Goodness-of-fit test of copula functions for semi-parametric univariate time series models

Abstract

In this paper, we propose a goodness-of-fit test, named pseudo “in-and-out-of-likelihood” (PIOL) ratio test, to check for misspecification in semi-parametric copula models for univariate time series. The proposed test extends the idea of the IOS test by Presnell and Boos (J Am Stat Assoc 99:216–227, 2004) and PIOS test by Zhang et al. (J Econom, 193:215–233, 2016), which are problematic for direct application to univariate time series. The PIOL test provides an integrated framework for both independent data and time series data. In addition, an approximation method is implemented to alleviate the computational burden of calculating the test statistics. Asymptotic properties of the proposed test statistics are discussed. The finite-sample performance is examined through simulation studies. We also demonstrate the proposed method through the analysis of a time series of daily transactions of Apple trade.

Appendix

1.1 Appendix I: Proof of theorems in Section 3

Proof of Theorem 1

First, by condition (A1), applying law of large number for stationary ergodic time series (DasGupta 2008), we have

$$\begin{aligned} \frac{1}{n}\sum _{t=1}^{n} \ddot{\ell }(U_{t-1},U_{t}; \theta ^*) {\mathop {\rightarrow }\limits ^{pr}} {\mathcal {S}}, \ \ \ n\rightarrow \infty . \end{aligned}$$

By conditions (A2), we have

$$\begin{aligned}&\left\| \ddot{\ell }({\hat{U}}_{t-1}, {\hat{U}}_{t}, {\hat{\theta }}) - \ddot{l} (U_{t-1},U_{t}; \theta ^*)\right\| \\&\quad \le \left\| \ddot{\ell }({\hat{U}}_{t-1}, {\hat{U}}_{t}, {\hat{\theta }}) - \ddot{\ell }({\hat{U}}_{t-1}, {\hat{U}}_{t}, \theta ^*)\right\| + \left\| \ddot{\ell }({\hat{U}}_{t-1}, {\hat{U}}_{t}, \theta ^*)- \ddot{l} (U_{t-1},U_{t}; \theta ^*)\right\| \\&\quad \le \sup _{(G,\theta )\in \mathcal F_{\delta }}\Vert \sum _{i,j=1}^{p}\frac{\partial }{\partial \theta }\ddot{\ell }( U_{t-1},U_{t}, \theta )\Vert \Vert {\hat{\theta }}-\theta ^*\Vert \\&\qquad + \sup _{(G,\theta )\in {\mathcal {F}}_{\delta }}\Vert \sum _{i = 1}^{2}\ddot{l}_i( U_{t-1},U_{t}, \theta )\Vert \Vert G_n - G^*\Vert _{\infty }. \end{aligned}$$

Since $\Vert {\hat{\theta }}-\theta ^*\Vert {\mathop {\rightarrow }\limits ^{pr}} 0$ and $\Vert G_n - G^*\Vert _{\infty } {\mathop {\rightarrow }\limits ^{pr}} 0$, as $n\rightarrow \infty$, we have

$$\begin{aligned} S_n= & {} -\frac{1}{n}\sum _{k = 1}^{n}\ddot{\ell }({\hat{U}}_{t-1}, {\hat{U}}_{t};{\hat{\theta }}){\mathop {\rightarrow }\limits ^{pr}}{\mathcal {S}}, \ \ \hbox {as} \ \ n\rightarrow \infty . \end{aligned}$$

Following similar arguments as above, we also can prove that

$$\begin{aligned} V_n= & {} \frac{1}{n}\sum _{k = 1}^{n}{\dot{\ell }}({\hat{U}}_{t-1}, {\hat{U}}_{t};{\hat{\theta }})\dot{l}^T({\hat{U}}_{t-1}, {\hat{U}}_{t};{\hat{\theta }})^{T}{\mathop {\rightarrow }\limits ^{pr}}{\mathcal {V}}, \ \ \hbox {as} \ \ n\rightarrow \infty . \end{aligned}$$

It follows from Assumption (A3) and Slutsky’s Theorem that

$$\begin{aligned} R = tr\left\{ S_n^{-1}V_n\right\} {\mathop {\rightarrow }\limits ^{pr}}tr\left\{ {\mathcal {S}}^{-1}{\mathcal {V}}\right\} , \ \ \hbox {as} \ \ n\rightarrow \infty . \end{aligned}$$

$\square$

To prove Theorem 2, we need the following Lemma 1, which gives the expansion of the estimated negative sensitivity matrix $S_n$ and estimated variability matrix $V_n$. Let $S_n({\hat{U}}_{t-1}, {\hat{U}}_{t}; {\hat{\theta }})_{ij}$ denote the (i, j)th element of $S_n$, and $V_n({\hat{U}}_{t-1}, {\hat{U}}_{t}; {\hat{\theta }})_{ij}$ denote the (i, j)th element of $V_n$.

