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Nonparametric estimation of general multivariate tail dependence and applications to financial time series

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Abstract

In order to analyse the entire tail dependence structure among random variables in a multidimensional setting, we present and study several nonparametric estimators of general tail dependence functions. These estimators measure tail dependence in different orthants, complementing the commonly studied positive (lower and upper) tail dependence. This approach is in line with the parametric analysis of general tail dependence. Under this unifying approach the different dependencies are analysed using the associated copulas. We generalise estimators of the lower and upper tail dependence coefficient to the general multivariate tail dependence function and study their statistical properties. Tail dependence measures come as a response to the incapability of the correlation coefficient as an extreme dependence measure. We run a Monte Carlo simulation study to assess the performance of the nonparametric estimators. We also employ selected estimators in two empirical applications to detect and measure the general multivariate non-positive tail dependence in financial data, which popular parametric copula models commonly applied in the financial literature fail to capture.

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Notes

  1. The authors refer to the exponent \(a\) as the stable tail dependence function.

  2. The authors refer to this copula as the Pareto copula.

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Acknowledgments

The authors would like to thank Aristidis Nikoloulopoulos, Nick Constantinou, John O’Hara, Andy Tremayne and participants of the International Finance Conference in Sydney for their valuable feedback for previous versions of this paper. The authors are also very grateful to the Editor and two anonymous referees for the helpful comments and suggestions that led to an improvement of the paper. Yuri Salazar acknowledges financial support from CONACYT Mexico.

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Correspondence to Wing Lon Ng.

Appendices

Appendix A: Proofs

1.1 Proof for Proposition 1

Proof

  1. (i)

    Note that if \(R_{i,n}^{(j)}\) is the rank of \(X_{i}^{(j)}\), then \( X_{i}^{(j)}\) is the \(X_{i}^{\left( R_{i,n}^{(j)},n\right) }\) order statistic, for \(i\in \{1,\ldots ,d\}\) and \(j\in \{1,\ldots ,n\}\). Hence, if \( X_{i}^{(j)}\) is the \(K_{i}\)th order statistic, \(K_{i}=R_{i,n}^{(j)}\). Considering the definition of \(F_{i,n}^{\leftarrow }(u_{i})\), we then have

    $$\begin{aligned} X_{i}^{(j)}\le F_{i,n}^{\leftarrow }(u_{i})\iff R_{i,n}^{(j)}\le K_{i} \text {,} \end{aligned}$$
    (30)

    where \(K_{i}\) satisfies \(\frac{K_{i}-1}{n}<u_{i}\le \frac{K_{i}}{n}\), for \( i\in \{1,\ldots ,d\}\) and \(j\in \{1,\ldots ,n\}\). Furthermore, from the definition of \(\overline{F}_{i,n}^{\leftarrow }(u_{i})\), we have

    $$\begin{aligned} X_{i}^{(j)}>\overline{F}_{i,n}^{\leftarrow }(u_{i})\iff R_{i,n}^{(j)}>K_{i} \text {,} \end{aligned}$$
    (31)

    where \(K_{i}\) satisfies \(n-K_{i}\le nu_{i}<n-K_{i}+1\), for \(i\in \{1,\ldots ,d\} \) and \(j\in \{1,\ldots ,n\}\). From their definition, the difference, for \( 0<u_{i}<1\), \(i\in \{1,\ldots ,d\}\), between the empirical and the rank estimators is

    $$\begin{aligned} \left| C_{\mathbf {D},n}^{rank}-C_{\mathbf {D},n}^{e}\right| =\frac{1}{ n}\left| \underset{j=1}{\overset{n}{\sum }}\left( 1_{\left\{ \underset{ i=1}{\overset{d}{\cap }}\mathcal {M}_{D_{i},i,n}^{(j)}(u_{i})\right\} }-1_{\{\overset{d}{\underset{i=1}{\cap }}\mathcal {B}_{i}^{(j)}(F_{D_{i},i,n}^{ \leftarrow }(u_{i}))\}}\right) \right| \text {.} \end{aligned}$$
    (32)

