Factor Modeling For Longitudinal Data

Substantial progress has been made over the last decade in the research of longi-tudinal data, which also attracts wide-spreading attention in the field of medicine and sociology. Common linear model can be adopted to analysis longitudinal data. However, there are always latent variables in real life, thus this article introduces the common factor models to analysis longitudinal data. Common factor model has been widely used to study the vector time series. As longitudinal data and vector time series has similarities in many aspects, so using common factor model to study longitudinal data seems reasonable. This article firstly proposes using the common factor model to study longitudinal data, and establishes two-dimension and three-dimension common factor models for analyzing longitudinal data. Also, these models can be used to study high-dimensional longitudinal data. For two-dimension common factor model, we propose a two-step estimation method. In particular, in the first step we estimate the fixed coefficients in the model by improved least square or lasso least square method, in the second step we adopt the factor analysis method to estimate the common factor and its factor loading matrix. For three-dimension common factor model, we adopt the factor analysis method to estimate the common factor and its factor loading matrix. On the theoretical properties, we establish the asymptotic normality for the improved least square estimator, the consistency for the lasso least square estimator and the consistency for the common factor estimators based on Principal component analysis. The simulation part demonstrates the good performance of the estimators in different data sets.