Lemma 1: Under the conditions (A2)–(A3) and (B1)–(B3), we have

$$\begin{aligned} S_n({\hat{U}}_{t-1}, {\hat{U}}_{t}; {\hat{\theta }})_{ij}= & {} \frac{1}{n}\sum _{t=1}^{n} \left[ \ddot{\ell }( U_{t-1},U_{t}, \theta ^*)_{ij} + W_S( U_{t-1},U_{t})_{ij} \right] \nonumber \\&+\,\frac{1}{n}\sum _{t=1}^{n}C^1_{ij}{{\mathcal {S}}}^{-1}\left[ \dot{\ell }( U_{t-1},U_{t}, \theta ^*)+W( U_{t-1},U_{t})\right] + o_p(n^{-1/2})\nonumber \\\triangleq & {} \frac{1}{n}\sum _{t=1}^{n}h_S(U_{t-1},U_{t}, \theta ^*)_{ij} + o_p(n^{-1/2}), \end{aligned}$$

(8)

and

$$\begin{aligned} V_n({\hat{U}}_{t-1}, {\hat{U}}_{t}; {\hat{\theta }})_{ij}= & {} \frac{1}{n}\sum _{t=1}^{n} \left[ {\dot{\ell }}( U_{t-1},U_{t}, \theta ^*)_i{\dot{\ell }}( U_{t-1},U_{t}, \theta ^*)_j + W_V( U_{t-1},U_{t})_{ij}\right] \nonumber \\&+\,\frac{1}{n}\sum _{t=1}^{n}C^2_{ij}{{\mathcal {S}}}^{-1}\left[ \dot{\ell }( U_{t-1},U_{t}, \theta ^*)+W( U_{t-1},U_{t}))\right] + o_p(n^{-1/2})\nonumber \\\triangleq & {} \frac{1}{n}\sum _{t=1}^{n}h_V(U_{t-1},U_{t}, \theta ^*)_{ij} + o_p(n^{-1/2}), \end{aligned}$$

(9)

where $C^1_{ij} \triangleq \mathbb {E}_{0}\left[ \frac{\partial \ddot{\ell }\left\{ U_0,U_1 ; \theta ^{*}\right\} _{i j}}{\partial \theta ^{T}}\right]$, $C^2_{ij} \triangleq 2\mathbb {E}_{0}\left[ \dot{\ell }( U_{t-1},U_{t}, \theta ^*)_i\frac{\partial {\dot{\ell }}\left\{ U_0,U_1 ; \theta ^{*}\right\} _{j}}{\partial \theta ^{T}}\right]$, and

$$\begin{aligned} W( U_{t-1},U_{t})= & {} \int _{0}^{1}\int _{0}^{1}Z_t(v_1,v_2)c(v_1,v_2;\theta ^*)dv_1dv_2, \\ W_s( U_{t-1},U_{t})= & {} \int _{0}^{1}\int _{0}^{1}Z^s_t(v_1,v_2)c(v_1,v_2;\theta ^*)dv_1dv_2,\\ W_v( U_{t-1},U_{t})= & {} \int _{0}^{1}\int _{0}^{1}Z^v_t(v_1,v_2)c(v_1,v_2;\theta ^*)dv_1dv_2, \end{aligned}$$

