Inference for vast dimensional elliptical distributions

Abstract

We propose a quantile–based method to estimate the parameters of an elliptical distribution, and a battery of tests for model adequacy. The method is suitable for vast dimensions as the estimators for location and dispersion have closed–form expressions, while estimation of the tail index boils down to univariate optimizations. The tests for model adequacy are for the null hypothesis of correct specification of one or several level contours. A Monte Carlo study to three distributions (Gaussian, Student–t and elliptical stable) for dimensions 20, 200 and 2000 reveals the goodness of the method, both in terms of computational time and finite samples. An empirical application to financial data illustrates the method.

Appendix: eigenvalue cleaning

Let $\hat{\varvec{\Gamma }}_N=diag(\hat{\varvec{\Sigma }}_N)^{-1/2}\hat{\varvec{\Sigma }}_Ndiag(\hat{\varvec{\Sigma }}_N)^{-1/2}$ be the estimated standardized dispersion matrix with spectral decomposition $\hat{\varvec{\Gamma }}_N=\hat{\mathbf{Q }}_N\hat{\varvec{\Lambda }}_N\hat{\mathbf{Q }}_N^{^{\prime }}$, where $\hat{\mathbf{Q }}_N$ is the orthonormal matrix of estimated eigenvectors and $\hat{\varvec{\Lambda }}_N$ is the diagonal matrix of estimated eigenvalues. Let $\hat{\lambda }_{(1)\,N} \ge \ldots \ge \hat{\lambda }_{(J)\,N}$ be the ordered eigenvalues (i.e. $\hat{\lambda }_{(1)\,N}$ is the largest and $\hat{\lambda }_{(J)\,N}$ is the smallest). Eigenvalue cleaning is based on replacing the eigenvalues less than a threshold $\lambda _{max}$ by the average of the positive eigenvalues below $\lambda _{max}$:

$$\begin{aligned} \tilde{\lambda }_N = \frac{\sum _{l=0}^{L} \max (0,\hat{\lambda }_{(J-l)\,N})}{L+1}, \end{aligned}$$

where $L+1$ corresponds to the position, from the right, of the largest eigenvalue smaller than $\lambda _{max}$.

The resulting estimated standardized dispersion matrix is $\tilde{\varvec{\Gamma }}_N=\hat{\mathbf{Q }}_N\tilde{\varvec{\Lambda }}_N\hat{\mathbf{Q }}_N^{^{\prime }}$ and the positive definite estimated dispersion matrix is obtained by un–standardizing $\tilde{\varvec{\Gamma }}_N$: $\tilde{\varvec{\Sigma }}_N=diag(\hat{\varvec{\Sigma }})_N^{1/2}\tilde{\varvec{\Gamma }}_N diag(\hat{\varvec{\Sigma }}_N)^{1/2}$.

The threshold is given by $\lambda _{max}=\left(1-\sum _{l=1}^{L^*}\hat{\lambda }_{(l)\,N}/J\right)\left(1+J/N+2\sqrt{J/N}\right)$, i.e. it is a function of the $L^*$ largest eigenvalues. The smaller $L^*$, the largest the difference between $\hat{\varvec{\Sigma }}_N$ and $\tilde{\varvec{\Sigma }}_N$. Laloux et al. (1999), Tola et al. (2008), and Hautsch et al. (2011) consider $L^*=1$ on the grounds that $\hat{\lambda }_{(1)\,N}$ represents the common dispersion. There is no reason however in our case to consider this value. After calibration we have found that $L^*=10$ is the best compromise. Results for the Monte Carlo study and the empirical application with other values of $L^*$ are available upon request.