    From Definition 8, we have

    $$\begin{aligned} \mathcal {M}_{D_{i},i,n}^{(j)}(u_{i})=\left\{ \begin{array}{ll} \{R_{i,n}^{(j)}\le nu_{i}\} &{} \quad \text {if }\quad D_{i}=L \\ \{R_{i,n}^{(j)}>n(1-u_{i})\} &{} \quad \text {if }\quad D_{i}=U \end{array} \right. , \end{aligned}$$

    and from equivalences (30) and (31) we have

    $$\begin{aligned} \mathcal {B}_{i}^{(j)}(F_{D_{i},i,n}^{\leftarrow }(u_{i}))\}=\left\{ \begin{array}{c} \{R_{i,n}^{(j)}\le K_{i}\}\quad \text { if }D_{i}=L \\ \{R_{i,n}^{(j)}>K_{i}\}\quad \text { if }D_{i}=U \end{array} \right. \text {.} \end{aligned}$$

    This holds for \(K_{i}\) such that \(K_{i}-1<nu_{i}\le K_{i}\), for \(i\in I_{L}^{\mathbf {D}}\), and \(n-K_{i}\le nu_{i}<n-K_{i}+1\), for \(i\in I_{U}^{ \mathbf {D}}\). It follows that the estimators coincide for \((u_{1},\ldots ,u_{d})\) on the grid \(g_{1}\). Further to this, note that

    $$\begin{aligned} \underset{i=1}{\overset{d}{\cap }}\mathcal {M}_{D_{i},i,n}^{(j)}(u_{i})= \left( \underset{i\in I_{L}^{\mathbf {D}}}{\cap }\{R_{i,n}^{(j)}\le nu_{i}\}\right) \cap \left( \underset{i\in I_{U}^{\mathbf {D}}}{\cap } \{R_{i,n}^{(j)}>n(1-u_{i})\}\right) \text {,} \end{aligned}$$

    and

    $$\begin{aligned} \overset{d}{\underset{i=1}{\cap }}\mathcal {B}_{i}^{(j)}(F_{D_{i},i,n}^{ \leftarrow }(u_{i}))=\left( \underset{i\in I_{L}^{\mathbf {D}}}{\cap } \{R_{i,n}^{(j)}\le K_{i}\}\right) \cap \left( \underset{i\in I_{U}^{\mathbf { D}}}{\cap }\{R_{i,n}^{(j)}>K_{i}\}\right) \,\text {.} \end{aligned}$$

    This implies the following equivalence of events

    $$\begin{aligned}&\left( \underset{i\in I_{L}^{\mathbf {D}}}{\cap }\{R_{i,n}^{(j)}\le K_{i}\}\right) \cap \left( \underset{i\in I_{U}^{\mathbf {D}}}{\cap } \{R_{i,n}^{(j)}>K_{i}\}\right) \nonumber \\&\quad \iff \left( \underset{i\in I_{L}^{\mathbf {D}}}{\cap }\{R_{i,n}^{(j)}\le nu_{i}\}\right) \cap \left( \underset{i\in I_{U}^{\mathbf {D}}}{\cap } \{R_{i,n}^{(j)}>n(1-u_{i})\}\right) \text {,} \end{aligned}$$
    (33)

    which holds except for, at most, when \(R_{i,n}^{(j)}=K_{i}\), \(i\in \{1,\ldots ,d\}\). No estimator is generally greater than or equal to the other one. For fixed \(j\), if \(R_{i,n}^{(j)}=K_{i}\), for \(i\in I_{L}^{\mathbf {D}}\), it is possible that only the right hand side holds (if \(nu\) is not an integer, the left hand side does not hold), and if \(R_{i,n}^{(j)}=K_{i}\), for \(i\in I_{U}^{\mathbf {D}}\), it is possible that only the left hand side holds (if \(nu\) is not an integer, the left hand side does not hold). In the first case we would have a (\(-1\)) in the sum of Eq. (32) and in the second case a (\(+1\)). This implies that \(|I_{L}^{\mathbf {D}}|\) (\(-1\)) and \(|I_{U}^{\mathbf {D}}|\) (\(+1\)) might appear in the sum, which yields equation (7). We now analyse the difference between the new associated estimators and their underlying estimators. In the case of the empirical estimator, to bound the difference, we use the following two functions

