Conditional quantiles and tail dependence

Accepted Manuscript Conditional quantiles and tail dependence Carole Bernard, Claudia Czado PII: DOI: Reference:

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Conditional Quantiles and Tail Dependence Carole Bernard∗ and Claudia Czado†.

‡

December 29, 2014

Abstract Conditional quantile estimation is a crucial step in many statistical problems. For example, the recent work on systemic risk relies on estimating risk conditional on an institution being in distress or conditional on being in a crisis (Adrian and Brunnermeier [2010], Brownlees and Engle [2011]). Specifically, the CoVaR systemic risk measure is based on a conditional quantile when one of the variable is in the tail of the distribution. In this paper, we study properties of conditional quantiles and how they relate to properties of the copula. In particular, we provide a new graphical characterization of tail dependence and intermediate tail dependence from plots of conditional quantiles with normalized marginal distributions (probit scale). A popular method to estimate conditional quantiles is the quantile regression (Koenker [2005], Koenker and Bassett [1978]). We discuss the properties and pitfalls of this estimation approach.

Key-words: Tail dependence, Intermediate Tail Dependence, Quantile regression, Conditional quantiles, Tawn copula, Cauchy copula, Gamma copula, Uniform copula.

∗

corresponding author: C. Bernard, Grenoble Ecole de Management, 12 rue Pierre S´emard, 38003 Grenoble cedex 01, France. [email protected] † C. Czado, Technische Universit¨ at M¨ unchen, Germany. [email protected] ‡ C. Bernard acknowledges support from NSERC, from the Alexander von Humboldt Research Foundation and from the Systemic Risk research project led by B. Maillet. We thank Don McLeish for helpful suggestions on an earlier draft of the paper. We are grateful to two referees for their insightful suggestions.

1

Conditional Quantiles and Tail Dependence

Abstract Conditional quantile estimation is a crucial step in many statistical problems. For example, the recent work on systemic risk relies on estimating risk conditional on an institution being in distress or conditional on being in a crisis (Adrian and Brunnermeier [2010], Brownlees and Engle [2011]). Specifically, the CoVaR systemic risk measure is based on a conditional quantile when one of the variable is in the tail of the distribution. In this paper, we study properties of conditional quantiles and how they relate to properties of the copula. In particular, we provide a new graphical characterization of tail dependence and intermediate tail dependence from plots of conditional quantiles with normalized marginal distributions (probit scale). A popular method to estimate conditional quantiles is the quantile regression (Koenker [2005], Koenker and Bassett [1978]). We discuss the properties and pitfalls of this estimation approach.

Key-words: Tail dependence, Intermediate Tail Dependence, Quantile regression, Conditional quantiles, Tawn copula, Cauchy copula.

1

Introduction

Conditional quantile estimation is a crucial step in many statistical problems and in quantitative risk management. For example, the recent work on systemic risk deals with estimating risk conditional on an institution or on the system being in distress. In this context, the conditional quantiles are conditional on some variables being in the tail of their distributions (see Adrian and Brunnermeier [2010]). Other financial applications include Chan and Lakonishok [1992] for robust beta estimation, Boucher and Maillet [2013] for macroeconomics risk, Glosten et al. [1993] for the relationship between expected value and volatility and Engle and Manganelli [2004], Brownlees and Engle [2011] for estimating conditional Value-at-Risk, respectively systemic risk. In other fields, conditional quantiles have been used to estimate wage (Buchinsky [1994]), economic growth (Castellano and Ho [2013]), or educational attainment (Eide and Showalter [1998]). One contribution of this paper is to study how informative conditional quantiles are about asymmetric dependence, tail dependence (McNeil et al. [2005]) and intermediate tail dependence (Heffernan [2000], Coles et al. [1999], Hua and Joe [2011], Coles and Tawn [1991]). In particular, we propose a new graphical method to detect and estimate tail dependence by making use of the probit scale (for which marginal distributions are standard normal). It complements the recent work of Hua and Joe [2014] and of Joe and Li [2011]. Our approach deals with conditional quantiles. It is an alternative to the existing methods, which are related to the study of tail dependence coefficients (McNeil and Frey [2000]), conditional expectations and strength of dependence in the tails (Hua and Joe [2014]) and it may also help to choose the appropriate dependence structure (or copula) (Bouy´e et al. [2000], Durrleman et al. [2000]). Compared to the existing literature, our approach of tail dependence estimation is non symmetric, in the sense that our measure of tail dependence through conditional quantiles will depend on the order of the variables and thus on the non-exchangeability between the variables. It is possible that X|Y displays tail dependence, whereas Y |X does not display tail dependence. This is relevant in the context of studying tail dependence when the two variables under study do not play symmetric roles and are potentially of different order of magnitude. Recall that tail dependence refers typically to a property of the copula and therefore it does not take into account the marginal distributions and thus the order of magnitude of the variables. For instance, if X is the return on a market index and Y is the return 2

on a small bank. Then, a risk manager will be concerned of tail dependence of Y given X while having less interest in studying the tail dependence of X given Y . Potentially, in the context of systemic risk estimation, a regulator may be more interested in studying the tail dependence of X given Y , where X and Y represent the return of the financial market as a whole and of a big player respectively. We will show that the study of conditional quantiles is thus very insightful for studying tail dependence of non-exchangeable copulas, which standard tail dependence coefficients fail to measure adequately. We will illustrate this point by the study of the Tawn copula among others. There are several methods to estimate conditional quantiles, including quantile regression (Koenker [2005], Koenker and Bassett [1978]), local quantile regression (Spokoiny et al. [2013]) and non parametric estimation of conditional quantiles (Li and Racine [2007]). Although quantile regression per se is only one of the possible approaches available to estimate a conditional quantile, it seems to be the most popular approach in recent studies that involve the estimation of conditional quantiles. The quantile regression method is a natural extension of the classical least squares estimation of conditional mean models to the estimation of the conditional quantile functions. In this paper, we explain what quantile regression consists of, why and when it can be used or should not be used. In particular, we give the conditions under which conditional quantiles are linear, and thus, under which (linear) quantile regression may be the appropriate tool to estimate a conditional quantile. An alternative approach can be found in Bouy´e and Salmon [2009] who propose to use non-linear quantile regression and design non-linear regressors linked to the dependence among the data. In Section 2, we define “normalized conditional quantiles”, i.e. conditional quantiles with standard normal margins (after transformation). We briefly describe the methodology behind quantile regression and illustrate graphically conditional quantiles for a wide range of possible dependence structure. This preliminary study already shows that conditional quantiles may be highly non-linear, and in particular that quantile regression fails to estimate conditional quantiles as soon as there is some tail dependence. Section 3 gives conditions under which conditional quantiles are linear and quantile regression is thus a suitable estimation approach. In Section 4, we then provide theoretical properties and explain these differences in shapes by properties of tail dependence and intermediate tail dependence.

2

Preliminary on Conditional Quantiles

Let (X1 , X2 , ..., Xn , Y ) be n + 1 random variables. We are interested in the conditional distribution of Y given X := (X1 , X2 , ..., Xn ). Let us denote by FY |X=x the conditional cdf FY |X=x (y) = P (Y 6 y|X1 = x1 , ..., Xn = xn ) and the conditional quantiles  (y) > α . FY−1 (α) = inf y ∈ R | F Y |X=x |X=x

(1)

Throughout the paper, we assume that X1 ,X2 ,...,Xn , Y are continuously distributed.

2.1

Conditional Quantiles for Bivariate Risks

Consider two continuously distributed risks X and Y. The conditional quantiles of Y given X = x is given by FY−1 |X=x (α). From Sklar’s theorem the joint distribution of X and Y is defined as P (X 6 x, Y 6 y) = C(FX (x), FY (y)) where FX and FY denote the marginal distributions of X and Y respectively and where C denotes the copula between X and Y (unique as FX and FY are continuous). Recall that the conditional

3

probability of V given U = u and U given V = v can be computed by the first derivatives of the copula with respect to each of the variable. We denote by C2|1 and C1|2 these derivatives1 . For example, C2|1 (v|u) := P (V 6 v|U = u) =

∂ C(u, v), ∂u

and the conditional quantile of Y given X = x at α ∈ (0, 1) is then given by   −1 −1 FY−1 (α) = F C (α|F (x)) . X Y |X=x 2|1

−1 The notation C2|1 (α|FX (x)) denotes the inverse of the function v 7→ C2|1 (v|FX (x)) =

(2)


∂C ∂u (u, v) u=FX (x) .

It is clear that FY (·) has an important impact on the conditional quantile as it controls in particular the range of values taken by the conditional quantile. On the contrary, FX (·) has no effect on the conditional quantile as FX (X) is uniformly distributed on (0, 1).2 In this paper, we show how to study conditional dependence and to detect tail dependence using conditional quantiles in the following “normalized” form. Definition 2.1 (Normalized Conditional Quantiles). A normalized conditional quantile for two continuously distributed risks X and Y with respective distributions FX and FY and copula C is defined as FΦ−1 (3) −1 (F (Y ))|Φ−1 (F (X))=x (α), Y X where Φ is the cdf of a N (0, 1) random variable. In the case when the margins are N (0, 1) i.e., FX = Φ and FY = Φ, formula (3) reduces to the expression of conditional quantiles given in (2). Normalized conditional quantiles are conditional quantiles of the transformed risk Φ−1 (FY (Y )) conditional on Φ−1 (FX (X)), where Φ−1 (FX (X)) and Φ−1 (FY (Y )) are distributed according to the standard normal distribution N (0, 1) (also known as the probit scale). Unless otherwise stated, all conditional quantiles in the paper are normalized, so that FX = FY = Φ. This representation will be useful to detect left and right tail dependence from conditional quantiles, to understand the implication of asymmetric dependence, and to identify the Gaussian copula (see characterization in Theorem 3.5). Such a transformation of marginal distributions is standard when studying dependence. Contour plots with standard normal margins are typically easier to interpret than contour plots with uniform U (0, 1) margins, see for example Czado [2010]. To study tail dependence, Coles et al. [1999] and Ledford and Tawn [1996, 1998] propose to use marginal distributions with unit Fr´echet distribution. In the context of conditional quantile estimation and quantile regression, the use of standard normal margins and normalized conditional quantiles is natural as it will become clearer in the remainder of the paper. Proposition 2.2 (Conditional Quantiles from Transformed Margins). Assume that X and Y with respective distributions FX and FY have been transformed such that the marginal distributions of X e := and Y are now FXe and FYe (by applying an increasing transformation to X and Y , respectively X −1 −1 e e F e ◦ FX (X) and Y := F e ◦ FY (Y )). Define the conditional quantiles of the transformed variables X X Y and Ye as e k(α, x e) := F e−1e (α). Y |X=e x

Then, the conditional quantiles of Y given X = x are given by     −1 −1 e FY−1 (α) = F F k α, F (F (x)) . X Y e |X=x Ye X

1

(4)

C2|1 (v|u) is also called the h function in the context of pair copula constructions. It is known explicitly for many bivariate copulas, see for example Aas et al. [2009]. 2 This was one of the key problem of CoVaR (risk measure for systemic risk) originally pointed out by Brownlees and Engle [2011] when they write that the volatility of the conditioning variable (here denoted by X) does not influence the CoVaR systemic risk measure. This point is further investigated in Bernard et al. [2013].

