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Estimation of conditional copulas: revisiting

asymptotic results in the i.i.d. case and extension to

serial dependence

Félix Camirand Lemyrea, Taoufik Bouezmarnia, Jean-François Quessyb

aDépartement de mathématiques, Université de Sherbrooke, Québec, Canada bDépartement de mathématiques et d’informatique, Université du Québec à

Trois-Rivières, Trois-Rivières, Canada

Abstract

Let (Y1, Y2, X) ∈ R3 be a random vector and consider the conditional distri- bution Hx(y1, y2) = P(Y1 ≤ y1, Y2 ≤ y2|X = x). The conditional copula in that context is the dependence structure that one extracts from Hx via the celebrated Sklar’s theorem. Specifically, if the conditional marginal distribu- tions F1x and F2x of Hx are continuous, then there exists a unique conditional copula Cx : [0, 1]

2 → [0, 1] such that Hx(y1, y2) = Cx{F1x(y1), F2x(y2) for all (y1, y2) ∈ R2. Then, Cx contains all the information on the form, and in particular on the strength, of the dependence between Y1 and Y2 for a given value of the covariate X. This paper considers the estimation of Cx when serially dependent copies of (Y1, Y2, X) are available, extending recent re- sults obtained by Veraverbeke et al. (2011) and Gijbels et al. (2011). Such observations naturally appear in financial contexts. It is shown that under appropriate conditions, the limiting behavior of suitably standardized ver- sions match those in the i.i.d. case. An application is given on measuring causality between Standard & Poor’s 500 index data and volume.

Keywords: α-mixing processes, conditional copula, empirical copula process, functional delta method, weak convergence

Email addresses: felix.camirand.lemyre@usherbrooke.ca (Félix Camirand Lemyre), taoufik.bouezmarni@usherbrooke.ca (Taoufik Bouezmarni), jean-francois.quessy@uqtr.ca (Jean-François Quessy)

Preprint submitted to Elsevier November 19, 2014

1. Introduction

Consider a random pair (Y1, Y2) from a joint distribution H having continuous marginal distributions F1 and F2. Then a celebrated theorem due to Sklar (1959) ensures the existence of a unique copula C : [0, 1]2 → [0, 1] such that H(y1, y2) = C{F1(y1), F2(y2)} holds for all (y1, y2) ∈ R2; see Nelsen (2006) for details on the theory of copulas. Based on (Y11, Y21), . . . , (Y1n, Y2n) i.i.d. H , the nonparametric estimation of C is usually performed by computing the empirical copula. A version that is asymptotically equivalent to that originally proposed by Rüschendorf (1976) is

Cn(u1, u2) = 1

n

n∑

i=1

I { Y1i ≤ F−1n1 (u1), Y2i ≤ F−1n2 (u2)

} ,

where Fn1 and Fn2 are the marginal empirical distribution functions. The asymptotic behavior of the empirical copula process Cn =

√ n(Cn −

C) has been investigated by Deheuvels (1979) under independence, i.e. when C(u, v) = uv is the independence copula. General weak convergence in the space D([0, 1]2) of càdlàg functions equipped with the Skorohod topology was investigated by Gaenßler & Stute (1987); van der Vaart & Wellner (1996) show weak convergence in the space `∞([a, b]2) of bounded functions on [a, b]2

for 0 < a < b < 1. The result was extended to the space `∞([0, 1]2) by Ferma- nian et al. (2004) while assuming the existence and continuity of the partial derivatives C [1](u1, u2) = ∂ C(u1, u2)/∂u1 and C

[2](u1, u2) = ∂ C(u1, u2)/∂u2 on [0, 1]2. In that case, Cn converges weakly with respect to the supremum distance to a limit process of the form

C(u1, u2) = α(u1, u2)− C [1](u1, u2)α(u1, 1)− C [2](u1, u2)α(1, u2), (1)

where α is a continuous and centered Gaussian process such that

E {α(u1, u2)α(u′1, u′2)} = C (u1 ∧ u′1, v1 ∧ v′1)− C(u1, u2)C(u′1, u′2).

Here and in the sequel, a ∧ b = min(a, b) for a, b ∈ R. As shown by Segers (2012), these requirements on the partial derivatives of C are not satisfied for many extensively used copula models. Fortunately, this author obtained that the result still holds if C [1] and C [2] exist and are continuous on the sets (0, 1)× [0, 1] and [0, 1]× (0, 1), respectively. The extension of this result to

2

serially dependent data has been considered recently by Bücher & Volgushev (2013) and Bücher & Ruppert (2013).

