#
Goodness-of-Fit Tests for Copulas of Multivariate Time Series

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## Abstract

**:**

## 1. Introduction

## 2. Weak Convergence of Empirical Processes of Residuals

**θ**, compute the residuals ${\mathbf{e}}_{i,n}={({e}_{1i,n},\dots ,{e}_{di,n})}^{\mathrm{\top}}$, where

**θ**. These terms are needed in order to be able to measure the difference between ${\mathbb{K}}_{n}$ and ${\alpha}_{n}$. Next, assume that for any $j\in \{1,\dots ,d\}$, and any $\mathbf{x}\in {\overline{\mathbb{R}}}^{d}$, the following properties hold:

- (A1)
- $\mathbf{\Gamma}}_{0,n}\left(s\right)=\frac{1}{n}\sum _{i=1}^{\lfloor ns\rfloor}{\mathit{\gamma}}_{0i}\stackrel{Pr}{\u27f6}s{\mathbf{\Gamma}}_{0$, $\mathbf{\Gamma}}_{1k,n}\left(s\right)=\frac{1}{n}\sum _{i=1}^{\lfloor ns\rfloor}{\mathit{\gamma}}_{1ki}\stackrel{Pr}{\u27f6}s{\mathbf{\Gamma}}_{1k$, uniformly in $s\in [0,1]$, where ${\mathbf{\Gamma}}_{0}$ and ${\mathbf{\Gamma}}_{1k}$ are deterministic, $k=1,\dots ,d$.
- (A2)
- $\frac{1}{n}\sum _{i=1}^{n}\mathrm{E}\left(\right)open="("\; close=")">\parallel {\mathit{\gamma}}_{0i}{\parallel}^{k}$ and $\frac{1}{n}\sum _{i=1}^{n}\mathrm{E}\left(\right)open="("\; close=")">\parallel {\mathit{\gamma}}_{1ji}{\parallel}^{k}$ are bounded, for $k=1,2$.
- (A3)
- There exists a sequence of positive terms ${r}_{i}>0$ so that ${\sum}_{i\ge 1}{r}_{i}<\infty $ and such that the sequence $\underset{1\le i\le n}{\mathrm{max}}\parallel {\mathbf{d}}_{i,n}\parallel /{r}_{i}$ is tight.
- (A4)
- ${\mathrm{max}}_{1\le i\le n}\parallel {\mathit{\gamma}}_{0i}\parallel /\sqrt{n}={o}_{P}\left(1\right)$ and ${\mathrm{max}}_{1\le i\le n}|{\epsilon}_{ji}|\parallel {\mathit{\gamma}}_{1ji}\parallel /\sqrt{n}={o}_{P}\left(1\right)$.
- (A5)
- $({\alpha}_{n},{\mathbf{\Theta}}_{n})\u21dd(\alpha ,\mathbf{\Theta})$ in $\mathcal{D}({[-\infty ,\infty ]}^{d})\times {\mathbb{R}}^{p}$.
- (A6)
- ${\partial}_{{x}_{j}}K\left(\mathbf{x}\right)$ and ${x}_{j}{\partial}_{{x}_{j}}K\left(\mathbf{x}\right)$ are bounded and continuous on ${\overline{\mathbb{R}}}^{d}={[-\infty ,+\infty ]}^{d}$. In addition, ${F}_{1},\dots ,{F}_{d}$ have continuous bounded densities ${f}_{1},\dots ,{f}_{d}$ respectively.
- (A7)
- For all $k\ne j$, ${f}_{j}\left({x}_{j}\right)E\left\{\right|{\epsilon}_{k1}|\mathbf{1}({\mathit{\epsilon}}_{1}\le x)|{\epsilon}_{j1}={x}_{j}\}$ and ${x}_{j}{f}_{j}\left({x}_{j}\right)E\left\{\right|{\epsilon}_{k1}|\mathbf{1}({\mathit{\epsilon}}_{1}\le x)|{\epsilon}_{j1}={x}_{j}\}$ are bounded and continuous on ${\overline{\mathbb{R}}}^{d}$.