4

Proof. To prove (4), observe that the copula is not affected by the increasing transformation of the margins. Thus, by definition, the conditional quantiles of the transformed variables of Ye given that e =x X e are 
 −1 e k(α, x e) = F e−1 C2|1 α|FXe (e x) Y   −1 (α|FX (x)) . The proof of (4) and the conditional quantiles of the original variables are FY−1 C2|1 e and Ye by their expressions. follows immediately by replacing X 

Our objective throughout the paper is to study tail dependence between random variables using conditional quantiles. It is motivated by the CoVaR risk measure proposed to assess systemic risk, which is based on a conditional quantile conditional on some extreme event (tail event). It was originally suggested to be estimated by quantile regression (Adrian and Brunnermeier [2010]). This method that we recall in the next paragraph (Section 2.2) approximates conditional quantiles by linear curves and by doing so, cannot capture tail dependence inherent in conditional quantiles (as illustrated graphically in Section 2.3).

2.2

Quantile regression

We briefly recall the methodology of Koenker [2005] to estimate (unconditional) quantiles. It is inspired by the fact that a quantile, similarly as the mean, is the solution to an optimization problem. Proposition 2.3 (Quantile Estimation (empirical)). Let α ∈ (0, 1). Assume that (yi )i=1..N are N observations of Y . Then, a solution to the following convex minimization ( ) N N α X 1−αX + + min (q − yi ) + (yi − q) (5) q∈R N N i=1

i=1

exists and is equal to the sample quantile of level α: FˆY−1 (α) = y(k) (k th element when the order is k k increasing) such that k−1 N < α 6 N . When α = n , the solution to (5) is not unique. 

Proof. The proof is recalled in Appendix A.1.

“Quantile regression” for conditional quantiles can be seen as an extension of regression and estimation of the conditional mean E[Y |X]. Recall indeed that the unconditional mean µ = E[Y ] is the solution to the following convex minimization: min µ∈R

N X i=1

(yi − µ)2 .

Then, assuming that E[Y |X] can be approximated by a + bX, E[Y |X] can be approximated by the solution to N X min (yj − a − bxj )2 . a,b

j=1

From (5) the (unconditional) quantile of Y at level α is a solution to the following convex minimization  min (1 − α)(q − yi )+ + α(yi − q)+ (6) q∈R

where the function to minimize is a convex function in the variable q. So instead of optimizing over all possible q ∈ R to find the unconditional quantile FY−1 (α) as done in Proposition 2.3 in (5), we can assume that the conditional quantiles are linear in a set of variables 1, X1 , X2 , ..., Xn and then solve for the best such conditional quantile. See Koenker [2005] and Koenker and Bassett [1978].

5

Proposition 2.4 (Conditional Quantile Regression (Koenker [2005])). Assuming that the conditional quantiles of Y |(X1 , X2 , ..Xn ) are linear in the set of variables (1, X1 , X2 , ..., Xn ), then the conditional quantile solves  !+ !+  n n N N  1 − α X X X X α βi xij − yj βi xij β0 + yj − β 0 − + min  N β∈Rn+1  N j=1

i=1

j=1

i=1

where for each observation yj , we observe (x1j , x2j , . . . , xnj ) for the n variables used in the conditioning.

In Section 3, we characterize the situations under which conditional quantiles are linear and thus, under which quantile regression can be applied. We first examine a few examples of conditional quantiles.

2.3

First Examples of Bivariate Dependence and their conditional quantiles

We start with a few examples of normalized conditional quantiles to illustrate how the shape of normalized conditional quantiles is affected by the choice of the dependence structure. We use a series of well-known bivariate copulas. For more details, see Joe [1997], Nelsen [2006] and Heffernan [2000]. Archimedean copulas Many of the formulas in Table 2 can be derived using the following observation for Archimedian copulas. Assume that Ψ is the generator of an Archimedian copula. We have that C(u, v) = Ψ−1 (Ψ(u) + Ψ(v)). Thus, after differentiating Ψ(C(u, v)) = Ψ(u) + Ψ(v) with respect to u and inverting, we find that       ′ −1 Ψ′ (u) −1 −1 C2|1 (α|u) = Ψ Ψ Ψ − Ψ(u) . (7) α This expression can also be found in Bouy´e and Salmon [2009].

For the Gumbel, Crowder and BBs copulas, we can compute everything in the expression (7) −1 except [Ψ′ ]−1 . The only two examples for which we can find closed-form expressions for C2|1 (α|u) are the Frank and the Clayton copulas. For all the other following examples of copulas, it is easy to obtain C2|1 (v|u) but there is no formula available for the inverse. To simulate from these copulas, and to compute conditional quantiles, one needs to invert numerically C2|1 (v|u) with respect to v. The expressions of the inverse of the first derivative (needed in the calculation of (2)) for the Gaussian, Student, Frank, Clayton and Gumbel copulas can also be found as h functions in Aas et al. [2009]. In −1 Table 2, we summarize the expressions of C2|1 (v|u) and C2|1 (α|u) for the Gaussian, Student3 copulas and the eight Archimedian copulas that have generators in Table 1.  −uδ  −1 Frank Ψ(u) = − ln ee−δ −1 Clayton Ψ(u) = 1δ (u−δ − 1)   1 Gumbel Ψ(u) = u−δ − 1 Crowder (BB9) Ψ(u) = exp δ − (δ θ + u) θ 1 
−δ

δ θ −1 BB6 Ψ(u) = − ln 1 − (1 − u)θ BB7 Ψ(u) = 1 − (1 − u)θ
1 1 BB3 Ψ(u) = exp 1δ log(1 + u) θ Joe (B5) Ψ(u) = 1 − (1 − exp(−u)) δ Table 1: Generators for the Archimedean Copulas studied in this paper

3

More properties of the Student copula can be found for instance in Demarta and McNeil [2005], Embrechts et al. [1999, 2002] and in Johnson, Kotz, and Balakrishnan [2002].

6

Gaussian copula ρ ∈ [−1, 1] Aas et al. [2009] Student T copula ρ ∈ [−1, 1], ν ∈ N\{0} tν is the cdf of the Student with ν degrees of freedom (d.f.) tρ,ν is the cdf of the bivariate Student (correlation ρ, ν d.f.) Frank copula δ ∈ R\{0} Clayton copula, δ > 0 Joe [1997] p. 141 Nelsen [2006], Table 4.1 p. 116. δ = 0: independence. Gumbel copula 16δ<∞ Crowder or BB9 copula Joe [1997] page 154 Heffernan [2000], δ > 0 and θ > 1 BB6 copula Joe [1997] page 152 δ > 1 and θ > 1. θ = 1: Gumbel copula. δ = 1: Joe copula. BB7 copula Joe [1997] page 153 δ > 0 and θ > 1. Table 2 in Czado et al. [2012]. θ = 1: Clayton copula. BB3 copula Joe [1997] page 151. δ > 0 and θ > 1. θ = 1: Clayton copula. Joe copula (or B5 copula) Joe [1997] page 141. θ > 1. θ = 1: independence copula.


C(u, v) = Φρ Φ−1 (u), Φ−1 (v) , where Φρ is the  cdf of a bivariate  standard normal with correlation ρ,

C2|1 (v|u) = Φ

Φ−1 (v)−ρΦ−1 (u)

√

1−ρ2

 1 − ρ2 + ρΦ−1 (u) .
−1 t−1 ν(u), tν (v) 



−1 (α|u) = Φ Φ−1 (α) C2|1

C(u, v) = tρ,ν

C2|1 (v|u) = tν+1  r

p

−1 t−1 ν (v)−ρtν (u)

 2 2 (ν+(t−1 ν (u)) )(1−ρ ) ! r ν+1 2 2 (ν+(t−1 ν (u)) )(1−ρ ) −1 −1 −1 + ρtν (u) . C2|1 (α|u) = tν tν+1 (α) ν+1   (e−δu −1)(e−δv −1) C(u, v) = − 1δ ln 1 + e−δ −1 e−δu (1−e−δv ) −e−δ −e−δ(u+v) +e−δu +e−δv  −δ ) = − 1δ ln 1 − e−δ uα(1−e . +α(1−e−δ u ) −1 −δ −δ δ

C2|1 (v|u) = −1 (α|u) C2|1

C(u, v) = (u

+v

− 1)


−1 −1 −δ−1 u C2|1 (v|u) = u + v − 1 δ  −δ   −1 δ −1 C2|1 (α|u) = α 1+δ − 1 u−δ + 1 . n  1 o CG (u, v) = exp − (− ln(u))δ + (− ln(v))δ δ  1−δ  δ δ δ δ−1 C2|1 (v|u) = u1 (− ln (u)) (− ln (u)) + (− ln (v)) CG (u, v)     θ1 θ θ +δ C(u, v) = exp − (δ − ln (u)) + (δ − ln (v)) − δ θ −δ

−δ

1

1 − (δ−ln(u))θ +(δ−ln(v))θ −δ θ ) θ +δ ((δ−ln(u))θ +(δ−ln(v))θ −δθ ) θ (δ−ln(u))θ e ( C2|1 (v|u) = u(δ−ln(u))((δ−ln(u))θ +(δ−ln(v))θ −δ θ )  θ1  1 −(Aδu +Bvδ ) δ BB6 (u, v) = 1 − 1 − e , 1

1−δ − Aδ +B δ δ (Aδu +Bvδ ) δ (Au )δ−1 (1−u)θ−1 e ( u v ) C2|1 (v|u) = , θ−1 θ  (1−(1−u) )(1−BB  6 (u,v))   θ θ where Au = − ln 1 − (1 − u) and Bv = − ln 1 − (1 − v) . − δ1 ! θ1  −δ  −δ θ θ 1 − (1 − u) + 1 − (1 − v) −1 BB7 (u, v) = 1 − 1 −



(1−(1−u)θ ) C2|1 (v|u) = 

BB3 (u, v) = exp −

−δ

+(1−(1−v)θ )

−δ

−1

δ

(1−BB7 (u,v)) 

ln e

δ (− ln(u))θ

(1−(1−u)θ )  !1

θ

δ

−δ−1

(1−u)θ−1

θ−1

+eδ (− ln(v)) −1

ln(BB3 (u,v))(− ln(u)) C2|1 (v|u) = − BB3 (u,v) δ (− ln(BB3 (u,v)))θ ue

− δ+1

θ

θ−1 δ (− ln(u))θ

e (− ln(BB3 (u,v)))θ



 1 θ θ θ θ θ CJ (u, v) = 1 − (1 − u) + (1 − v) − (1 − u) (1 − v)   θ 1−θ θ−1 1 − (1 − v) C2|1 (v|u) = (1 − CJ (u, v)) (1 − u)

−1 Table 2: Exchangeable Bivariate Copulas: C2|1 (α|u) must be found numerically for the last six copulas.