    $$\begin{aligned} e_{i}(u_{i})&= \frac{1}{n}\underset{j=1}{\overset{n}{\sum }} 1_{\{X_{i}^{(j)}\le F_{i,n}^{\leftarrow }(u_{i})\}} \\ h_{i}(u_{i})&= \frac{1}{n}\underset{j=1}{\overset{n}{\sum }} 1_{\{X_{i}^{(j)}>\overline{F}_{i,n}^{\leftarrow }(u_{i})\}}\text {,} \end{aligned}$$

    for \(i\in \{1,\ldots ,d\}\). Given that we defined \(\overline{F} _{i,n}^{\leftarrow }(u)=F_{i,n}^{\leftarrow }(1-u)\), we have \( h_{i}(u_{i})=1-e_{i}(1-u_{i})\) and \(e_{i}(u_{i})=\frac{K_{i}}{n}\), for \( \frac{K_{i}-1}{n}<u_{i}\le \frac{K_{i}}{n}\). This implies

    $$\begin{aligned} 0&\le e_{i}(u_{i})-u_{i}<\frac{1}{n} \end{aligned}$$
    (34)
    $$\begin{aligned} 0&\le u_{i}-h_{i}(u_{i})<\frac{1}{n}\text {,} \end{aligned}$$
    (35)

    for \(i\in \{1,\ldots ,d\}\) and \(0<u_{i}<1\). Note that the lower bound of the inequalities is reached when \(nu_{i}\) is an integer and in this case \(e_{i}(u_{i})=h_{i}(u_{i})=u_{i}\). These inequalities allow us to bound difference between the \(\mathbf {D}^{+}\) -associated estimator of \(C_{\mathbf {D}^{*}}\), considering the empirical as underlying estimator of Definition 9, and the empirical estimator of \(C_{\mathbf {D}^{*}}\), denoted as \(C_{\mathbf {D}^{*},n}^{ \mathbf {D}^{+}(e)}\) and \(C_{\mathbf {D}^{*},n}^{e}\), respectively. From its definition, using the same notations as in Definition 9, we have

    $$\begin{aligned} C_{\mathbf {D}^{*},n}^{\mathbf {D}^{+}(e)}(u_{1},\ldots ,u_{d})=\underset{j=0}{\overset{d_{2}}{\sum }}(-1)^{j}\underset{k=1}{\overset{\left( {\begin{array}{c}d_{2}\\ j\end{array}}\right) }{\sum }}C_{\mathbf {D}^{+},n}^{e}(\mathbf {W}_{j,k})\text { .} \end{aligned}$$
    (36)

    For a dependence \(\mathbf {D}\), the empirical estimator of \(C_{\mathbf {D}}\) is defined as (see Definition 7)

    $$\begin{aligned} C_{\mathbf {D},n}^{e}(u_{1},\ldots ,u_{d})=\frac{1}{n}\underset{j=1}{\overset{n}{ \sum }}1_{\left\{ \overset{d}{\underset{i=1}{\cap }}\mathcal {B} _{i}^{(j)}\left( F_{D_{i},i,n}^{\leftarrow }(u_{i})\right) \right\} }\text {,} \end{aligned}$$
    (37)

    with

    $$\begin{aligned} \mathcal {B}_{i}^{(j)}\left( F_{D_{i},i,n}^{\leftarrow }(u_{i})\right) =\left\{ \begin{array}{ll} \{X_{i}^{(j)}\le F_{i,n}^{\leftarrow }(u_{i})\}\quad &{} \quad \text {if } D_{i}=L \\ \{X_{i}^{(j)}>\overline{F}_{i,n}^{\leftarrow }(u_{i})\}\quad &{} \quad \text {if } D_{i}=U \end{array} \right. \text { .} \end{aligned}$$

