The generalized dynamic factor model: Representation theory

This paper, along with the companion paper Forni, Hallin, Lippi, and Reichlin (2000, Review of Economics and Statistics 82, 540-554), introduces a new model-the generalized dynamic factor model-for the empirical analysis of financial and macroeconomic data sets characterized by a large number of observations both cross section and over time. This model provides a generalization of the static approximate factor model of Chamberlain (1983, Econometrica 51, 1181-1304) and Chamberlain and Rothschild (1983, Econometrica 51, 1305-1324) by allowing serial correlation within and across individual processes and of the dynamic factor model of Sargent and Sims (1977, in C.A. Sims (ed.), New Methods in Business Cycle Research, pp. 45-109) and Geweke (1977, in D.J. Aigner & A.S. Goldberger (eds.), Latent Variables in Socio-Economic Models, pp. 365-383) by allowing for nonorthogonal idiosyncratic terms. Whereas the companion paper concentrates on identification and estimation, here we give a full characterization of the generalized dynamic factor model in terms of observable spectral density matrices, thus laying a firm basis for empirical implementation of the model. Moreover, the common factors are obtained as limits of linear combinations of dynamic principal components. Thus the paper reconciles two seemingly unrelated statistical constructions.