7

2.4

Graphical Comparison of Conditional Quantiles

Figure 1 illustrates the conditional quantiles for the bivariate copulas listed in Table 2. They are all exchangeable copulas (C(u, v) = C(v, u)). We observe that conditional quantiles are fundamentally different across various dependence structures. The next sections will give some theoretical background on how to interpret the shape of the normalized conditional quantiles plotted in Figure 1. Gauss Copula with ρ=0.71

Clayton Copula with δ=2

0

−2

−2

0 x

2

2

0

−2

−4 −4

4

Frank Copula with δ=5.7

0 x

2 1 0 −1 −2 −3 −2

0

2

1 0 −1 −2 −3

4

−2

0

2

−2

2

1 0 −1 −2 −3 −2

1 0 −1 −2 −3 −2

0

2

1 0 −1 −2 −3 −2

−1 −2 −3 2

4

2

4

BB7 Copula with δ=1.5, θ=1.6 4

3 2 1 0 −1 −2 −3 −4 −4

0

x

Conditional Quantile of Y|X=x

Conditional Quantile of Y|X=x

0

4

2

BB7 Copula with δ=0.15, θ=2.7

1

2

3

−4 −4

4

4

2

0

BB6 Copula with δ=1.5, θ=1.6

x

3

4

4

2

Joe Copula with δ=2.8

2

x

3

−4 −4

4

0

2

−4 −4

4

Conditional Quantile of Y|X=x

Conditional Quantile of Y|X=x

Conditional Quantile of Y|X=x

0

4

Conditional Quantile of Y|X=x

−2

3

BB3 Copula with δ=0.25, θ=1.7

2

0

−3

x

4

−2

−2

Student Copula, ρ=0.71, ν=15

2

Crowder Copula with δ=0.5, θ=3.3

−4 −4

0 −1

x

3

−4 −4

4

0 x

1

4

x

−2

2

−4 −4

4

Conditional Quantile of Y|X=x

3

−4 −4

2

4

Conditional Quantile of Y|X=x

Conditional Quantile of Y|X=x

−2

3

Student Copula, ρ=0.71, ν=2

4

−4 −4

4

Conditional Quantile of Y|X=x

2

−4 −4

Gumbel Copula with δ=2

4

Conditional Quantile of Y|X=x

Conditional Quantile of Y|X=x

4

−2

0

x

x

2

4

3 2 1 0 −1 −2 −3 −4 −4

−2

0

2

4

x

Figure 1: Normalized Conditional Quantiles at level 1% (lowest curve), 10%, 50% (middle curve), 90% and 99% (highest curve) for N (0, 1) margins and various exchangeable copulas that all have the same Kendall’s tau equal to 0.5. The dotted line corresponds to the line Y = X, which will play a key role in detecting tail dependence.

8

2.5

Study of the Conditional Quantiles for a Non-exchangeable Copula

The Tawn copula or asymmetric logistic copula (introduced by Tawn [1988]) is an extreme value copula with Pickands dependence function 1

A(t) = (ψ2 − ψ1 )t + (1 − ψ2 ) + ([ψ2 (1 − t)]δ + (ψ1 t)δ ) δ

for t ∈ [0, 1], 0 6 ψ1 6 1, 0 6 ψ2 6 1 and δ > 1. Recall that the copula can be written as a function of the Pickands dependence function as follows4 C(u, v) = (uv)A(ω) ,

ω=

ln (u) ln (u v)

Then, the copula is non-exchangeable and  1 (uv)A(ω)  ∂C(u, v) 1 − ψ1 + K δ −1 ψ1 δ ω δ−1 , = ∂u u

(8)

 1 (uv)A(ω)  ∂C(u, v) = 1 − ψ2 + K δ −1 ψ2 δ (1 − ω)δ−1 , ∂v v

(9)

with K = (ψ2 (1 − ω))δ + (ψ1 ω)δ . When δ → +∞, the Tawn copula is the Marshall-Olkin copula with parameter (ψ1 , ψ2 ). In Figure 2, we represent theoretical conditional quantiles for the Tawn copula for different sets of parameters. Tawn Copula with δ=2, ψ1=1, ψ2=1 Gumbel Copula with δ=2 and τ=0.5

Tawn Copula with δ=2, ψ1=0.1, ψ2=1, τ=0.08

0

−2

−2

0 x

2

2

0

−2

−4 −4

4

Tawn Copula with δ=6, ψ =1, ψ =0.1, τ=0.1 1

0 x

2

1

0

−2

0 x

2

−2

4

0 x

2

4

4

2

0

−2

−4 −4

−2

Tawn Copula with δ=1, ψ =1, ψ =1 1 2 Independence copula, τ=0

2

Conditional Quantile of Y|X=x

2

−2

0

−4 −4

4

4

Conditional Quantile of Y|X=x

Conditional Quantile of Y|X=x

−2

2

Tawn Copula with δ=6, ψ =0.1, ψ =1, τ=0.1

2

4

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Figure 2: Normalized Conditional Quantiles FY−1 |X=x (α) at level 1% (lowest curve), 10%, 50% (middle curve), 90% and 99% (highest curve) (for N (0, 1) margins) and Tawn copula for various parameters. A simulated value of the Kendall’s tau is indicated in each panel. In the last panel of Figure 2, δ = ψ1 = ψ2 = 1, the Tawn copula coincides with the independence copula. In this case, conditional quantiles of Y given X = x do not depend on x and are therefore horizontal lines (the Kendall’s tau is equal to 0). We also observe in Figure 2 (middle panels) that the Using the notation for extreme value copulas, C(u, v) = exp (−ℓ∗ (− ln(u), − ln(v))) where ℓ∗ (x1 , x2 ) =  standard  1 (x1 + x2 )A x1x+x . See Segers and Gudendorf [2010]. 2 4

9

Tawn copula displays “partial” upper tail independence (for conditional quantiles with low level of α). It means that the conditional quantiles for low level of α are horizontal and thus asymptotically similar to the independence copula. In the next section, we will formally define this corresponding notion of tail independence. For high level of α, in the upper tail, in all panels (except the last one), there is some tail dependence. On the upper left corner, the conditional quantiles correspond to the case when ψ1 = ψ2 = 1, which is also the Gumbel copula. Both the quantiles of Y |X = x and X|Y = y display the same shape because it is an exchangeable copula. However, for the other cases, this is not true anymore. For instance, if ψ1 = 0.1 and ψ2 = 1, then the quantiles of Y |X = x are represented in the middle panel of the upper graphs in Figure 2. If we would represent the quantiles of X|Y = y, they would look like the right panel of the upper graphs (obtained with ψ1 = 1 and ψ2 = 0.1).5 Similarly the left panel of the lower graphs in Figure 2 gives the conditional quantiles of Y |X = x for ψ1 = 1 and ψ2 = 0.1. The quantiles of X|Y = y would look similarly as the middle panel of the lower graphs in Figure 2. Thus, the tail behavior of Y |X can be very different from the tail behavior of X|Y . The interpretation of the conditional quantiles is thus very instructive in the analysis of tail dependence of non-exchangeable copulas.

3

Linearity of Conditional Quantiles

From Figure 1, we observe that the conditional quantiles curves for different values of α seem to be parallel for the Gauss copula, asymptotically parallel in both tails for the Frank copula, or asymptotically parallel in the right tail for the Clayton or Crowder copulas (i.e. when x is sufficiently large) and asymptotically parallel in the left tail for the Joe and BB6 copulas (i.e. when x is sufficiently small). In this section, we formalize these observations.

3.1

(Asymptotic) Deterministic Dependence from Conditional Quantiles

The conditional quantiles are asymptotically parallel if there exist deterministic functions a(·) and g(·) such that the conditional quantiles are asymptotically equal to a(α) + g(X). Proposition 3.1 (Linearity of Conditional Quantiles). Assume that Xi takes values in the real line (such as N (0, 1) margins). If for all α ∈ (0, 1), the conditional quantiles of Y given (X1 , ..., Xn ) = x are linear in xi for i = 1, ..., n with deterministic slopes bi then FY−1 |X=x (α) = a(α) +

n X

bi xi

i=1

where a(·) is a deterministic non-decreasing function of α and bi is constant and does not depend on α. In addition, R1 P E[Y |X = x] = 0 a(α)dα + ni=1 bi xi , 2 R R1 (10) 1 var[Y |X = x] = 0 a(α)2 dα − 0 a(α)dα .

Assume that the supports of Xi are the non-negative real numbers, then FY−1 |X=x (α) = a(α) +

n X

bi (α)xi

i=1

where a and bi are deterministic non-decreasing functions of α. 5

This observation follows from the symmetry in the roles of ψ1 and ψ2 for Y |X and X|Y that can be seen from the expression of the conditional copulas (8) and (9).

