13.1 600.3 2.48 5 602.9 9.25 16.7 604.4 0.665 31.1 610.8 0.52 66.7 614.2 3.58 5 629.5
74.9 74.9 74.9 75.6 75.9 76.6 76.4
84.9 84.9 84.7 82.9 83.5 83.4 85.1
X = 1:000656016; cX = 0:030403482; Y = 1:000225799; cY = 0:035275623: 2.25 30.7 592.8 6.9 32.5 593 2.23 11.6 594.9 1.57 11 595.1 2.03 9.5 596 0.619 5.5 611.5 0.474 25.7 612.7
75.4 75.6 75.1 75 75.1 75.9 76.7
93.8 93.4 93.4 93.3 93.3 93.5 93.6
Credit Suisse parameters: Gumbel–Hougaard Galambos H(usler–Reiss Frank Cuadras–AugEe Linear Spearman Clayton Novartis parameters: Clayton Frank Gumbel–Hougaard Galambos H(usler–Reiss Cuadras–AugEe Linear Spearman
p-value
(−ln L)min
is also of interest. It is de,ned by −ln L = n ln [1 − FX (v0 ) − FY (v0 ) + F(v0 ; w0 )]; 6 6 − Zi; j ln [F(vi ; wj ) − F(vi−1 ; wj ) − F(vi ; wj−1 ) + F(vi−1 ; wj−1 )]:
(5.4)
i=1 j=1
Following the IFM method described above, one obtains numerical values of , which minimize −ln L under a p-value of at least 5%. Our examples show that the IFM method reduces the p-value of 32 sometimes rather drastically. For this reason, we prefer the proposed bivariate minimum chi-square or maximum p-value estimation method. To illustrate, we start with a comparison of ,t based on the IFM method. Table 1 reports copula ,tting results for daily cumulative returns between the SMI index and a single stock for the 1-year period between 29 September 1998 and 24 September 1999. For the Credit Suisse Group stock the Gumbel–Hougaard copula maximizes the bivariate log-likelihood, the linear Spearman copula yields the highest p-value. For the Novartis stock the Clayton copula maximizes the bivariate log-likelihood, the Frank copula yields the highest p-value. The Table 2 reports comparisons of ,t using the bivariate minimum chi-square estimation method for 6 pairs of daily cumulative returns between the SMI index and a single stock for the same 1-year period. Except for the NestlEe stock, whose ,tted marginal distribution is rejected, we ,nd in most cases satisfactory ,ts, in particular for the SMI stocks Credit Suisse Group, Sulzer, Novartis and UBS. The rejected ,t of the
366
W. Hurlimann / Computational Statistics & Data Analysis 45 (2004) 355 – 372
Table 2 Bivariate maximum p-value method for some stocks in the SMI index
Model
(p-value)max
−ln L
pX
pY
76.6 76 74.7 74.6 75.5 74.5 76.4
83.4 83.3 85.4 85.5 83 85.5 85.1
614.3 593 592.8 612.9 594.9 595.1 596.5
76.6 75.6 75.3 76 75.1 75 75
93.5 93.4 93.8 93.4 93.4 93.3 93.3
X = 1:000656016; cX = 0:030403482; Y = 1:0016621; cY = 0:0477182: Linear Spearman 0.403 38 Clayton 2.42 25.3 Frank 7.43 14.5 Gumbel–Hougaard 2.38 14.1 Galambos 1.69 14 H(usler–Reiss 2.34 13.8 Cuadras–AugEe 0.55 5.5
629.6 648.2 633 626.9 629 638.6 634
77.1 76.7 76.7 76.8 76.8 76.8 76.7
52.3 49.6 54.2 52.6 52.6 52.5 54
X = 1:000656016; cX = 0:030403482; Y = 1:00096518; cY = 0:03362619: Cuadras–AugEe 0.5 34 Gumbel–Hougaard 2.11 32.1 Linear Spearman 0.377 31.5 Galambos 1.42 31.2 H(usler–Reiss 2.01 29.7 Frank 6.34 29.3 Clayton 2.03 2.5
604.1 579.9 606.1 579.5 578.9 584.9 592.2
79.1 80.8 78.8 80.8 80.9 75.6 75.8
0 0 0 0 0 0.1 0
X = 1:000656016; cX = 0:030403482; Y = 1:0014062; cY = 0:0466025: Linear Spearman 0.264 69.7 Gumbel–Hougaard 1.77 64.9 Galambos 1.09 64.9 H(usler–Reiss 1.62 64.8 Frank 4.72 50.8 Cuadras–AugEe 0.382 46.2 Clayton 1.45 24.3
