Maximum Likelihood Dynamic Factor Modeling for Arbitrary N and T Using SEM

Abstract

This article has 3 objectives that build on each other. First, we demonstrate how to obtain maximum likelihood estimates for dynamic factor models (the direct autoregressive factor score model) with arbitrary T and N by means of structural equation modeling (SEM) and compare the approach to existing methods. Second, we go beyond standard time series analysis (T large and N = 1) and conventional SEM (N large and T = 1 or small) by integrating both approaches. The resulting combined model offers a variety of new modeling options including a direct test of the ergodicity hypothesis, according to which the factorial structure of an individual observed at many time points is identical to the factorial structure of a group of individuals observed at a single point in time. Third, we illustrate the flexibility of SEM time series modeling by extending the approach to account for complex error structures. We end with a discussion of current limitations and future applications of SEM-based time series modeling for arbitrary T and N.

Keywords:

• dynamic factor analysis
• factorial invariance
• maximum likelihood estimation
• time series analysis

Notes

¹More generally speaking, S is singular; that is, |S| cannot be computed, as long as the number of independent cases is smaller than the number of time points multiplied by the number of observed variables (i.e., N < Tq).

^aReproduced with permission.

^bResults are based on 90 replications.

^cResults are based on 97 replications.

²For a comparison to the three other estimation techniques, the reader is referred to Table 2 to Table 4 in Zhang et al. (2008).

³Of course, the approach also permits covariances among measurement errors within each time point.

⁴Another potential advantage of the simultaneous estimation of between- and within-person models might be an improved initializing of the individual time series by borrowing information on the between-person model at the first time point. This remains to be explored in future research.