10

Proof. By the monotonicity of the quantile function, the quantiles for different α cannot cross. When the support of Xi is the real line, then the coefficient bi cannot depend on α, otherwise quantiles at different values of α would cross (by taking xi sufficiently large or sufficiently small). If Xi takes non-negative values only, then bi is a non-decreasing function of α (so that the lines may cross when xi is negative, which is not possible). It is also clear that when xi = 0 for all i, then a(·) must be non-decreasing so that the quantiles are non-decreasing as a function of α. To establish (10), we use R R1 2 2 |X = x] = 1 (F −1 (α)dα and E[Y the fact that E[Y |X = x] = 0 FY−1 0 Y |X=x (α)) dα. The expressions |X=x of the conditional mean and variance in (10) follow.  The following necessary condition to ensure linearity of the conditional quantiles is a straightforward corollary of Proposition 3.1. Corollary 3.2 (Necessary Condition for Linearity of the Conditional Quantiles). Assume that Xi takes values in the real line for i = 1, ...n. If the conditional variance var[Y |X = x] depends on x, then the conditional quantiles cannot be linear in x (with a deterministic slope). The result of this corollary does not hold when Xi takes non-negative values only as the formula for the conditional variance may depend on x when the slope b depends on α. Proposition 3.3 (Necessary and Sufficient Condition for FY−1 |X=x (α) = a(α) + g(x) when a is non-decreasing). Let Y be a random variable and X be a random vector then the two following conditions are equivalent for all α ∈ (0, 1): (a) Y = g(X) + ε where ε is a random variable independent of X,

(b) FY−1 |X (α) = a(α) + g(X) where a is non-decreasing and right-continuous.

Proof. Proof of (a) ⇒ (b) : If Y = g(X) + ε where ε is independent of X then we solve for y such that P (Y 6 y|X = x) = α, i.e. P (g(x) + ε 6 y) = α, we find that y = g(x) + Fε−1 (α), which proves (b). Proof of (b) ⇒ (a) : Reciprocally, assume (b), then P (Y − g(X) 6 a(α)|X = x) = α. Thus the quantile function of Y − g(X) is equal to a(α) and does not depend on X. Thus, ε := Y − g(X) is independent of X and (a) is proved.  Note that g may depend on α and in this case, ε may also depend on α but the proof holds. A characterization of linearity of conditional quantiles follows by choosing g(X) = bX in Proposition 3.3. Pn Corollary 3.4 (Necessary and Sufficient Condition for FY−1 i=1 bi xi ). Let Y be a |X=x (α) = a(α) + P −1 random variable and X be a random vector. We have that FY |X=x (α) = a(α) + ni=1 bi xi if and only P if Y = ni=1 bi Xi + ε where ε is independent of X. In addition, a(α) = Fε−1 (α). P The only copulas that ensure linear conditional quantiles are copulas between X and ni=1 bi Xi + ε for some independent variable Pn ε. In other words, the dependence between X and Y can be reduced to a linear factor model, Y = i=1 bi Xi + ε if and only if the conditional quantiles of Y |X = x are linear. Factor models are used in credit risk modeling, where dependence is introduced typically through a multivariate vector of latent variables (Glasserman and Li [2005] for instance). Note that when X and Y have the same marginal distributions with finite variance, using var(Y ) = b2 var(X) + var(ε), it is clear that b ∈ (−1, 1). From Figure 1, the conditional quantiles of Y |X appear linear when both variables are distributed with N (0, 1) and their dependence is the Gaussian copula. This result is well-known theoretically as p p in this case, Y = ρX + 1 − ρ2 Z where Z is an independent N (0, 1) variable, so that ε = 1 − ρ2 Z. Conditional quantiles are also almost linear for N (0, 1) variables with the Student copula with a high degree of freedom (Figure 1, Panel 6, when the degree of freedom for the Student distribution is ν = 15). The above results already show that strong conditions on the copula are needed to ensure the linearity of conditional quantiles (so that quantile regression introduced in Section 2.2 is suitable). Next, we first discuss the case of normalized quantiles (with N (0, 1) margins). We then examine the case of other margins for which we are able to find the explicit copula that ensures the linearity of conditional quantiles, and we illustrate the method with Cauchy, Gamma and Uniform margins. 11

3.2

Linearity of Normalized Conditional Quantiles

We now show that assuming normal marginal distributions (and thus including the case of normalized conditional quantiles) the only dependence for which conditional quantiles are linear (with deterministic slopes) is the Gaussian copula. To do so, we first recall well-known results and notation for the multivariate normal distribution. Assume that (Y, X1 , ..., Xn ) follows a multivariate Gaussian distribution, such that Y ∼ N (µY , σY2 ), Xi ∼ N (µi , σi2 ) for i = 1, ..., n, and    2  µY σY Σ1n (Y, X1 , ..., Xn ) ∼ M V N , µ Σn1 Σnn where µ=(µ1 , µ2 , ...µn )T , Σnn is the (invertible) covariance matrix of (X1 , ..., Xn ), Σn1 is a n × 1 matrix such that Σn1 = (cov(Xi , Y))i=1..n and Σ1n = ΣTn1 . It is well-known that Y |X = x is normally distributed with conditional mean and variance as follows  µ∗ (x) := E[Y |X = x] = µY + Σ1n Σ−1 nn (x − µ) σ∗2 := var[Y |X = x] = σY2 − Σ1n Σ−1 nn Σn1 where X denotes the vector (X1 , X2 , ...Xn )T , x = (x1 , x2 , ..., xn )T . Thus, the conditional quantiles (1) are equal to −1 FY−1 (11) |X=x (α) = µ∗ (x) + σ∗ Φ (α), where Φ−1 (α) is the α-quantile of N (0, 1). Then (11) can also be rewritten as −1 FY−1 |X=x (α) = a0 + Σ1n Σnn x

(12)

−1 where a0 := µY − Σ1n Σ−1 nn µ + σ∗ Φ (α) does not depend on x. The conditional quantiles in (12) for the multivariate Gaussian are thus clearly linear in the conditioning variables. In the case when n = 1, we can write explicitly this conditional quantile as a function of the correlation between X and Y . p ρσY (x − µ) + FY−1 (α) = µ + 1 − ρ2 σY Φ−1 (α). (13) Y |X σX

In particular, the slope is equal to ρσY /σX . This feature appears clearly in the top panel of Figure 1 in the case of the Gaussian copula. In the figure, σX = σY = 1 and the slope of the conditional quantiles of Y given X is exactly 0.71 (which corresponds to the correlation ρ such that the Kendall’s tau is 0.5). Theorem 3.5 (Necessary and Sufficient Condition for linearity of normalized conditional quantiles). Assume that Y ∼ N (µY , σY2 ), 2 ) then F −1 (α) is linear in X with a deterministic slope for all α if and only if (a) if X ∼ N (µX , σX Y |X (X, Y ) has a bivariate normal distribution.

(b) if X ∼ MVN (µX , ΣX 2 ) then FY−1 |X (α) is linear in X for all α with deterministic slopes if and only if (X, Y ) has a multivariate normal distribution. The proof is given in Appendix A.2, it builds on Corollary 3.4. It relies on the characterization of the bivariate normal distribution by its characteristic function. A natural question concerns a possible generalization of Theorem 3.5 to other margins. Specifically, can the conditional quantiles be exactly linear with other margins, for all α ∈ (0, 1)? The answer to this question is yes and Proposition 3.6 in the next section provides a construction of the dependence that makes the conditional quantiles linear. 12

3.3

Linearity of Conditional Quantiles for Any Margins

The dependence between Y and X in the context of Proposition 3.3 and Corollary 3.4 is essentially deterministic with some independent noise ε. Given some marginal distributions for X and Y , we can use the characterization in Corollary 3.4 to construct the copula such that the conditional quantiles of Y given X would be linear in X. Proposition 3.6 (Copula for Given Marginals to achieve linear conditional quantiles). Given two random variables X and Y with characteristic functions ΨX and ΨY , respectively. Assume that (t) Ψε (t) := ΨΨXY (bt) is a characteristic function. Then, the conditional quantiles of Y |X are linear of the form bX + a(α) if and only if the copula between X and Y is equal to the copula between bX + ε and X, where ε has characteristic function Ψε (·). Proof. Assume that ΨY (t)/ΨX (bt) is a characteristic function. Then, when the conditional quantiles are linear, Y = bX +ε for some independent variable ε (Proposition 3.3). Therefore Ψε (t) = ΨY (t)/ΨX (bt) is a characteristic function and uniquely determines the distribution of ε. The copula between X and Y is then equal to the copula between X and bX + ε where X and ε are independent. Reciprocally, if the copula between X and Y is equal to the copula of X and bX + ε for some independent variable (t) . We can then obtain ΨY which uniquely determines the ε with characteristic function Ψε (t) = ΨΨXY (bt) distribution of Y , so that Y has the same distribution as bX + ε. Therefore (X, bX + ε) has the same −1 joint distribution as (X, Y ) and thus the same conditional quantiles. It is clear that FbX+ε|X=x (α) = −1  bx + Fε (α), and thus that the conditional quantiles of Y |X = x are linear.

Note that Proposition 3.6 does not require any assumptions on the support of X and Y and can be applied to non-negative variables or bounded variables. We illustrate this proposition with Cauchy variables with support R, with Gamma variables with support R+ and with uniform variables (bounded support).

Example (Cauchy Copula). The Cauchy distribution with parameters µ ∈ R and θ > 0 has cdf
1 F (z) = π1 arctan z−µ + , and characteristic function Ψ(t) = eitµ−θ|t| . Let X be a Cauchy(µX , θX ) and θ 2 Y be a Cauchy(µY , θY ) with θX 6 θY . We call “Cauchy copula” the dependence structure that ensures that the conditional quantiles of Y |X are linear. From Proposition 3.6, we find that Y = bX + ε where b ∈ (−1, 1) such that θY − |b|θX > 0, and where ε and X are independent so that their characteristic functions satisfy Ψε (t)ΨbX (t) = ΨY (t) and thus Ψε (t) =

eitµY −θY |t| ΨY (t) = ibtµ −θ |b||t| , ΨX (bt) e X X

(14)

so that ε is a Cauchy(µY − bµX , θY − |b|θX ).     1 1 −1 −1 FX (α) = µX + θX tan π(α − ) , FY (α) = µY + θY tan π(α − ) 2 2   1 1 t − (µY − bµX ) Fε (t) = arctan + . π θY − |b|θX 2


After some straightforward computations, we find that C2|1 (v|u) = Fε FY−1 (v) − bFX−1 (u) , thus 1 C2|1 (v|u) = arctan π

−1 C2|1 (α|u)

1 1 = + arctan 2 π



! θY tan π(v − 21 ) − bθX tan π(u − 21 ) 1 + , θY − |b|θX 2 cot (α π) (|b| θX − θY ) + bθX tan π u − θY

13

1 2



!