    Now, let \(q(\mathbf {W}_{j,k})=\{i|W_{j,k,i}\ne 1\}\) and, in the case \( |q(W_{j,k})|=1\), let \(i_{*}=q(\mathbf {W}_{j,k})\). Considering this and Eq. (37), using the notation of Definition 9, \(C_{ \mathbf {D}^{*},n}^{e}\) can be rewritten as

    $$\begin{aligned} C_{\mathbf {D}^{*},n}^{e}(u_{1},\ldots ,u_{d})=\underset{j=0}{\overset{d_{2}}{ \sum }}(-1)^{j}\underset{k=1}{\overset{\left( {\begin{array}{c}d_{2}\\ j\end{array}}\right) }{\sum }}C_{\mathbf {D} ^{+},n}^{^{\prime }}(\mathbf {W}_{j,k})\text {,} \end{aligned}$$
    (38)

    with

    $$\begin{aligned} C_{\mathbf {D}^{+},n}^{^{\prime }}(\mathbf {W}_{j,k})=\left\{ \begin{array}{ll} C_{\mathbf {D}^{+},n}^{e}(\mathbf {W}_{j,k})\quad &{} \quad \text {if }|q(\mathbf {W}_{j,k})|\ne 1 \\ e_{i}(W_{j,k,i_{*}})\quad &{} \quad \text {if }|q(\mathbf {W}_{j,k})|=1 \text { and } D_{i_{*}}^{+}=L \\ h_{i}(W_{j,k,i_{*}})\quad &{} \quad \text {if }|q(\mathbf {W}_{j,k})|=1 \text { and }D_{i_{*}}^{+}=U \end{array} \right. \text { .} \end{aligned}$$

    In the case \(d_{1}\ge 2\), from the way \(\mathbf {W}_{j,k}\) is defined, \(|q( \mathbf {W}_{j,k})|\ge 2\) for all \(\mathbf {W}_{j,k}\). Hence, from Eqs. ( 36) and (38), it follows that

    $$\begin{aligned} C_{\mathbf {D}^{*},n}^{\mathbf {D}^{+}(e)}=C_{\mathbf {D}^{*},n}^{ \mathbf {e}}\text { .} \end{aligned}$$
    (39)

    In the case \(d_{1}=1\), let \(i_{*}=I_{1}\). Using equations (36) and (38), the only case where the estimators are different is when \( j=0\), and it follows that

    $$\begin{aligned} C_{\mathbf {D}^{*},n}^{e}(u_{1},\ldots ,u_{d})-C_{\mathbf {D}^{*},n}^{ \mathbf {D}^{+}(e)}(u_{1},\ldots ,u_{d})=\left\{ \begin{array}{ll} e_{i}(u_{i_{*}})-u_{i_{*}}\quad &{} \quad \text {if } D_{i_{*}}^{+}=L\\ h_{i}(u_{i_{*}})-u_{i_{*}}\quad &{} \quad \text {if } D_{i_{*}}^{+}=U \end{array} \right. \text { .} \end{aligned}$$

    Using Eqs. (34) and (35), this implies

    $$\begin{aligned} \left| C_{\mathbf {D}^{*},n}^{e}-C_{\mathbf {D}^{*},n}^{\mathbf {D} ^{+}(e)}\right| <\frac{1}{n}\text { .} \end{aligned}$$
    (40)

    In the case \(d_{1}=0\), we have \(\mathbf {D}^{+}=\mathbf {D}^{*\complement }\). Define

    $$\begin{aligned} o_{i}(u_{i})=\left\{ \begin{array}{ll} e_{i}(u_{i})\quad &{} \quad \text {if } D_{i}^{+}=L\\ h_{i}(u_{i})\quad &{} \quad \text {if } D_{i}^{+}=U \end{array} \right. , \end{aligned}$$

    for \(i\in \{1,\ldots ,d\}\). Using Eqs. (36) and (38), the only case where the estimators are different is when \(j=1\). It follows that

    $$\begin{aligned} C_{\mathbf {D}^{*},n}^{e}(u_{1},\ldots ,u_{d})-C_{\mathbf {D}^{*},n}^{\mathbf {D}^{+}(e)}(u_{1},\ldots ,u_{d})=\underset{i=1}{\overset{d}{\sum }}(1-u_{i})-o_{i}(1-u_{i})\text { .} \end{aligned}$$
    (41)

    Furthermore, from Eqs. (34) and (35), we have that the elements of the sum are bounded in the following way

    $$\begin{aligned} \begin{array}{ll} -\frac{1}{n}<(1-u_{i})-o_{i}(1-u_{i})\le 0\quad &{} \quad \text {if } D_{i}^{+}=L\\ 0\le (1-u_{i})-o_{i}(1-u_{i})<\frac{1}{n}\quad &{} \quad \text {if } D_{i}^{+}=U \end{array} \text {.} \end{aligned}$$