649.1 636.9 637.8 639.9 641.8 649.9 634.9
76.8 77.1 77.1 77 76.3 76.7 76.2
46.7 47.9 47.9 48 48.7 47.8 42.7
X = 1:000656016; cX = 0:030403482; Y = 1:001624799; cY = 0:051511031: 0.495 71.8 614.5 0.64 36.9 611.3 3.05 28.8 601.4 2.37 28.2 602.8 10.6 26.4 605.7 3.15 26.4 609.5 3.83 5.7 632.2
Credit Suisse group parameters: Linear Spearman Cuadras–AugEe Gumbel–Hougaard Galambos Frank H(usler–Reiss Clayton Novartis parameters: Linear Spearman Frank Clayton Cuadras–AugEe Gumbel–Hougaard Galambos H(usler–Reiss
X = 1:000656016; cX = 0:030403482; Y = 1:000225799; cY = 0:035275623: 0.399 56.4 6.9 32.5 2.18 31.1 0.547 15.6 2.23 11.6 1.56 11.1 2.17 10.3
UBS parameters:
Nestl:e parameters:
Sulzer parameters:
W. Hurlimann / Computational Statistics & Data Analysis 45 (2004) 355 – 372
367
Table 2 (continued) Model Swisscom parameters: Cuadras–AugEe Frank Linear Spearman Gumbel–Hougaard Galambos H(usler–Reiss Clayton
(p-value)max
X = 1:000656016; cX = 0:030403482; Y = 1:00071313; cY = 0:05098643: 0.151 15.9 2.09 12.8 0.094 9.1 1.22 8.5 0.47 7.9 0.84 7.7 0.36 2.5
−ln L
pX
pY
700.6 695.7 703.7 696.3 696.6 697 701.9
75.6 75.3 75.2 75.7 75.9 75.9 72
18.2 18.1 18.3 18 18 17.9 18.4
NestlEe marginal distribution may be due to the fact that the extreme values of the daily returns were rather moderate between −6% and 6%. The ,t of the Swisscom stock is not rejected but less satisfactory. Rather surprisingly, the highest bivariate p-value is obtained for the very tractable linear Spearman copula (except for the Swisscom stock). Except for the Cuadras–AugEe (UBS) and Clayton copulas (Credit Suisse Group and Swisscom), the p-values seem suIciently high. In our view, the linear Spearman, Frank and Gumbel–Hougaard copulas provide the best overall ,ts for the analyzed pairs. Despite of the obtained good ,t for the linear Spearman copula, we do not claim that the proposed model explains the joint distribution inferred by the data. Indeed, from a modelling viewpoint, the linear Spearman copula with positive dependence is a mixture of the independent copula and the FrEechet upper bound. This implies that the data can be split into two populations, one independent and the other with perfect positive dependence. Therefore, a scatter plot should reveal a sub-population of data, where one variable is a monotonically increasing function of the other. This property does not seem to be exhibited easily if at all. 6. Covariance estimation with the linear Spearman copula As an application of our estimation results, let us compare standard empirical values of the linear correlation coeIcient with the estimated linear correlation coeIcient from a ,tted copula. This is of interest because for heavy tailed data the standard product– moment correlation estimator has a very bad performance, and there is a need for more robust estimators (e.g. Lindskog, 2000). In view of the importance of correlation measures in modern ,nance (portfolio theory, CAPM models), this is a highly relevant issue. Extreme, synchronized rises and falls of indices and stocks occur infrequently, but more often than what is predicted by a bivariate normal model. Since the linear Spearman copula has a simple tail dependence structure (end of Section 4) and our copula ,tting was quite satisfactory, we focus on the evaluation of the covariance for this copula.
