# Dynamic factor analysis vs factor analysis on differences

I'm trying to wrap my head around dynamic factor analysis. So far, my understanding is that DFA is just factor analysis plus a time series model on the scores (the loadings remain fixed). However, in the cases that I've seen, the model on the scores is just a random walk with a diagonal correlation matrix. This seems identical to normal factor analysis applied to the differences. What am I missing?

If you know of any good references to get me started, I'd appreciate them. I'd actually like to find something that allows the loadings to be slowly-varying; my context for thinking about that is West&Harrison-style DLMs, which hasn't got me far.

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If your loadings slowly vary and your factor scores also vary its not immediately clear how you'd identify the model. Covariates on the factor scores perhaps? –  conjugateprior Jul 6 at 18:01
@conjugateprior Check this out –  bfoste01 Jul 16 at 2:31
After an (admittedly brief) skim of the paper my point is that one could not index both the loadings $\lambda$ and the factor scores $f$ with $t$. At most one of them. –  conjugateprior Jul 17 at 11:44

## 1 Answer

Here goes:

In my field (developmental science) we apply DFA to intensive multivariate time-series data for an individual. Intensive small samples are key. DFA allows us to examine both the structure and time-lagged relationships of latent factors. Model parameters are constant across time, so stationary time-series (i.e., probability distributions of stationarity of stochastic process is constant) is really what you are looking at with these models. However, researchers have relaxed this a bit by including time-varying covariates. There are many ways to estimate the DFA, most of which involve the Toeplitz matrices: maximum likelihood (ML) estimation with block Toeplitz matrices (Molenaar, 1985), generalized least squares estimation with block Toeplitz matrices (Molenaar & Nesselroade, 1998), ordinary least squares estimation with lagged correlation matrices (Browne & Zhang, 2007), raw data ML estimation with the Kalman filter (Engle & Watson, 1981; Hamaker, Dolan, & Molenaar, 2005), and the Bayesian approach (Z. Zhang, Hamaker, & Nesselroade, 2008).

In my field DFA has become an essential tool in modeling nomothetic relations at a latent level, while also capturing idiosyncratic features of the manifest indicators: the idiographic filter.

The P-technique was a precursor to DFA, so you might want to check that out, as well as what came after... state-space models.

Read any of the references in the list for estimation procedures for nice overviews.

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