Abstract

Modelling and estimation of multivariate time series remains a challenging task, in particular for signals of high dimension and a time-varying second order structure. This can be found in numerous fields of applications such as in the situation of modelling the covariance matrix of multivariate financial data. This work represents a first approach for developing non-parametric models, accompanied by methods of local estimation and hypothesis testing, for multivariate time series with a time-changing covariance structure. The concept of local stationarity shall be used to prove asymptotic properties of the estimators, such as consistency and asymptotic normality.

More specifically, we may estimate the time-evolutive covariance matrix by a kernel regression smoother with a common bandwidth. The drawback of this approach is that in order to have a positive definite estimator of the matrix, it is not allowed to model and estimate the different degree of smoothness of each covariance function (i.e. each entry of the covariance matrix) separately. We aim at applying tests on reducing the rank of the true underlying matrix, and we show that the outcome of this test depends on the quality of the estimator, i.e. using different smoothing parameters for the different entries of the estimated covariance matrix leads to a more precise result for the final dimension reduction.

In view of the problem that the estimator obtained with differently smoothed entries is no more (in general) nonnegative definite, we show the approach of using a preliminary Cholesky decomposition of the matrix to be a promising possibility. In this decomposition the idea is to be able to estimate the different elements of the covariance matrix separately of each other and still getting a positive-definite estimator.

This is a joint work with Rainer von Sachs, Institut de Statistique, Louvain-la-Neuve, Belgium and Christian Hafner, Econometric Institute, Erasmus University Rotterdam, The Netherlands.