with

$$\begin{aligned} Z_t(v_1,v_2)= & {} \left[ I\{U_{t-1}\le v_1\}-v_1\right] \dot{l}_1(v_1,v_2;\theta ^*)+\left[ I\{U_{t}\le v_2\}-v_2\right] \dot{l}_2(v_1,v_2;\theta ^*),\\ Z^s_t(v_1,v_2)= & {} \left[ I\{U_{t-1}\le v_1\}-v_1\right] \ddot{l}_1(v_1,v_2;\theta ^*)+\left[ I\{U_{t}\le v_2\}-v_2\right] \ddot{l}_2(v_1,v_2;\theta ^*),\\ Z^v_t(v_1,v_2)= & {} \left[ I\{U_{t-1}\le v_1\}-v_1\right] \dot{l}_1(v_1,v_2;\theta ^*)\dot{l}(v_1,v_2;\theta ^*)^T\\&+\,\left[ I\{U_{t}\le v_2\}-v_2\right] \dot{l}_2(v_1,v_2;\theta ^*)\dot{l}(v_1,v_2;\theta ^*)^T. \end{aligned}$$

Proof of Lemma 1

First, we prove Eq. (8). Denote the pathwise derivative of $\ddot{\ell }( G(Y_{t-1}), G(Y_{t}); \theta ^*)$ w.r.t. G in the direction of dG as $\nabla \ddot{\ell }( G(Y_{t-1}),G(Y_{t}); \theta ^*)[dG]$, where $dG = G-G^*$. Following similar arguments of Proposition 4.3 in Chen and Fan (2006b), we have

$$\begin{aligned} \nabla \ddot{\ell }( U_{t-1},U_{t}; \theta ^*)[dG]= & {} \frac{1}{n}\sum _{s=1}^{n}\left\{ \ddot{l}_1( U_{t-1},U_{t})[I\{U_{s-1}\le U_{t-1}\} -U_{t-1}]\right. \\&\left. +\,\ddot{l}_2( U_{t-1},U_{t})[I\{U_s\le U_{t}\} -U_{t}]\right\} . \end{aligned}$$

Expanding $\ddot{\ell }( U_{t-1},U_{t};\theta ^*)$ around $\theta ^*$ and $G^*$ in the direction dG, we obtain

$$\begin{aligned} S_n({\hat{U}}_{t-1}, {\hat{U}}_{t};{\hat{\theta }})_{ij}= & {} \frac{1}{n}\sum _{t = 1}^{n}\ddot{\ell }( U_{t-1},U_{t};\theta ^*)_{ij}+ \frac{1}{n}\sum _{t = 1}^{n}\frac{\partial }{\partial \theta ^{T}}\ddot{\ell }(U_{t-1},U_{t};\theta ^*)_{ij}\left( {\hat{\theta }}-\theta ^*\right) \\&+\,\frac{1}{n}\sum _{t = 1}^{n}\nabla \ddot{\ell }( U_{t-1},U_{t})[dG]_{ij} + o_p(n^{-1/2}). \end{aligned}$$

By Proposition 4.3 in Chen and Fan (2006b), we have

$$\begin{aligned} {\hat{\theta }} - \theta ^* = \frac{1}{n}\sum _{t=1}^{n}{\mathcal S}^{-1}\left[ {\dot{\ell }}( U_{t-1},U_{t};\theta ^*)+ W( U_{t-1},U_{t})\right] + o_p(n^{-1/2}). \end{aligned}$$

By condition (B3), employing similar arguments in Theorem 1, we have

$$\begin{aligned} \frac{1}{n}\sum _{t = 1}^{n}\frac{\partial }{\partial \theta }\ddot{\ell }(U_{t-1},U_{t};\theta )\rightarrow C^1_{ij},\ \ \ \hbox {as}\ \ n\rightarrow \infty . \end{aligned}$$

By proposition 4.3 in Chen and Fan (2006b), we obtain

$$\begin{aligned} \nabla \ddot{\ell }( U_{t-1},U_{t}; \theta ^*)[dG]= & {} \int _0^{1}\int _0^1\left\{ \ddot{l}_1(v_1,v_2)[I\{U_{t-1}\le v_1\} -v_1]\right\} c(v_1,v_2;\theta ^*)dv_1dv_2 \\&+\,\int _0^{1}\int _0^1\left\{ \ddot{l}_2(v_1,v_2)[I\{ U_{t-1},U_{t}\le v_2\} -v_2]\right\} \\&\quad \times \,c(v_1,v_2;\theta ^*)dv_1dv_2 + o_p(n^{-1/2}). \end{aligned}$$