,

where cot(x) = 1/ tan(x). The density of the Cauchy copula can be computed explicitly as follows,


θY 1 + tan2 (π(v − 21 )) θY tan π(v − 12 ) − bθX tan π(u − 21 ) ∂2C (u, v) = , h(u, v) := . (15) ∂u∂v (θY − |b|θX )(1 + h2 (u, v)) θY − |b|θX We use this expression of the density to plot the theoretical contours in Figure 3. A: Scatterplot

B: Contours Cauchy

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Cauchy Copula, θX=2, θY=2, b=0.22

Cauchy Copula, θX=2, θY=2, b=0.22

Conditional Quantile of Y|X=x

C: Contours T copula

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Figure 3: All six panels are with N (0, 1) margins. Panel A: Sample of 10,000 couples of the Cauchy copula with parameters θX = θY = 2 and b = .22. Panel B: Contour shapes of this Cauchy copula using the expression of the density in (15). Panel C: Contour shapes of the Student copula, ρ = 0.404 and ν = 1 degree of freedom. Panels D, E, F: Normalized Conditional Quantiles at level 1% (lowest curve), 10%, 50% (middle curve), 90% and 99% (highest curve). Panels D and E represent respectively the conditional quantiles of Y |X = x and X|Y = y for the Cauchy copula with parameters θX = θY = 2 and b = .22. Panel F displays the conditional quantiles of Y |X = x for the Student copula with ρ = 0.404 and ν = 1 degree of freedom. Recall that contours of a Gaussian copula would be ellipses with N (0, 1) margins. Other contour shapes with N (0, 1) margins can be found in Czado [2010] for comparisons. Kendall’s tau τ = 0.26 for the Student copula and for the Cauchy copula (estimated by Monte Carlo with 500,000 simulated couples). All panels of Figure 3 assume N (0, 1) margins. From the scatterplot in Panel A, this copula may be realistic to model strong dependence in the tails and low dependence in the middle of the 14

4

distribution. Two stocks from different industry sectors may have that property. Indeed, they will be mostly independent unless a major event in the market triggers a big change in both sectors. This copula is an example of non-exchangeable copula as it appears clearly on the scatterplot and contours in Figure 3, Panels A and B. In Figure 3, Panels D and E represent the normalized conditional quantiles for Y |X and X|Y respectively. Note that by construction, panel D of Figure 3 would display linear conditional quantiles of Y |X = x if we were to use Cauchy margins instead of N (0, 1) margins. The standard Cauchy distribution (θ = 1, µ = 0) coincides with the Student’s t-distribution with one degree of freedom. A comparison between the Cauchy copula and the Student copula with 1 degree of freedom when marginal distributions are N (0, 1) is displayed in Figure 3 (panels C and F). The normalized conditional quantiles for the Student copula and for the Cauchy copula have different shapes. The Student copula is symmetric: conditional quantiles of X|Y = y and Y |X = x are the same, which is not true for the Cauchy copula (Panels D and E). We now provide a second example for which the support of the marginal distributions is R+ . Example (Gamma Copula). Assume that Y follows a Gamma distribution with shape parameter kY > 0 and scale parameter θ > 0. We write Y ∼ Γ(kY , θ). Then, Y has support on R+ and characteristic function equal to 1 ΨY (t) = (1 − itθ)kY The distribution and density are defined for y > 0 and given by
γ kY , yθ y kY −1 exp(−y/θ) fY (y) = . FY (y) = Γ(kY ) Γ(kY )θkY

Assume that X ∼ Γ(kX , θX ), where kX < kY and θX := θb . We call “Gamma copula” the dependence structure that ensures that the conditional quantiles of Y |X are linear. To find an expression of it, we apply Proposition 3.6. Then Y = bX + ε, where b ∈ (0, 1), ε and X are independent, and where the characteristic function of ε satisfies Ψε (t)ΨbX (t) = ΨY (t). Thus Ψε (t) =

ΨY (t) (1 − itbθX )kX 1 = = , k ΨX (bt) (1 − itθ) Y (1 − itθ)kY −kX

(16)

so that ε is a Gamma distribution with shape parameter kY − kX > 0 and scale θ. For the ease of exposition, we choose kY = 2 and kX = 1 (exponential distribution) and θ = b. Then, for x > 0, FX (x) = 1 − e−x , fX (x) = e−x , and Ψε is the characteristic function of an exponential variable with parameter 1θ = 1b . The copula C between X and bX + ε is equal to C(u, v) = P (FX (X) 6 u, FY (bX + ε) 6 v) = P (X 6 FX−1 (u), ε 6 FY−1 (v) − bX)  Z min(F −1 (u),F −1 (v)/b)  F −1 (v)−bx Y X − Y b 1−e = e−x dx 0

F −1 (v)
Y = 1 − exp − min(FX−1 (u), FY−1 (v)/b) − e− b min(FX−1 (u), FY−1 (v)/b)   −F −1 (v)/b Y 1{v>FY (−b ln(1−u))} so that for all A straightforward computation gives C2|1 (v|u) = 1 − e 1−u

−1 α ∈ (0, 1), C2|1 (α|u) = FY (−b ln((1 − α)(1 − u))).

In general, there is no closed-form expression for the copula that ensures linearity of conditional quantiles (as (14) or (16) may not be easily simplified and interpreted). Mostly, one could solve the problem by simulation: Assume a cdf and a characteristic function for X and for Y , for a given b, simulate the independent variable ε such that Y = bX + ε from its characteristic function (14) or (16) (see for instance Bernard et al. [2012]). 15

5

Gamma Copula, θ=0.3, b=0.3

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Figure 4: The three panels are with N (0, 1) margins. First Panel: Sample of 10,000 couples of the Gamma copula with parameter b = .22. Middle panel: Contour shapes of the Gamma copula. Right Panel: Normalized Conditional Quantiles at level 1% (lowest curve), 10%, 50% (middle curve), 90% and 99% (highest curve) for N (0, 1) margins with the Gamma copula. Kendall’s tau τ ≈ 0.5 (estimated by Monte Carlo with 500,000 simulated couples). Example (Uniform margins). Assume that X ∼ U (0, 1) and ε ∼ U (0, 1), b ∈ (0, 1) and X is independent of ε. Then let FY be the cdf of X + bε, then for y ∈ (0, 1 + b),  2 y   √ if 0 6 y < b   if 0 6 y < 2b  2b b  2bα if b 6 y < 1 y−2 FY (y) = FY−1 (α) = if 2b 6 α < 1 − 2b α + 2b p (y−(b+1))2   1 − if 1 6 y < 1 + b  2b  1 + b − 2b(1 − α) if 1 − 2b 6 α 6 1  1 if y > 1 + b And the density of Y is equal to fY (y) = yb 1y∈[0,b] + 1b
1+b−y 1y∈[1,1+b] . b

We refer to this copula

C(u, v) = P (FX (X) 6 u, FY (bX + ε) 6 v) ! ! −1 −1 F (v) F (v) − 1 = FY−1 (v) min u, Y + (1 − FY−1 (v)) max 0, Y b b  !2 !2  F −1 (v) F −1 (v) − 1  b − max 0, Y − min u, Y 2 b b

−1 (α|u) = A straightforward computation gives C2|1 (v|u) = (FY−1 (v) − bu)1{v>FY (bu)} so that C2|1 FY (α + bu). We refer to it as the “Uniform copula”. The copula density is useful to determine the contours of this copula analytically (middle panel in Figure 5)

∂2C 1 1  F −1 (v)−1 F −1 (v)  . (u, v) = ∂u∂v fY (FY−1 (v)) u∈ Y b , Yb Remark. Assume that X and Y have a copula C. In general, there exist no increasing transformations of the individual variables X and Y that preserve the copula and would ensure that the conditional quantiles of Y |X are linear. Such a transformation exists if and only if the copula C is also the copula between h(Y ) = bg(X) + ε and g(X) for some increasing functions g and h and for some variable ε that is independent of X. 16

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Figure 5: All panels are with N (0, 1) margins. First Panel: Sample of 10,000 couples of the Uniform copula with parameter b = .22. Middle panel: Contour shapes of the Uniform copula. Right Panel: Normalized Conditional Quantiles at level 1% (lowest curve), 10%, 50% (middle curve), 90% and 99% (highest curve) for N (0, 1) margins with the Uniform copula. Kendall’s tau τ ≈ 0.17 (estimated by Monte Carlo with 500,000 simulated couples). Proposition 3.7 (Clayton copula). Assuming the dependence of (X, Y ) is the Clayton copula, then there are no distributions with support on the real line such that the conditional quantiles are linear. In other words, there exist no increasing functions f and g such that f (Y ) = g(X) + ǫ where ǫ is independent of X and the dependence between X and Y is the Clayton copula. The proof is given in Appendix A.3.

3.4

Pitfalls of Quantile Regression

As shown from the previous examples and from the theoretical analysis on conditional normalized quantiles, there are situations in which the quantile regression methodology is well suited and in which it will fail. First, the linearity of conditional quantiles with Normal margins is characterized by the Gaussian dependence (Theorem 3.5). Therefore, if the dependence in the data comes from a Gaussian copula then it is natural to transform the marginal distributions to N (0, 1) and work on a multivariate normal distribution. In this case, conditional quantiles are linear and therefore a quantile regression is appropriate to estimate conditional quantiles. A typical example for which quantile regression will perform well is given by n X β i Xi + U j ∀j ∈ {1, ..., N }, Yj = β0 + i=1

where the Uj are supposed to be i.i.d. errors with cdf Fu . This is a linear regression model. Then FY−1 |X=x (α) = β0 +

n X

βi xi + Fu−1 (α).

i=1

However, for non-Gaussian types of dependence, the high non-linearity of conditional quantiles that can be seen from Figure 1 may obviously lead to inaccurate estimation of conditional quantiles 17

5

obtained from the standard quantile regression methodology. When the marginal distribution of Y is not normal, then the linearity of quantiles may happen for dependence that is not the Gaussian copula. See for example the copulas that make the conditional quantile linear for Cauchy margins, Gamma margins or bounded uniform margins (Figures 3, 4 and 5). Second, it seems that quantile regression may be well suited when the conditioning variable is in the “middle” of the distribution and thus to predict the conditional quantiles of Y given X for X being between [-1.5, 1.5] (when the standard deviation of X is 1 and its mean is 0). However quantile regression seems to be inappropriate to approximate conditional quantiles when the values for X are taken outside of the middle range as the conditional quantiles are highly non linear in the tails (see Figures 1 and 3). An obvious problem of quantile regression outside the middle area is that (unless the conditional quantiles at different levels are perfectly parallel), the regression lines may have different slopes and thus will inevitably cross (He [1997],Gouri´eroux and Jasiak [2008]). Conditional quantiles at different levels (as plotted in Figure 1 from theoretical distributions) should never cross! However, a direct estimation of conditional quantiles using quantile regression leads for instance to very bad estimation of conditional quantiles as in Figure 6, where theoretical conditional quantiles are superposed with conditional quantiles estimated with quantile regression. Frank Copula with δ=5.7363

Clayton Copula with δ=2

4

3

3

3

2

2

2

1 0 −1 −2 −3 −4 −3

Conditional Quantile of Y|X=x

4

Conditional Quantile of Y|X=x

Conditional Quantile of Y|X=x

Gauss Copula with ρ=0.70711

4

1 0 −1 −2 −3

−2

−1

0

1

x

Panel A: Gauss

2

3

−4 −3

1 0 −1 −2 −3

−2

−1

0

1

x

Panel B: Frank

2

3

−4 −3

−2

−1

0

1

2

3

x

Panel C: Clayton

Figure 6: Example of quantile regression on 1,000 couples simulated from Clayton, Frank and Gauss copulas and N (0, 1) margins. We superpose the quantile regression lines at 1%, 10%, 50%, 90%, 99% with the theoretical quantiles as computed in Section 2. Surprisingly, little attention in recent applications using conditional quantiles has been devoted to finding alternative semi-parametric estimators for conditional quantiles. There are yet important theoretical and practical benefits in estimating conditional quantiles in situations in which quantile regression fails (such as when the conditioning variable is in the tail of its distribution as it is the case for example in conditional quantiles used to estimate systemic risk or in stress testing).