    Hence, if \(i_{1}\) and \(i_{2}\) satisfy \(D_{i_{1}}^{+}\ne \) \(D_{i_{2}}^{+}\),

    $$\begin{aligned} |(1-u_{i_{1}})-o_{i_{1}}(1-u_{i_{1}})+(1-u_{i_{2}})-o_{i_{2}}(1-u_{i_{2}})|< \frac{1}{n}\text { .} \end{aligned}$$

    Then Eq. (41) implies equation (8). Also, because \( e_{i}(u_{i})=h_{i}(u_{i})=u_{i}\), when \(nu_{i}\) is an integer, the estimators always coincide on the grid \(g_{1}\). The case of rank estimator is analogous to the empirical estimator. In this case, instead of \(e_{i}\) and \(h_{i}\), the following two functions should be used

    $$\begin{aligned} s_{i}(u_{i})&= \frac{1}{n}\underset{j=1}{\overset{n}{\sum }} 1_{\{R_{i,n}^{(j)}\le nu_{i}\}}\\ t_{i}(u_{i})&= \frac{1}{n}\underset{ j=1}{\overset{n}{\sum }}1_{\{R_{i,n}^{(j)}>n(1-u_{i})\}}\text {,} \end{aligned}$$

    for \(i\in \{1,\ldots ,d\}\). We have

    $$\begin{aligned} t_{i}(u_{i})&= 1-s_{i}(1-u_{i}) \\ s_{i}(u_{i})&= \frac{1}{n}\lfloor nu_{i}\rfloor \text { ,} \end{aligned}$$

    where \(\lfloor \cdot \rfloor \) represents the greatest integer less than or equal to. We obtain equation (9) following the same procedure as in the previous case. Also, because \(s_{i}(u_{i})=t_{i}(u_{i})=u_{i}\), when \(nu_{i}\) is an integer, the estimators always coincide on the grid \(g_{1}\). By taking the limit when \(n\rightarrow \infty \) in the three inequalities we just proved, the asymptotic equivalence follows.

  2. (ii)

    As pointed out by (Segers (2012), Example 1.1), if a copula model \(C\) has lower tail dependence, it implies a discontinuity for its partial derivatives around \(\mathbf {0}\). Similarly, if it has upper tail dependence, it implies a discontinuity around \(\mathbf {1}\). Given that we defined general tail dependence in terms of the associated copula \(C_{ \mathbf {D}}\), we can generalise this in a simple way. If general tail dependence exists for dependence \(\mathbf {D}\), then all of the partial derivatives of \(C_{\mathbf {D}}\) are discontinuous in \(\mathbf {0}\). From the relationship between associated copulas, this implies that all partial derivatives of \(C\) are discontinuous in \((1_{\{D_{1}=U\}},\ldots ,1_{\{D_{d}=U \}})\). However, according to Segers (2012) no restriction is needed around boundary points for weak convergence to exist. Hence, the hypothesis is sufficient for weak convergence of \(C_{\mathbf {D},n}^{e}\). Given that this is an asymptotic property, (i) implies weak convergence for the other estimators.

\(\square \)

1.2 Proof for Proposition 2

Proof

  1. (i)

    Note that the estimators of the tail dependence function considered here are simple replacement estimators based on the copula estimators of Proposition 1. Simple replacement estimators are copula estimators multiplied by the factor \(\frac{n}{k}\) and evaluated in \( \frac{k\mathbf {w}}{n}\). Regardless of the value where they are evaluated, these copula estimators satisfy Proposition 1. Hence, by multiplying both sides of the inequalities in that proposition by \(\frac{n}{k }\), we get the desired result. It follows that, if \(\underset{n\rightarrow \infty }{\lim }k(n)=\infty \), they are asymptotically equivalent. Now, if \( \mathbf {w}\) is in the grid \(g_{2}\), \(\frac{k\mathbf {w}}{n}\) is in the grid \( g_{1}\) and the estimators coincide. In particular taking \(l_{1}=\cdots =l_{d}=n\), we have that the estimators of the TDC are the same.