Hence, we have

$$\begin{aligned} S_n({\hat{U}}_{t-1}, {\hat{U}}_{t}; {\hat{\theta }})_{ij} = \frac{1}{n}\sum _{t=1}^{n}h_S(U_{t-1},U_{t}, \theta ^*)_{ij} + o_p(n^{-1/2}). \end{aligned}$$

Applying similar arguments as above, we can prove

$$\begin{aligned} V_n({\hat{U}}_{t-1}, {\hat{U}}_{t}; {\hat{\theta }})_{ij} = \frac{1}{n}\sum _{t=1}^{n}h_S(U_{t-1},U_{t}, \theta ^*)_{ij} + o_p(n^{-1/2}). \end{aligned}$$

$\square$

Proof of Theorem 2

Under the null hypothesis of the copula model being correctly specified, the Bartlett identity holds, hence, we have ${{\mathcal {S}}} = {{\mathcal {V}}}$, i.e. ${{\mathcal {S}}}^{-1}{{\mathcal {V}}} =I_p$. The test statistic R given in Eq. (7) can be represented as follows:

$$\begin{aligned} \sqrt{n}\left( R-p\right)= & {} \sqrt{n}\hbox {tr}\left\{ S_n^{-1}V_n - I_p\right\} =\sqrt{n}\hbox {tr} \left\{ S_n^{-1}V_n - {\mathcal S}^{-1}{{\mathcal {V}}}\right\} \\= & {} \hbox {tr}\left\{ {{\mathcal {S}}}^{-1}\sqrt{n}\left( V_n - {\mathcal V}\right) \right\} + \hbox {tr}\left\{ {{\mathcal {S}}}^{-1}V_n{\mathcal S}^{-1}\sqrt{n}\left( {{\mathcal {S}}}-S_n\right) \right\} + o_p(1). \end{aligned}$$

By Lemma 1, we have

$$\begin{aligned} \sqrt{n}\left\{ S_n - {{\mathcal {S}}}\right\}= & {} \frac{1}{\sqrt{n}}\sum \limits _{t = 1}^{n}\left\{ h_S( U_{t-1},U_{t};\theta ^*) - {{\mathcal {S}}}\right\} +o_p(1) \end{aligned}$$

and

$$\begin{aligned} \sqrt{n}\left( V_n - {{\mathcal {V}}}\right)= & {} \frac{1}{\sqrt{n}}\sum \limits _{t = 1}^{n}\left\{ h_V( U_{t-1},U_{t};\theta ^*)-{{\mathcal {V}}}\right\} +o_p(1). \end{aligned}$$

Thus, we can write $\sqrt{n}\left( R-p\right) = \frac{1}{\sqrt{n}}\sum \limits _{t = 1}^{n} h_R(U_{t-1},U_{t};\theta ^*)$, where

$$\begin{aligned} h_R(U_{t-1},U_{t};\theta ^*)= & {} \hbox {tr}\left\{ {\mathcal S}^{-1}\left[ h_V( U_{t-1},U_{t};\theta ^*)-{\mathcal V}\right] \right\} \\&-\,\hbox {tr}\left\{ {{\mathcal {S}}}^{-1}V_n{\mathcal S}^{-1}\left[ h_S( U_{t-1},U_{t};\theta ^*) - {\mathcal S}\right] \right\} . \end{aligned}$$

Hence, by condition (B3) and central limit theorem for time series, we obtain

$$\begin{aligned} \sqrt{n}\left( R- p\right) {\mathop {\rightarrow }\limits ^{d}} N(0,\sigma _R^2), \end{aligned}$$

where asymptotic variance $\sigma _R^2 = Var\left[ h_R(U_{t-1},U_{t};\theta ^*)\right]$, which can be computed consistently via the parametric bootstrap method in Sect. 4.2. $\square$

To prove Theorem 3, we need the following Lemma 2, which states the limiting difference between the parameter estimates ${\hat{\theta }}$ and ${\hat{\theta }}_{-t}$ or $\tilde{\theta }_{-t}$.

