Ellen L. Hamaker, Conditions for the Equivalence of the Autoregressive Latent Trajectory Model and a Latent Growth Curve Model With Autoregressive Disturbances, Sociological Methods & Research 2005 33: 404-416.

Curran and Bollen combined two models for longitudinal panel data: the latent growth curve model and the autoregressive model. In their model, the autoregressive relationships are modeledbetween the observed variables. This is a different model than a latent growth curve model with autoregressive relationships between the disturbances. However, when the autoregressive parameter is invariant over time and lies between-1 and 1, it can be shown that these models are algebraically equivalent. This result can be shown to generalize to the multivariate case. When theautoregressive parameters in the autoregressive latent trajectory model vary over time, the equivalence between the autoregressive latent trajectory model and a latent growth curve model with autoregressive disturbances no longer holds. However, a latent growth curve model with time-varying autoregressive parameters for the disturbances could be considered an interesting alternative to the autoregressive latent trajectory model with time-varying autoregressive parameters.

Key Words: latent growth curve model • autoregressive latent trajectory model • autoregression • structural equation modeling

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