**Remark**

**1.**

- Model 1. In this model, for the sake of identifiability, we assume that the correlation matrix $E\left(\right)open="("\; close=")">{\mathit{\epsilon}}_{i}{\mathit{\epsilon}}_{i}^{\mathrm{\top}}$ is the identity matrix. This means that the conditional covariance matrix of the observations relative to ${\mathcal{F}}_{i-1}$ is ${\mathit{\sigma}}_{i}{\mathit{\sigma}}_{i}^{\mathrm{\top}}$. In particular, the conditional correlation is not necessarily constant and it is estimated.
- Model 2. In this model, ${\mathit{\sigma}}_{i}$ is a diagonal matrix for any i, and there is no restriction on the correlation matrix of the innovations. This means ${\mathit{\sigma}}_{i}{\mathit{\sigma}}_{i}^{\mathrm{\top}}$ is diagonal and for any $j\in \{1,\dots ,d\}$, ${\left(\right)}_{{\mathit{\sigma}}_{i}}$ is the conditional variance of the observations ${X}_{ji}$ relative to ${\mathcal{F}}_{i-1}$. In particular, this implies that the conditional correlation between the observations is constant, does not depend on
**θ**, and it is implicitly incorporated in the copula C of the innovations. This model appears naturally in practice when the parameters of univariate time series are estimated separately as in Chen and Fan [10]. This model is not the same as the diagonal representation in Engle and Kroner [17], which is included in Model 1.

**Remark**

**2.**

**Theorem**

**1.**

**Remark**

**3.**

**ϕ**. Goodness-of-fit tests could also be based on the so-called Rosenblatt transform of K. See, e.g., Genest and Rémillard [18] and Rémillard [19] for details.

**Corollary**

**1.**

**Remark**

**4.**

#### Empirical Processes Related to the Copula

**Condition**

**1.**

**Corollary**

**2.**

**Remark**

**5.**

**Θ**, even if $\mathbb{K}$ does. This important property will play a major role in the next section, where specification tests for the copula are discussed. Also, recall that $\stackrel{\u02c7}{\mathbb{C}}(1,\xb7)$ is the asymptotic limit of the empirical copula process constructed from innovations if they were observable; see, e.g., Gänssler and Stute [23], Fermanian et al. [24], Tsukahara [25]. In fact, setting ${\stackrel{\u02c7}{\mathbf{U}}}_{i,n}={\mathbf{R}}_{i}/(n+1)$, where ${\mathbf{R}}_{1},\dots ,{\mathbf{R}}_{n}$ are the associated rank vectors of ${\mathbf{U}}_{1},\dots ,{\mathbf{U}}_{n}$, one easily obtains the result that $\stackrel{\u02c7}{\mathbb{C}}$ is the asymptotic limit of ${\stackrel{\u02c7}{\mathbb{C}}}_{n}(s,\mathbf{u})=\frac{1}{\sqrt{n}}{\sum}_{i=1}^{\lfloor ns\rfloor}\{\mathbf{1}({\stackrel{\u02c7}{\mathbf{U}}}_{i,n}\le \mathbf{u})-C\left(\mathbf{u}\right)\}$, $(s,\mathbf{u})\in {[0,1]}^{1+d}$.

#### Semiparametric Estimation of the Copula

## 3. Specification Tests for the Copula

**θ**is either estimated by maximum likelihood, which requires assuming parametric families for the marginal distributions ${F}_{j}$ of ${\epsilon}_{ji}$, or by quasi maximum likelihood. The dependence between the innovations components ${\u03f5}_{1i},\dots ,{\u03f5}_{di}$ is then modeled by the copula C of ${\mathit{\epsilon}}_{i}$.