Lemma 2 : Under the conditions (B3), we have

$$\begin{aligned}&\sup _{1\le t\le n}\Vert {\hat{\theta }}-{\hat{\theta }}_{-t}\Vert = O_p(n^{-1})\quad \text {and} \end{aligned}$$

(10)

$$\begin{aligned}&\sup _{1\le t\le n}\Vert {\hat{\theta }}-{\tilde{\theta }}_{-t}\Vert = O_p(n^{-1}). \end{aligned}$$

(11)

Proof of Lemma 2

First, we prove Eq. (10). By Eq. (3), ${\hat{\theta }}_{-t}$ solves the following equation

$$\begin{aligned} 0 = \sum _{s\ne t, s=1}^{n}{\dot{\ell }}({\hat{U}}_{s-1},{\hat{U}}_{s}; {\hat{\theta }}_{-t}). \end{aligned}$$

Expanding ${\dot{\ell }}({\hat{U}}_{s-1},{\hat{U}}_{s};{\hat{\theta }}_{-t})$ around ${\hat{\theta }}$ leads to

$$\begin{aligned} 0 = \sum _{s\ne t, s=1}^{n}{\dot{\ell }}({\hat{U}}_{s-1},{\hat{U}}_{s};{\hat{\theta }}_{-t}) = -{\dot{\ell }}({\hat{U}}_{t-1},{\hat{U}}_{t};{\hat{\theta }}) + \sum _{s\ne t, s=1}^{n}\ddot{\ell }({\hat{U}}_{s-1},{\hat{U}}_{s};{\check{\theta }}_{-t})({\hat{\theta }}_{-t} - {\hat{\theta }}), \end{aligned}$$

where ${\check{\theta }}_{-t}$ lies between ${\hat{\theta }}$ and ${\hat{\theta }}_{-t}$. It follows that

$$\begin{aligned} {\hat{\theta }}_{-t} - {\hat{\theta }} = \left\{ \frac{1}{n}\sum _{s\ne t, s=1}^{n}\ddot{l}\left( {\hat{U}}_{s-1},{\hat{U}}_{s};{\tilde{\theta }}_{-t}\right) \right\} ^{-1} \frac{1}{n}\dot{\ell }({\hat{U}}_{t-1},{\hat{U}}_{t};{\hat{\theta }}). \end{aligned}$$

By conditions (B3),

$$\begin{aligned} \sup _{1\le t\le n}\frac{1}{n}\left\| {\dot{\ell }}({\hat{U}}_{t-1}, {\hat{U}}_{t};{\hat{\theta }})\right\| \le \frac{1}{n} \sup _{1\le t\le n}\sup _{(\theta ,G)\in {{\mathcal {F}}}_{\delta }}\left\| {\dot{\ell }}( U_{t-1},U_{t};\theta )\right\| =O_p(n^{-1}). \end{aligned}$$

In addition, by condition (B3), using the similar arguments in the proof of Theorem 1, we can show

$$\begin{aligned} \frac{1}{n}\sum _{s\ne t, s=1}^{n}\ddot{\ell }({\hat{U}}_{s-1},{\hat{U}}_{s};{\check{\theta }}_{-t}) {\mathop {\rightarrow }\limits ^{pr}} {{\mathcal {S}}}. \end{aligned}$$

Hence,

$$\begin{aligned} \sup _{1\le t\le n}\Vert {\hat{\theta }}_{-t} - {\hat{\theta }} \Vert\le & {} \sup _{1\le t\le n}\left\| \left\{ \frac{1}{n}\sum _{s\ne t, s=1}^{n}\ddot{\ell }({\hat{U}}_{s-1},{\hat{U}}_{s};{\check{\theta }}_{-t}) \right\} ^{-1}\right\| \\&\quad \times \,\sup _{1\le t\le n}\left\| \frac{1}{n}\sum _{t = 1}^{n}{\dot{\ell }}({\hat{U}}_{t-1},{\hat{U}}_{t};{\hat{\theta }})\right\| = O_p(n^{-1}). \end{aligned}$$

To approximate ${\hat{\theta }}_{-t}$, shown in Eq. (5), we use ${\tilde{\theta }}_{-t}$ which is defined as

$$\begin{aligned} {\tilde{\theta }}_{-t} = {\hat{\theta }}+\left\{ \frac{1}{n}\sum _{s = 1}^{n}\ddot{l}\left( {\hat{U}}_{s-1},{\hat{U}}_{s};{\hat{\theta }}\right) \right\} ^{-1} \frac{1}{n}{\dot{\ell }}({\hat{U}}_{t-1},{\hat{U}}_{t};{\hat{\theta }}). \end{aligned}$$

