This paper introduces a generalized dynamic factor model (GDFM) that allows for cross-correlation among idiosyncratic components, making it more realistic and applicable to a wide range of economic and financial data. The model is particularly useful for analyzing large systems with many variables and short time series, as it provides a parsimonious representation of the dynamics. The authors propose identification and estimation methods, showing that the model is identified under certain conditions as the number of variables tends to infinity. They also develop an estimator based on principal components, which is effective even when the cross-sectional dimension is large. The paper includes theoretical results, simulation studies, and an empirical illustration using US state-level data to estimate the "national component" of the business cycle. The method is shown to recover the dynamic structure of the data, providing insights into regional business cycles and their synchronization.This paper introduces a generalized dynamic factor model (GDFM) that allows for cross-correlation among idiosyncratic components, making it more realistic and applicable to a wide range of economic and financial data. The model is particularly useful for analyzing large systems with many variables and short time series, as it provides a parsimonious representation of the dynamics. The authors propose identification and estimation methods, showing that the model is identified under certain conditions as the number of variables tends to infinity. They also develop an estimator based on principal components, which is effective even when the cross-sectional dimension is large. The paper includes theoretical results, simulation studies, and an empirical illustration using US state-level data to estimate the "national component" of the business cycle. The method is shown to recover the dynamic structure of the data, providing insights into regional business cycles and their synchronization.