Estimation of α-stable Sub-Gaussian distributions for Asset Returns. Contributions to Economics

1 Fitting multivariate α-stable distributions to data is still not feasible in higher dimensions since the (non-parametric) spectral measure of the characteristic func-tion is extremely difficult to estimate in dimensions higher than 2. This was shown by Chen and Rachev (1995) and Nolan, Panorska and McCulloch (1996). α-stable sub-Gaussian distributions are a particular (parametric) subclass of the multivariate α-stable distributions. We present and extend a method based on Nolan (2005) to estimate the dispersion matrix of an α-stable sub-Gaussian dis-tribution and estimate the tail index α of the distribution. In particular, we de-velop an estimator for the off-diagonal entries of the dispersion matrix that has statistical properties superior to the normal off-diagonal estimator based on the covariation. Furthermore, this approach allows estimation of the dispersion ma-trix of any normal variance mixture distribution up to a scale parameter. We demonstrate the behaviour of these estimators by fitting an α-stable sub-Gaussia