By the similar arguments above, we also can show that

$$\begin{aligned} \sup _{1\le t\le n}\left\| {\tilde{\theta }}_{-t} - {\hat{\theta }}\right\| = O_p(n^{-1}). \end{aligned}$$

$\square$

Proof of Theorem 3

By definition, $T=\sum _{t = 1}^{n}\ell ({\hat{U}}_{t-1},{\hat{U}}_{t}; {\hat{\theta }}) - \sum _{t = 1}^{n}\ell ({\hat{U}}_{t-1},{\hat{U}}_{t}; {\hat{\theta }}_{-t})$. Expanding $\ell ({\hat{U}}_{t-1},{\hat{U}}_{t}; {\hat{\theta }}_{-t})$ around ${\hat{\theta }}$ leads to

$$\begin{aligned} T= & {} -\sum _{t = 1}^{n}{\dot{\ell }}({\hat{U}}_{t-1},{\hat{U}}_{t}; {\hat{\theta }})^T({\hat{\theta }} - {\hat{\theta }}_{-t} ) - \frac{1}{2} \sum _{t = 1}^{n}({\hat{\theta }} - {\hat{\theta }}_{-t} )^{T}\ddot{\ell }({\hat{U}}_{t-1},{\hat{U}}_{t}; {\check{\theta }}_{-t})({\hat{\theta }} - {\hat{\theta }}_{-t} )\\= & {} -\sum _{t = 1}^{n}{\dot{\ell }}({\hat{U}}_{t-1},{\hat{U}}_{t}; {\hat{\theta }})^T \left\{ \frac{1}{n}\sum _{s= 1}^{n}\ddot{\ell }({\hat{U}}_{s-1},{\hat{U}}_{s};{\hat{\theta }})\right\} ^{-1} \frac{1}{n} {\dot{\ell }}({\hat{U}}_{t-1},{\hat{U}}_{t};{\hat{\theta }}) \\&-\,\frac{1}{2} \sum _{t = 1}^{n}({\hat{\theta }} - {\hat{\theta }}_{-t} )^{T}\ddot{\ell }({\hat{U}}_{t-1},{\hat{U}}_{t}; {\check{\theta }}_{-t})({\hat{\theta }} - {\hat{\theta }}_{-t} )\\= & {} R -\frac{1}{2} \sum _{t = 1}^{n}({\hat{\theta }} - {\hat{\theta }}_{-t} )^{T}\ddot{\ell }({\hat{U}}_{t-1},{\hat{U}}_{t}; {\check{\theta }}_{-t})({\hat{\theta }} - {\hat{\theta }}_{-t} ), \end{aligned}$$

where ${\check{\theta }}_{-t}$ lies between $\theta$ and ${\hat{\theta }}_{-t}$. Lemma 1 implies that

$$\begin{aligned}&\sup _{1\le t\le n}\Vert \sum _{t = 1}^{n}({\hat{\theta }} - {\hat{\theta }}_{-t} )^{T}{\dot{\ell }}({\hat{U}}_{t-1},{\hat{U}}_{t}; {\check{\theta }}_{-t})({\hat{\theta }} - {\hat{\theta }}_{-t} )\Vert \\&\quad \le n \times \sup _{1\le t\le n}\Vert \theta -{\hat{\theta }}_{-t}\Vert ^2 \sup _{1\le t\le n}\sup _{\theta \in \Theta }\Vert {\dot{\ell }}({\hat{U}}_{t-1},{\hat{U}}_{t}; \theta )\Vert = O_p(n^{-1}). \end{aligned}$$

In summary, we prove that $T - R = O_p(n^{-1})$. Similarly, we can prove ${\tilde{T}} - R = O_p(n^{-1})$. $\square$

1.2 Appendix II: Copula families

For the sake of self-containedness, below we present some brief descriptions of the four copula families: Clayton, Frank, Gaussian and Student’s t, considered in the simulation studies. Figure S11 shows the scatter plots of 1000 observations generated from the four copula families with Kendall’s $\tau = 0.5$. In addition, we provide the explicit expression of $C_{2 | 1}^{-1}\left( x | u_1, \theta \right)$, which is used to generate the pseudo-observations of a univariate time series from the assumed copula. Note that this function is the inverse of the function $C_{2 | 1}\left( u_2 | u_1, \theta \right)$ described in Sect. 4.1, which is the conditional cdf of $U_{t}$ given $U_{t-1}=u$.

