Dynamic Factor Models

Abstract

In this chapter, we deal with linear dynamic factor models and related topics, such as dynamic principal component analysis (dynamic PCA). The main motivation for the use of such models is the so-called “curse of dimensionality” plaguing modeling of high-dimensional time series by “ordinary” multivariate AR or ARMA models. For instance, consider an AR system for, a say, 20-dimensional time series. Then each of the coefficient matrices contains 400 “free” parameters, if no additional restrictions on the parameter space have been imposed, i.e. in such a case the parameter spaces grow with the square of the output dimension n, whereas the data, for given sample size, grow linearly with n. Thus for moderate sample size and large n (as is the case, e.g. in many situations faced in macroeconomics), reliable parameter estimation in “fully parametrized” AR(X) or ARMA(X) models is hardly possible. On the other hand, e.g. in macroeconomics, for analysis and in particular for short-term forecasting, modeling of “comovement” and of “cross-sectional dependencies” between a large number of univariate time series recently has received increasing attention and appropriate tools for modeling of high-dimensional time series have been developed. Correspondingly, during the last 25 years, a substantial literature has emerged, dealing with such models, methods and applications, in particular for factor models in this context. The first section introduces a general framework for linear dynamic factor models. In the second section, we describe dynamic principal component analysis, which is a generalization of the well-known static principal component analysis to the dynamic case. In practical applications often the generalized dynamic factor model is used, which allows for cross-sectionally weakly dependent noise and assumes strong dependence in latent variables. This model class is suited for very high-dimensional time series.