- (B1)
- For every $\mathit{\varphi}\in \mathcal{P}$, the density ${c}_{\mathit{\varphi}}$ of ${C}_{\mathit{\varphi}}$ admits first and second order derivatives with respect to all components of
**ϕ**. The gradient (column) vector with respect to**ϕ**is denoted ${\dot{c}}_{\mathit{\varphi}}$, and the Hessian matrix is represented by ${\ddot{c}}_{\mathit{\varphi}}$. - (B2)
- For arbitrary $\mathbf{u}\in {(0,1)}^{d}$ and every ${\mathit{\varphi}}_{0}\in \mathcal{P}$, the mappings $\mathit{\varphi}\mapsto {\dot{c}}_{\mathit{\varphi}}\left(\mathbf{u}\right)/{c}_{\mathit{\varphi}}\left(\mathbf{u}\right)$ and $\mathit{\varphi}\mapsto {\ddot{c}}_{\mathit{\varphi}}\left(\mathbf{u}\right)/{c}_{\mathit{\varphi}}\left(\mathbf{u}\right)$ are continuous at ${\mathit{\varphi}}_{0}$.
- (B3)
- For every ${\mathit{\varphi}}_{0}\in \mathcal{P}$, there exist a neighborhood $\mathcal{N}$ of ${\mathit{\varphi}}_{0}$ and ${C}_{{\mathit{\varphi}}_{0}}$-integrable functions ${h}_{1},{h}_{2}:{\mathbb{R}}^{d}\to \mathbb{R}$ such that for every $\mathbf{u}\in {(0,1)}^{d}$,$$\underset{\mathit{\varphi}\in \mathcal{N}}{\mathrm{sup}}\left(\right)open="\parallel "\; close="\parallel ">\frac{{\dot{c}}_{\mathit{\varphi}}\left(\mathbf{u}\right)}{{c}_{\mathit{\varphi}}\left(\mathbf{u}\right)}\le {h}_{2}\left(\mathbf{u}\right).$$

**ϕ**. Specification tests based on ${\mathbb{P}}_{n}$ are described in Section 3.1 together with a bootstrapping method to estimate p-values. In Section 3.2, one describes processes constructed from the Rosenblatt transform; their asymptotic behavior is studied, tests statistics are proposed together with a bootstrapping method. All these tests use the estimation ${\mathit{\varphi}}_{n}$, hence the importance of finding the asymptotic behavior of ${\mathbf{\Phi}}_{n}=\sqrt{n}({\mathit{\varphi}}_{n}-\mathit{\varphi})$. In Section 3.3, we consider the most common estimation methods based on ranks, i.e., ${\mathit{\varphi}}_{n}={\mathcal{T}}_{n}({\mathbf{U}}_{1,n},\dots ,{\mathbf{U}}_{n,n})$, for some deterministic function ${\mathcal{T}}_{n}$. Finally, estimation methods based on common dependence measures are described in Section 3.4, and arguments in favor of the Rosenblatt transform vs. the copula are given in Section 3.5.

#### 3.1. Test Statistics Based on the Empirical Copula

**Remark**

**6.**

**ϕ**if $({\alpha}_{n},{\mathbb{W}}_{n},{\mathbf{\Phi}}_{n})\u21dd(\alpha ,\mathbb{W},\mathbf{\Phi})$ where the latter is centered Gaussian with $E\left(\right)open="("\; close=")">\mathbf{\Phi}{\mathbb{W}}^{\mathrm{\top}}$, and

**Φ**does not depend on

**θ**or

**Θ**. It is an immediate consequence of the delta method that the property of being regular is preserved by homeomorphisms. The basic result for testing goodness-of-fit using ${\mathbb{P}}_{n}$ is stated next and its proof is given in the Appendix B.

