An EML Estimator for Multivariate t Copulas

Author

Jin Zhang1

Keywords

Student t copula, Exact Maximum Likelihood Estimator, Inference for Margins, Differential Evolution

Review Status

Unreviewed

Abstract

In recent years, copulas have become very popular in financial research and actuarial science as they are more flexible in modelling the comovements and relationships of risk factors as compared with the conventional linear correlation coefficient by Pearson. However, a precise estimation of the copula parameters is vital in order to correctly capture the (possibly nonlinear) dependence structure and joint tail events. Since the exact maximum likelihood (EML) method can be computationally burdensome if the common Newton-Raphson algorithm is applied to optimize the objective function, the literature suggests the inference for margins (IFM) approach as this two-step procedure can obtain the estimates faster — but at the cost of lower efficiency and a higher bias, especially in higher dimensions.

In this study, we employ a Differential Evolution algorithm to tackle the parameter estimation of multivariate t distribution models in the EML approach. The approach can be applied to more complicated copula models using EML since the derivative-free optimizer overcomes the curse of dimensionality problem. Our experimental study shows that the proposed method provides more robust and more accurate estimates than compared with the IFM approach.

1 Introduction

Nowadays, copulas have been widely applied by market practitioners to model the dependence structure of financial risk factors, such as equity and exchange rate returns. The popularity of copulas is mainly due to their flexibility as they can be used to model both the linear and non-linear dependence structure of a multivariate distribution. The linear correlation by Pearson is not only insufficient in describing the dependence of risk factors when moving away from elliptical distributions, but also inconsistent under nonlinear strictly increasing transformations of risk factors. Therefore, using copula-based dependence measures will be more robust in capturing the dependence structure than calculating the linear correlation.

This work employs an evolutionary method in heuristic optimization, namely Differential Evolution (DE) to tackle the parameter estimation of multivariate t copula models in the EML framework. The approach is a one-step estimation procedure, and it does not require any starting guess of the decision variables. It employs a derivative-free optimization method to overcome the curse of dimensionality problem. Therefore, the proposed approach is particularly suitable for the inference of large and complicated copula models by using EML, while traditional optimization procedures tend to stop at local optima in such cases.

2 The Model

The probability density function $f(.)$ of general Student t distributions can be written as

(1)
\begin{align} f_{\nu_m,\mu_m,\sigma_m}^{t}(x) = \frac{\Gamma(\frac{\nu_m+1}{2})} {\Gamma(\frac{\nu_m}{2})} \frac{1}{\sqrt{\nu_m\pi\sigma_m^2}} \left(1+\frac{1}{\nu_m} \frac{(x-\mu_m)^2}{\sigma_m^2}\right)^{-\frac{\nu_m+1}{2}}, \end{align}

where $\Gamma(.)$ is the Gamma function, $\nu_m$ denotes the marginal degrees of freedom (DOF), $\mu_m$ and $\sigma_m$ represent the location and dispersion of the distribution.

The t-copula density $c(u_1,...,u_n)=\frac{\partial^n C(u_1,...,u_n)}{\partial u_1 ... \partial u_n}$ can be written as

(2)
\begin{align} c_{\nu,\rho}^{t}(u_1,...u_n) = \frac{1}{\sqrt{|\rho|}} \frac{\Gamma(\frac{\nu+n}{2}) \Gamma(\frac{\nu}{2})^{n-1}}{\Gamma(\frac{\nu+1}{2})^n} \frac{\prod_{k=1}^{n}(1+\frac{y_k^2}{\nu})^{\frac{\nu+1}{2}}}{(1+\frac{y'\rho^{-1}y}{\nu})^{\frac{\nu+n}{2}}}. \end{align}

The log-likelihood function $\ell^m$ of the Student t marginal distribution $f(.)$ can be written as

