Derivatives and Fisher information of bivariate copulas

Abstract

Data sets with complex relationships between random variables are increasingly studied in statistical applications. A popular approach to model their dependence is the use of copula functions. Our contribution is to derive expressions for the observed and expected information for several bivariate copula families, in particular for the Student’s $t$-copula. Further likelihood derivatives which are required for numerical implementations are computed and a numerically stable implementation is provided in the R-package VineCopula. Using a real world data set of stock returns, we demonstrate the applicability of our approach for the routinely calculation of standard errors. In particular, we illustrate how this prevents overestimating the time-variation of dependence parameters in a rolling window analysis.

Proofs

In this section we give proofs for Theorem 1 and Corollary 1 in Sect. 3.1.

proof

(Theorem 1) This proof is based on an earlier attempt by Nadarajah (2006). (1) Fisher information with respect to $\rho$: For the transformed variables $U$ and $V$, corresponding to $\rho = 0$,

$$\begin{aligned} \begin{pmatrix}X \\ Y\\ \end{pmatrix} = \frac{1}{2}\begin{pmatrix} \sqrt{1+\rho }+\sqrt{1-\rho }&\sqrt{1+\rho }-\sqrt{1-\rho } \\ \sqrt{1+\rho }-\sqrt{1-\rho }&\sqrt{1+\rho }+\sqrt{1-\rho } \\ \end{pmatrix}\begin{pmatrix}U \\ V \\ \end{pmatrix}, \end{aligned}$$

the moments are given as

$$\begin{aligned} E_N[U^pV^q] \!=\! \frac{m^{(p+q)/2}(N\!-\!1)}{\pi } B\left(\frac{p+q}{2}\!+\!1,N\!-\!\frac{p+q}{2}\!-\!1\right)B\left(\frac{p\!+\!1}{2},\frac{q\!+\!1}{2}\right),\nonumber \\ \end{aligned}$$

(1)

if both $p$ and $q$ are even integers (Nadarajah 2006). If either $p$ or $q$ is odd, Expression (1) is equal to zero.

For the calculation of the Fisher Information of the Pearson VII distribution with respect to $\rho$ we need the second partial derivative of the log-density with respect to $\rho$.

$$\begin{aligned} \frac{\partial \log L}{\partial \rho }&= \frac{\rho }{1-\rho ^2} - \frac{N}{\left(1+\frac{x^2+y^2-2\rho xy}{m(1-\rho ^2)}\right)} \cdot \frac{-2xy(1-\rho ^2) + 2\rho (x^2+y^2-2\rho xy)}{m(1-\rho ^2)^2},\\ \frac{\partial ^2 \log L}{\partial ^2 \rho }&= \frac{1+\rho ^2}{(1-\rho ^2)^2} - \left(1+\frac{x^2+y^2-2\rho xy}{m(1-\rho ^2)}\right)^{-1} \\&\cdot \left[ \frac{2N(x^2+y^2-2\rho xy)}{m(1-\rho ^2)^2} + \frac{8N(\rho x^2 + \rho y^2 -(1-\rho ^2)xy) \rho }{m(1-\rho ^2)^3} \right] \\&+ \left(1+\frac{x^2+y^2-2\rho xy}{m(1-\rho ^2)}\right)^{-2} \frac{4N (\rho x^2 + \rho y^2 -(1-\rho ^2)xy)^2}{m^2(1-\rho ^2)^4} \end{aligned}$$

Using the equations in the Appendix of Nadarajah (2006) for $E[X^pY^q]$ in terms of $E[U^pV^q]$ and Expression (1), we can determine the Fisher Information of Pearson VII distribution with respect to $\rho$:

$$\begin{aligned} \mathcal{I }_{\rho }&= E_N\left[ -\frac{\partial ^2 \log L}{\partial ^2 \rho } \right] = \frac{1+\rho ^2}{(1-\rho ^2)^2} - E_N\left[ \left(1+\frac{x^2+y^2-2\rho xy}{m(1-\rho ^2)}\right)^{-1} \right. \\&\cdot \frac{2N ( (x^2+y^2-2\rho xy)(1-\rho ^2) + 4\rho (\rho x^2+\rho y^2-(1+\rho ^2)xy)}{m(1-\rho ^2)^3} \\&+ \left. \left(1+\frac{x^2+y^2-2\rho xy}{m(1-\rho ^2)}\right)^{-2} \frac{4N\left(\rho x^2 + \rho y^2 - (1-\rho ^2)xy\right)}{m^2(1-\rho ^2)^4} \right] \\&= \frac{1+\rho ^2}{(1-\rho ^2)^2} + E_{N+2}\left[ \frac{4N(N-1)\left(\rho x^2 + \rho y^2 - (1-\rho ^2)xy\right)}{(N+1)m^2(1-\rho ^2)^4} \right] \\&- E_{N+1}\left[ \frac{2(N\!-\!1) ( (x^2\!+\!y^2\!-\!2\rho xy)(1\!-\!\rho ^2) \!+\! 4\rho (\rho x^2\!+\!\rho y^2\!-\!(1\!+\rho ^2)xy)}{m(1-\rho ^2)^3} \right] \\ \!&= \!\frac{1\!+\!\rho ^2}{(1-\rho ^2)^2} \!+\! \frac{N(N\!-\!1)\rho ^2}{(1\!-\!\rho ^2)^2} B(3,N-1) \!+\! \frac{N(N-1)(2-3\rho ^2+\rho ^6)}{4(1\!-\!\rho ^2)^4} B(3,N\!-\!1) \\&- \frac{2N(N-1)(1+\rho ^2)}{(1-\rho ^2)^2} B(2,N-1), \end{aligned}$$

where $B(a,b)$ is the Beta-function.

(2) and (3) of Theorem 1: (Nadarajah (2006), pp .198–200) $\square$

Proof

(Corollary 1) The $t$-distribution with parameters $\rho \in (-1,1)$ and $\nu >0$ results from the Pearson VII distribution for $N=\frac{m+2}{2}$ and $m=\nu$. The log density is

$$\begin{aligned} l(x,y;\rho ,\nu ) = -\log (2\pi ) - \frac{1}{2}\log (1-\rho ^2) - \frac{\nu +2}{2}\log \left(1+\frac{x^2+y^2-2\rho xy}{\nu (1-\rho ^2)}\right). \end{aligned}$$

Thus, (1) follows directly from Theorem 1 (1).

Since the Fisher Information for the Pearson VII distribution with respect to $N, m$ and $\rho$ is known from Theorem 1 ,we can compute the elements for (2) and (3) of the Fisher Information matrix easily. Denoting the Fisher Information of Pearson VII by $\mathcal{I }^{P7}$ and the Fisher Information of the $t$-distribution by $\mathcal{I }^{t}$ we obtain

$$\begin{aligned} \mathcal{I }^{t}_m&= \mathcal{I }^{P7}_m + 2\mathcal{I }^{P7}_{mN} \underbrace{\left(\frac{\partial N}{\partial m}\right)}_{1/2} + \mathcal{I }^{P7}_N \underbrace{\left(\frac{\partial N}{\partial m}\right)^2}_{1/4} \\&= \frac{1}{m} B\left(2,\frac{m}{2}\right) - \frac{m+2}{4m} B\left(3,\frac{m}{2}\right) \end{aligned}$$

and

$$\begin{aligned} \mathcal{I }^{t}_{\rho m}&= \mathcal{I }^{P7}_{\rho m} + \mathcal{I }^{P7}_{\rho N} \underbrace{\left(\frac{\partial N}{\partial m}\right)}_{1/2} \\&= -\frac{\rho }{2(1-\rho ^2)} \left( B\left(2,\frac{m}{2}\right) - \frac{m+2}{2} B\left(3,\frac{m}{2}\right) \right). \end{aligned}$$

Here, $\mathcal{I }_{m} = E[-\frac{\partial ^2 \log l}{\partial ^2 m}], \mathcal{I }_{mN} = E[-\frac{\partial ^2 \log l}{\partial N\partial m}], \mathcal{I }_{N} = E[-\frac{\partial ^2 \log l}{\partial ^2 N}], \mathcal{I }_{\rho m} = E[-\frac{\partial ^2 \log l}{\partial \rho \partial m}]$ and $\mathcal{I }_{\rho N} = E[-\frac{\partial ^2 \log l}{\partial \rho \partial N}]$. $\square$