**Proposition**

**1.**

**ϕ**, then ${\mathbb{P}}_{n}\u21dd\mathbb{P}$, and ${S}_{n}\u21ddS={\int}_{{[0,1]}^{d}}{\mathbb{P}}^{2}\left(\mathbf{u}\right)dC\left(\mathbf{u}\right)$, where $\mathbb{P}\left(\mathbf{u}\right)=\stackrel{\u02c7}{\mathbb{C}}(1,\mathbf{u})-\dot{C}{\left(\mathbf{u}\right)}^{\mathrm{\top}}\mathbf{\Phi}$, $\mathbf{u}\in {[0,1]}^{d}$. In fact, if ψ is a continuous function on the space $C\left(\right[0,1\left]\right)$, then ${T}_{n}=\psi \left({\mathbb{P}}_{n}\right)\u21ddT=\psi \left(\mathbb{P}\right)$. Moreover, the parametric bootstrap algorithm described next or the two-level parametric bootstrap proposed in Genest et al. [33] can be used to estimate p-values of ${S}_{n}$ or ${T}_{n}$.

#### Parametric Bootstrap for ${S}_{n}$

Algorithm 1: Parametric bootstrap for the empirical copula process. |

For some large integer N, do the following steps:- 1.-
- Compute ${C}_{n}$ and estimate
**ϕ**with ${\mathit{\varphi}}_{n}={\mathcal{T}}_{n}\left(\right)open="("\; close=")">{\mathbf{U}}_{1,n},\dots ,{\mathbf{U}}_{n,n}$. - 2.-
- Compute the value of ${S}_{n}$, as defined by (8).
- 3.-
- Repeat the following steps for every $k\in \{1,\dots ,N\}$:
- (a)
- Generate a random sample ${\mathbf{Y}}_{1,n}^{\left(k\right)},\dots ,{\mathbf{Y}}_{n,n}^{\left(k\right)}$ from distribution ${C}_{{\mathit{\varphi}}_{n}}$ and compute the pseudo-observations ${\mathbf{U}}_{i,n}^{\left(k\right)}={\mathbf{R}}_{i,n}^{\left(k\right)}/(n+1)$, where ${\mathbf{R}}_{1,n}^{\left(k\right)},\dots ,{\mathbf{R}}_{n,n}^{\left(k\right)}$ are the associated rank vectors of ${\mathbf{Y}}_{1,n}^{\left(k\right)},\dots ,{\mathbf{Y}}_{n,n}^{\left(k\right)}$.
- (b)
- Set$${C}_{n}^{\left(k\right)}\left(\mathbf{u}\right)=\frac{1}{n}\sum _{i=1}^{n}\mathbf{1}\left(\right)open="("\; close=")">{\mathbf{U}}_{i,n}^{\left(k\right)}\le \mathbf{u}$$
**ϕ**by ${\mathit{\varphi}}_{n}^{\left(k\right)}={\mathcal{T}}_{n}\left(\right)open="("\; close=")">{\mathbf{U}}_{1,n}^{\left(k\right)},\dots ,{\mathbf{U}}_{n,n}^{\left(k\right)}$. - (c)
- Compute$${S}_{n}^{\left(k\right)}=\sum _{i=1}^{n}{\left(\right)}^{{C}_{n}^{\left(k\right)}}-{C}_{{\mathit{\varphi}}_{n}^{\left(k\right)}}\left(\right)open="("\; close=")">{\mathbf{U}}_{i,n}^{\left(k\right)}2$$
An approximate p-value for the test is then given by ${\sum}_{k=1}^{N}\mathbf{1}\left(\right)open="("\; close=")">{S}_{n}^{\left(k\right)}{S}_{n}$. |

**θ**and

**ϕ**each time. This is possible only because $\mathbb{P}$ does not depend on

**Θ**or

**θ**. However, for Model 1 where $\mathbb{P}$ depends on

**Θ**and possibly on margin parameters, fitting the copula requires as much work as fitting K since one needs to generate the whole process ${\mathbf{X}}_{i}$ each time!