368
W. Hurlimann / Computational Statistics & Data Analysis 45 (2004) 355 – 372
Theorem 6.1. Let (X; Y ) be distributed as F(x; y) = C [FX (x); FY (y)], where C (u; v) is the linear Spearman copula, and the continuous and strictly increasing marginal distributions are de;ned on the open supports (aX ; bX ); (aY ; bY ). For an arbitrary di
(y)FY (y) = 0;
lim
(y) E[X ]FY (y) −
y→aY
y→bY
y
aY
FX−1 [FY (t)] dFY (t) = 0:
(RA)
Then one has the covariance formula Cov[X; (Y )] = sgn()E (FX−1 FY (Y ) − E[X ]) (Y ) ; where one sets FY (y)
=
FY (y);
¿ 0;
FW Y (y);
¡0
(6.1)
(6.2)
with the abbreviation FW Y (y) = 1 − FY (y). Proof. Let us ,rst derive the regression function E[X | Y = y]. The conditional distribution of X given Y = y equals for ¿ 0 FX (x) + FW X (x); x ¿ FX−1 [FY (y)]; @C [FX (x); FY (y)] = F(x | y) = (6.3) @v x ¡ FX−1 [FY (y)] (1 − )FX (x); and for ¡ 0
F(x | y) =
(1 + )FX (x);
x ¡ FX−1 [FW Y (y)];
FX (x) − FW X (x);
x ¿ FX−1 [FW Y (y)]
Through calculation one obtains the regression formula ∞ 0 E[X | y] = [1 − F(x | y)] d x − F(x | y) d x 0
−∞
=
E[X ] − (E[X ] − FX−1 [FY (y)]);
¿ 0;
E[X ] + (E[X ] − FX−1 [FW Y (y)]);
60
(6.4)
(6.5)
(6.6)
which is a weighted average of the mean E[X ] and the quantile FX−1 [FY (y)], respectively, FX−1 [FW Y (y)]. To obtain the stated covariance formula, one uses the well-known formula in HoeMding (1940) and Lehmann (1966, Lemma 2), to get the expression ∞ ∞ Cov[X; (Y )] = [F(x; y) − FX (x)FY (y)[ (y) d x dy −∞
=
∞
−∞
−∞
∞
−∞
FY (y)[F(x | Y 6 y) − FX (x)] (y) d x dy
W. Hurlimann / Computational Statistics & Data Analysis 45 (2004) 355 – 372
=
∞
−∞
+ =
0 ∞
−∞
FY (y) ∞
0
−∞
F(x | Y 6 y) d x −
FW X (x) d x −
0
∞
0
−∞
369
FX (x) d x
[1 − F(x | Y 6 y) d x]
(y) dy
(E[X ] − E[X | Y 6 y])FY (y) (y) dy:
(6.7)
Furthermore, from (6.6) one obtains E[X ] − E[X | Y 6 y] = sgn()(E[X ] − E[FX−1 [FY (Y )]Y 6 y]): Inserted in (6.7), a partial integration yields by E[X ]FY (y) − Cov[X; (Y )] = sgn()
y
aY
aY
(6.8)
FX−1 [FY (t)] dFY (t)
(y) dy
bY y −1 FX [FY (t)] dFY (t) (y) E[X ]FY (y) − aY aY = sgn() b y + (y)(FX−1 [FY (y)] − E[X ]) dFY (y) aY
; (6.9)
which implies (6.1) by the regularity assumption. The application of this result to margins from a symmetric location-scale family is simple. Corollary 6.1. Under the assumptions from Theorem 6.1 suppose that FX (x)=FZ (x − X =cX ); FY (x) = FZ (x − Y =cY ); X = E[X ]; Y = E[Y ]; and FW Z (−z) = FZ (z). Then one has cX Cov[X; (Y )] = Cov[Y; (Y )]: (6.10) cY Proof. The result follows from (6.1) noting that FX−1 [FY (y)] = X + sgn()(cX =cY ) (y − Y ). Example 6.1. In case X ∼ NI(X ; cX ; ); Y ∼ NI(Y ; cY ; ); ¿ 1, one has cX = 2( − 1)Var[X ]; cY = 2( −√1)Var[Y ]. For (y) = y the regularity assumption (RA) holds, hence Cov[X; Y ] = Var[X ]Var[Y ]. In this special situation Spearman’s correlation coeIcient coincides with Pearson’s linear correlation coeIcient. It is therefore possible to compare the standard product–moment correlation estimator with the estimated obtained from the linear Spearman copula ,tting. Results for some stocks are found in Table 3. The discrepancy between both estimators is considerable. However, on a relative scale both estimators rank the strength of dependence similarly.