4

Tail Dependence from Graphs of Conditional Quantiles

In this section, we present some theoretical properties of tail dependence explaining the various shapes of conditional quantiles observed in Figures 1 and 2 other than linear.

4.1

Asymptotic Conditional Tail Independence from Conditional Quantiles

For the independence copula, conditional quantiles horizontal lines (see the last graph of Figure 2), −1 as X = x does not affect Y, and thus FY−1 |X=x (α) = FY (α), which is independent of x. Thus, conditional tail independence (specifically asymptotic conditional tail independence) can be modelled 18

as flat conditional quantiles in the tail. We will see hereafter that this property is not equivalent to define tail independence with the tail dependence coefficients. Definition 4.1 (RCTI: Right Conditional Tail Independence). Y is conditionally independent of X in the right tail if ∀α ∈ (0, 1), lim FY−1 |X=x (α) = a(α) x→+∞

where a(·) does not depend on x. Definition 4.2 (LCTI: Left Conditional Tail Independence). Y is conditionally independent of X in the left tail if ∀α ∈ (0, 1), lim FY−1 |X=x (α) = a(α) x→−∞

where a(·) does not depend on x. For instance, from Figure 1 we observe flat conditional quantiles in both tails of the Frank copula, in the right tail of the Clayton and Crowder copulas and in the left tail of the Joe and BB6 copulas. Note that this observation does not depend on the fact that we use normalized quantiles. Formally, we have the following two propositions, which can easily be proved using (2) and replacing the copula C by the chosen copula. Proposition 4.3 (Asymptotic behavior of conditional quantiles for the Frank copula). For any cdf FX , FY , and the Frank copula, for α ∈ (0, 1),   1  −1 −1 −δ −δ , FY |X=x (α) = FY 1 + ln α(1 − e ) + e lim −1 − δ (1 ) x→FX   1  −1 −δ ln 1 − α(1 − e ) . FY−1 (α) = F − Y |X=x −1 + δ x→FX (0 ) lim

The Frank copula displays both right and left conditional tail independence.

Note that this asymptotic behavior does not depend on the marginal distribution FX . This means that for large values of x or small values of x, the conditional quantiles are “independent” of x: they are “horizontal lines” that are fully characterized by FY , α and δ. Proposition 4.4 (Asymptotic behavior of conditional quantiles for the Clayton copula). For any cdf FX , FY , and the Clayton copula with parameter δ > 0, for α ∈ (0, 1)  1  −1 −1 + α 1+δ , lim FY−1 lim FY−1 |X=x (α) = FY |X=x (α) = FY (0 ). −1 − x→FX (1 )

−1 + x→FX (0 )

The Clayton copula displays right conditional tail independence. Propositions 4.3 and 4.4 have been also checked numerically using simulations from Frank and Clayton copulas. Such explicit expressions for the right and left limits as in Propositions 4.3 and 4.4 seem hard to obtain for the Joe, BB6 and Crowder copulas given that the quantiles are not known in closed-form.

4.2

Tail dependence Coefficients

In general, tail dependence is studied using the lower and upper tail dependence coefficients:6 λL = lim P (Y < FY−1 (v)|X < FX−1 (v)) = lim

v→0+

v→0+

6

Joe [1997], McNeil et al. [2005].

19

C(v, v) v

(17)

1 − 2v + C(v, v) (18) 1−v An issue with these two coefficients is that λU and/or λL may be equal to 0 even if the variables are not tail independent according to definitions 4.1 and 4.2. For example, λL = λU = 0 for the Gaussian dependence with ρ ∈ (−1, 1) and the convergence to 0 requires v to be very close to 0 or 1 (respectively). See Figure 6 in Coles et al. [1999]. Coles et al. [1999] and Heffernan [2000] also define the following two coefficients of intermediate tail dependence (or weak dependence),     2 ln(v) 2 ln(1 − v) ¯ ¯ λL = lim − 1 , λU = lim −1 . (19) v→0+ ln(C(v, v)) v→1− ln(1 − 2v + C(v, v)) ¯ L and λ ¯ U take values in [−1, 1]. In particular, if λU > 0 then λ ¯ U = 1, similarly if λL > 0 then Note that λ 7 ¯ L = 1. For independent variables, λ ¯ U = λU = 0 and λ ¯ U = λU = 0. There are some intermediary λ ¯U = λ ¯L = ρ interesting cases such as for example for the Gaussian copula, λU = λL = 0 and λ λU = lim P (Y > FY−1 (v)|X > FX−1 (v)) = lim v→1−

v→1−

(Coles et al. [1999], Ledford and Tawn [1996]). See Table 3 for the tail dependence coefficients of the families of copulas introduced so far. The definition of tail dependence coefficients λL and λU recalled in (17) and (18) are clearly not affected by the exchangeability of the copula (property of the copula that C(u, v) = C(v, u) for all u and v). These tail dependence coefficients can also be written as λL = lim P (V 6 v|U 6 v) = lim P (U 6 v|V 6 v) v→0+

v→0+

λU = lim P (V > v|U > v) = lim P (U > v|V > v) v→1−

v→0+

We have seen from Figure 2 on the conditional quantiles of the Tawn copula that the conditional quantiles of X|Y and Y |X can look very different, and thus the tail dependence behaviour can be explained by one strong dependence either of X|Y or Y |X. If the copula is not exchangeable, then ∂C(u,v) −1 is possibly different from ∂C(u,v) and thus the conditional quantiles FY−1 ∂u ∂v |X=x (α) and FX|Y =y (α) may behave differently. To illustrate this point, we discuss the example of the 3-parameter Tawn copula that was introduced in Section 2.5. The upper (resp. lower) tail dependence coefficient of the Tawn copula can be computed from l’Hopital’s rule8 as
lim 2 − C2|1 (v|v) − C1|2 (v|v) − v→1
(resp. limv→0+ C2|1 (v|v) + C1|2 (v|v) ). We find that λL = 0,

¯L = λ

2

1

2 − ψ1 − ψ2 + (ψ1δ + ψ2δ ) δ 1

λu = (ψ1 + ψ2 ) − (ψ1δ + ψ2δ ) δ ,

− 1 (if δ 6= 1)

¯u = 1 λ

(if δ 6= 1) ¯ U = 0 and λL = λ ¯ L = 0. In the case of δ = 1, we find the independence copula, therefore λU = λ 1 In particular when ψ1 = ψ2 = ψ then λu = (2 − 2 δ )ψ which is ψ times the upper tail dependence coefficient of the Gumbel copula. Table 3 summarizes the tail dependence coefficients and intermediate tail dependence coefficients for all the copulas discussed so far. For non-exchangeable copulas, as the Tawn copula or the Cauchy copula, the tail coefficients are not so informative because their definition is symmetric in both variables. Tail dependence functions can be defined for non-exchangeable copulas (see for instance Eschenburg [2013]). The tail dependence coefficients for the Tawn and the Cauchy copulas are computed in Appendices A.4 and A.5 respectively. It is straightforward to prove that limv→0+ C2|1 (v|v) = 0, limv→0+ C1|2 (v|v) = 0, limv→1− C2|1 (v|v) = 1 and limv→0+ C1|2 (v|v) = 1 for the Uniform copula and we omit the proof. We were unable to derive the tail coefficients of the Gamma copula in a simple way. 7 8

This property follows from l’Hopital’s rule and the differentiability of v 7→ C(v, v) at 0. See details in Appendix A.4.

20

Gauss

Student

Frank

Clayton

−1 < ρ < 1

δ>0

δ>0

δ>0

0 ρ

2tν+1 (Λ) 1

0 0

2− δ 1

λL ¯L λ

Crowder  δ>0 θ>1

1

 BB3 δ>0 θ>1

0

1− θ1

2

1 1

−1

λU ¯U λ

0 ρ

2tν+1 (Λ) 1

0 0

0 0

0 0

LCTI

no

no

yes

no

no

no

RCTI

no

no

yes

yes

yes

no

λL ¯L λ λU ¯U λ LCTI RCTI

 BB7 δ>0 θ>1 1

2− δ 1

Joe

Gumbel

Tawn

δ>1

δ>1

δ 6= 1

0

0

0

1− 1δ

0 1

2 − 2θ 1 no no

2 1

2 − 2δ 1 yes no

−1 1

2 − 2δ 1 no no

2−2 1



2

1

2−ψ1 −ψ2 +(ψ1δ +ψ2δ ) δ

 BB6 δ>1 θ>1 0

1− 1δ

2 1 θ

1

2 − 2 δθ 1  yes if δ = 1 no, otherwise. no

Cauchy b, θX , θY > 0 θY − |b|θX > 0

−1 1

(ψ1 + ψ2 ) − (ψ1δ + ψ2δ ) δ 1 no no

−1

Uniform b ∈ (0, 1)

bθX θY

0

1

0

bθX θY

0 0 yes yes

1 no no

¯ L λU λ ¯ U can be found in Heffernan [2000]. For Table 3: Some of the√ results in this table for λL , λ √ 1+ν 1−ρ and tν+1 is the cdf of a Student with ν + 1 degrees of freedom (see the T-copula, Λ = − √1+ρ Demarta and McNeil [2005]). Note that the Clayton case is obtained as BB3 for θ = 1 and BB7 for ¯ U = 0 in this case (unlike what is written in Table 1 of Heffernan [2000]). Similarly, θ = 1 and that λ one needs to restrict δ > 1 for the Gumbel or Joe copula as the independence case δ = 1 satisfies ¯ U = 0. LCTI and RCTI correspond to left conditional tail independence and right conditional λU = λ tail independence, respectively, as in definitions 4.1 and 4.2.

