Developments in the Factor Analysis of Individual Time Series
Michael W. Browne and Guangjian Zhang
Ohio State University

The development of methodology for the factor analysis of data from a single subject over repeated observations on a battery of variables arose out of the P-technique recommended by Cattell. In this approach the data are treated as if the repeated observations are independent and subjected to a standard factor analysis. Comments by T. W. Anderson that the P-technique should be supplemented by the modeling of time dependence in the data were influential on later modifications of Cattell's approach.

Work on the P-technique is first reviewed and associated developments are presented. It is seen that two types of model have been used. In one, the process factor analysis model, the factors follow a VARMA process. This model has a single factor matrix and additional time series parameter matrices. In the other, the shock factor analysis model, the factors are independently distributed random shock vectors or, equivalently, white noise. This model has several factor matrices, each representing the effect of the latent shock vector on the vector of manifest variables at one of a number of lags. It has no time series parameter matrices. Different perspectives on the same situation are provided by the two models and it is instructive to employ them concurrently.

Two classes of estimation methodology are also considered. One uses full information maximum likelihood based on the original observations. The other is a moment-based approach. It involves the initial computation of lagged correlation matrices, followed by the fitting of appropriate correlation structures using a minimum discrepancy approach.

A modern development is then described. The process factor analysis model is estimated using a moment-based approach. Both confirmatory, and exploratory variants with rotation of factors, are available. In addition, the shock factor representation for any process factor analysis is provided. Examples are employed to illustrate capabilities of the model.