**Remark**

**7.**

#### 3.2. Tests Statistics Based on the Rosenblatt Transform

**ϕ**if $({\mathbb{B}}_{n},{\mathbb{W}}_{n},{\mathbf{\Phi}}_{n})\u21dd(\mathbb{B},\mathbb{W},\mathbf{\Phi})$ where the latter is centered Gaussian with $E\left(\right)open="("\; close=")">\mathbf{\Phi}{\mathbb{W}}^{\mathrm{\top}}$, and

**Φ**does not depend on

**θ**or

**Θ**.

**ϕ**calculated with ${\stackrel{\u02c7}{\mathbf{U}}}_{i,n}={\mathbf{R}}_{i}/(n+1)$, $i\in \{1,\dots ,n\}$. Further set

**Theorem**

**2.**

**ϕ**, then ${\mathbb{D}}_{n}-{\stackrel{\u02c7}{\mathbb{D}}}_{n}\u21dd0$ and ${\stackrel{\u02c7}{\mathbb{D}}}_{n}\u21dd\stackrel{\u02c7}{\mathbb{D}}$, with $\stackrel{\u02c7}{\mathbb{D}}$ given by

**θ**, as if

**θ**were known. The following result is then a direct application of the continuous mapping theorem.

**Proposition**

**2.**

#### A Parametric Bootstrap for ${S}_{n}^{\left(B\right)}$

Algorithm 2: Parametric bootstrap for the empirical Rosenblatt process. |

For some large integer N, do the following steps:- 1.
- 2.
- For some large integer N, repeat the following steps for every $k\in \{1,\dots ,N\}$:
- (a)
- Generate a random sample ${\mathbf{Y}}_{1,n}^{\left(k\right)},\dots ,{\mathbf{Y}}_{n,n}^{\left(k\right)}$ from distribution ${C}_{{\mathit{\varphi}}_{n}}$ and compute the pseudo-observations ${\mathbf{U}}_{i,n}^{\left(k\right)}={\mathbf{R}}_{i,n}^{\left(k\right)}/(n+1)$, where ${\mathbf{R}}_{1,n}^{\left(k\right)},\dots ,{\mathbf{R}}_{n,n}^{\left(k\right)}$ are the associated rank vectors of ${\mathbf{Y}}_{1,n}^{\left(k\right)},\dots ,{\mathbf{Y}}_{n,n}^{\left(k\right)}$.
- (b)
- Estimate
**ϕ**by ${\mathit{\varphi}}_{n}^{\left(k\right)}={\mathcal{T}}_{n}\left(\right)open="("\; close=")">{\mathbf{U}}_{1,n}^{\left(k\right)},\dots ,{\mathbf{U}}_{n,n}^{\left(k\right)}$, and compute and compute ${\mathbf{E}}_{i,n}^{\left(k\right)}={\mathcal{R}}_{{{\mathit{\varphi}}_{n}}^{\left(k\right)}}\left(\right)open="("\; close=")">{\mathbf{U}}_{i,n}^{\left(k\right)}$, $i\in \{1,\dots ,n\}$. - (c)
- Let$${\mathbb{D}}_{n}^{\left(k\right)}\left(\mathbf{u}\right)=\frac{1}{\sqrt{n}}\phantom{\rule{0.166667em}{0ex}}\sum _{i=1}^{n}\left(\right)open="\{"\; close="\}">\mathbf{1}\left(\right)open="("\; close=")">{\mathbf{E}}_{i,n}^{\left(k\right)}\le \mathbf{u},\phantom{\rule{1.em}{0ex}}\mathbf{u}\in {[0,1]}^{d}$$$${S}_{n,k}^{\left(B\right)}={\int}_{{[0,1]}^{d}}{\left(\right)}^{{\mathbb{D}}_{n}^{\left(k\right)}}2$$
An approximate p-value for the test is then given by ${\sum}_{k=1}^{N}\mathbf{1}\left(\right)open="("\; close=")">{S}_{n,k}^{\left(B\right)}{S}_{n}^{\left(B\right)}$. |

#### 3.3. Estimation of Copula Parameters

#### 3.3.1. Maximum Pseudo-Likelihood Estimators

**θ**required for the evaluation of the residuals! In fact, it has the same representation as the estimator studied by Genest et al. [11] in the serially independent case, i.e., if

