Copula and Markov Models

Abstract

This chapter introduces the basic concepts on copulas and Markov models. We review the formal definition of copulas with its fundamental properties. We then introduce Kendall’s tau as a measure of dependence structure for a pair of random variables, and its relationship with a copula. Examples of copulas are reviewed, such as the Clayton copula, the Gaussian copula, the Frank copula, and the Joe copula. Finally, we introduce the copula-based Markov chain time series models and their fundamental properties.

Appendices

Appendix A: The Proof of $\overline{C}\left({u,v} \right)\; = \;u\; + \;v\; - \;1\; + \;C\left({1\; - \;u,\;1\; - \;v} \right)$ being a Copula

We verify the conditions (C1) and (C2) for $\bar{C}(u,v).$ Recall that $C(u,v)$ is a copula by assumption. The condition (C1) holds since

$$ \bar{C}(u,0) = u + 0 - 1 + C(1 - u,1) = u - 1 + 1 - u = 0, $$

$$ \bar{C}(0,v) = 0 + v - 1 + C(1,1 - v) = v - 1 + 1 - v = 0, $$

$$ \bar{C}(u,1) = u + 1 - 1 + C(1 - u,0) = u + 0 = u, $$

$$ \bar{C}(1,v) = 1 + v - 1 + C(0,1 - v) = v + 0 = v. $$

The condition (C2) holds since

$$ \begin{aligned} & \bar{C}(u_{2},v_{2} ) - \bar{C}(u_{2},v_{1} ) - \bar{C}(u_{1},v_{2} ) + \bar{C}(u_{1},v_{1} ) \\ & = u_{2} + v_{2} - 1 + C(1 - u_{2},1 - v_{2} ) - [u_{2} + v_{1} - 1 + C(1 - u_{2},1 - v_{1} )] \\ & \,\,\,\,\,- [u_{1} + v_{2} - 1 + C(1 - u_{1},1 - v_{2} )] + u_{1} + v_{1} - 1 + C(1 - u_{1},1 - v_{1} ) \\ & = C(1 - u_{1},1 - v_{1} ) - C(1 - u_{2},1 - v_{1} ) - C(1 - u_{1},1 - v_{2} ) + C(1 - u_{2},1 - v_{2} ) \\ & = C(\bar{u}_{1},\bar{v}_{1} ) - C(\bar{u}_{2},\bar{v}_{1} ) - C(\bar{u}_{1},\bar{v}_{2} ) + C(\bar{u}_{2},\bar{v}_{2} ) \ge 0 \\ \end{aligned} $$

Here, $\bar{u}_{i} \equiv 1 - u_{i}$ and $\bar{v}_{i} \equiv 1 - v_{i}$ for $i =$ 1 and 2. ∎

Appendix B: Proofs of Copulas Approaching to the Independence

We first consider the Clayton copula. As $\alpha \to 0,$

$$ \begin{aligned} \mathop {\lim }\limits_{{\alpha \to 0}} C_{\alpha } (u,\;v) = & \mathop {\lim }\limits_{{\alpha \to 0}} \exp \{\log (u^{{ - \alpha }} + v^{{ - \alpha }} - 1)^{{ - 1/\alpha }} \} \; \\ & = \exp [\mathop {\lim }\limits_{{\alpha \to 0}} \{ - \log (u^{{ - \alpha }} + v^{{ - \alpha }} - 1)/\alpha \} ]\; \\ & = \exp [\mathop {\lim }\limits_{{\alpha \to 0}} \{ - (u^{{ - \alpha }} \log u + v^{{ - \alpha }} \log v)/(u^{{ - \alpha }} + v^{{ - \alpha }} - 1)\} ]\;\;\;\;\;\;\;\;\;\;({\text{L'Hopital's rule}})\; \\ & = \exp [\log u + \log v]\; \\ & = uv. \\ \end{aligned} $$

The last expression is the independence copula. An alternative proof is to consider the generator $\phi_{\alpha } (t) = (t^{ - \alpha } - 1)/\alpha$ for $\alpha > 0.$ It follows that

$$ \mathop {\lim }\limits_{\alpha \to 0} \phi_{\alpha } (t) = \mathop {\lim }\limits_{\alpha \to 0} \frac{{t^{ - \alpha } - t^{ - 0} }}{\alpha } = \frac{d}{d\alpha }\left. {t^{ - \alpha } } \right|_{\alpha = 0} = - \log (t). $$

The last expression is the generator of the independence copula. Similarly, the Frank copula reduces to the independence copula because

$$ \mathop {\lim }\limits_{\alpha \to 0} \phi_{\alpha } (t) = \mathop {\lim }\limits_{\alpha \to 0} \left[ { - \log \left({\frac{{e^{ - \alpha t} - 1}}{{e^{ - \alpha } - 1}}} \right)} \right] = \mathop {\lim }\limits_{\alpha \to 0} \left[ { - \log \left({\frac{{te^{ - \alpha t} }}{{e^{ - \alpha } }}} \right)} \right] = - \log (t). $$

