Return Models and Covariance Matrices

Sammanfattning: Return models and covariance matrices of return series have been studied. In particular, GARCH and SV models are compared with respect to their forecasting accuracy when applied to intraday return series. SV models are found to be considerably more accurate and more consistent in accuracy in forecasting. Covariance matrices formed from Gaussian and GARCH return series, and in particular, return series auto-correlated as an AR(1) process, have been studied. In the case of Gaussian returns, the largest eigenvalue is found to approximately follow a gamma distribution also when the returns are auto-correlated. Expressions relating the mean and the variance of the asymptotic Gaussian distribution of the matrix elements are derived. In the case of GARCH returns, both the largest and the smallest eigenvalues of the covariance matrix are seen to increase with increasing auto-correlation. The matrix elements are found to follow Levy distributions with different Levy indexes for the diagonal and the non-diagonal elements. Localization of eigenvectors of covariance matrices of returns from GARCH processes has been investigated. It is found that the localization is reduced as the auto-correlation is increased. Quantitatively, the number of localized eigenvectors decreases approximately as a quadratic function with the auto-correlation strength, i.e. the autoregressive coefficient of the AR(1) process.