  2. (ii)

    Considering the conditions stated in this part, we prove asymptotic normality using a functional delta method as in Schmidt and Stadtmüller (2006), Theorem 6. In order to make this generalisation we use the same notation and concepts. To obtain the result, we need to redefine and adjust some of the concepts as we now explain. We split the set \(\overline{\mathbb {R}}_{+}^{d}\) into the subsets \(S_{j}=[s_{1}\times \ldots .s_{d}]\), with \(s_{i}\) either \([0,\infty )\) or \(\infty \), \(i\in \{1,\ldots d\}\) and \(j\in \{1,\ldots 2^{d-1}\}\); we exclude the case when all \(s_{i}\) are \(\infty \). Given that we exclude this case, in each subset there is \(s_{i_{j}}=[0,\infty )\). For \(\mathbf {w}=(w_{1},\ldots ,w_{d})\) define \(\mathbf {w}_{i}:=(\infty ,\ldots ,\infty ,w_{i},\infty ,\ldots ,\infty )\), \(i\in \{1,\ldots ,d\}\). Now, for a function \(\gamma \) and \(\mathbf {w}\in S_{j}\), define the corresponding map as

    $$\begin{aligned}&\phi :\gamma (\mathbf {w})\overset{\phi _{1}}{\longmapsto }\left\{ \gamma ( \mathbf {w}),\gamma (\mathbf {v}_{1}),\ldots ,\gamma (\mathbf {v}_{d})\right\} \overset{\phi _{2}}{\longmapsto }\\&\left\{ \gamma (\mathbf {w}),\gamma ^{-}(\mathbf {v}_{1}),\ldots ,\gamma ^{-}(\mathbf {v}_{d})\right\} \overset{\phi _{3}}{\longmapsto }\gamma (v_{1},\ldots ,v_{d})\text {,} \end{aligned}$$

    with

    $$\begin{aligned} (\mathbf {v}_{i},v_{i})=\left\{ \begin{array}{ll} (\mathbf {w}_{i},\gamma ^{-}(\mathbf {w}_{1}))\quad &{} \quad \text {if }s_{i}=[0,\infty )\\ (\mathbf {w}_{i_{j}},\infty )\quad &{} \quad \text {if } s_{i}=\infty \end{array} \right. , \end{aligned}$$

    for \(i\in \{1,\ldots ,d\}\). The conditions of the delta method are then satisfied by \(\phi \). In the case of the derivative of \(\phi _{3}\), we take \(\alpha (\mathbf {w})=(\gamma (\mathbf {w}_{1}),\ldots ,\gamma (\mathbf {w}_{d}))\), \(\beta (\mathbf {w})=\gamma (\mathbf {w})\), \(B(\mathbf {w})=b_{\mathbf {D}}(\mathbf {w})\) and \(A(\mathbf {w})=(b_{\mathbf {D}}^{-}(\mathbf {w}_{1}),\ldots ,b_{\mathbf {D}}^{-}(\mathbf {w}_{d}))\) in Schmidt and Stadtmüller (2006), Lemma 1. We then evaluate the derivative of \(\phi \) at \(b_{\mathbf {D},n}\) in the direction of \(\mathbb {G}_{b_{\mathbf {D},n}^{*}}\) to obtain

    $$\begin{aligned} \mathbb {G}_{b_{\mathbf {D},n}}(\mathbf {w})=\mathbb {G}_{b_{\mathbf {D},n}^{*}}(\mathbf {w})-\underset{i=1}{\overset{d}{\sum }} \frac{\partial b_{\mathbf {D}}(\mathbf {w})}{\partial u_{i}}\mathbb {G}_{b_{\mathbf {D},n}^{*}}(\mathbf {w}_{i})\text {,} \end{aligned}$$
    (42)

    with \(\mathbb {G}_{b_{\mathbf {D},n}^{*}}\) as a centred tight continuous Gaussian field which satisfies

    $$\begin{aligned}&E\left( \mathbb {G}_{b_{\mathbf {D},n}^{*}}(w_{1},\ldots ,w_{d}) \mathbb {G}_{b_{\mathbf {D},n}^{*}}(\overline{w}_{1},\ldots ,\overline{w}_{d})\right) \nonumber \\&\quad =b_{ \mathbf {D}}\left( \min (w_{1},\overline{w}_{1}),\ldots ,\min (w_{d},\overline{w}_{d})\right) \text { .} \end{aligned}$$
    (43)