**θ**were known. More precisely, one has

**ϕ**since $({\mathbf{\Phi}}_{n},{\mathbb{W}}_{n})\u21dd(\mathbf{\Phi},\mathbb{W})$ which is centered Gaussian, and $E\left(\right)open="("\; close=")">\mathbf{\Phi}{\mathbb{W}}^{\mathrm{\top}}$. Note also that $({\mathbb{B}}_{n},{\mathbb{W}}_{n},{\mathbf{\Phi}}_{n})\u21dd(\mathbb{B},\mathbb{W},\mathbf{\Phi})$ where the latter is centered Gaussian, so the assumptions of Theorem 2 are also met.

**Remark**

**8.**

#### 3.3.2. Two-Stage Estimators

**ϕ**since $({\mathbf{\Phi}}_{n},{\mathbb{W}}_{n})\u21dd(\mathbf{\Phi},\mathbb{W})$ which is a centered Gaussian vector with $E\left(\mathbf{\Phi}{\mathbb{W}}^{\mathrm{\top}}\right)=I$. Also if ${\mathbf{\Phi}}_{1}$ does not depend on

**θ**, then

**Φ**does not either. This is the case if ${\mathit{\varphi}}_{1,n}$ is a function of ${C}_{n}$.

#### 3.4. Estimators Based on Measures of Dependence

#### 3.4.1. Kendall’s Tau

**Proposition**

**3.**

#### 3.4.2. Spearman’s Rho

**Proposition**

**4.**

#### 3.4.3. Van der Waerden’s Coefficient

**Proposition**

**5.**

#### 3.4.4. Blomqvist’s Coefficient

**Proposition**

**6.**

#### 3.5. Copula vs. Rosenblatt Transform

## 4. Example of Application

## 5. Conclusions

## Acknowledgments

## Conflicts of Interest

## Appendix A. Proofs of the Main Results

#### Appendix A.1. Proof of Theorem 1

#### Appendix A.2. Proof of Theorem 2

**ϕ**, implying that $E\left(\right)open="("\; close=")">\mathbf{\Phi}{\mathbb{W}}^{\mathrm{\top}}$. It then follows from Genest and Rémillard [18] that the parametric bootstrap work for ${\stackrel{\u02c7}{\mathbb{D}}}_{n}$. To complete the proof, it only remains to show that ${\mathbb{D}}_{n}-{\stackrel{\u02c7}{\mathbb{D}}}_{n}\u21dd0$. To this end, note that ${\mathbf{V}}_{i,n}={H}_{n}\left({\mathbf{e}}_{i,n}\right)$, where ${H}_{n}={\mathcal{R}}_{{\mathit{\varphi}}_{n}}\circ {\mathbf{F}}_{n}$, so if ${\mathbb{H}}_{n}=\sqrt{n}({H}_{n}-H)$, then ${\mathbb{H}}_{n}\u21dd\mathbb{H}$, where, for all $j\in \{1,\dots ,d\}$,

## Appendix B. Other Proofs

**Lemma**

**B.1.**

**Proof.**

#### Appendix B.1. Proof of Proposition 1

#### Appendix B.2. Proof of Propositions 3–6

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Rémillard, B.
Goodness-of-Fit Tests for Copulas of Multivariate Time Series

. *Econometrics* **2017**, *5*, 13.
https://doi.org/10.3390/econometrics5010013

**AMA Style**

Rémillard B.
Goodness-of-Fit Tests for Copulas of Multivariate Time Series

. *Econometrics*. 2017; 5(1):13.
https://doi.org/10.3390/econometrics5010013

**Chicago/Turabian Style**

Rémillard, Bruno.
2017. "Goodness-of-Fit Tests for Copulas of Multivariate Time Series

" *Econometrics* 5, no. 1: 13.
https://doi.org/10.3390/econometrics5010013