4.3

Asymptotic Linearity and Tail Dependence

We are now interested in the “slope” of normalized conditional quantiles in the tail. To do so we study (x) asymptotic behavior. Recall that two functions f and g are equivalent at a if limx→a fg(x) = 1. We denote it by f (x) ∼ g(x). x→a

Lemma 4.5. Let X and Y be N (0, 1) variables   −1 ln C2|1 (α|z) ∼ ln(z)

⇔

z→0+

FY−1 |X=x (α)

∼

x→−∞

x.

−1 Lemma 4.6. If C2|1 (α|z) ∼ kα z for kα > 0 then the normalized conditional quantiles satisfy z→0+

FY−1 |X=x (α)

∼

x→−∞

x.

−1 Similarly, if C2|1 (α|z) ∼ kα z for kα 6= 0 then FY−1 |X=x (α) ∼ x. x→∞

z→1−

The proofs of Lemmas 4.5 and 4.6 are given in Appendix A.6 and are based on the asymptotic behavior of the quantile function of a standard normal distribution. 21

Theorem 4.7 (Asymptotic Linearity of conditional quantiles and Tail Dependence). Assume that X and Y are N (0, 1) and that the conditional quantiles satisfy for some α ∈ (0, 1), FY−1 |X=x (α) = xLα (x)

(20)

where limx→−∞ Lα (x) = 1 (resp. limx→+∞ Lα (x) = 1) and Lα (x) > 1 for all x < a where a ∈ R (resp. ¯ L = 1 (lower tail dependence), resp. λU > 0 and Lα (x) 6 1 for x > a where a ∈ R), then λL > 0 and λ ¯ λU = 1 (upper tail dependence). The proof is in Appendix A.7. Note that the conditions in Theorem 4.7 are not necessary. For instance, the conditional quantiles of the Cauchy copula in Figure 3 (left panel) are above (resp. below) the line y = x for all α for small x (resp. for large x). However, the tail dependence coefficients, λU > 0 and λL > 0. The Gaussian copula with ρ ∈ (−1, 1) displays an intermediate tail dependence equal to ρ as well as linear conditional quantiles of slope ρ. This relation may be sufficient but is not necessary as for example the Gumbel copula has intermediate tail dependence in the lower tail but no obvious asymptotic linearity. Example (Clayton copula). Using Lemmas 4.5 and 4.6, we study the asymptotic behavior of conditional quantiles of the Clayton copula. From Proposition 4.4, the normalized conditional quantiles in the left tail of Y for the Clayton copula are diverging (FY−1 (0+ ) = −∞ as FY = Φ). Specifically, for α ∈ (0, 1)   −1  −δ δ −1 −1 −δ ∼ x. (21) FY |X=x (α) = Φ (α 1+δ − 1)Φ(x) + 1 x→−∞



 −1 δ δ −1 (21) follows from Lemma 4.6 and from C2|1 (α|z) = z α− 1+δ − 1 + z δ ∼ kα z where kα = z→0+    −1  −1  −δ  δ δ δ > 0. In addition, Φ−1 (α 1+δ − 1)Φ(x)−δ + 1 6 x for α small enough. This α− 1+δ − 1

can be seen from the expression of the conditional quantiles of a Clayton copula.

Theorem 4.8 (Asymptotic Tail Independence). Assume that X and Y are N (0, 1) with an exchange∂C able copula C so that ∂C ∂u (v, v) = ∂v (v, v). If the normalized conditional quantiles satisfy for all α ∈ (0, 1), lim FY−1 lim FY−1 (22) |X=x (α) = a(α) (resp. |X=x (α) = a(α)) x→+∞

x→−∞

then λU = 0 (no upper tail dependence), resp. λL = 0 (no lower tail dependence). The proof is given in Appendix A.8. From Figure 1, the conditional quantiles for the upper tail of the Clayton and Crowder copulas, for both tails of the Frank copula, for the left tail of the Joe copula and the BB6 copula (with δ = 1), all display asymptotic independence as defined by (22) and all have their corresponding tail dependence coefficient λL = 0 or λU = 0 as reported in Table 3. This is consistent with Theorem 4.8. Theorem 4.8 gives a sufficient condition for asymptotic tail independence in the sense of Definitions 4.1 and 4.2 but it is not a necessary condition: λL = 0 does not imply asymptotic independence as defined in (22). For example, both the Gumbel or Crowder copulas both have λL = 0 but the conditional quantiles depend on x in the limit.

5

Conclusions

We study “tail dependence” and “tail independence” using conditional quantiles, conditionally on one of the variable. This approach allows to study tail dependence when the copula is potentially nonexchangeable. In practice, risks are not exchangeable and have causal effects that are not symmetric. 22

It is thus important to develop tools to study conditional risks that are not based on the assumption that the underlying copula is exchangeable. In addition, existing tools for tail dependence assessment are essentially based on the “diagonal” and thus are symmetric. Our approach allows to detect tail dependence for X given Y , but no tail dependence for Y given X and thus takes into account the potential non-exchangeability of the dependence between X and Y . This paper shows that the specification of the copula and its tail dependence is of utmost importance in conditional quantile estimation when conditioning involves the tail of one of the marginal distribution. Estimating Value-at-Risk at a very high confidence level (i.e. a quantile in the tail of the distribution) is known to be a challenge (McNeil et al. [2005], Embrechts et al. [1997], Embrechts et al. [2013]) and estimating a conditional Value-at-Risk is thus even more challenging. In particular, such estimation is needed in the calculation of the systemic risk measure CoVaR when conditioning on an extreme event. Quantile regression is a popular method but it is based on the assumption that conditional quantiles are linear in the conditioning variables. However, throughout our study, we have seen that couples with normal margins may have linear conditional quantiles only when the dependence structure between the variables is the Gaussian copula. Other techniques to estimate conditional quantiles include the fully non-parametric approach of Li and Racine [2007] using Kernel smoothing. But it may not perform well in the tail as the behavior of the conditional quantile in the tail is driven by the choice of the kernel only. Another approach to estimate conditional quantiles may be to use the local quantile regression proposed by Spokoiny et al. [2013] or to fit a flexible multivariate model (Aas et al. [2009], Czado [2010], Kurowicka and Joe [2011]) in order to compute the conditional quantiles in that fitted model.

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25

A

Proofs

A.1

Proof of Proposition 2.3

Proof. We start by recalling a theoretical result on quantiles. Lemma A.1 (Quantile Estimation (theory)). Let α ∈ (0, 1) and assume that Y has finite mean. Then, a solution to the following convex minimization min {(α − 1)E [(Y − q)1Y 6q ] + αE [(Y − q)1Y >q ]}

(23)

q∈R

exists and is equal to the quantile FY−1 (α) = inf {q ∈ R | FY (q) > α} . Proof. The objective to minimize in (23) can be rewritten as h(q) := αE[Y ] − qα − E[Y 1Y 6q ] + qP (Y 6 q). Define h′ (q) as the right derivative of h (that is the limit when ε goes to 0, ε > 0, of (h(q+ε)−h(q))/ε. When Y has a density value fY (q) at q, then h′ (q) = −α−qfY (q)+P (Y 6 q)+qfY (q) = P (Y 6 q)−α. When Y has a mass point P (Y = q) > 0 at q, then9 P (Y < q) − α 6 h′ (q) 6 P (Y 6 q) − α

(24)

To prove (24), note that h(q) = αE[Y ] − qα + E[Y 1Y − limε↓0 P (q 6 Y < q + ε) = −P (Y = q). Therefore, equation (24) follows. In all cases, h′ (q) is non-decreasing in q and thus h is convex. In addition, limq→−∞ h(q) = +∞ and limq→+∞ h(q) = +∞. An absolute minimum exists but may not be unique if Y has some mass points. However, it is clear that FY−1 (α) is a zero of h′ and a solution to the convex minimization.  1 PN ˆ We then replace in Lemma A.1 the knowledge of Y by its empirical cdf FY (y) = N i=1 1y>yi . The problem of minimization (23) becomes (5). Using the expression of the quantile in Lemma A.1 for instance, the sample quantile of level α is an optimum. Note that it is unique if N α is not an integer. When N α = k ∈ N, the objective function has a flat part between [yk , yk+1 [ and any value in [yk , yk+1 [ minimizes (5). 

A.2

Proof of Theorem 3.5

Proof. From Proposition 3.1, since the conditional quantiles of Y |X are linear in X (with deterministic slope), there exist a(·) and b ∈ R such that FY−1 |X (α) = a(α)+bX. The characteristic functions of X and 1

2

1

2

2

2

Y satisfy ΨX (u) = E[eiuX ] = eiµX u− 2 σX u , and ΨY (u) = E[eiuY ] = eiµY u− 2 σY u . The joint distribution between (X, Y ) is characterized by the marginal distributions and the conditional distribution of Y |X (then the joint distribution can be obtained easily). Let us compute the characteristic function of Y |X = x. To do so, let u ∈ R, Z iuY ΨY |X=x (u) = E[e |X = x] = eiuy fY |X=x (y)dy.

R1 R 1 iuF −1 (α) Y |X=x dα = 0 eiua(α)+iubx dα. Let y = FY−1 |X=x (α), then dα = fY |X=x (y)dy and thus ΨY |X=x (u) = 0 e R1 We find that ΨY |X=x (u) = eiubx 0 eiua(α) dα. Observe that ΨY |X is a random variable as a function of 9

We thank one of the referees for correcting the proof.

26

X. We use the tower property for the expectations and E[ΨY |X (u)] = E[E[eiuY |X]] = E[eiY u ] = ΨY (u). Thus after integrating the previous equation for x ∈ (−∞, +∞) for all possible values of X, it is clear that Z Z Z ΨY (u) =

+∞

−∞

2 i(µY −bµX )u− 21 (σY

1

eiubx fX (x)dx

1

1 2 2 2 σX u

eiua(α) dα = eibµX u− 2 b

eiua(α) dα.