One is often interested in the behavior of a random couple conditional on the value taken by some covariate. Specifically, let (Y1, Y2, X) be a random vector taking values in R3 and for a fixed x ∈ R, consider the conditional joint distribution of (Y1, Y2) given X = x, that is

Hx(y1, y2) = P (Y1 ≤ y1, Y2 ≤ y2|X = x) . (2)

The conditional marginal distributions of Y1 and Y2 given X = x obtains from Hx via F1x(y) = limw→∞Hx(y, w) and F2x(y) = limw→∞Hx(w, y). If F1x and F2x are continuous, then Sklar’s theorem ensures that there exists a unique copula Cx : [0, 1]

2 → [0, 1] such that Hx(y1, y2) = Cx{F1x(y1), F2x(y2)}. Con- versely, the copula associated to the bivariate conditional distribution Hx can be extracted from the formula

Cx(u1, u2) = Hx { F−11x (u1), F2x(u2)

} . (3)

The bivariate function Cx is called the conditional copula. The latter con- tains all the dependence feature of (Y1, Y2) given a fixed value taken by the covariate. For that reason, it is an important task to be able to estimate Cx.

The estimation of Cx from i.i.d. observations has been considered by Veraverbeke et al. (2011) and Gijbels et al. (2011). Specifically, assuming the availability of independent random triplets W1, . . . ,Wn, where Wi = (Y1i, Y2i, Xi), two empirical versions of Cx were proposed. First consider the estimator of the joint conditional distribution Hx given by

Hxh(y1, y2) =

n∑

i=1

whi(x) I (Y1i ≤ y1, Y2i ≤ y2) , (4)

where wh1, . . . , whn are weight functions that smooth the covariate space and h is a bandwidth parameter. The latter typically depends on the sample size, but the subscript n is omitted in the sequel for notational simplic- ity. The conditional empirical marginal distributions are simply F1xh(y) = limw→∞Hxh(y, w) and F2xh(y) = limw→∞Hxh(w, y). From representation (3), a natural plug-in estimator of Cx is given by

Cxh(u1, u2) = Hxh { F−11xh(u1), F

−1 2xh(u2)

}

=

n∑

i=1

whi(x) I { Y1i ≤ F−11xh(u1), Y2i ≤ F−12xh(u2)

} , (5)

3

where for j = 1, 2, F−1jxh(u) = inf{y ∈ R : Fjxh(y) ≥ u} is the left-continuous generalized inverse of Fjxh. As noted by Gijbels et al. (2011), the estimator Cxh may be severely biased if any of the two marginal distributions is strongly influenced by the covariate. For that reason, they proposed a second esti- mator that aims at removing this effect of the covariate on the margins and hopefully obtain a smaller bias. To this end, define for each i ∈ {1, . . . , n} the pseudo-uniformized observations (Ũ1i, Ũ2i) = (F1Xih1(Y1i), F2Xih2(Y2i)), where h1, h2 are bandwidth parameters that may differ from h. Then, let

G̃xh(v1, v2) =

n∑

i=1

whi(x) I {F1Xih1(Y1i) ≤ v1, F2Xih2(Y2i) ≤ v2} (6)

and estimate Cx with

C̃xh(u1, u2) = G̃xh

{ G̃−11xh(u1), G̃

−1 2xh(u2)

} , (7)

where G̃1xh and G̃2xh are the marginals extracted from G̃xh. The investigation of the asymptotic properties of the conditional empirical copula processes

Cxh = √ nh ( Cxh − Cx

) and C̃xh =

√ nh ( C̃xh − Cx

)

as random elements in the space `∞([0, 1]2) of bounded functions on [0, 1]2

endowed with the supremum norm have been done by Veraverbeke et al. (2011). Specifically, they obtained that Cxh and C̃xh both converge weakly to Gaussian processes whose stochastic behavior differ only on their respective bias. The main goals of this paper are to

(i) re-derive the above-mentioned asymptotic results of Veraverbeke et al. (2011) in a simpler way using the functional delta method;

(ii) extend these results to serially dependent observations, i.e. time series.

The paper is organized as followed. Section 2 revisits some results on classi- cal and conditional empirical copula processes in the light of the functional delta method. Section 3 extends these results to conditional empirical cop- ula processes under serial dependence. Some simulation results that aim at studying the accuracy of these estimators are provided in Section 4. An ap- plication is given in Section 5 on measuring causality in the bivariate time series of returns and volume of the Standard & Poor’s 500 index. The proofs of the main results are to be found in Section 6 and auxiliary results needed for these theoretical results to hold are relegated to an Appendix.

4

2. Revisiting conditional empirical copulas in the i.i.d. case

2.1. A Hadamard functional

Let D be the space of bivariate distribution functions on R2 and consider the mapping Φ : D → `∞([0, 1]2) defined by

Φ(H) = H ◦ ( F−11 , F

−1 2

) , (8)

where F1(y) = limw→∞H(y, w) and F2(y) = limw→∞H(w, y). Then if F1 and F2 are continuous, the unique copula of H writes C = Φ(H). The Hadamard differentiability of Φ has been first investigated in van der Vaart & Wellner (1996) under strong assumptions on the partial derivatives of C. These assumptions were recently relaxed by Bücher & Volgushev (2013) and match those identified by Segers (2012).

Definition 2.1. In the sequel, D is the class of copulas C such that the partia