370
W. Hurlimann / Computational Statistics & Data Analysis 45 (2004) 355 – 372
Table 3 Linear correlation estimators for the linear Spearman copula
Stock
Product–moment estimator
Fitted estimator
Credit Suisse Group UBS NestlEe Novartis Sulzer Swisscom
0.829 0.708 0.777 0.798 0.663 0.285
0.495 0.403 0.377 0.399 0.264 0.094
Acknowledgements The author is grateful to the referees for corrections and helpful comments. References BarndorM-Nielsen, O.E., 1977. Exponentially decreasing distributions for the logarithm of particle size. Proc. Royal Soc. London Ser. A 353, 401–419. BarndorM-Nielsen, O.E., 1978. Hyperbolic distributions and distributions on hyperbolae. Scandinavian J. Statist. 5, 151–157. BarndorM-Nielsen, O.E., 1998. Processes of normal inverse Gaussian type. Finance Stochastics 2, 41–68. BarndorM-Nielsen, O.E., Blaesild, P., 1981. Hyperbolic distributions and rami,cations: contributions to theory and application. In: Taillie, D., Patil, G., Baldessari, B. (Eds). Statistical Distributions in Scienti,c Work, Vol. 4. Reidel, Dordrecht, pp. 19 – 44. BarndorM-Nielsen, O.E., Halgreen, O., 1977. In,nite divisibility of the hyperbolic and generalized inverse Gaussian distributions. Z. Wahrscheinlichkeitstheorie und verwandte Gebiete 38, 309–312. BarndorM-Nielsen, O.E., Shephard, N., 2001. Modelling by LEevy Processes for Financial Econometrics. In: BarndorM-Nielsen, O.E., Mikosch, T., Resnick, S. (Eds.), LEevy Processes: Theory and Applications. Birkh(auser, Boston, pp. 283–318. Benes, V., Stepen, J. (Eds.), 1997. Distributions with Given Marginals and Moment Problems. Kluwer Academic Publishers, Dordrecht. Blattberg, R.C., Gonedes, N.J., 1974. A comparison of the stable and Student distributions as statistical models for stock prices. J. Bus. 47, 244–280. BouyEe, E., Nikeghbali, A., Riboulet, G., Roncalli, T., 2000. Copulas for Finance. A Reading Guide and some Applications. Groupe de Recherche OpEerationnelle, Bercy-Expo, Paris, preprint. Breckling, J., Eberlein, E., Kokic, P., 2000. A tailored suit for risk management: hyperbolic model. In: Franke, J., H(ardle, W., Stahl, G. (Eds.). Measuring Risk in Complex Stochastic Systems. Lecture Notes in Statistics, Vol. 147, pp. 189 –200 (www.xplore-stat.de/ebooks/ebooks.html). Carriere, J., 1994. Dependent decrement theory. Trans. Soc. Actuaries XLVI, 45–74. Clayton, D., 1978. A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence. Biometrika 65, 141–151. Cohen, J., 1960. A coeIcient of agreement for nominal scales. Edu. Psychol. Meas. 20, 37–46. Cook, R.D., Johnson, M.E., 1981. A family of distributions for modeling non-elliptically symmetric multivariate data. J. Roy. Statist. Soc. Ser. B 43, 210–218. Cuadras, C.M., AugEe, J., 1981. A continuous general multivariate distribution and its properties. Comm. Statist.—Theory Methods Ser. A 10 (4), 339–353. Dall’ Aglio, G., Kotz, S., Salinetti, G. (Eds.), 1991. Advances in Probability Distributions with Given Marginals. Kluwer Academic Publishers, Dordrecht.
