Generalized Dynamic Factor Models Structure Theory and Estimation for Single Frequency and Mixed Frequency Data

The thesis deals with the theory of Generalized Dynamic Factor Models. These models have been introduced about 10 years ago by two groups simultaneously and have been discussed intensively since then. The reason why this model class attracts so many different research groups is, that it is, up to now, the most general class of factor models. The new aspect of these models is, that the spectrum of the noise does not have to be diagonal anymore. This is possible as asymptotics, in both the time and the cross-section, are regarded simultaneously. The divergence of the cross-section dimension is reasonable as these models are designed for high dimensional data sets. An important part of this thesis is the so called structure theory for the latent variables. It is, at least to our knowledge, the most general way to model these variables. This has a price, as we are faced with singular AR processes. These processes have an autoregressive representation where the driving white noise has a singular covariance matrix. Theoretically we only need singular AR(1) models to describe the dynamics of a minimal state. Unfortunately we cannot assume that the whole state can be reconstructed by the latent variables. Nevertheless we have shown, that if the dimension of a minimal static factor is larger than the dimension of the driving white noise, the minimal static factor has generically a singular autoregressive representation. Consequently we present a detailed description of singular autoregressive processes, which have been poorly discussed in the existing literature. As the regularity of the covariance matrix of the driving white noise plays a central role in the theory of regular autoregressive processes, the generalization to the singular case is much more complicated than one might expect. An interesting fact is, that not only purely linearly regular stationary autoregressive processes exist, contrary to the regular case. Additionally we discuss the estimation of the coefficients and the minimal order of singular AR processes. Again the singularity of the covariance matrix of the noise destroys several results from the regular case. Finally we discuss the problem of mixed frequency sampling data, which means that not all variables