• 1. Clayton copula (Clayton 1978)

  $$\begin{aligned} C(u_1,u_2; \theta ) = \left( u_1^{-\theta }+u_2^{-\theta }-1\right) ^{-1/\theta }, \end{aligned}$$

  where the dependence coefficient $\theta \ge 0$, which corresponds to Kendall’s $\tau = \theta /(\theta +2)$. This copula represents lower tail dependency and is of particular interest in modeling the value-at-risk data. In addition, $C_{2 | 1}^{-1}\left( x | u_{1} ; \theta \right) =\left[ \left( x^{-\theta /(1+\theta )}-1\right) u_{1}^{-\theta }+1\right] ^{-1 / \theta }$.

• 2. Frank copula (Genest 1987)

  $$\begin{aligned} C(u_1,u_2; \theta ) = -\theta ^{-1}\log \left( 1+ \frac{[e^{-\theta u_1}-1][e^{-\theta u_2}-1]}{e^{-\theta }-1}\right) , \end{aligned}$$

  (12)

  where $\theta \ne 0$ and the corresponding Kendall’s tau is $\tau = 1-4(D(\theta ) -1)/\theta$, $D(\theta ) = \frac{1}{\theta }\int _{0}^{\theta }\frac{t}{e^t-1}dt$ for $\theta >0$ and $D(\theta ) = \frac{1}{\theta }\int _{0}^{\theta }\frac{t}{e^t-1}dt + 0.5\theta$ for $\theta <0$. In addition, $C_{2 | 1}^{-1}\left( x | u_{1} ; \theta \right) = - \theta \log \left[ 1-\frac{1-e^{-\theta }}{1+e^{-\theta u_1}(1/x-1)}\right]$.

• 3. The bivariate Gaussian copula is defined as

  $$\begin{aligned} C(u_1,u_2;\theta ) = \Phi _2(\Phi ^{-1}(u_1), \Phi ^{-1}(u_2); \rho ), \end{aligned}$$

  where $\Phi (x)$ is the standard univariate Gaussian distribution, $\Phi _2(x_1,x_2; \rho )$ is the bivariate Gaussian distribution with zeros mean and unit variance, and $\rho \in [-1,1]$ is the correlation coefficient. Here, the copula parameter $\theta =\rho$, corresponding to Kendall’s $\tau = \frac{2}{\pi }\arcsin (\theta )$. In addition, $C_{2 | 1}^{-1}\left( x | u_{1} ; \theta \right) = \Phi \left( \theta \Phi ^{-1}(u_1)+\sqrt{1-\theta ^2}x\right)$.

• 4. The Student’s t copula is a two-parameter copula, which is given by

  $$\begin{aligned} C(u_1,u_2; \nu ; \theta )= & {} \int _{-\infty }^{t_{\nu }^{-1}(u_1)}\int _{-\infty }^{t_{\nu }^{-1}(u_2)}\frac{1}{\sqrt{2\pi (1-\rho ^2)^{1/2}}}\\&\times \,\left( 1+\frac{x^2-2\rho xy + y^2}{\nu (1-\rho ^2)}\right) ^{-\nu /2-1}dxdy, \end{aligned}$$

  where the copula parameter $\theta =(\rho ,\nu )$ with the correlation coefficient $\rho$ and the d.f. $\nu$. The Kendall’s $\tau = \frac{2}{\pi }\arcsin (\rho )$. Student’s t copula has received increasing attention in the modeling of multivariate financial time series. Many empirical studies have shown that the Student’s t copulas fit better than Gaussian copulas since the former better captures the phenomenon of dependent extreme values, which is often observed in the financial asset return data (Breymann et al. 2003; Demarta and McNeil 2005). In addition, $C_{2 | 1}^{-1}\left( x | u_{1} ; \theta \right) = F_{t_{\nu }}(\rho F^{-1}_{t_\nu }(u_1)+\sigma x)$, where $F_{t_v}$ is the cdf of a t random variable with d.f. $\nu$, and $\sigma =\sqrt{\left[ \nu +F^{-1}_{t_\nu }(u_1)^2\right] (1-\rho ^2)/(\nu +1)}$.