W. Hurlimann / Computational Statistics & Data Analysis 45 (2004) 355 – 372
371
Eberlein, E., 2001. Application of generalized hyperbolic LEevy motions to ,nance. In: BarndorM-Nielsen, O.E., Mikosch, T., Resnick, S. (Eds.), LEevy Processes: Theory and Applications. Birkh(auser, Boston, pp. 319–336. Eberlein, E., Keller, U., 1995. Hyperbolic distributions in ,nance. Bernoulli 1, 281–299. Eberlein, E., Prause, K., 1998. The generalized hyperbolic model: ,nancial derivatives and risk measures. Preprint 56, Freiburg Center for Data Analysis and Modelling. Eberlein, E., Keller, U., Prause, K., 1998. New insights into smile, mispricing and value at risk: the hyperbolic model. J. Bus. 71, 371–406. Embrechts, P., McNeil, A., Straumann, D., 2002. Corrrelation and dependency in Risk Management: properties and pitfalls. In: Dempster, M.A.H. (Ed.), Risk Management: Value at Risk and Beyond. Cambridge University Press, Cambridge, pp. 176–223. Frank, M.J., 1979. On the simultaneous associativity of F(x; y) and x+y−F(x; y). Aequationes Mathematicae 19, 194–226. FrEechet, M., 1958. Remarques au sujet de la note prEecEedente. C. R. l’Acad. Sci. Paris. SEer. I Math. 246 2719 –2220. Frees, E.W., Valdez, E.A., 1998. Understanding relationships using copulas. North American Actuarial J. 2, 1–25. Frees, E.W., Carriere, J., Valdez, E.A., 1996. Annuity valuation with dependent mortality. J. Risk Insur. 63, 229–261. Galambos, J., 1975. Order statistics of samples from multivariate distributions. J. Amer. Statist. Assoc. 70, 674–680. Gumbel, E.J., 1960. Distributions des valeurs extrˆemes en plusieurs dimensions. Publ. Inst. Statist. Univ. Paris 9, 171–173. HoeMding, W., 1940. Massstabinvariante Korrelations-theorie. Schriften Math. Instituts. Univ. Berlin 5, 181–233. Hogg, R., Klugman, S., 1984. Loss Distributions. Wiley, New York. Hougaard, P., 1986. A class of multivariate failure time distributions. Biometrika 73, 671–678. H(urlimann, W., 1997. Is there a rational evidence for an in,nite variance asset pricing model? Proceedings of the Seventh International AFIR Colloquium, Cairns, Australia. H(urlimann, W., 2001. Financial data analysis with two symmetric distributions. (First Prize G. Benktander ASTIN Award Competition, 2000). ASTIN Bull. 31, 187–211. H(urlimann, W., 2002. Hutchinson–Lai’s conjecture for bivariate extreme value copulas. Statist. Probab. Lett., 61 (2), 191–198. H(usler, J., Reiss, R.-D., 1989. Maxima of normal random vectors: between independence and complete dependence. Statist. Probab. Lett. 7, 283–286. Hutchinson, T.P., Lai, C.D., 1990. Continuous Bivariate Distributions, Emphasizing Applications. Rumsby, Sydney, Australia. Joe, H., 1997. Multivariate Models and Dependence Concepts. In: Monographs on Statistics and Applied Probability, Vol. 73. Chapman & Hall, London. Kimeldorf, G., Sampson, A., 1975a. One-parameter families of bivariate distributions with ,xed marginals. Comm. Statist. 4, 293–301. Kimeldorf, G., Sampson, A., 1975b. Uniform representations of bivariate distributions. Comm. Statist. 