Multiscale methods for the analysis of high-dimensional locally stationary time series

This thesis deals with multiscale modelling of the covariance pattern of discrete time series with time-varying autocovariance function. We propose a novel class of locally stationary wavelet (LSW) processes which allows for correlations between the increment terms at the different scales of resolution. The first part of the thesis focuses on univariate time series modelling. We prove that the autocovariance representation of the extended class of LSW processes is asymptotically unique and propose a consistent estimator of the corresponding evolutionary wavelet spectrum (EWS). The second part deals with multivariate time series modelling. As for the univariate setting, we prove that the new class of LSW processes has an asymptotically unique covariance representation and we derive a consistent estimation procedure of the EWS matrix. In the third part, we use the extended class of multivariate LSW processes to build a dynamic factor model. We construct a consistent estimator of the number of common factors and the standardized factor loadings, based on an eigenvalue decomposition of the estimated EWS matrix of the multivariate LSW model. In the fourth part we propose a multiscale estimator of covariance matrices in high-dimensional settings using unbalanced Haar wavelets. We prove consistency in operator norm of the estimator, given conditions on the degree of sparsity of the wavelet transform of the population covariance matrix. Each of the four parts, contains applied sections in which we use the proposed modelling techniques to study various economic and financial phenomena.