(3)
\begin{align} \notag \ell^m = \ell_{\mu_j,\sigma_j,\nu_j} = -N \cdot\left[ \log(\sigma_j)+\log(\sqrt{\nu_j})+ \log(\sqrt{\pi}) + \log\left(\Gamma\left(\frac{\nu_j}{2}\right)\right) + \log\left(\Gamma\left(\frac{1+\nu_j}{2}\right)\right) \right]\\ -\left(\frac{\nu_j+1}{2}\right)\cdot\sum_{i=1}^{N}\log\left(1+\frac{(x_{j,i}-\mu_j)^2}{\sigma_j^2\cdot\nu_j}\right), \end{align}

where $N$ is the observation number; and $\mu_j,\sigma_j,\nu_j$ denote the location, dispersion and degrees of freedom of the $j$-th marginal distribution, respectively. The log-likelihood function $\ell^C$ of the t copula density can be
written as

(4)
\begin{align} \notag \ell^C = \ell_{\rho,\nu}= N\cdot\left[-\frac{1}{2}\cdot\log(|\bm{\rho}|)-2\cdot\log\left(\Gamma\left(\frac{\nu+1}{2}\right)\right) + \log\left(\Gamma\left(\frac{\nu+2}{2}\right)\right) + \log\left(\Gamma\left(\frac{\nu}{2}\right)\right)\right] + \sum_{j=1}^{n}\sum_{i=1}^{N} \frac{\nu+1}{2}\cdot\log\left(1+\frac{y_{j,i}^{2}}{\nu}\right) - \frac{\nu+2}{2}\cdot \sum_{i=1}^{N}\log\left[1+\frac{1}{\nu} \bm{y_i}'\cdot\bm{\rho}^{-1}\cdot\bm{y_i}\right], \end{align}

where $n$ denotes the dimension of the risk factors; $y_j$ represents the inverse transform of Student t with $\nu$ degrees of freedom for the $j$-th risk factor's observations after a strictly increasing transform, i.e., the Student t cumulative distribution function.

The parameters of the cumulative distribution function are estimated jointly with parameters of the copula model. The fitness of the final objective function is defined as the sum of log-likelihood values of both the marginal and copula density functions. To estimate the copula model, we adopt a population based evolutionary method to optimize parameters of copula models while taking the marginal distribution and dependent structure into account simultaneously.

3 Results

To assess the performance of the proposed EML estimation with the DE, we first simulate a set of $200 \times 2$ random variables with bivariate student t distribution at a total iteration number of $N=5,000$. The true distribution parameters are set as $\mu_1=0$, $\mu_2=0,\sigma_1=0.2548,\sigma_2=0.2250,\rho=0.43$ and $\nu=6$. After that, we estimate the parameters of the bivariate t-copula with t-margins by using the proposed approach and a traditional approach respectively. Since the standard hill-climbing algorithm such as the Newton-Raphson approach for the EML method did not generate any results but only for the IFM framework, we only compare the latter one with the DE procedure applied on EML.

EML_IFM_den.jpg

The above figure shows the kernel densities of the estimated distribution parameters for both estimation procedures. In total, it can be seen that the parameters responsible centered moments of the distribution, i.e. $\sigma_1,\sigma_2$ and $\nu$ can be better estimated with the EML approach. These results are indeed important as they reveal that the IFM approach often preferred in the financial literature is more likely to provide less reliable estimators of the underlying joint distribution and, hence, less able to correctly capture the dependence structure and the tail dependence of risk factors.

4 Conclusion

This paper suggests implementing an evolutionary algorithm in the exact maximum likelihood estimation of multivariate copula models as standard hill-climbing procedure. Usually, the Newton-Raphson algorithm fails to optimize the objective function when the number of dimensions turns out to be high, while a derivative-free optimizer can overcome this problem. Through a simple Monte-Carlo simulation study, we show that the proposed methodology already provide reasonably good results in a straightforward 2-dimensional setting with a bivariate Student t-copula. As expected, the estimates obtained by the EML approach enhanced with the Differential Evolution are often closer to the true values as compared with the IFM alternatives. The Differential Evolution is competent for the EML inference of more complicated copula models than the bivariate Student t copula studied.

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