Generalized Factor Models

The analysis of high-dimensional time series, when the number of single time series is relatively large in relation to sample size, has recently attracted substantial attention. The idea is to compress information in both, the time and the cross-sectional dimension, and to learn from adding new time series. Here we consider generalized linear dynamic factor models (GDFM`s) as proposed and analysed in Forni, Hallin, Lippi, and Reichlin (2000), Forni and Lippi (2001), Stock and Watson (2002a). These models generalize and combine linear dynamic factor models with strictly idiosyncratic noise (Geweke (1977), Sargent and Sims (1977), Scherrer and Deistler (1998)) and static generalized factor models as introduced in Chamberlain (1983), Chamberlain and Rothschild (1983). Important areas of application are analysis and forecasting of high dimensional financial time series, such as returns of financial assets, and of high dimensional macroeconomic series, for instance for cross country studies. This research project consists of the following parts: 1. To develop a "structure theory" for GDFM`s based on the assumption that the latent variables are stationary with a singular rational spectral density. Here those properties of the relation between the population second moments of the observations and the parameters of a state space- or an ARMA system generating the latent variables, which are relevant for estimation, are analysed. 2. Based on the structure theoretic results, a "direct" estimation procedure will be proposed and analysed. In addition maximum likelihood type estimation procedures will be treated. 3. In applications often the single time series are defined over different time segments or they are sampled with different frequency.We intend to develop appropriate estimation procedures for such cases. 4. Finally we plan to consider model selection procedures for GDFM`s. Here in particular estimation of the dimension of the dynamic and the static factors, and estimation of the state dimension will be considered.

Modelling and forecasting of high dimensional time series is an important and prevailing problem in fields such as econometrics, meteorology, genomics, chemometrics, biological and environmental research and finance.Generalized dynamic factor models (GDFMs) belong to the most important model class for high dimensional time series. GDFMs avoid the so-called curse of dimensionality plaguing traditional multivariate time series modelling, by compressing the information contained in the data in both the cross-sectional dimension and in the time dimension.GDFMs have been introduced approximately fifteen years ago and have been successfully applied since. Nevertheless, there is still a number of important problems to be solved. The focus of our work was on structure theory and estimation. In particular we have analysed so-called singular autoregressive models as models for the latent variables and the static factors. Here in particular the genericity of such models and the properties of Yule Walker parameter estimates have been analysed.Another focus of our work is in identifiability and estimation of such models from mixed frequency data, i.e. from data where the univariate time series are sampled at different frequencies, for instance, some time series, like GDP, are sampled quarterly whereas other time series, such as unemployment, are sampled monthly. The main results derived here are generic identifiability and consistent estimation of the parameters of the underlying system generating all data at the highest frequency.