    Equation (16) follows from the fact that \(\mathbb {G}_{b_{\mathbf {D},n}^{*}}\) is centred. Note that \(b_{\mathbf {D}}(\infty ,\ldots ,\infty ,\) \(w_{i},\infty ,\ldots ,\infty ,\) \(w_{j},\infty ,\ldots \infty )\) is equal to the \((D_{i},D_{j})\)-tail dependence function of the marginal copula \(C_{i,j}\) of \(C\), denoted as \(b_{(D_{i},D_{j})}(w_{i},w_{j})\). Then, squaring (42) and using (43) yields equation (16).

  3. (iii)

    For the strong consistency, as proved in (Krajina (2010), Chapter 4), under the conditions of this case, the empirical exponent \(a_{\mathbf {D},n}\) is strongly consistent.Footnote 1 The relationship between \(b_{\mathbf {D}}\) and the \(a_{\mathbf {D}}\) is given in Eq. (13). Also, from the proof of (Salazar (2012), Proposition 26, equations (A.43) and (A.44)), we have

    $$\begin{aligned} C_{\mathbf {D}}(u_{1},\ldots ,u_{d})=\underset{k=1}{\overset{d}{\sum }}u_{k}+\underset{j=2}{\overset{d}{\sum }}(-1)^{j+1}\underset{k=1}{\overset{\left( {\begin{array}{c}d\\ j\end{array}}\right) }{\sum }}\left\{ 1-C_{\mathbf {D}^{\complement }(S_{j,k})}(1-u_{i}|i\in S_{j,k})\right\} \text { .} \end{aligned}$$
    (44)

    Using Eq. (13), \(a_{\mathbf {D},n}\) defines an estimator of \(b_{\mathbf {D}}\), that we denote as \(b_{\mathbf {D},n}^{*}\), which is, by construction, strongly consistent. Consider the simple replacement estimator \(b_{\mathbf {D},k,n}^{\mathbf {D}^{\complement }(rank)}\). This estimator is based on Eq. (44) and on the rank estimator. Given that \(a_{\mathbf {D},n}\) in Krajina (2010) is defined in terms of ranks, we have that \(\left| a_{\mathbf {D},n}-\left( 1-\frac{n}{k(n)}C_{\mathbf {D}^{\complement },n}^{rank}\right) \right| <\frac{d}{k(n)}\), and it follows that

    $$\begin{aligned} \left| b_{\mathbf {D},n}^{*}-b_{\mathbf {D},k,n}^{\mathbf {D}^{\complement }(rank)}\right| <\frac{2^{d}\cdot d}{k(n)}\text { .} \end{aligned}$$

    Taking the limit \(n\rightarrow \infty \) and using (i), the result follows. \(\square \)

Appendix B: Confidence intervals

The confidence intervals we present for estimator \(b_{\mathbf {D},k,n}\) are based on the asymptotic normality of the estimators and its variance as stated in Proposition 2, Eq. (16). In order to estimate this variance of estimators of the bivariate lower TDC, Schmidt and Stadtmüller (2006) presented a method based on the assumption of the distributional copula being equal to the MTCJ copula \(C_{\theta }^{MTCJ}\).Footnote 2 Under this assumption, the estimate of the variance is given by the bivariate version of Corollary 1 and an estimate of the parameter \(\theta \). The authors found that this method provides accurate estimates of the variance, even when the data does not have an MTCJ copula (see also Joe et al. 2010; Nelsen 2006). Following this approach for \(b_{\mathbf {D},k,n}\), we assume that the \(\mathbf {D}\)-associated copula is the multivariate MTCJ copula, that is

$$\begin{aligned} C_{\mathbf {D}}(u_{1},\ldots u_{d})=C_{\theta }^{MTCJ}=(u_{1}^{-\theta }+\cdots +u_{d}^{-\theta }-d+1)^{\frac{-1}{\theta }}\text { ,} \end{aligned}$$

for \(0<\theta <\infty \). Then, its \(\mathbf {D}\)-tail dependence function is

$$\begin{aligned} b_{\mathbf {D},\theta }(w_{1},\ldots ,w_{d})=b_{L,\theta }^{MTCJ}=(w_{1}^{-\theta }+\cdots +w_{d}^{-\theta })^{-\frac{1}{\theta }}\text { .} \end{aligned}$$