0

0

R1

2 )u2 −b2 σX

Therefore e = 0 eiua(α) dα. a(α) is non-decreasing and can be interpreted as 1 2 2 2 2 the quantile function of some random variable T so that E[eiuT ] = ei(µY −bµX )u− 2 (σY −b σX )u . By uniqueness of the characteristic function, T must have a normal density with mean µY − bµX and 2 . Thus a(α) = F −1 (α) is given by variance10 σY2 − b2 σX T q 2 Φ−1 (α). a(α) = µY − bµX + σY2 − b2 σX ρσY We find back the expression (13) for FY−1 |X with b = σX ensuring that the dependence between Y and X is coming from the Gaussian copula. The proof of the multivariate case is similar and omitted. 

A.3

Proof of Proposition 3.7

Proof. From the linearity of conditional quantiles and from the expression of a conditional quantile for the Clayton copula given in Table 2, 

FY (y) = 1 +

FX−δ



y − a(α) b



 −δ  where y = bx + a(α) and c(α) := α 1+δ − 1 . It implies that y→

FXδ

c(α) 

y−a(α) b

c(α)

−1/δ



does not depend on α. By differentiating with respect to α, we find that    ′  a (α) y − a(α) ′ y − a(α) ′ + c(α)g =0 c (α)g b b b where g(z) = FXδ (z). Bydividing bythe term in g ′ (which is always positive as F has the real line as  support), y 7→ g y−a(α) /g ′ y−a(α) does not depend on y for all α. Thus z 7→ g (z)/g ′ (z) does not b b

depend on z. Thus (g ′ )2 − gg ′′ = 0 and g must be of the form kez , for some k ∈ R which is impossible as F is a cdf over the real line and g(z) = FXδ (z). 

A.4

Intermediate tail dependence coefficients of the Tawn copula

¯ L , and using l’Hopital’s rule, From (19) of λ   2C(v, v) 1 ¯ λL = lim −1 v C1|2 (v|v) + C2|1 (v|v) 0+ 

10

2A(w) ¯ L = lim  2v λ v 0+

2v 2A(w) v



1 1

1

2 − ψ1 − ψ2 + K δ −1 ψ1δ wδ−1 + K δ −1 ψ2δ (1 − w)δ−1

 − 1

2 2 σY2 − b2 σX > 0 because σY2 = var[Y ] = E[var[Y |X]] + var[E[Y |X]] = E[var[Y |X]] + b2 var[X] > b2 σX .

27



Observe that w = then

ln(v) 2 ln(v)

=

1 2

¯ L does not depend on v anymore. Use the fact that K = so that λ ¯L = λ

2 2 − ψ1 − ψ2 + ψ1δ + ψ2δ

ψ1δ 2δ

+

ψ2δ 2δ


1 − 1 δ

When ψ1 = ψ2 , we find back the expression of the intermediate tail dependence coefficient for the ¯ L 6= 1). Gumbel copula. In particular we find that λL = 0 (as λ For the upper tail dependence coefficient, λU , its expression has been derived in the literature. It is equal to 2(1 − A(1/2)) where A is the Pickands function. See for instance see Eschenburg [2013]. ¯ U = 1. Since λU > 0, λ 

A.5

Tail dependence coefficients of the Cauchy copula for b > 0

Proof. By l’Hopital’s rule, λU = lim

v→1−

For the Cauchy copula, recall that



2 − C2|1 (v|v) − C1|2 (v|v) , λL = lim C2|1 (v|v) + C1|2 (v|v) . v→0+



! θY tan π(v − 21 ) − bθX tan π(u − 21 ) 1 + . θY − |b|θX 2

1 C2|1 (v|u) = arctan π

thus, it can be simplified with u = v (since b > 0) to the following expression C2|1 (v|v) = v,

(25)

which obviously converges to 0 when v → 0+ and to 1 when v → 1− . But the Cauchy copula is not an exchangeable copula, and unfortunately, there is no closed-form ∂2C given in (15). Then, expression for C1|2 (u|v). We obtain an expression for it by integrating ∂u∂v Z v 2 ∂ C C1|2 (v|v) = (x, v)dx. (26) 0 ∂u∂v

We now prove that

lim C1|2 (v|v) = b

v→0+

θX θY

(27)

θX X ¯ Thus λL = bθ θY , λL = 1. Let r = b θY . To prove (27), use (26) and observe that the limit limv→0+ C1|2 (v|v) can then be rewritten as follows 1 Z 1 +1 tan2 (π(v− 12 )) (1 − r) lim   vdw tan(π(wv− 21 )) 2 v→0+ 0 (1−r)2 + 1 − r tan(π(v− 1 )) tan2 (π(v− 1 )) 2

where we so that 0

(1 − r) lim

v→0+

tan(π(wv− 12 )) tan(π(v− 21 ))

define w = x/v that the above

2

limit is also equal to

because

2

divided the numerator and the denominator by tan2 (π(v − 12 )) and where we
1 2 v 2 + 2 π 4 v 4 + o v 4 → 0 so = π < w < 1, dx = vdw. Note that tan2 (π(v− 1 3 ))

=

1 w

Z

1

1 0

+ v2

(1 − r)



1 π2 3 w

2

π2v2

+

2 4 4 3π v

+

o (v 4 )

 − 13 π 2 w + o(v 3 ) →

1 w



+ 1−

 vdw tan(π(wv− 21 )) 2 r tan(π(v− 1 )) 2

as v → 0, w > 0 so the term in

1 tan2 (π(v− 21 ))

in the numerator converges to 0 and is omitted. Observe that Z 1 Z 1 1 w2 (1 − r) lim v dw = (1 − r) lim v
2 2 2 2 2 dw 2 r 2 v→0+ v→0+ 0 (1 − r) π 2 v 2 + 1 − w 0 (1 − r) w π v + (w − r) Z 1 w2 = (1 − r) lim v dw 2 2 2 2 2 v→0+ 0 w (1 + (1 − r) π v ) − 2rw + r 28

Using Maple for instance, this expression can then easily be integrated in closed form involving the function arctan. A Taylor expansion of the resulting expression shows for 0 < r < 1 that Z 1 r w2 dw = v + o(v). 2 2 2 2 2 1−r 0 w (1 + (1 − r) π v ) − 2rw + r
Then, since 23 π 4 v 4 + o v 4 is bounded in [−ε, +ε] for v < η (for some η > 0), it is clear from the above computations that Z 1 1 (1 − r) lim v
dw = r 2 2 2 r 2 2 4 4 v→0+ 0 (1 − r) π v + 3 π v + o (v 4 ) + 1 − w

which ends the proof for the lower tail coefficient. To study λU , observe from (25) that limv→1− C2|1 (v|v) = bθX X ¯ 1 and that limv→1− C1|2 (v|v) = 1 − bθ  θY so that λU = θY and λU = 1.

A.6

Proof of Lemmas 4.5 and 4.6

Proof. Recall from Dominici [2003] that 11 Φ−1 (x) ∼ − 

−1 Φ−1 C2|1 (α|u) + u→0 Φ−1 (u)

x→0+



q ln

1 2πx2




. Observe that limx→−∞

FY−1 (α) |X=x x

−1 (α|u) = 0, so that we can apply . From this equality, we have that limu→0+ C2|1 lim −1 the equivalence at 0 for Φ , we find that v   u u   u  1  −1 Φ−1 C2|1 (α|u) ∼ −u 2  tln   u→0+ −1 2π C2|1 (α|u) −1

Φ then ln



1

−1 2π C2|1 (α|u)

2

!

∼ ln



1 2π(u)2

−1 C2|1 (α) ∼ kα z for kα 6= 0 then



s  (u) ∼ − ln u→0+

 1 , 2π (u)2

. This ends the proof of Lemma 4.5. To prove Lemma 4.6,

z→0+

1

∼ −1 2π(C2|1 (α))2 z→0+

1 2πkα2 z 2

Recall that if f (x) ∼ g(x) and f (x) → ∞ when x → ∞ then ln(f (x)) ∼ ln(g(x)). Thus x→∞

x→∞

ln

1 −1 2π(C2|1 (α))2

!

∼ ln

z→0+

  −1 Finally Φ−1 C2|1 (α) ∼ Φ−1 (kα z) . Note also that



1 2πkα2 z 2



z→0+

Φ−1 (kα z) Φ−1 (z)

∼

z→0+

s p ln(1/(kα2 2πz 2 )) ln(kα2 ) p = 1+ →1 ln(2πz 2 ) ln(1/(2πz 2 ))

−1 −1 when z → 0+ . Thus FY−1 |X=x (α) = Φ (C2|1 (α)) the first part of Lemma 4.6 is proved by replacing z → 0+ by Φ(x) with x → −∞. The second part is similar and thus the proof is omitted.  11

Precisely, Dominici [2003] proves that Φ−1 (x)

solves the equation W (x)e

W (x)

∼

x→0+

q − W

1 2πx2

= x and satisfy W (x) ∼ ln(x). x→∞

29




where W is known as the Lambert function which

=

A.7

Proof of Theorem 4.7

Proof. Let us concentrate on the left tail coefficient as the proof for the upper tail is similar. By definition of Lα , FY−1 |X=x (α) = xLα (x)

where Lα (x) → 1 when x → −∞, and there exists a ∈ R such that Lα (x) 6 1. λL = = Therefore, for all x 6 a, λL >

A.8

Rx

−∞

lim

v→0+

lim

P (X < Φ−1 (v), Y < Φ−1 (v)) v Rx −∞ P (Y < x|X = z)φ(z)dz Φ(x)

x→−∞

P (Y
= α > 0 as zLα (z) 6 z 6 x.



Proof of Theorem 4.8

Proof. Assume that limx→−∞ FY−1 |X=x (α) = a(α). Define u = Φ(x), U = Φ(X) and V = Φ(Y ),

so that for all u ∈ (0, 1) and α ∈ (0, 1),

  vα,u := Φ FY−1 (α) |X=x

P (V 6 vα,u |U = u) = α Then limx→−∞ FY−1 |X=x (α) = a(α) can also be written as limu→0 vα,u = Φ(a(α)) > 0. It is clear that there exists u0 ∈ (0, 1), such that for all u < u0 , u 6 vα,u and thus that limu→0+ P (V 6 u|U = u) 6 α. This property holds for all α > 0, which implies that limu→0+ P (V 6 u|U = u) = 0. For exchangeable  variables, it means also that limu→0+ P (U 6 u|V = u) = 0, so that limv→0+ C1|2 (v|v) + C2|1 (v|v) = 0 and thus λL = 0 (by l’Hopital’s rule). 

30