∎

Appendix C: Derivations of Kendall’s Tau

Kendall’s tau for the Clayton copula:

$$ \begin{aligned} \tau_{\theta } \,=\, & 1 - 4\int_{0}^{\infty } {s\left\{ {\,\frac{d}{ds}\left({\,1 + \alpha s\,} \right)^{{ - \frac{1}{\alpha }}} \,} \right\}^{2} ds} = 1 - 4\int_{0}^{\infty } {s\left\{ {{\kern 1pt} - \left({{\kern 1pt} 1 + \alpha s} \right)^{{ - \frac{1}{\alpha } - 1}} {\kern 1pt} } \right\}^{2} ds} \\ = & 1 - 4\int_{0}^{\infty } {s\left({\,1 + \alpha s\,} \right)^{{ - \frac{2}{\alpha } - 2}} ds} = 1 - 4\frac{1}{{\alpha^{2} }}\int_{1}^{\infty } {\left({\,t - 1\,} \right)t^{{ - \frac{2}{\alpha } - 2}} dt} \\ = & 1 - 4\frac{1}{{\alpha^{2} }}\left\{ {{\kern 1pt} \left. { - \frac{\alpha }{2}t^{{ - \frac{2}{\theta }}} } \right|_{1}^{\infty } + \left. {\frac{\alpha }{2 + \alpha }t^{{ - \frac{2}{\theta } - 1}} } \right|_{1}^{\infty } {\kern 1pt} } \right\} \\ = & \frac{\alpha }{\alpha + 2} \\ \end{aligned} $$

Kendall’s tau for the Joe copula:

$$ \begin{aligned} \tau _{\alpha } \,=\, & 1 - 4\int_{0}^{\infty } {s\left[ {\frac{\partial }{{\partial s}}\left\{ {1 - (1 - e^{{ - s}} )^{{\frac{1}{\alpha }}} } \right\}} \right]^{2} ds} \\ = & 1 - 4\int_{0}^{\infty } {s\left\{ {{\kern 1pt} - \frac{1}{\alpha }\left({{\kern 1pt} 1 - e^{{ - s}} {\kern 1pt} } \right)^{{\frac{1}{\alpha } - 1}} e^{{ - s}} {\kern 1pt} } \right\}^{2} ds} \\ = & 1 - \frac{4}{{\alpha ^{2} }}\int_{0}^{\infty } {s(1 - e^{{ - s}} )^{{\frac{2}{\alpha } - 2}} e^{{ - 2s}} ds}. \\ \end{aligned} $$

Kendall’s tau for the Frank copula:

Kendall’s tau can be derived by

$$ \begin{aligned} \tau \,=\, & 4\int_{0}^{1} {\int_{0}^{1} {C(u,\;v)dC(u,\;v)} } - 1 \\ = \,& 4\int_{0}^{1} {\int_{0}^{1} {C(u,\;v)C^{[1,1]} (u,\;v)dudv} } - 1 \\ =\, & 4\int_{0}^{1} {\left\{ {C(u,\;v)\left. {C^{[0,1]} (u,\;v)} \right|_{u = 0}^{u = 1} - \int_{0}^{1} {C^{[1,0]} (u,\;v)C^{[0,1]} (u,\;v)du} } \right\}dv} - 1 \\ =\, & 4\int_{0}^{1} {\left\{ {v - \int_{0}^{1} {C^{[1,0]} (u,\;v)C^{[0,1]} (u,\;v)du} } \right\}dv} - 1 \\ =\, & 2 - 4\int_{0}^{1} {\int_{0}^{1} {C^{[1,0]} (u,\;v)C^{[0,1]} (u,\;v)dudv} } - 1 \\ =\, & 1 - 4\int_{0}^{1} {\int_{0}^{1} {C^{[1,0]} (u,\;v)C^{[0,1]} (u,\;v)dudv} } \\ \end{aligned} $$

after integration by parts. According to Nelsen (1986), under the Frank copula,

$$ C_{\alpha }^{[1,\;0]} (u,\;v) = \frac{{e^{ - \alpha u} (e^{ - \alpha v} - 1)}}{{e^{ - \alpha } - 1 + (e^{ - \alpha u} - 1)(e^{ - \alpha v} - 1)}} $$

and similarly

$$ C_{\alpha }^{[0,\;1]} (u,\;v) = \frac{{e^{ - \alpha v} (e^{ - \alpha u} - 1)}}{{e^{ - \alpha } - 1 + (e^{ - \alpha u} - 1)(e^{ - \alpha v} - 1)}}. $$

Thus,

$$ \begin{aligned} \tau & = 1 - 4\int_{0}^{1} {\int_{0}^{1} {\frac{{e^{ - \alpha u} e^{ - \alpha v} (e^{ - \alpha u} - 1)(e^{ - \alpha v} - 1)}}{{\{ e^{ - \alpha } - 1 + (e^{ - \alpha u} - 1)(e^{ - \alpha v} - 1)\}^{2} }}dudv} } \\ & = 1 - \frac{4}{\alpha }\left({1 - \frac{1}{\alpha }\int_{0}^{\alpha } {\frac{t}{{e^{t} - 1}}dt} } \right) \\ \end{aligned}. $$

Nelsen (1986) provided the above derivation, but the derivation of the last equality is unclear to us. To prove the last equality, we give a method of numerical verification by R codes.