Considering equation (16), the asymptotic variance for the estimator \(b_{\mathbf {D},k,n}\) is

$$\begin{aligned} \sigma _{b_{\mathbf {D},k,n}}^{2}(\theta )&= b_{L,\theta }^{MTCJ}+\left( b_{L,\theta }^{MTCJ}\right) ^{2\theta +2}\overset{d}{\underset{i=1}{\sum }}\left( w_{i}\right) ^{-(2\theta +1)}\nonumber \\&\quad -\,2\left( b_{L,\theta }^{MTCJ}\right) ^{\theta +2}\overset{d}{\underset{i=1}{\sum }}\left( w_{i}\right) ^{-(\theta +1)} \nonumber \\&\quad +\,2\left( b_{L,\theta }^{MTCJ}\right) ^{2\theta +2}\underset{i\ne j}{\sum } \left( \frac{1}{w_{i}w_{j}}\right) ^{\theta +1}b_{L,i,j}^{MTCJ}(w_{i},w_{j})\text { } \end{aligned}$$
(45)

(note that for notational convenience we omitted the \((w_{1},\ldots ,w_{d})\)). In the case of the tail dependence coefficient, considering Corollary 1, this becomes

$$\begin{aligned} \sigma _{\lambda _{\mathbf {D},k,n}}^{2}(\theta )=d^{-\frac{1}{\theta }}-\left( 2-d^{-1}\right) d^{-\frac{2}{\theta }}+\left( {\begin{array}{c}d\\ 2\end{array}}\right) 2^{-\frac{1}{\theta }+1}\cdot d^{-2-\frac{2}{\theta }}\text { .} \end{aligned}$$
(46)

The parameter \(\theta \) is estimated from the data via maximum likelihood. We then estimate the variance of the estimators as \(\sigma _{\lambda _{ \mathbf {D},k,n}}^{2}(\widehat{\theta })\).

Appendix C: ARMA-GARCH modelling

It is widely known in the econometric literature that financial asset returns exhibit so-called stylised facts, in particular volatility clusters, which is at odds with the i.i.d. assumption. All raw returns considered throughout the paper present serial autocorrelation in the second moment. To account for potential autocorrelation of financial returns, ARMA-GARCH models have proven to be effective in obtaining “standardised returns”. A popular approach for multivariate data is to first use a GARCH(1,1) model for the individual variables and then fit a copula [e.g. (Nikoloulopoulos et al. (2012), Section 4) and Christoffersen et al. (2012)]. Considering similar financial data for estimations of copulas, Czado et al. (2012) and Min and Czado (2010) also first obtained non-autocorrelated marginals using an ARMA(1,1)-GARCH(1,1) model, while Patton (2006) suggested an AR(1)-\(t\)-GARCH(1,1) in his study.

Hansen and Lunde (2005) have shown in a comprehensive comparison study of over 300 models that the GARCH(1,1) specification is in most scenarios the best choice. We first also study different model specifications recommended in the above literature and then apply a Portmanteau test on the corresponding model residuals to check their serial autocorrelation. In our application, the ARMA(1,1)-GARCH(1,1) specification appears to be sufficient to remove the time dependence in the raw data. The test results for a small selection of lags are presented in Table 13. According to this table, the model is suitable to account for the iid assumption, as almost all \(p\)-values are higher than the conventional 10 % significance level. In the majority of cases, the null hypothesis of no serial autocorrelation can not be rejected for the “standardised returns”.

Table 13 Summary of \(p\) values of the Portmanteau test for the residuals and squared residuals of the ARMA-GARCH models (with lags \(l=\{5, 22,65\}\), corresponding to one week, one month and one quarter)

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Salazar, Y., Ng, W.L. Nonparametric estimation of general multivariate tail dependence and applications to financial time series. Stat Methods Appl 24, 121–158 (2015). https://doi.org/10.1007/s10260-014-0274-7

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