4, 617–627. Kon, S.J., 1984. Models of stock returns—a comparison. J. Finance 39, 147–165. Konijn, H.S., 1959. Positive and negative dependence of two random variables. SankhyEa 21, 269–280. Klugman, S.A., Prasa, R., 1999. Fitting bivariate loss distributions with copulas. Insur.: Math. Econom. 24, 139–148. Klugman, S., Panjer, H., Willmot, G., 1998. Loss Models. From Data to Decisions. Wiley, New York. Lehmann, E.L., 1966. Some concepts of dependence. Ann. Math. Statist. 37, 1137–1153. Li, D.X., 1999. On default correlation: a copula function approach. Risk Metrics Group. Lindskog, F., 2000. Linear correlation estimation. Preprint, RiskLab ETH Z(urich, www.math.ethz/ ∼lindskog. McLeish, D.L., Small, C.G., 1988. The Theory and Applications of Statistical Inference Functions. In: Lecture Notes in Statistics, Vol. 44. Springer, New York.
372
W. Hurlimann / Computational Statistics & Data Analysis 45 (2004) 355 – 372
Moore, D., 1978. Chi-square tests. In: Hogg, R. (Ed.), Studies in Statistics, Vol. 19. Mathematical Association of America, pp. 453– 463. Moore, D., 1986. Tests of chi-squared type. In: D’Agostino, R., Stephens, M. (Eds.), Goodness-of-Fit Techniques. Marcel Dekker, New York, pp. 63–95. Nelsen, R.B., 1991. Copulas and association. In: Dall’Aglio, G., Kotz, S., Salinetti, G. (Eds.), Advances in Probability Distributions with Given Marginals. Kluwer, Academic Publishers, Dordrecht, pp. 51–74. Nelsen, R.B., 1999. An Introduction to Copulas. Springer, New York. Praetz, P.D., 1972. The distribution of share price changes. J. Bus. 45, 49–65. Prause, K., 1997. Modelling ,nancial data using generalized hyperbolic distributions. Preprint 48, Freiburg Center for Data Analysis and Modelling. Prause, K., 1999. The generalized hyperbolic model: estimation, ,nancial derivatives, and risk measures. Dissertation, University of Freiburg. Predota, M., 2002. The hyperbolic model: Option pricing using approximation and Quasi-Monte Carlo methods. Dissertation, www.finanz.math.tugraz.at/∼predota. Technical University of Graz. Quesenberry, C., 1986. Some transformation methods in goodness-of-,t. In: D’Agostino, R., Stephens, M. (Eds.), Goodness-of-Fit Techniques. Marcel Dekker, New York, pp. 235–277. Resnick, S.I., 1987. Extreme Values, Regular Variation and Point Processes. Springer, Berlin. R(uschendorf, L., Schweizer, B., Taylor, M.D. (Eds.), 1996. Distributions with Fixed Marginals and Related Topics. Institute of Mathematical Statistics, Hayward. Schwartz, G., 1978. Estimating the dimension of a model. Ann. Statist. 6, 461–464. Schweizer, B., Sklar, A., 1983. Probabilistic Metric Spaces. North-Holland, Amsterdam. Sibuya, M., 1961. Bivariate extreme statistics. Ann. Math. Statist. 11, 195–210. Sklar, A., 1959. Fonctions de rEepartition et leurs marges. Publ. l’Inst. Statist. l’Univ. Paris 8, 229–231. Spearman, C., 1904. The proof and measurement of association between two things. Amer. J. Psychol. 15, 72–101. Taylor, S.J., 1992. Modeling Financial Time Series (3rd reprint). Wiley, New York. Xu, J.J., 1996. Statistical modelling and inference for multivariate and longitudinal discrete response data. Ph.D. Thesis, University of British Columbia.