No matter how we choose the value of “alpha,” the two numerical integrations give the identical value. ∎

Appendix D: Derivation of $C_{\alpha }^{{\left[ {1,1} \right]}} \;\left({u,v} \right)$ Under the Frank Copula

$$ \Pr (V \le v|U = u) = C_{\alpha }^{[1,\;0]} (u,\;v) = \frac{{e^{ - \alpha u} (e^{ - \alpha v} - 1)}}{{e^{ - \alpha } - 1 + (e^{ - \alpha u} - 1)(e^{ - \alpha v} - 1)}} $$

$$ \begin{aligned} & C_{\alpha }^{{[1,1]}} (u,\,v)\, = \,\frac{\partial }{{\partial v}}C_{\alpha }^{{[1,0]}} (u,\,v) \\ & = \,\frac{{ \frac{\partial }{{\partial v}}\{ e^{{ - \alpha u}} (e^{{ - \alpha v}} \, - \,1)\} \{ e^{{ - \alpha }} \, - \,1\, + \,(e^{{ - \alpha u}} \, - \,1)\,e^{{ - \alpha v}} \, - \,1)\} }} {{\{ e^{{ - \alpha }} \, - \,1\, + \,(e^{{ - \alpha u}} \, - \,1)(e^{{ - \alpha v}} \, - \,1)\} ^{2} }} \\ & \quad\quad \, - \, \frac{{ \{ e^{{ - \alpha u}} (e^{{ - \alpha v}} \, - \,1)\} \frac{\partial }{{\partial v}}\{ e^{{ - \alpha }} \, - \,1\, + \,(e^{{ - \alpha u}} \, - \,1)(e^{{ - \alpha v}} \, - \,1)\} }} {{\{ e^{{ - \alpha }} \, - \,1\, + \,(e^{{ - \alpha u}} \, - \,1)(e^{{ - \alpha v}} \, - \,1)\} ^{2} }} \\ & = \,\frac{{ - \alpha e^{{ - \alpha u}} e^{{ - \alpha v}} \{ e^{{ - \alpha }} \, - \,1\, + \,(e^{{ - \alpha u}} \, - \,1)(e^{{ - \alpha v}} \, - \,1)\} \, + \,\alpha e^{{ - \alpha u}} e^{{ - \alpha v}} (e^{{ - \alpha u}} \, - \,1)(e^{{ - \alpha v}} \, - \,1)}}{{\{ e^{{ - \alpha }} \, - \,1\, + \,(e^{{ - \alpha u}} \, - \,1)(e^{{ - \alpha v}} \, - \,1)\} ^{2} }} \\ & = \,\frac{{\alpha (1\, - \,e^{{ - \alpha }} )e^{{ - \alpha u}} e^{{ - \alpha v}} }}{{\{ e^{{ - \alpha }} \, - \,1\, + \,(e^{{ - \alpha u}} \, - \,1)(e^{{ - \alpha v}} \, - \,1)\} ^{2} }} \\ \end{aligned} $$

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Appendix E: Derivation of $C_{\rho }^{{[1,0]}} \;(u,\;v)$ Under the Gaussian Copula

Let $X \equiv \varPhi^{ - 1} (U)$ and $Y \equiv \varPhi^{ - 1} (V).$ Then, $X$ and $Y$ jointly follow a bivariate normal distribution with $E[X] = E[Y] = 0,$ $Var[X] = Var[Y] = 1,$ and $Cov(X,Y) = \rho \in (- 1,1).$ By the property of the bivariate normal distribution, $Y|X = x\sim N(\rho x,\;1 - \rho^{2} ).$ Thus,

$$ \Pr (Y \le y|X = x) = \int\limits_{0}^{y} {\frac{1}{{\sqrt {2\pi (1 - \rho^{2} )} }}\exp \left[ { - \frac{{(t - \rho x)^{2} }}{{2(1 - \rho^{2} )}}} \right]} dt. $$

The expression can be re-expressed as

$$ \Pr (V \le v|U = u) = \int\limits_{0}^{{\frac{{\varPhi^{ - 1} (v) - \rho \varPhi^{ - 1} (u)}}{{\sqrt {1 - \rho^{2} } }}}} {\frac{1}{{\sqrt {2\pi } }}\exp \left[ { - \frac{{z^{2} }}{2}} \right]} dz = \varPhi \left[ {\frac{{\varPhi^{ - 1} (v) - \rho \varPhi^{ - 1} (u)}}{{\sqrt {1 - \rho^{2} } }}} \right], $$

where a transformation $z = \frac{t - \rho x}{{\sqrt {1 - \rho^{2} } }}$ is applied.∎